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https://mathoverflow.net/questions/268809/covariation-of-the-stochastic-integral-and-the-wiener-process | # Covariation of the stochastic integral and the Wiener process
Let$^1$
• $T>0$
• $U,H$ be separable $\mathbb R$-Hilbert spaces
• $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint operator with finite trace $\operatorname{tr}Q$
• $(e^n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe^n=\lambda_ne^n\;\;\;\text{for all }n\in\mathbb N\tag1$$ for some $(\lambda_n)_{n\in\mathbb N}\subseteq(0,\infty)$ and $$e_0^n:=Q^{1/2}e^n=\sqrt{\lambda_n}e^n\;\;\;\text{for }n\in\mathbb N$$
• $U_0:=Q^{1/2}U$ be equipped with $$\langle u_0,v_0\rangle_{U_0}:=\langle Q^{-1/2}u_0,Q^{-1/2}v_0\rangle_U\;\;\;\text{for }u_0,v_0\in U_0$$
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $\Phi:\Omega\times[0,T]\to\operatorname{HS}(U_0,H)$ be predictable with $$\int_0^T\operatorname E\left[\left\|\Phi_t\right\|_{\operatorname{HS}(U_0,\:H)}^2\right]{\rm d}t<\infty\tag2$$
Let $u\in U$ and $h\in H$. Then, $$\langle\Phi_s(\omega)Qu,h\rangle_H=\sum_{n\in\mathbb N}\lambda_n\langle u,e^n\rangle_U\langle\Phi_s(\omega)e^n,h\rangle_H\;\;\;\text{for all }(\omega,s)\in\Omega\times[0,T]\tag3\;.$$ I want to conclude that $$\operatorname E\left[\int_0^t\langle\Phi_sQu,h\rangle_H\:{\rm d}s\mid\mathcal F_r\right]=\operatorname E\left[\sum_{n\in\mathbb N}\lambda_n\langle u,e^n\rangle_U\int_0^t\langle\Phi_se^n,h\rangle_H\:{\rm d}s\mid\mathcal F_r\right]\tag4$$ for all $r,t\in[0,T]$ with $r\le t$.
My problem is that I'm not even able to show that, for fixed $s\in[0,T]$, $$S_N:=\sum_{n=1}^N\lambda_n\langle u,e^n\rangle_U\langle\Phi_se^n,h\rangle_H\;\;\;\text{for }N\in\mathbb N$$ converges for $N\to\infty$ to $\langle\Phi_sQu,h\rangle_H$ in $L^1(\operatorname P)$. This would follow from $(3)$ by Lebesgue's dominated convergence theorem, but the best bound I obtain is $$\left|S_N(\omega)\right|\le\left\|u\right\|_U\left\|h\right\|_H\left\|\Phi_s(\omega)\right\|_{\operatorname{HS}(U_0,\:H)}\sum_{n=1}^N\sqrt{\lambda_n}\;\;\;\text{for all }\omega\in\Omega\text{ and }N\in\mathbb N\;.\tag5$$ This is not sufficient, since I don't see how we should bound $\sum_{n=1}^N\sqrt{\lambda_n}$. Note that $$\sum_{n\in\mathbb N}\lambda_n=\operatorname{tr}Q<\infty\tag6\;.$$
$^1$ Let $\mathfrak L(A,B)$ and $\operatorname{HS}(A,B)$ denote the space of bounded linear Operators and Hilbert-Schmidt operators, respectively. Moreover, let $\mathfrak L(A):=\mathfrak L(A,A)$.
• Please note that I know that this is not a "research level" question. However, I've asked several stochastic analysis questions on MSE and almost never got an answer. – 0xbadf00d May 2 '17 at 14:36
Since $$\lambda_n\left\|\Phi_s(\omega)e^n\right\|_H\le\begin{cases}\lambda_n&\text{, if }\left\|\Phi_s(\omega)e^n\right\|_H\le1\\\left\|\Phi_s(\omega)e_0^n\right\|_H^2&\text{, if }\left\|\Phi_s(\omega)e^n\right\|_H\ge1\end{cases}\tag7$$ for all $n\in\mathbb N$, $\sum_{n\in\mathbb N}\lambda_n=\operatorname{tr}Q$ and $\sum_{n\in\mathbb N}\left\|\Phi_s(\omega)e_0^n\right\|_H^2=\left\|\Phi_s(\omega)\right\|_{\operatorname{HS}(U_0,\:H)}^2$, we obtain $$\sum_{n\in\mathbb N}\lambda_n\left\|\Phi_s(\omega)e^n\right\|_H<\infty\tag8$$ by the comparison test for $(\operatorname P\otimes\lambda^1)$-almost all $(\omega,s)\in\Omega\times[0,T]$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9972103834152222, "perplexity": 97.652497399498}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107902683.56/warc/CC-MAIN-20201029010437-20201029040437-00315.warc.gz"} |
https://www.physicsforums.com/threads/integral-equations-picard-method-of-succesive-approximation.931607/ | # A Integral equations -- Picard method of succesive approximation
1. Nov 14, 2017
### LagrangeEuler
Equation
$$\varphi(x)=x+1-\int^{x}_0 \varphi(y)dy$$
If I start from $\varphi_0(x)=1$ or $\varphi_0(x)=x+1$ I will get solution of this equation using Picard method in following way
$$\varphi_1(x)=x+1-\int^{x}_0 \varphi_0(y)dy$$
$$\varphi_2(x)=x+1-\int^{x}_0 \varphi_1(y)dy$$
$$\varphi_3(x)=x+1-\int^{x}_0 \varphi_2(y)dy$$
...
Then solution is given by
$$\varphi(x)=\lim_{n \to \infty}\varphi_n(x)$$.
When I could say that this sequence will converge to solution of integral equation. How to see if there is some fixed point? I know how to use this method, but I am not sure from the form of equation, when I can use this method. Thanks for the answer.
2. Nov 14, 2017
### mathman
$$\phi_0(x)=1\ results \ in \ \phi_1(x)=1$$, you're done!
3. Nov 15, 2017
### LagrangeEuler
This is not my question. I know how to solve this. I am not sure when I can use this method. When sequence of functions $\varphi_0(x)$, $\varphi_1(x)$... will converge to $\varphi(x)$.
4. Nov 15, 2017
### mathman
Since all $$\phi_n(x)=1$$ are the same, the sequence trivially converges to $$\phi(x)=1.$$ I am not sure what you are looking for.
5. Nov 15, 2017
### WWGD
I think s/he is looking for general conditions for convergence, not just for this particular problem.
6. Nov 16, 2017
### LagrangeEuler
Yes. Thanks.
7. Nov 16, 2017
### MathematicalPhysicist
You can use this method when you have: $\int \lim_{n\to\infty} \varphi_n(y)dy = \lim_{n\to \infty} \int \varphi_n(y)dy$.
8. Nov 16, 2017
### WWGD
Isn't this equivalent to dominated or monotone convergence?
9. Nov 16, 2017
### mathman
Dominated convergence is a sufficient condition, but not necessary.
10. Nov 16, 2017
### WWGD
Ah, yes, Dominated, no reason for Monotone here. Need some caffeine.
11. Nov 17, 2017
### LagrangeEuler
Yes but if I have for example equation in the form
$$\varphi(x)=f(x)+\lambda \int^x_0K(x,t)\varphi(t)dt$$
could I see this just for looking in kernel $K(x,t)$ and parameter $\lambda$?
12. Nov 17, 2017
### MathematicalPhysicist
@LagrangeEuler in your last post this is an eigenvalue problem: if we denote by: $K\varphi(x) = \int_0^x K(x,t)\varphi(t)dt$
Then you want to solve the equation: $(I-\lambda K)\varphi = f$; you need to solve the equation $\det |I-\lambda K| \ne 0$ and then you have a solution: $\varphi(x) = (I-\lambda K)^{-1}f(x)$; how to find the inverse, check any functional analysis textbook or Courant's and Hilbert's first volume. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9700987935066223, "perplexity": 914.92570072079}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221217006.78/warc/CC-MAIN-20180820195652-20180820215652-00166.warc.gz"} |
https://it.mathworks.com/help/robust/ref/diskmargin.html | # diskmargin
Disk-based stability margins of feedback loops
## Syntax
``[DM,MM] = diskmargin(L)``
``MMIO = diskmargin(P,C)``
``___ = diskmargin(___,sigma)``
## Description
example
````[DM,MM] = diskmargin(L)` computes the disk-based stability margins for the SISO or MIMO negative feedback loop `feedback(L,eye(N))`, where `N` is the number of inputs and outputs in `L`. The `diskmargin` command returns loop-at-a-time stability margins in `DM` and multiloop margins in `MM`. Disk-based margin analysis provides a stronger guarantee of stability than the classical gain and phase margins. For general information about disk margins, see Stability Analysis Using Disk Margins.```
example
````MMIO = diskmargin(P,C)` computes the stability margins when considering independent, concurrent variations at both the plant inputs and plant outputs the negative feedback loop of the following diagram. ```
example
````___ = diskmargin(___,sigma)` specifies an additional skew parameter that biases the modeled gain and phase variation toward gain increase (positive `sigma`) or gain decrease (negative `sigma`). You can use this argument to test the relative sensitivity of stability margins to gain increases versus decreases. You can use this argument with any of the previous syntaxes.```
## Examples
collapse all
`diskmargin` computes both loop-at-a-time and multiloop disk margins. This example illustrates that loop-at-a-time margins can give an overly optimistic assessment of the true robustness of MIMO feedback loops. Margins of individual loops can be sensitive to small perturbations in other loops, and loop-at-a-time margins ignore such loop interactions.
Consider the two-channel MIMO feedback loop of the following illustration.
The plant model `P` is drawn from MIMO Stability Margins for Spinning Satellite and `C` is the static output-feedback gain [1 -2;0 1].
```a = [0 10;-10 0]; b = eye(2); c = [1 10;-10 1]; P = ss(a,b,c,0); C = [1 -2;0 1]; ```
Compute the disk-based margins at the plant output. The negative-feedback open-loop response at the plant output is `Lo = P*C`.
```Lo = P*C; [DMo,MMo] = diskmargin(Lo);```
Examine the loop-at-a-time disk margins returned in the structure array `DM`. Each entry in `DM` contains the stability margins of the corresponding feedback channel.
`DMo(1)`
```ans = struct with fields: GainMargin: [0 Inf] PhaseMargin: [-90 90] DiskMargin: 2 LowerBound: 2 UpperBound: 2 Frequency: Inf WorstPerturbation: [2x2 ss] ```
`DMo(2)`
```ans = struct with fields: GainMargin: [0 Inf] PhaseMargin: [-90 90] DiskMargin: 2 LowerBound: 2 UpperBound: 2 Frequency: 0 WorstPerturbation: [2x2 ss] ```
The loop-at-a-time margins are excellent (infinite gain margin and 90° phase margin). Next examine the multiloop disk margins `MMo`. These consider independent and concurrent gain (phase) variations in both feedback loops. This is a more realistic assessment because plant uncertainty typically affects both channels simultaneously.
`MMo`
```MMo = struct with fields: GainMargin: [0.6839 1.4621] PhaseMargin: [-21.2607 21.2607] DiskMargin: 0.3754 LowerBound: 0.3754 UpperBound: 0.3762 Frequency: 0 WorstPerturbation: [2x2 ss] ```
The multiloop gain and phase margins are much weaker than their loop-at-a-time counterparts. Stability is only guaranteed when the gain in each loop varies by a factor less than 1.46, or when the phase of each loop varies by less than 21°. Use `diskmarginplot` to visualize the gain and phase margins as a function of frequency.
`diskmarginplot(Lo)`
Typically, there is uncertainty in both the actuators (inputs) and sensors (outputs). Therefore, it is a good idea to compute the disk margins at the plant inputs as well as the outputs. Use `Li = C*P` to compute the margins at the plant inputs. For this system, the margins are the same at the plant inputs and outputs.
```Li = C*P; [DMi,MMi] = diskmargin(Li); MMi```
```MMi = struct with fields: GainMargin: [0.6839 1.4621] PhaseMargin: [-21.2607 21.2607] DiskMargin: 0.3754 LowerBound: 0.3754 UpperBound: 0.3762 Frequency: 0 WorstPerturbation: [2x2 ss] ```
Finally, you can also compute the multiloop disk margins for gain or phase variations at both the inputs and outputs of the plant. This approach is the most thorough assessment of stability margins, because it this considers independent and concurrent gain or phase variations in all input and output channels. As expected, of all three measures, this gives the smallest gain and phase margins.
```MMio = diskmargin(P,C); diskmarginplot(MMio.GainMargin)```
Stability is only guaranteed when the gain varies by a less than 2 dB or when the phase varies by less than 13°. However, these variations take place at the inputs and the outputs of P, so the total change in I/O gain or phase is twice that.
By default, `diskmargin` computes a symmetric gain margin, with `gmin = 1/gmax`, and an associated phase margin. In some systems, however, loop stability may be more sensitive to increases or decreases in open-loop gain. Use the skew parameter `sigma` to examine this sensitivity.
Compute the disk margin and associated disk-based gain and phase margins for a SISO transfer function, at three values of `sigma`. Negative `sigma` biases the computation toward gain decrease. Positive `sigma` biases toward gain increase.
```L = tf(25,[1 10 10 10]); DMdec = diskmargin(L,-2); DMbal = diskmargin(L,0); DMinc = diskmargin(L,2); DGMdec = DMdec.GainMargin```
```DGMdec = 1×2 0.4013 1.3745 ```
`DGMbal = DMbal.GainMargin`
```DGMbal = 1×2 0.6273 1.5942 ```
`DGMinc = DMinc.GainMargin `
```DGMinc = 1×2 0.7717 1.7247 ```
Put together, these results show that in the absence of phase variation, stability is maintained for relative gain variations between 0.4 and 1.72. To see how the phase margin depends on these gain variations, plot the stable ranges of gain and phase variations for each `diskmargin` result.
```diskmarginplot([DGMdec;DGMbal;DGMinc]) legend('sigma = -2','sigma = 0','sigma = 2') title('Stable range of gain and phase variations')```
This plot shows that the feedback loop can tolerate larger phase variations when the gain decreases. In other words, the loop stability is more sensitive to gain increase. Although `sigma` = –2 yields a phase margin as large as 30 degrees, this large value assumes a small gain increase of less than 3 dB. However, the plot shows that when the gain increases by 4 dB, the phase margin drops to less than 15 degrees. By contrast, it remains greater than 30 degrees when the gain decreases by 4 dB.
Thus, varying the skew `sigma` can give a fuller picture of sensitivity to gain and phase uncertainty. Unless you are mostly concerned with gain variations in one direction (increase or decrease), it is not recommended to draw conclusions from a single nonzero value of `sigma`. Instead use the default `sigma` = 0 to get unbiased estimates of gain and phase margins. When using nonzero values of `sigma`, use both positive and negative values to compare relative sensitivity to gain increase and decrease.
## Input Arguments
collapse all
Open-loop response, specified as a dynamic system model. `L` can be SISO or MIMO, as long as it has the same number of inputs and outputs. `diskmargin` computes the disk-based stability margins for the negative-feedback closed-loop system `feedback(L,eye(N))`.
To compute the disk margins of the positive feedback system `feedback(L,eye(N),+1)`, use `diskmargin(-L)`.
When you have a controller `P` and a plant `C`, you can compute the disk margins for gain (or phase) variations at the plant inputs or outputs, as in the following diagram.
• To compute margins at the plant outputs, set `L = P*C`.
• To compute margins at the plant inputs, set `L = C*P`.
`L` can be continuous time or discrete time. If `L` is a generalized state-space model (`genss` or `uss`) then `diskmargin` uses the current or nominal value of all control design blocks in `L`.
If `L` is a frequency-response data model (such as `frd`), then `diskmargin` computes the margins at each frequency represented in the model. The function returns the margins at the frequency with the smallest disk margin.
If `L` is a model array, then `diskmargin` computes margins for each model in the array.
Plant, specified as a dynamic system model. `P` can be SISO or MIMO, as long as `P*C` has the same number of inputs and outputs. `diskmargin` computes the disk-based stability margins for a negative-feedback closed-loop system. To compute the disk margins of the system with positive feedback, use `diskmargin(P,-C)`.
`P` can be continuous time or discrete time. If `P` is a generalized state-space model (`genss` or `uss`) then `diskmargin` uses the current or nominal value of all control design blocks in `P`.
If `P` is a frequency-response data model (such as `frd`), then `diskmargin` computes the margins at each frequency represented in the model. The function returns the margins at the frequency with the smallest disk margin.
Controller, specified as a dynamic system model. `C` can be SISO or MIMO, as long as `P*C` has the same number of inputs and outputs. `diskmargin` computes the disk-based stability margins for a negative-feedback closed-loop system. To compute the disk margins of the system with positive feedback, use `diskmargin(P,-C)`.
`C` can be continuous time or discrete time. If `C` is a generalized state-space model (`genss` or `uss`) then `diskmargin` uses the current or nominal value of all control design blocks in `C`.
If `C` is a frequency-response data model (such as `frd`), then `diskmargin` computes the margins at each frequency represented in the model. The function returns the margins at the frequency with the smallest disk margin.
Skew of uncertainty region used to compute the stability margins, specified as a real scalar value. This parameter biases the uncertainty used to model gain and phase variations toward gain increase or gain decrease.
• The default `sigma` = 0 uses a balanced model of gain variation in a range `[gmin,gmax]`, with ```gmin = 1/gmax```.
• Positive `sigma` uses a model with more gain increase than decrease (`gmax > 1/gmin`).
• Negative `sigma` uses a model with more gain decrease than increase (`gmin < 1/gmax`).
Use the default `sigma` = 0 to get unbiased estimates of gain and phase margins. You can test relative sensitivity to gain increase and decrease by comparing the margins obtained with both positive and negative `sigma` values. For an example, see Sensitivity of Disk-Based Margins to Gain Increase and Decrease. For more detailed information about how the choice of `sigma` affects the margin computation, see Stability Analysis Using Disk Margins.
## Output Arguments
collapse all
Disk margins for each feedback channel with all other loops closed, returned as a structure for SISO feedback loops, or an N-by-1 structure array for a MIMO loop with N feedback channels. The fields of `DM(i)` are:
FieldValue
`GainMargin`Disk-based gain margins of the corresponding feedback channel, returned as a vector of the form `[gmin,gmax]`. These values express in absolute units the amount by which the loop gain in that channel can decrease or increase while preserving stability. For example, if ```DM(i).GainMargin = [0.8,1.25]``` then the gain of the ith loop can be multiplied by any factor between 0.8 and 1.25 without causing instability. When `sigma` = 0, `gmin = 1/gmax`. If the open-loop gain can change sign without loss of stability, `gmin` can be less than zero for large enough negative `sigma`. If the nominal closed-loop system is unstable, then `DM(i).GainMargin = [1 1]`.
`PhaseMargin`Disk-based phase margin of the corresponding feedback channel, returned as a vector of the form `[-pm,pm]` in degrees. These values express the amount by which the loop phase in that channel can decrease or increase while preserving stability. If the closed-loop system is unstable, then `DM(i).PhaseMargin = [0 0]`.
`DiskMargin`Maximum ɑ compatible with closed-loop stability for the corresponding feedback channel. ɑ parameterizes the uncertainty in the loop response (see Algorithms). If the closed-loop system is unstable, then ```DM(i).DiskMargin = 0```.
`LowerBound`Lower bound on disk margin. This value is the same as `DiskMargin`.
`UpperBound`Upper bound on disk margin. This value represents an upper limit on the actual disk margin of the system. In other words, the disk margin is guaranteed to be no worse than `LowerBound` and no better than `UpperBound`.
`Frequency`Frequency at which the weakest margin occurs for the corresponding loop channel. This value is in rad/`TimeUnit`, where `TimeUnit` is the `TimeUnit` property of `L`.
`WorstPerturbation`
Smallest gain and phase variation that drives the feedback loop unstable, returned as a state-space (`ss`) model with N inputs and outputs, where N is the number of inputs and outputs in `L`. The system `F(s) = WorstPerturbation` is such that the following feedback loop is marginally stable, with a pole on the stability boundary at the frequency `DM(i).Frequency`.
This state-space model is a diagonal perturbation of the form `F(s) = diag(f1(s),...,fN(s))`. Each `fj(s)` is a real-parameter dynamic system that realizes the worst-case complex gain and phase variation applied to each channel of the feedback loop. For the loop-at-a-time margin of the kth feedback loop, only the kth entry `fk(s)` of `DM(k).WorstPerturbation` differs from unity.
For more information on interpreting `WorstPerturbation`, see Disk Margin and Smallest Destabilizing Perturbation
When analyzing a linear approximation of a nonlinear system, it can be useful to inject `WorstPerturbation` into the nonlinear simulation to further analyze the destabilizing affect of this worst-case gain and phase variation. For an example, see Robust MIMO Controller for Two-Loop Autopilot.
When `L = P*C` is the open-loop response of a system comprising a controller and plant with unit negative feedback in each channel, `DM` contains the stability margins for variations at the plant outputs. To compute the stability margins for variations at the plant inputs, use `L = C*P`. To compute the stability margins for simultaneous, independent variations at both the plant inputs and outputs, use ```MMIO = diskmargin(P,C)```.
When `L` is a model array, `DM` has additional dimensions corresponding to the array dimensions of `L`. For instance, if `L` is a 1-by-3 array of two-input, two-output models, then `DM` is a 2-by-3 structure array. `DM(j,k)` contains the margins for the jth feedback channel of the kth model in the array.
Multiloop disk margins, returned as a structure. The gain (or phase) margins quantify how much gain variation (or phase variation) the system can tolerate in all feedback channels at once while remaining stable. Thus, `MM` is a single structure regardless of the number of feedback channels in the system. (For SISO systems, `MM` = `DM`.) The fields of `MM` are:
FieldValue
`GainMargin`Multiloop disk-based gain margins, returned as a vector of the form `[gmin,gmax]`. These values express in absolute units the amount by which the loop gain can vary in all channels independently and concurrently while preserving stability. For example, if ```MM.GainMargin = [0.8,1.25]``` then the gain of all loops can be multiplied by any factor between 0.8 and 1.25 without causing instability. When `sigma` = 0, `gmin = 1/gmax`.
`PhaseMargin`Multiloop disk-based phase margin, returned as a vector of the form `[-pm,pm]` in degrees. These values express the amount by which the loop phase can vary in all channels independently and concurrently while preserving stability.
`DiskMargin`Maximum ɑ compatible with closed-loop stability. ɑ parameterizes the uncertainty in the loop response (see Algorithms).
`LowerBound`Lower bound on disk margin. This value is the same as `DiskMargin`.
`UpperBound`Upper bound on disk margin. This value represents an upper limit on the actual disk margin of the system. In other words, the disk margin is guaranteed to be no worse than `LowerBound` and no better than `UpperBound`.
`Frequency`Frequency at which the weakest margin occurs. This value is in rad/`TimeUnit`, where `TimeUnit` is the `TimeUnit` property of `L`.
`WorstPerturbation`
Smallest gain and phase variation that drives the feedback loop unstable, returned as a state-space (`ss`) model with N inputs and outputs, where N is the number of inputs and outputs in `L`. The system `F(s) = WorstPerturbation` is such that the following feedback loop is marginally stable, with a pole on the stability boundary at `MM.Frequency`.
This state-space model is a diagonal perturbation of the form `F(s) = diag(f1(s),...,fN(s))`. Each `fj(s)` is a real-parameter dynamic system that realizes the worst-case complex gain and phase variation applied to each channel of the feedback loop.
For more information on interpreting `WorstPerturbation`, see Disk Margin and Smallest Destabilizing Perturbation
When analyzing a linear approximation of a nonlinear system, it can be useful to inject `WorstPerturbation` into the nonlinear simulation to further analyze the destabilizing affect of this worst-case gain and phase variation. For an example, see Robust MIMO Controller for Two-Loop Autopilot.
When `L = P*C` is the open-loop response of a system comprising a controller and plant with unit negative feedback in each channel, `MM` contains the stability margins for variations at the plant outputs. To compute the stability margins for variations at the plant inputs, use `L = C*P`. To compute the stability margins for simultaneous, independent variations at both the plant inputs and outputs, use ```MMIO = diskmargin(P,C)```.
When `L` is a model array, `MM` is a structure array with one entry for each model in `L`.
Disk margins for independent variations applied simultaneously at input and output channels of the plant `P`, returned as a structure having the same fields as `MM`.
For variations applied simultaneously at inputs and outputs, the `WorstPerturbation` field is itself a structure with fields `Input` and `Output`. Each of these fields contains a state-space model such that for ```Fi(s) = MMIO.WorstPerturbation.Input``` and ```Fo(s) = MMIO.WorstPerturbation.Output```, the system of the following diagram is marginally unstable, with a pole on the stability boundary at the frequency `MMIO.Frequency`.
These state-space models `Input` and `Output` are diagonal perturbations of the form `F(s) = diag(f1(s),...,fN(s))`. Each `fj(s)` is a real-parameter dynamic system that realizes the worst-case complex gain and phase variation applied to each channel of the feedback loop.
## Tips
• `diskmargin` assumes negative feedback. To compute the disk margins of a positive feedback system, use `diskmargin(-L)` or `diskmargin(P,-C)`.
• To compute disk margins for a system modeled in Simulink®, first linearize the model to obtain the open-loop response at a particular operating point. Then, use `diskmargin` to compute stability margins for the linearized system. For more information, see Stability Margins of a Simulink Model.
• To compute classical gain and phase margins, use `allmargin`.
• You can visualize disk margins using `diskmarginplot`.
## Algorithms
collapse all
`diskmargin` computes gain and phase margins by applying a disk-based uncertainty model to represent gain and phase variations, and then finding the largest such disk for which the closed-loop system is stable.
### Gain and Phase Uncertainty Model
For SISO L, the uncertainty model for disk-margin analysis incorporates a multiplicative complex uncertainty F into the loop transfer function as follows:
`$F=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]\delta }{1-\alpha \left[\left(1+\sigma \right)/2\right]\delta }.$`
Here,
• δ is a gain-bounded dynamic uncertainty, normalized so that it always varies within the unit disk (|δ| < 1).
• α sets the amount of gain and phase variation modeled by F. For fixed σ, the parameter ɑ controls the size of the disk. For α = 0, the multiplicative factor is 1, corresponding to the nominal L.
• σ, called the skew, biases the modeled uncertainty toward gain increase or gain decrease. (For details about the effect of skew on the uncertainty model, see Stability Analysis Using Disk Margins.)
For MIMO systems, the model allows the uncertainty to vary independently in each channel:
`${F}_{j}=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]{\delta }_{j}}{1-\alpha \left[\left(1+\sigma \right)/2\right]{\delta }_{j}}.$`
The model replaces the MIMO open-loop response L with L*F, where
`$F=\left(\begin{array}{ccc}{F}_{1}& 0& 0\\ 0& \ddots & 0\\ 0& 0& {F}_{N}\end{array}\right).$`
### Disk-Margin Computation
For a given `sigma`, the disk margin is the largest ɑ for which the closed-loop system `feedback(L*F,1)` (or `feedback(L*F,eye(N))` for MIMO systems) is stable for all values of F. To find this value, `diskmargin` solves a robust stability problem: Find the largest α such that the closed-loop system is stable for all F in the uncertainty disk Δ(α,σ) described by
`$\Delta \left(\alpha ,\sigma \right)=\left\{F=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]\delta }{1-\alpha \left[\left(1+\sigma \right)/2\right]\delta }\text{\hspace{0.17em}}\text{\hspace{0.17em}}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\delta |<1\right\}.$`
In the SISO case, the robust stability analysis leads to
`${\alpha }_{max}={‖\frac{1}{S+\left(\sigma -1\right)/2}‖}_{\infty },$`
where S is the sensitivity function (1 + L)–1 .
In the MIMO case, the robust stability analysis leads to
`${\alpha }_{max}=\frac{1}{{\mu }_{\Delta }\left(S+\frac{\left(\sigma -1\right)I}{2}\right)}.$`
Here, μΔ is the structured singular value (`mussv`) for the diagonal structure
`$\Delta =\left(\begin{array}{ccc}{\delta }_{1}& 0& 0\\ 0& \ddots & 0\\ 0& 0& {\delta }_{N}\end{array}\right),$`
and δj is the normalized uncertainty for each Fj.
For more details about the margin computation, see [2].
## Compatibility Considerations
expand all
Behavior changed in R2020a
## References
[1] Blight, James D., R. Lane Dailey, and Dagfinn Gangsaas. “Practical Control Law Design for Aircraft Using Multivariable Techniques.” International Journal of Control 59, no. 1 (January 1994): 93–137. https://doi.org/10.1080/00207179408923071.
[2] Seiler, Peter, Andrew Packard, and Pascal Gahinet. "An Introduction to Disk Margins." IEEE Control Systems Magazine (forthcoming). https://arxiv.org/abs/2003.04771. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9220920205116272, "perplexity": 1207.6470503410335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107905965.68/warc/CC-MAIN-20201029214439-20201030004439-00093.warc.gz"} |
https://daviddalpiaz.github.io/stat400fa17/homework/hw03-assign.html | Please see the detailed homework policy document for information about homework formatting, submission, and grading.
## Exercise 1
The label on a small package of Bertie Bott’s Every Flavour Beans claims that 3 beans are caramel flavored, 6 are butterscotch, and 4 are earwax. Unable to tell them apart just by looking at them, Ron Weasley selects 5 beans at random. Find the probability that Ron ends up with $$\ldots$$
(a) $$\ldots$$ 1 caramel flavored, 2 butterscotch, and 2 earwax beans.
(b) $$\ldots$$ no earwax flavored beans.
(c) $$\ldots$$ at least 2 caramel flavored beans.
## Exercise 2
Let $$X$$ denote the number of times Ron Weasley manages to irritate Professor Snape in one day. Then $$X$$ has the following probability distribution:
$$x$$ $$f(x)$$
0 0.15
1 0.20
2 0.20
3 0.30
4 0.15
(a) Find the probability that Ron will get in trouble with Professor Snape at least two times in one day.
(b) Find the expected number of times Ron will get in trouble with Professor Snape, $$\text{E}[X]$$.
(c) Find the standard deviation of the number of times Ron will get in trouble with Professor Snape, $$\text{SD}[X]$$.
(d) Each day, Professor Snape takes 20 points from Gryffindor, simply because he can. Additionally, Professor Snape takes 10 points from Gryffindor each time Ron Weasley irritates him. If these are the only two sources of point deductions for Gryffindor, what is the expected point loss for Gryffindor each day?
(e) What is the standard deviation of points lost for Gryffindor each day?
## Exercise 3
Consider a random variable $$X$$ with the probability mass function
$f(x) = \frac{6}{3^x}, \quad x = 2, 3, 4, 5, \ldots$
Calculate the expected value of $$X$$.
## Exercise 4
Consider a random variable $$Y$$ with the probability mass function
$f(y) = c \cdot \frac{2^y}{y!}, \quad y = 2, 3, 4, 5, \ldots$
where $$c = \displaystyle\frac{1}{e^2 - 3}$$.
Calculate the expected value of $$Y$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8154257535934448, "perplexity": 2466.7209279327813}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250590107.3/warc/CC-MAIN-20200117180950-20200117204950-00321.warc.gz"} |
https://socratic.org/questions/what-is-the-cartesian-form-of-44-5pi-12 | Calculus
Topics
# What is the Cartesian form of (-44,(5pi)/12))?
Jan 16, 2016
$\left(11.388 , 42.501\right)$
#### Explanation:
Note the Cartesian form cannot have the modulus negative so I assume the $- 44$ should be $44$.
To convert polar form $\left(r , \theta\right)$ to rectangular form $\left(x , y\right)$, we use the following transformations :
$x = r \cos \theta = 44 \cos \left(\frac{5 \pi}{12}\right) = 11.388$.
$y = r \sin \theta = 44 \sin \left(\frac{5 \pi}{12}\right) = 42.501$.
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http://ned.ipac.caltech.edu/level5/March01/Franceschini/Fran9.html | ### 9. DEEP SKY SURVEYS WITH THE INFRARED SPACE OBSERVATORY (ISO)
ISO has been the most important IR astronomical mission of the 1990s. Launched by ESA, it consisted of a 60 cm telescope operative in a highly eccentric 70000 Km orbit. It included two instruments of cosmological interest (in addition to two spectrographs): a mid-IR 32×32 camera (ISOCAM, 4 to 18 µm), and a far-IR imaging photometer (ISOPHOT, with small 3×3 and 2×2 detector arrays from 60 to 200 µm). The whole payload was cooled to 2 K by a He3 cooling system so performant to allow ISO to operate for 30 months (Nov 1995 to Apr 1998), instead of the nominal 18 months. An excellent review of the extragalactic results from ISO can be found in Genzel & Cesarsky (2000).
While designed as an observatory-type mission, the vastly improved sensitivity offered by ISO with respect to the previous IRAS surveys motivated to spend a relevant fraction of the observing time to perform a set of deep sky explorations at mid- and far-IR wavelengths. The basic argument for this was to parallel optical searches of the deep sky with complementary observations at wavelengths where, in particular, the effect of dust is far less effective in extinguishing optical light. This could have been particularly relevant for investigations of the distant universe, given the large uncertainties implied by the (pobably large) extinction corrections in optical spectra of high redshift galaxies (e.g. Meurer et al. 1997).
Observations in the mid- and far-IR also sample the portion of the e.m. spectrum dominated by dust re-processed light, and are then ideally complementary to optical surveys to evaluate the global energy output by stellar populations and active nuclei.
Organized in parallel with the discovery of the CIRB, a major intent of the deep ISO surveys was to start to physically characterize the distant sources of the background and to single out the fraction contributed by nuclear non-thermal activity in AGNs.
Finally, exploring the sky to unprecedented sensitivity limits should have provided an obvious potential for discoveries of new unexpected phenomena from our local environment up to the most distant universe.
Deep surveys with ISO have been performed in two wide mid-IR (LW2: 5-8.5µm and LW3: 12-18µm) and two far-IR ( = 90 and 170 µm) bands. The diffraction-limited spatial resolutions were 5 arcsec at 10 µm and 50 arcsec at 100 µm. Mostly because of the better imaging quality, ISO sensitivity limits in the mid-IR are 1000 times better than at the long wavelengths (0.1 mJy versus 100 mJy). At some level the confusion problem will remain a fundamental limitation also for future space missions (SIRTF, FIRST, ASTRO-F). A kind of compensation to these different performances as a function of derives from the typical FIR spectra of galaxies and AGNs, which are almost typically one order of magnitude more luminous at 100 µm than at 10 µm. We detail in the following the most relevant programs of ISO surveys.
Five extragalactic surveys with the LW2 and LW3 filters have been performed in the ISOCAM GT (GITES, P.I. C. Cesarsky), including large-area shallow surveys and small-area deep integrations. A total area of 1.5 square degrees in the Lockman Hole and the "Marano" southern field have been surveyed, where more than one thousand sources have been detected (Elbaz et al. 1999). These two areas were selected for their low zodiacal and cirrus emissions and because of the existence of data at other wavelengths (optical, radio, X).
ELAIS is the most important program in the ISO Open Time (377 hours, P.I. M. Rowan-Robinson, see Oliver et al. 2000a). A total of 12 square degrees have been surveyed at 15 µm with ISOCAM and at 90 µm with ISOPHOT, 6 and 1 sq. degrees have been covered with the two instruments at 6.7 and 170 µm. To reduce the effects of cosmic variance, ELAIS was split into 4 fields of comparable size, 3 in the north, one in the south, plus six smaller areas. While data analysis is still in progress, a source list of over 1000 (mostly 15 µm) sources is being published, including starburst galaxies and AGNs (type-1 and type-2), typically at z < 0.5, with several quasars (including various BAL QSOs) found up to the highest z.
Very successful programs by the Hubble Space Telescope have been the two ultradeep exposures in black fields areas, one in the North and the other in the South, called the Hubble Deep Fields (HDF). These surveys promoted a substantial effort of multi-wavelength studies aimed at characterizing the SEDs of distant and high-z galaxies. These areas, including the Flanking Fields for a total of 50 sq. arcmin, have been observed by ISOCAM (P.I. M. Rowan-Robinson) at 6.7 and 15 µm, achieving completness to a limiting flux of 100 µJy at 15 µm.
These have been among the most sensitive surveys of ISO and have allowed to discover luminous starburst galaxies over a wide redshift interval up to z = 1.5 (Rowan-Robinson et al. 1997; Aussel et al, 1999). In the inner 10 sq. arcmin, the exceptional images of HST provided a detail morphological information for ISO galaxies at any redshifts (see Figure 4). Furthermore, these two fields benefit by an almost complete redshift information (Cohen et al. 1999), allowing a very detailed characterization of the faint distant IR sources.
Two fields from the Canada-France Redshift Survey (CFRS) have been observed with ISOCAM to intermediate depths: the '14+52' field (observed at 6.7 and 15 µm) and the '03+00' field (with only 15µm data, but twice as deep). The CFRS is, with the HDFs, one of the best studied fields with multi-wavelength data. Studies of the galaxies detected in both fields have provided the first tentative interpretation of the nature of the galaxies detected in ISOCAM surveys (Flores et al. 1999).
FIRBACK is a set of deep cosmological surveys in the far-IR, specifically aimed at detecting at 170 µm the sources of the far-IR background (P.I. J.L. Puget, see Puget et al. 1999). Part of this survey was carried out in the Marano area, and part in collaboration with the ELAIS team in ELAIS N1 and N2, for a total of 4 sq. degrees. This survey is limited by extragalactic source confusion in the large ISOPHOT beam (90 arcsec) to S170 100 mJy. Some constraints on the counts below the confusion limit obtained from a fluctuation analysis of one Marano/FIRBACK field are discussed by Lagache & Puget (2000) (Sect. 9.4). The roughly 300 sources detected are presently targets of follow-up observations, especially using deep radio exposures of the same area to help reducing the large ISO errorbox and to identify the optical counterparts. Also an effort is being made to follow-up these sources with sub-mm telescopes (IRAM, SCUBA): this can provide significant constraints on the redshift of sources which would be otherwise very difficult to measure in the optical (Sect 12.2).
Three lensing galaxy clusters, Abell 2390, Abell 370 and Abell 2218, have received very long integrations by ISOCAM (Altieri et al 1999). The lensing has been exploited to achieve even better sensitivities with respect to ultra-deep blank-field surveys (e.g. the HDFs), and allowed detection of sources between 30 and 100 µJy at 15 µm. However this was obviously at the expense of distorting the areal projection and ultimately making uncertain the source count estimate.
An ultra-deep survey of the Lockman Hole in the 7µm ISOCAM band was performed by Taniguchi et al. (1997; the survey field is different from that of the GITES Lockman survey). Another field, SSA13, was covered to a similar depth (P.I. Y. Taniguchi). The Lockman region was also surveyed with ISOPHOT by the same team: constraints on the source counts at 90 and 175 µ were derived by Matsuhara et al. (2000) based on a fluctuation analysis.
ISOCAM data need particular care to remove the effects of glitches induced by the frequent impacts of cosmic rays on the detectors (the 960 pixels registered on average 4.5 events/sec). This badly conspired with the need to keep them cryogenically cooled to reduce the instrumental noise, which implied a slow electron reaction time and longterm memory effects. For the deep surveys this implied a problem to disentangle faint sources from trace signals by cosmic ray impacts.
To correct for that, tools have been developed by various groups for the two main instruments (CAM and PHOT), essentially based on identifying patterns in the time history of the response of single pixels, which are specific to either astrophysical sources (a jump above the average background flux when a source falls on the pixel) or cosmic ray glitches (transient spikes followed by a slow recovery to the nominal background). The most performant algorithm for CAM data reduction is PRETI (Stark et al. 1999), a tool exploiting multi-resolution wavelet transforms (in the 2D observable plane of the position on the detector vs. time sequence). An independent method limited to brighter flux sources, developed by Désert et al. (1999), has been found to provide consistent results with PRETI, in the flux range in common. Other methods have been used by Oliver et al. (2000a) and Lari et al. (2000). These various detection schemes and photometry algorithms have been tested by means of very sophisticated Monte Carlo simulations, including all possible artifacts introduced by the analyses.
With simulations it is has been possible to control as a function of the flux threshold: the detection reliability, the completeness, the Eddington bias and photometric accuracy (10% where enough redundancy was available, as for CAM HDFs and Ultradeep surveys). Also the astrometric accuracy is good (of order of 1-2 arcsec for deep highly-redundant images), allowing straightforward identification of the sources (Aussel et al. 1999, see Fig. 4). The quality of the results for the CAM surveys is proven by the very good consistency of the counts from independent surveys (see Fig. [5] below).
Figure 4. ISOCAM LW3 map ( = 15 µm, yellow contours) of the Hubble Deep Field North by Aussel et al. (1999), overimposed on the HST image. The (green) circles are the LW2 ( = 6.7 µm) sources. The figure illustrates the spatial accuracy of the ISO deep images with LW3, allowing a reliable identification of the IR sources [courtesy of H. Aussel].
Longer wavelength ISOPHOT observations also suffered from similar problems. The 175µm counts from PHOT C200 surveys are reliable above the confusion limit S170 100 mJy, and required only relatively standard procedures of baseline corrections and "de-glitching". More severe are the noise problems for the C100 90µm channel, which would otherwise benefit by a better spatial resolution than C200. The C100 PHOT survey dataset is still presently under analysis.
IR-selected galaxies have typically red colors, because of the dust responsible for the excess IR emission. The most distant are also quite faint in the optical. For this reason the redshift information is available only for very limited subsamples (e.g. in the HDF North and CFRS areas). In this situation, the source number counts, compared with predictions based on the local luminosity function, provide important constraints on the evolution properties.
Particularly relevant information comes from the mid-IR samples selected from the CAM GITES and HDF surveys in the LW3 (12-18 µm) filter, because they include the faintest, most distant and most numerous ISO-detected sources. They are also easier to identify because of the small ISO error box for redundant sampling at these wavelengths.
Surveys of different sizes and depths are necessary to cover a wide dynamic range in flux with enough source statistics, which justified performing a variety of independent surveys at different flux limits. The differential counts based on these data, shown in Fig. 5, reveal an impressive agreement between so many independent samples. Including ELAIS and IRAS survey data, the range in fluxes would reach four orders of magnitude. The combined 15 µm differential counts display various remarkable features (Elbaz et al. 1999): a roughly euclidean slope from the brightest IRAS observed fluxes down to S15 5 mJy, a sudden upturn at S15 < 3 mJy, with the counts increasing as dN S-3.1dS to S15 0.4mJy, and evidence for a flattening below S15 0.3mJy (where the slope becomes quickly sub-Euclidean, N S-2).
Figure 5. Differential counts at = 15 µm normalized to the Euclidean law (N[S] S-2.5; the differential form is preferred here because all data points are statistically independent). The data come from an analysis of the GITES surveys by Elbaz et al. (1999). The dotted line corresponds to the expected counts for a population of non-evolving spirals. The dashed line comes from our modelled population of strongly evolving starburst galaxies, while the long-dashes are AGNs. The shaded region at S15 > 10 mJy comes from an extrapolation of the faint 60 µm IRAS counts by Mazzei et al. (2000).
The areal density of ISOCAM 15µm sources at the limit of 50-80 µJy is 5 arcmin-2. This is nominally the ISO confusion limit at 15 µm, if we consider that the diffraction-limited size of a point-source is 50 arcsec2: from eq. (8.26) and for = - 2, confusion sets in at a source areal density of 0.1/resolution element, or 7/arcmin2 in our case. The IR sky is so populated at these wavelengths that ISO was confusion limited longwards of = 15 µm. This will also be the case for NASA's SIRTF (due to launch in mid 2002), in spite of the moderately larger primary collector (85cm).
Obviously, far-IR selected samples are even more seriously affected by confusion. The datapoints on the 175µm integral counts reported in Fig. 6 come from the FIRBACK survey. Similarly deep observations at 90, 150 and 175 µm are reported by Iuvela, Mattila & Lemke (2000). Given the moderate depth of these direct counts, background fluctuation analyses were used to constrain their continuation below the survey detection limit. The analysis of small-scale fluctuations in one FIRBACK field by Lagache & Puget (2000) produced 0.07 MJy/sr with a beam of size 6 10-4 sr. From eq. [8.25], this may be used to constrain the continuation of the counts in Fig. 6 fainter than 100 mJy.
Figure 6. Integral counts based on the ISOPHOT FIRBACK survey (Sect.10.2.5) at = 175 µm (filled circles, from Dole et al. 2000) and on the ISOPHOT Serendipitous survey. The dashed and dot-dashed lines correspond to the non-evolving and the strongly evolving populations as in Fig. 5. The lowest curve is the expected (negligible) contribution of AGNs. The horizontal lines mark the confusion limits for three telescope sizes (based on eq. 8.26): the lines marked "60cm" and "360cm" correspond to the ISO and FIRST limits for faint source detection.
The 15µm counts in Fig. 5 display a remarkable convergence below S15 0.2 mJy, proven by at least three independent surveys. The asymptotic slope flatter than -1 in integral count units implies a modest contribution to the integrated CIRB flux by sources fainter than this limit, unless a sharp upturn of the counts would happen at much fainter fluxes with very steep number count distributions, a rather unplausible situation. A meaningful estimate of the CIRB flux can then be obtained from direct integration of the observed mid-IR counts (the two datapoints at 15 and 7 µm in Fig. 3). If we further consider how close these are to the upper limits set by the observed TeV cosmic opacity (Fig. 3), the ISOCAM surveys appear to have resolved a significant (50-70%) fraction of the CIRB in the mid-IR. On the other hand, the depth of the ISO far-IR surveys (FIRBACK) is not enough to resolve more than ten percent of the CIRB at its peak wavelenth. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8596123456954956, "perplexity": 3158.94504232802}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00494-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://stats.stackexchange.com/questions/5703/are-there-bounds-on-the-spearman-correlation-of-a-sum-of-two-variables | # Are there bounds on the Spearman correlation of a sum of two variables?
Given $n$-vectors $x, y_1, y_2$ such that the Spearman correlation coefficient of $x$ and $y_i$ is $\rho_i = \rho(x,y_i)$, are there known bounds on the Spearman coefficient of $x$ with $y_1 + y_2$, in terms of the $\rho_i$ (and $n$, presumably)? That is, can one find (non-trivial) functions $l(\rho_1,\rho_2,n), u(\rho_1,\rho_2,n)$ such that $$l(\rho_1,\rho_2,n) \le \rho(x,y_1+y_2) \le u(\rho_1,\rho_2,n)$$
edit: per @whuber's example in the comment, it appears that in the general case, only the trivial bounds $l = -1, u = 1$ can be made. Thus, I would like to further impose the constraint:
• $y_1, y_2$ are permutations of the integers $1 \ldots n$.
-
Only knowing $\rho_{1}, \rho_{2}$, the interval containing $\rho(x, y_{1} + y_{2})$ must include $\rho_{1}$ and $\rho_{2}$: for each $y_{1}, y_{2}$ could have very small values (while having any rank-order), and thus simply "jitter" the values in $y_{1}$ when added to $y_{1}$. Thus the rank-order of $y_{1}$ wouldn't be affected. I don't know if the interval can exceed the $\rho_{i}$. – caracal Dec 22 '10 at 20:36
@caracal Good observations. The interval definitely can be wider than the $\rho_i$: just consider the case where both correlations are zero. The correlation with the sum can easily be nonzero--it can range all the way from -1 to 1. E.g., x = (1,2,3,4,5); y1 = (3,-10,2,10,1); y2 = (-8,9,-2,-9,4); y1+y2 = (-5,-1,0,1,5) has $\rho_1=\rho_2=0$ but $\rho=1$. – whuber Dec 22 '10 at 20:57
@whuber: this seems to imply only trivial bounds exist (i.e. $l = -1, u = 1$). Perhaps I have to throw another constraint at the problem. – shabbychef Dec 22 '10 at 21:13
@shabbychef No, you have posted a nice problem: it's not trivial. In case $\rho_1 = \rho_2 = 1$, for instance, the only possibility is $\rho = 1$. I suspect the bounds are nontrivial except when $\rho_1 = \rho_2 = 0$; they must get narrower as $\rho_1$ and $\rho_2$ approach $\pm 1$. – whuber Dec 22 '10 at 21:21
Here’s another pathological case. Suppose that $x = y_1$ and $y_1 = -y_2$. Then $\rho(x, y_1 + y_2) = 0$, but $\rho_1 = 1$ and $\rho_2 = −1$. It might be enlightening to think about a simpler, probabilistic version of the problem. Let $X$, $Y_1$, and $Y_2$ be random variables, each with marginally Uniform distributions. Now let $G$ be the CDF of $Y_1 + Y_2$. What can we say about $Cov(X, G(Y_1 + Y_2))$ based on $Cov(X,Y_1)$ and $Cov(X,Y_2)$? – vqv Dec 22 '10 at 23:56
Spearman's rank correlation is just the Pearson product-moment correlation between the ranks of the variables. Shabbychef's extra constraint means that $y_1$ and $y_2$ are the same as their ranks and that there are no ties, so they have equal standard deviation $\sigma_y$ (say). If we also replace x by its ranks, the problem becomes the equivalent problem for the Pearson product-moment correlation.
By definition of the Pearson product-moment correlation, \begin{align} \rho(x,y_1+y_2) &= \frac{\operatorname{Cov}(x,y_1+y_2)} {\sigma_x \sqrt{\operatorname{Var}(y_1+y_2)}} \\ &= \frac{\operatorname{Cov}(x,y_1) + \operatorname{Cov}(x,y_2)} {\sigma_x \sqrt{\operatorname{Var}(y_1)+\operatorname{Var}(y_2) + 2\operatorname{Cov}(y_1,y_2)}} \\ &= \frac{\rho_1\sigma_x\sigma_y + \rho_2\sigma_x\sigma_y} {\sigma_x \sqrt{2\sigma_y^2 + 2\sigma_y^2\rho(y_1,y_2)}} \\ &= \frac{\rho_1 + \rho_2} {\sqrt{2}\left(1+\rho(y_1,y_2)\right)^{1/2}}. \\ \end{align} For any set of three variables, if we know two of their three correlations we can put bounds on the third correlation (see e.g. Vos 2009, or from the formula for partial correlation): $$\rho_1\rho_2 - \sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2} \leq \rho(y_1,y_2) \leq \rho_1\rho_2 + \sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2}$$ Therefore $$\frac{\rho_1 + \rho_2} {\sqrt{2}\left(1+\rho_1\rho_2 + \sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2}\right)^{1/2}} \leq \rho(x,y_1+y_2) \leq \frac{\rho_1 + \rho_2} {\sqrt{2}\left(1+\rho_1\rho_2 - \sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2}\right)^{1/2}}$$ if $\rho_1 + \rho_2 \geq 0$; if $\rho_1 + \rho_2 \le 0$ you need to switch the bounds around.
@vqv but if $y_1$ and $y_2$ are permutations of the integers $1\ldots n$ then they are exactly the same as their ranks. – onestop Dec 23 '10 at 20:52
The ranked values of $y_1 + y_2$ are in general a nonlinear function of $y_1 + y_2$ — even if $y_1$ and $y_2$ are each a permutation of the integers $1,\ldots,n$. Here’s an example: $y_1 = (1,2,3,4)$ and $y_2 = (2,3,1,4)$. Then $y_1+y_2 = (3,5,4,8)$ and $rank(y_1+y_2) = (1,3,2,4)$. Plot $y_1+y_2$ against $rank(y_1+y_2)$ and you’ll see that there is no linear relationship between the two. The above assertion that $\rho(x,y_1+y_2) = Cov(x,y_1+y_2) / \cdots$ is in general false, even under the assumption that $y_1$ and $y_2$ are permutations of the integers. – vqv Dec 26 '10 at 6:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9985426664352417, "perplexity": 336.27694746688735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164740723/warc/CC-MAIN-20131204134540-00072-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://www.varsitytutors.com/act_math-help/how-to-find-the-perimeter-of-a-right-triangle | # ACT Math : How to find the perimeter of a right triangle
## Example Questions
### Example Question #1 : How To Find The Perimeter Of A Right Triangle
In the figure below, right triangle has a hypotenuse of 6. If and , find the perimeter of the triangle .
Explanation:
# How do you find the perimeter of a right triangle?
There are three primary methods used to find the perimeter of a right triangle.
1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.
## Method 1:
This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:
If we know the lengths of sides , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:
## Method 2:
In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the angle and they are labeled and . The side of the triangle that is opposite of the angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .
If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:
We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .
Rearrange and take the square root of both sides.
Simplify.
Now, let's use the Pythagorean theorem to solve for one of the legs, .
Subtract from both sides of the equation.
Take the square root of both sides.
Simplify.
Last, let's use the Pythagorean theorem to solve for the adjacent leg, .
Subtract from both sides of the equation.
Take the square root of both sides.
Simplify.
It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:
After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.
## Method 3:
This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:
When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:
After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.
## Solution:
Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.
First, we need to use the Pythagorean theorem to solve for .
Because we are dealing with a triangle, the only valid solution is because we can't have negative values.
After you have found , plug it in to find the perimeter. Remember to simplify all square roots!
### Example Question #1 : How To Find The Perimeter Of A Right Triangle
Find the perimeter of the triangle below.
Explanation:
# How do you find the perimeter of a right triangle?
There are three primary methods used to find the perimeter of a right triangle.
1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.
## Method 1:
This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:
If we know the lengths of sides , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:
## Method 2:
In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the angle and they are labeled and . The side of the triangle that is opposite of the angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .
If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:
We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .
Rearrange and take the square root of both sides.
Simplify.
Now, let's use the Pythagorean theorem to solve for one of the legs, .
Subtract from both sides of the equation.
Take the square root of both sides.
Simplify.
Last, let's use the Pythagorean theorem to solve for the adjacent leg, .
Subtract from both sides of the equation.
Take the square root of both sides.
Simplify.
It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:
After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.
## Method 3:
This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:
When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:
After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.
## Solution:
Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem. The perimeter of a triangle is simply the sum of its three sides. Our problem is that we only know two of the sides. The key for us is the fact that we have a right triangle (as indicated by the little box in the one angle). Knowing two sides of a right triangle and needing the third is a classic case for using the Pythagorean theorem. In simple (sort of), the Pythagorean theorem says that sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse.
Every right triangle has three sides and a right angle. The side across from the right angle (also the longest) is called the hypotenuse. The other two sides are each called legs. That means in our triangle, the side with length 17 is the hypotenuse, while the one with length 8 and the one we need to find are each legs.
What the Pythagorean theorem tells us is that if we square the lengths of our two legs and add those two numbers together, we get the same number as when we square the length of our hypotenuse. Since we don't know the length of our second leg, we can identify it with the variable .
This allows us to create the following algebraic equation:
which simplified becomes
To solve this equation, we first need to get the variable by itself, which can be done by subtracting 64 from both sides, giving us
From here, we simply take the square root of both sides.
Technically, would also be a square root of 225, but since a side of a triangle can only have a positive length, we'll stick with 15 as our answer.
But we aren't done yet. We now know the length of our missing side, but we still need to add the three side lengths together to find the perimeter.
### Example Question #3 : How To Find The Perimeter Of A Right Triangle
Given that two sides of a right triangle are and and the hypotenuse is unknown, find the perimeter of the triangle.
Explanation:
# How do you find the perimeter of a right triangle?
There are three primary methods used to find the perimeter of a right triangle.
1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.
## Method 1:
This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:
If we know the lengths of sides , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:
## Method 2:
In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the angle and they are labeled and . The side of the triangle that is opposite of the angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .
If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:
We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .
Rearrange and take the square root of both sides.
Simplify.
Now, let's use the Pythagorean theorem to solve for one of the legs, .
Subtract from both sides of the equation.
Take the square root of both sides.
Simplify.
Last, let's use the Pythagorean theorem to solve for the adjacent leg, .
Subtract from both sides of the equation.
Take the square root of both sides.
Simplify.
It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:
After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.
## Method 3:
This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:
When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:
After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.
## Solution:
Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.
In order to calculate the perimeter we need to find the length of the hypotenuse using the Pythagorean theorem.
Rearrange.
Substitute in known values.
Now that we have found the missing side, we can substitute the values into the perimeter formula and solve. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.930959939956665, "perplexity": 168.34461177152318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886946.21/warc/CC-MAIN-20180117142113-20180117162113-00630.warc.gz"} |
http://methods.sagepub.com/Reference/encyclopedia-of-survey-research-methods/n515.xml | # Sampling Variance
Encyclopedia
Edited by: Published: 2008
• ## Subject Index
Sampling variance is the variance of the sampling distribution for a random variable. It measures the spread or variability of the sample estimate about its expected value in hypothetical repetitions of the sample. Sampling variance is one of the two components of sampling error associated with any sample survey that does not cover the entire population of interest. The other component of sampling error is coverage bias due to systematic nonobservation. The totality of sampling errors in all possible samples of the same size generates the sampling distribution for a given variable. Sampling variance arises because only a sample rather than the entire population is observed. The particular sample selected is one of a large number of possible samples of the same size that could ...
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## Methods Map
Research Methods
Copy and paste the following HTML into your website | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9442132115364075, "perplexity": 3134.48460058554}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998879.63/warc/CC-MAIN-20190619003600-20190619025600-00213.warc.gz"} |
https://dsp.stackexchange.com/questions/16423/time-varying-waveform/16428#16428 | # Time-varying waveform
My situation is as follows: i am trying to generate a waveform the hard way, by constructing the samples one by one and then saving the result to a .wav file using Python.
When the frequency is constant, everything is fine: i use $y(t) = \sin(2 \pi \cdot f \cdot t)$. However, if i change the frequency to be a function of time, things go wrong. If the function is a linear one of the form $f(t) = a + bt$, it still works. But if i choose, for example, $f(t) = 40 + 10 \sin(t)$, the max frequency increases over time, reaching a maximum higher than the expect 50Hz.
I have read something about instantaneous frequency, namely, this: Why does a wave continuously decreasing in frequency start increasing its frequency past the half of its length?. But doing the integral evaluation makes the sound even weirder. And the method i currently have works for linear function of time, so i figured there must be something else wrong.
I also tried to generate chunks of sound in the frequency i need, at each time, and then glue them together. I calculated the period of the oscillation, so that a chunk has as many samples as necessary to make a whole period on that frequency, so that there are no "jumps" between different frequencies. But generates a cracking sound in the sample.
Here is an example of the kind of a JavaScript implementation of the kind of sound i need: http://jsfiddle.net/m7US6/4/.
Any help would be appreciated. Thanks.
• Can you post the bit of your Python code that generates the waveform? May 22 '14 at 9:10
Consider the function $\sin(2\pi f t)$. When $2\pi f t$ goes from $0$ to $2\pi$ you get oscillation of the sine wave, $2\pi$ to $4\pi$, another, and so on. So every time the argument changes by $2\pi$ you get one oscillation.
Now lets plot $2\pi f(t) t$ where $f(t) = 10 + 10 \sin(2 \pi t)$. (I've modified it a bit to show the effect more):
As $t$ increases, this value changes faster and faster, meaning $\sin(2\pi f(t) t)$ will have higher and higher frequency.
Another way of thinking about it - the frequency is the derivative of $f(t) t$. For the example this is $20 \pi (\sin(2 \pi t)+2 \pi \mathbf{t} \cos(2 \pi t)+1)$. Note the $t$ multiplier in there - $t$ increases thus the frequency increases.
So what you really want is $\tfrac{d}{dt} f(t) t = 40 + 10 \sin(t)$ which gives $f(t) t = 40t - 10\cos(t) + c$ and for your example try $\sin(2\pi(40t - 10 \cos(t)))$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8100590705871582, "perplexity": 280.05785685839436}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363149.85/warc/CC-MAIN-20211205065810-20211205095810-00611.warc.gz"} |
https://dsp.stackexchange.com/questions/33666/autocorrelation-matrix-derivation | # Autocorrelation matrix derivation
Hi I am trying to derive the autocorrelation matrix and I am unsure about how exactly it works. I can see that the $4\times 1$ matrices result in the Hermitiain and Toeplitz matrix? Surely the only non-zero values in each of these is at $n = 0$ which is the variance?
Any help would be greatly appreciated!
You get a diagonal matrix because the values of $\eta(n)$ at different times (sample indices $n$) are uncorrelated, as shown in your first equation. This means that all off-diagonal elements of the auto-correlation matrix must be zero (they equal $E[\eta(n)\eta(n+m]$ for $m\neq 0$). Only the diagonal elements are non-zero, and they are equal to the variance. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.882736325263977, "perplexity": 179.9617551596172}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347385193.5/warc/CC-MAIN-20200524210325-20200525000325-00360.warc.gz"} |
https://www.physicsforums.com/threads/modulus-of-elasticity.332138/ | # Homework Help: Modulus of Elasticity
1. Aug 23, 2009
### boyblair
1. A Strain gauge records the following strain when a block of material is pulled:
Force(N) Strain(%)
100 0.01
1000 0.1
2000 0.179
3100 0.9
Area of block = 10cmx10cm
Work out the modulus of elasticity and when the material fails?
3. The attempt at a solution
Area over which force acts = 3.14x5x5
= 78.5cm2
In inches = 78.5/2.54
= 30.9 inch2
Stress = force/area
= 3100/30.9
=100.3 psi
E = Stress/Strain
= 100.3/0.9
= 111.4psi
2. Aug 23, 2009
### nvn
Perhaps first study how to compute the area of a square. If the material cross-sectional area is not a square, please clarify the problem statement. Also, try to avoid converting to a nondecimal, nonstandard, incoherent measurement system. Just convert cm to mm, and all stresses will be in MPa. Also study how to compute the slope of a straight line.
3. Aug 26, 2009
### boyblair
Area = 10x10
= 100cm2 or 10000mm2
E = Force applied x original length/area of cross section x change in length
= 1700 x 100/10000 x 0.17
= 100 Mpa
Not sure where I should use slope of straight line calculation.
4. Aug 26, 2009
### nvn
boyblair: Yes, use 10 000 mm^2 for the area. You could compute the stress (force divided by area) at point 1 or 2. Then, to obtain E, in your particular case, you could divide stress by strain at point 1 or 2. I don't quite understand from where you got 1700 N and 0.17 %, but you somehow got the correct answer, nonetheless (except the unit symbol for megapascal is spelled MPa).
5. Aug 26, 2009
### PhanthomJay
You are going to have to clarify the problem. Is the length of the bar given? Strain is a dimensionless quantity (change in length divided by original length). You indicate it as a percent; if that's the case, under the 100N load, for example, the strain is 0.0001, and the change in length (elongation) is 0.01 mm, if the bar is 100 mm long. The stress strain curve is linear for the first 2 load cases. I'd use one of those values for determining E. Then the strain goes way up (non linear) under the 3100N load. Is that the failure load? This value of strain under that load condition should not be used for determining E. Then be consistent in determining E. You can use either of your formulas
(E = stress/strain or E = FL/A(elongation)), but watch your units and values to use. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8739410638809204, "perplexity": 1754.6019847006705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593302.74/warc/CC-MAIN-20180722135607-20180722155607-00070.warc.gz"} |
http://export.arxiv.org/abs/2103.04397v2 | math.MG
(what is this?)
# Title: Intrinsic metrics under conformal and quasiregular mappings
Authors: Oona Rainio
Abstract: The distortion of six different intrinsic metrics and quasi-metrics under conformal and quasiregular mappings is studied in a few simple domains $G\subsetneq\mathbb{R}^n$. The already known inequalities between the hyperbolic metric and these intrinsic metrics for points $x,y$ in the unit ball $\mathbb{B}^n$ are improved by limiting the absolute values of the points $x,y$ and the new results are then used to study the conformal distortion of the intrinsic metrics. For the triangular ratio metric between two points $x,y\in\mathbb{B}^n$, the conformal distortion is bounded in terms of the hyperbolic midpoint and the hyperbolic distance of $x,y$. Furthermore, quasiregular and quasiconformal mappings are studied, and new sharp versions of the Schwarz lemma are introduced.
Comments: 25 pages, 1 figure Subjects: Metric Geometry (math.MG) MSC classes: 51M10 (Primary) 30C35 (Secondary) Cite as: arXiv:2103.04397 [math.MG] (or arXiv:2103.04397v2 [math.MG] for this version)
## Submission history
From: Oona Rainio [view email]
[v1] Sun, 7 Mar 2021 16:39:29 GMT (198kb,D)
[v2] Mon, 29 Mar 2021 08:49:45 GMT (20kb,D)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8833348155021667, "perplexity": 2034.442049434116}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103328647.18/warc/CC-MAIN-20220627043200-20220627073200-00552.warc.gz"} |
http://www.maa.org/publications/maa-reviews/complex-analysis-5 | # Complex Analysis
###### Elias M. Stein and Rami Shakarchi
Publisher:
Princeton University Press
Publication Date:
2003
Number of Pages:
392
Format:
Hardcover
Series:
Princeton Lectures in Analysis II
Price:
49.95
ISBN:
0-691-11385-8
Category:
General
[Reviewed by
Fernando Q. Gouvêa
, on
08/27/2005
]
See the reviews of the other volumes in the "Princeton Lectures on Analysis" series, on Fourier Analysis and on Real Analysis: Measure Theory, Integration, & Hilbert Spaces.
Foreword vii
Introduction xv
Chapter 1. Preliminaries to Complex Analysis 1
1 Complex numbers and the complex plane 1
1.1 Basic properties 1
1.2 Convergence 5
1.3 Sets in the complex plane 5
2 Functions on the complex plane 8
2.1 Continuous functions 8
2.2 Holomorphic functions 8
2.3 Power series 14
3 Integration along curves 18
4 Exercises 24
Chapter 2. Cauchy's Theorem and Its Applications 32
1 Goursat's theorem 34
2 Local existence of primitives and Cauchy's theorem in a disc 37
3 Evaluation of some integrals 41
4 Cauchy's integral formulas 45
5 Further applications 53
5.1 Morera's theorem 53
5.2 Sequences of holomorphic functions 53
5.3 Holomorphic functions defined in terms of integrals 55
5.4 Schwarz reflection principle 57
5.5 Runge's approximation theorem 60
6 Exercises 64
7 Problems 67
Chapter 3. Meromorphic Functions and the Logarithm 71
1 Zeros and poles 72
2 The residue formula 76
2.1 Examples 77
3 Singularities and meromorphic functions 83
4 The argument principle and applications 89
5 Homotopies and simply connected domains 93
6 The complex logarithm 97
7 Fourier series and harmonic functions 101
8 Exercises 103
9 Problems 108
Chapter 4. The Fourier Transform 111
1 The class F 113
2 Action of the Fourier transform on F 114
3 Paley-Wiener theorem 121
4 Exercises 127
5 Problems 131
Chapter 5. Entire Functions 134
1 Jensen's formula 135
2 Functions of finite order 138
3 Infinite products 140
3.1 Generalities 140
3.2 Example: the product formula for the sine function 142
4 Weierstrass infinite products 145
6 Exercises 153
7 Problems 156
Chapter 6. The Gamma and Zeta Functions 159
1 The gamma function 160
1.1 Analytic continuation 161
1.2 Further properties of T 163
2 The zeta function 168
2.1 Functional equation and analytic continuation 168
3 Exercises 174
4 Problems 179
Chapter 7. The Zeta Function and Prime Number Theorem 181
1 Zeros of the zeta function 182
1.1 Estimates for 1/s(s) 187
2 Reduction to the functions v and v1 188
2.1 Proof of the asymptotics for v1 194
Note on interchanging double sums 197
3 Exercises 199
4 Problems 203
Chapter 8. Conformal Mappings 205
1 Conformal equivalence and examples 206
1.1 The disc and upper half-plane 208
1.2 Further examples 209
1.3 The Dirichlet problem in a strip 212
2 The Schwarz lemma; automorphisms of the disc and upper half-plane 218
2.1 Automorphisms of the disc 219
2.2 Automorphisms of the upper half-plane 221
3 The Riemann mapping theorem 224
3.1 Necessary conditions and statement of the theorem 224
3.2 Montel's theorem 225
3.3 Proof of the Riemann mapping theorem 228
4 Conformal mappings onto polygons 231
4.1 Some examples 231
4.2 The Schwarz-Christoffel integral 235
4.3 Boundary behavior 238
4.4 The mapping formula 241
5 Exercises 248
6 Problems 254
Chapter 9. An Introduction to Elliptic Functions 261
1 Elliptic functions 262
1.1 Liouville's theorems 264
1.2 The Weierstrass p function 266
2 The modular character of elliptic functions and Eisenstein series 273
2.1 Eisenstein series 273
2.2 Eisenstein series and divisor functions 276
3 Exercises 278
4 Problems 281
Chapter 10. Applications of Theta Functions 283
1 Product formula for the Jacobi theta function 284
1.1 Further transformation laws 289
2 Generating functions 293
3 The theorems about sums of squares 296
3.1 The two-squares theorem 297
3.2 The four-squares theorem 304
4 Exercises 309
5 Problems 314
Appendix A: Asymptotics 318
1 Bessel functions 319
2 Laplace's method; Stirling's formula 323
3 The Airy function 328
4 The partition function 334
5 Problems 341
Appendix B: Simple Connectivity and Jordan Curve Theorem 344
1 Equivalent formulations of simple connectivity 345
2 The Jordan curve theorem 351
2.1 Proof of a general form of Cauchy's theorem 361
Notes and References 365
Bibliography 369
Symbol Glossary 373
Index 375 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8394814729690552, "perplexity": 2285.528562436373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246656168.61/warc/CC-MAIN-20150417045736-00127-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.intellecquity.com/what-is-a-coefficient-in-math/ | # What is a Coefficient in Math?
What is a coefficient in math? A coefficient is a number used to multiply a variable.
Example: 6z means 6 times z, and “z” is a variable, so 6 is a coefficient.
Variables with no number have a coefficient of 1.
Example: x is really 1x.
Sometimes a letter stands in for the number.
Example: In ax2 + bx + c, “x” is a variable, and “a” and “b” are coefficients.
STRUGGLING WITH MATH? Get the assistance, on-going support and tutors you need to learn faster, make fewer mistakes and get the grades you deserve. Learn More.
## Wikipedia Describes a Coefficient as;
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables.
For example, in
the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant coefficient. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term.
Often coefficients are numbers as in this example, although they could be parameters of the problem or any expression in these parameters. In such a case one must clearly distinguish between symbols representing variables and symbols representing parameters. Following René Descartes, the variables are often denoted by x, y, …, and the parameters by a, b, c, …, but it is not always the case. For example, if y is considered as a parameter in the above expression, the coefficient of x is −3y, and the constant coefficient is 1.5 + y.
When one writes,
it is generally supposed that x is the only variable and that a, b and c are parameters; thus the constant coefficient is c in this case.
Similarly, any polynomial in one variable x can be written as
for some positive integer , where are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient. For the largest (if any), is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial
is 4.
Some specific coefficients that occur frequently in mathematics have received a name. This is the case of the binomial coefficients, the coefficients which occur in the expanded form of, and are tabulated in Pascal’s triangle.
If you have any other Math related questions, drop us a comment and we will endeavour to get back to you or check if we already have answered it for you.
STRUGGLING WITH MATH? Get the assistance, on-going support and tutors you need to learn faster, make fewer mistakes and get the grades you deserve. Learn More. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9168024063110352, "perplexity": 545.2623785446058}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153816.3/warc/CC-MAIN-20210729043158-20210729073158-00362.warc.gz"} |
http://proceedings.mlr.press/v75/mangoubi18a.html | # Convex Optimization with Unbounded Nonconvex Oracles using Simulated Annealing
Oren Mangoubi, Nisheeth K. Vishnoi ;
Proceedings of the 31st Conference On Learning Theory, PMLR 75:1086-1124, 2018.
#### Abstract
We consider the problem of minimizing a convex objective function $F$ when one can only evaluate its noisy approximation $\hat{F}$. Unless one assumes some structure on the noise, $\hat{F}$ may be an arbitrary nonconvex function, making the task of minimizing $F$ intractable. To overcome this, prior work has often focused on the case when $F(x)-\hat{F}(x)$ is uniformly-bounded. In this paper we study the more general case when the noise has magnitude $\alpha F(x) + \beta$ for some $\alpha, \beta > 0$, and present a polynomial time algorithm that finds an approximate minimizer of $F$ for this noise model. Previously, Markov chains, such as the stochastic gradient Langevin dynamics, have been used to arrive at approximate solutions to these optimization problems. However, for the noise model considered in this paper, no single temperature allows such a Markov chain to both mix quickly and concentrate near the global minimizer. We bypass this by combining “simulated annealing" with the stochastic gradient Langevin dynamics, and gradually decreasing the temperature of the chain in order to approach the global minimizer. As a corollary one can approximately minimize a nonconvex function that is close to a convex function; however, the closeness can deteriorate as one moves away from the optimum. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9552269577980042, "perplexity": 329.40612547892215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593586.54/warc/CC-MAIN-20180722194125-20180722214125-00187.warc.gz"} |
http://tex.stackexchange.com/questions/34203/where-to-put-sty-and-cls-file-for-project/114919 | # where to put .sty and .cls file for project
I want to use my own .sty and .cls file in a project. I know of the following two possibilities of placing them:
• in my local tex tree (or the general tex tree, which is worse)
• in the same directory as the .tex file
Both of these I find unsatisfactory for the following reasons:
The local tex tree is unsatisfactory since the .sty file is specific for the project, so I want it to be close to the project, so I easily remember where it is etc. Also, I am using dropbox to synchronize the project files between different computers automatically, so the .sty file should be in the dropbox folder.
The 'same directory' is unsatisfactory since the project is in a directory with many subdirectories. The .tex files in all these subdirectories should be able to access the same .sty file.
What I would like to do is: create a subdirectory 'style files' in the project directory, where I put style and class files (there are several for the project), then tell tex somehow where to find them.
How can I do this?
I am using a recent TexLive on Mac OS X.6
I know there have been questions about placement of .sty files, but I did not find an answer to this question.
-
there is a third possibility and for projects the best one: put all in a texmf directory which is located inside your documents directory. Then do a export TEXMFHOME=texmf before running pdflatex or something else and the texmf tree will be searched first.
Inside this local texmf you must have the same TeX Directory Structure as usual: http://tug.org/tds For a local texmf tree you do not need to run texhash because files are searched recursively in that tree
-
Modern file systems know symbolic links. So you may have the class and package files at a subfolder of your project and link this subfolder to a tex/latex/ subfolder at the local (or private) TEXMF tree. With such a symbolic link you needn't change any configurations or environment variables.
-
Thanks. Using symbolic links like that seems the best solution to me. However, it doesn't work. I get a 'Missing \begin{document}' error message where tex starts to read the .sty file. I put a link to the actual style file inside $HOME/Library/texmf/tex/latex. How can I make it work? – Daniel Nov 13 '11 at 20:01 If files at $HOME/Library/texmf/tex/latex work, symbolic links should also. Note: You should use subdirectories at $HOME/Library/texmf/tex/latex, e.g. $HOME/Library/texmf/tex/latex/myproject/myclass.cls. Using files like $HOME/Library/texmf/tex/latex/myclass.cls is not recommended (but should nevertheless work). Does kpsewhich -var-value=TEXMFHOME list $HOME/Library/texmf/tex/latex? Does kpsewhich <yourfile> find the file? – Schweinebacke Nov 14 '11 at 6:51
Thanks again, Schweinebacke. If I put the .cls or .sty file at $HOME/Library/texmf/tex/latex , it works. Also, if I put the .cls file somewhere else and a link into$HOME/Library/texmf/tex/latex then it's ok. However, if I put the .sty file somewhere else and a link in \$HOME/Library/texmf/tex/latex , then latex says: Invalid character book^^@^^@^ etc., so apparently it cannot deal with the link correctly. kpsewhich does find the files. Any ideas? Merry Christmas! – Daniel Dec 24 '11 at 10:56
I would organize the project like this:
/project/files/....
/texmf/tex/latex/myclass/...
And then I would register /project/texmf as a new local texmf tree.
In miktex registering such a new root can be done either with miktex-settings or on the command line with initexmf --register-root=path\to\project\texmf.
In TeXLive you could e.g. add it to the TEXMFLOCAL variable in your texmf.cnf.
-
At the top level of the TeX Live distribution there is a texmf.cnf file that you can edit, if you don't want to set environment variables; the usual value for TEXMFHOME is, with an vanilla TeX Live
TEXMFHOME = ~/texmf
which stands for a texmf folder in your home. With the MacTeX installed TeX Live it is
TEXMFHOME = ~/Library/texmf
This kpathsea variable can be set to whatever you prefer:
TEXMFHOME = {~/Library/texmf,/Volumes/Dropbox/texmf}
would make TeX programs search also the texmf folder (which should be organized as a TeX tree) in the disk called Dropbox. You should know the precise path to give. Assuming this, you can put your classes and packages inside
/Volumes/Dropbox/texmf/tex/latex/myproject
(choose a better name) and all users that modify accordingly the texmf.cnf file on their machines will be able to access the tree.
Launching TeX programs as, say,
TEXMFHOME=:/Volumes/Dropbox/texmf// pdflatex filename
would be equivalent (notice the initial colon that means "append" the new tree after the value stated in texmf.cnf and the trailing // to mean "search recursively). Such a setting of TEXMFHOME can be of course stated in the overall environment. The texmf.cnf way is safer, as it doesn't depend on shell setup; the extra tree will be ignored if not found on the system.
-
Probably the easiest trick to do is to create aliases of every .sty and .cls file (located in your 'style files' directory) and put them wherever you need them. This is not (only) for saving space, but for unity.
-
Yes, ~/texmf` is a convinient solution for me after a deep searching of this topic. Here is my solution to use a single copy of preamble file, templates, and bib files:
https://github.com/caesar0301/CTexCustom
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9831314086914062, "perplexity": 3236.445126271162}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802765698.11/warc/CC-MAIN-20141217075245-00086-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://drorbn.net/index.php?title=AKT-09/HW3 | # AKT-09/HW3
Problem 1 With $T$, $B^+$ and $R$ as below, write $B^+$ as a composition of $T$ and two $R$'s, using he basic TG operations $d_e$, $u_e$, and $\#$.
Problem 2 Show that the "topological boundary" operator $\partial_T$ and the "crossing change" operator $x_T$ of the class of November 5 are compositions of the basic TG operations $d_e$, $u_e$, and $\#$ (you are also allowed to use "nullary" operations, otherwise known as "constants").
Problem 3 Write the third Reidemeister move R3 as a relation on $Z(B_+)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 28, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9637391567230225, "perplexity": 610.7156944133318}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583795042.29/warc/CC-MAIN-20190121152218-20190121174218-00315.warc.gz"} |
http://www.computer.org/csdl/trans/tc/1976/08/01674698-abs.html | Subscribe
Issue No.08 - August (1976 vol.25)
pp: 801-807
P.B. Andrews , Carnegie-Mellon University
ABSTRACT
Occurrences of literals in the initial clauses of a refutation by resolution (with each clause-occurrence used only once) are mated iff their descendants are resolved with each other. This leads to an abstract notion of a mating as a relation between: occurrences of literals in a set of clause-occurrences. The existence of many refutations with the same mating leads to wasteful redundancy in the search for a refutation, so it is natural to focus on the essential problem of finding appropriate matings.
INDEX TERMS
Automatic theorem proving, clause-occurrence, cycle, first-order logic, mating, merge, refutation, resolution, unsatisfiability.
CITATION
P.B. Andrews, "Refutations by Matings", IEEE Transactions on Computers, vol.25, no. 8, pp. 801-807, August 1976, doi:10.1109/TC.1976.1674698 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8871369957923889, "perplexity": 3977.341160579903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131295619.33/warc/CC-MAIN-20150323172135-00210-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/59322/how-can-we-pave-the-multiplicative-semigroup-mathbb-n-cdot | # How can we pave the multiplicative semigroup $(\mathbb N,\cdot)$?
Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s_1,...s_k\in S$ such that the sets $s_i\cdot W$ are pairwise disjoint and cover $S$. For instance, tiles for the semigroup $(\mathbb N,+)$ are given by the multiples of some fixed natural number. Here is my question: is there any explicit example of tile for the semigroup $(\mathbb N,\cdot)$?
A literature remark: I do not know if the notion of "tile" is already defined/used somewhere. I know the existence of "syndetic sets" that are pretty similar but different. I hope you like the name "tile"!
If someone of you wants to know more specifically my problem: Let $(S,\cdot)$ be a countable amenable semigroup and let $W\subset S$ be a subset. Let me denote by $\chi_W$ the characteristic function of $W$ and define the following numbers $$W^-=\inf [f(\chi_W),f \text{ bi-invariant mean}]$$ $$W^+=\sup[f(\chi_W), f \text{ bi-invariant mean}]$$ I am first of all interested in the case $W^-=W^+$ (this is why I'm interested in tiles). That would be the case when one can define some notion of intrinsic probability of the set $W$. Then it would be nice to find some description for the number $W^+-W^-$ and hopefully finding how the numbers $f(\chi_W)$ are distributed into the interval $[W^-,W^+]$.
-
Since you ask, a note regarding terminology: problems of this type (in particular for the integers but also other groups) can be found under the name 'cover' or 'covering system'. This is not exactly your 'tile', but I believe an 'exact covering' or 'exact 1-cover' would be; (exact) m-covers are also studied for other values of m. See, e.g., en.wikipedia.org/wiki/Covering_system – quid Mar 23 '11 at 17:30
So if there are $k$ integers $s_1,\dots,s_k \in \mathbb{N}$ so that the sets $s_i\ \cdot W$ are disjoint and cover $\mathbb{N}$, does that mean that $f(\chi_W)=\frac{1}{k}$? – Aaron Meyerowitz Mar 26 '11 at 4:02
Yes, if $f$ is an invariant mean.. – Valerio Capraro Apr 2 '11 at 16:02
Prime factorization provides an isomorphism between the semigroup $(\mathbb N,\cdot)$ and an infinite direct sum of copies of $(\mathbb N,+)$. So you can reduce your problem to the case that you already know how to solve.
Warning: my first $\mathbb N$ does not contain the element zero, whereas my second $\mathbb N$ does!
Concretely, here is an example of a tile of $(\mathbb N,\cdot)$: the set of all $n\in\mathbb N$ whose $p$-valuation is a multiple of $k$, where $p$ is a prime number, and $k$ is arbitrary.
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Consider $W$ consisting of those natural numbers in which a given prime $p$ occurs in the prime factorization with exponent divisible by some constant $k$. For example, if $W$ is numbers in which the exponent of 2 is even, then $W \cap 2W = \emptyset$ and $W \cup 2W = \mathbb{N}$.
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Thanks a lot. You know if there are any others? You're a statistic and so you might understand what I have in mind: sets such that $W^+=W^-$ have some kind of intrinsic probability. They should be the analogue of Haar-measurable subset of a compact group. It would be great to find a description of them. – Valerio Capraro Mar 24 '11 at 11:20
For each $b \gt 1$ there is a (unique) solutions to $\lbrace 1,b\rbrace W=W \cup bW=\mathbb{N}.$ So far the case of a prime power $b=p^k$ (particularly $b=2$) has been mentioned. The cases $b=6$ and $b=12$ are worth a look. I will call these 2-dimensional.
In a factorization $\mathbb{N}=VW,$ either of $V$ and $W$ uniquely determines the other. The 1-dimensional cases $V=\lbrace 1,p,p^2,p^3\rbrace$ , $V=\lbrace 1,p^{6},p^{12}\rbrace$ , $V=\lbrace 1,p,p^{10},p^{11},p^{20},p^{21}\rbrace$ hint at the most general 1-dimensional case.
The problem asks when we can factor the positive integers into two subsets $V,W$ so that each $n \in \mathbb{N}$ can be uniquely expressed as $n=vw$ i.e. $\mathbb{N}=VW$ where $V$ is finite. Since either set determines the other, it is sometimes easier to consider $V$ . Define the dimension of the factorization to be the number $d$ of primes dividing the members of $V$. Then factorizations correspond to tilings by translation (additive tilings) of ${(\mathbb{N}_{0})}^d$.
Here $W$ is thought of as the tile and $V$ as a finite set of placements or dilations. Some reflection shows that $V \cap W=\lbrace 1 \rbrace$. In particular $W=XY$ where $Y$ is the set of integers relatively prime to all the (prime divisors of) members of $V$ and $VX$ is the set of integers whose prime factors are divisors of members of $V$.
So $VW=\mathbb{N}$ is a factorization which we have refined to $VXY=\mathbb{N}$. It is worthwhile to consider factorizations of $\mathbb{N}$ into several (or even infinitely many) parts.
The one dimensional case was completely described by N. G. DeBruijn. Each coreesponds to a sequence of integers (exponents) each dividing the next (alternately, a mixed radix numeral system). An example will suffice to illustrate: Replace $b$ by $p$ to remember that we have a prime. and consider $1 \mid 2 \mid 10 \mid 30 \mid 90 \mid 180$ From this we build
$V=\lbrace 1,p \rbrace \lbrace 1,p^{10},p^{20} \rbrace\lbrace 1,p^{90} \rbrace=\lbrace 1,p,{p}^{10},{p}^{11},{p}^{20},{p}^{21},{p}^{90},{p}^{91},{p} ^{100},{p}^{101},{p}^{110},{p}^{111} \rbrace$ Then $VU$ is all the powers of $p$ from $p^0$ up to $p^{179}$ where $$U=\lbrace1,p^2,{p}^{4},{p}^{6},{p}^{8}\rbrace \lbrace 1,p^{30},p^{60} \rbrace=\lbrace 1,{p}^{2},{p}^{4},{p}^{6},{p}^{8},{p}^{30},{p}^{32},{p}^{34}, {p}^{36},{p}^{38},{p}^{60},{p}^{62},{p}^{64},{p}^{66},{p}^{68} \rbrace$$
• This could be sumarized as $$\frac{1-p^2}{1-p}\frac{1-p^{10}}{1-p^2}\frac{1-p^{30}}{1-p^{10}}\frac{1-p^{90}}{1-p^{30}}\frac{1-p^{180}}{1-p^{90}}\frac{1}{1-p^{180}}=\frac{1}{1-p}$$.
• This views things as working with polynomials (later in several variables) with integer $0,1$ coeffcients. Since things depend on the prime factorization of the exponents and later we have several variables corresponding to different prime bases, that notation seems too confusing.
• Additively, we have a 12 tile $\lbrace 0,1,10,11,20,21,90,91,100,101,110,111 \rbrace$ and 15 translations of it make an interval of length $180$. We could as well consider that 12 translations of the 15-tile $\lbrace 0,2,4,6,8,30,32,34,36,38,60,62,64,66,68 \rbrace$ make the same interval.
For dimension $d=2$ there are more cases. The article below by I. Niven classifies them all. The example above with $V=\lbrace 1,6 \rbrace$ relates to the factorization $$\lbrace 1,pq,p^2q^2,\cdots\rbrace \lbrace1,p,p^2,p^3,\cdots,q,q^2,q^3,\cdots \rbrace=\lbrace p^iq^j \mid i,j,\ge 0 \rbrace$$
The following references are from the excellent article Closed factors of normal Z-semimodules by Daniel A Marcus which classifies the most general case.and has some very worthwhile examples.
References
N. G. DeBruijn, On bases for the set of integers, Publ. Math., (Debrecen) 1 (1950), 232-242.
N. G. DeBruijn, On number systems, Nieuw Arch. Wisk., 4 (1956), 15-17.
N. G. DeBruijn, Some direct decompositions of the set of integers, Math. Comp., 18 (1964), 537-546.
R. T. Hansen, Complementing pairs of subsets of the plane, Duke Math. J., 36 (1969), 441-449.
C. T. Long, Addition theorems for sets of integers, Pacific J. Math., 23 (1967), 107-112.
I. Niven, A characterization of complementing sets of pairs of integers, Duke Math. J., 38 (1971)
S. K. Stein, Algebraic tiling, Amer. Math. Monthly, 81 (1974), 445-462.
S. K. Stein, Factors of some direct products, Duke Math. J., 41 (1974), 537-539.
C. Swenson, Direct sum subset decompositions of Z, Pacific J. Math., 53 (1974), 629-632.
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http://mathhelpforum.com/algebra/4325-quadradic-equation-print.html | • Jul 26th 2006, 10:29 AM
Brooke
Urgent!!!
Solve by using the quadratic formula $x^2-13x+30=0$
• Jul 26th 2006, 12:50 PM
galactus
Surely, Brooke, you can use the quadratic formula to solve this yourself.
Please say you can. It's just a matter of plugging and chugging.
That's why the formula is so handy. Just be careful with your signs.
• Jul 26th 2006, 01:26 PM
ThePerfectHacker
Quote:
Originally Posted by Brooke
Solve by using the quadratic formula $x^2-13x+30=0$
Note,
$a=1,b=-13,c=30$
Thus,
$x=\frac{b\pm \sqrt{b^2-4ac} }{2a}$
Thus,
$x=\frac{-13\pm \sqrt{169-4(1)(30)}}{2(1)}$
Thus,
$x=\frac{-13\pm \sqrt{169-120}}{2}$
Thus,
$x=\frac{-13\pm 7}{2}$
Thus,
$x=\frac{-13+7}{2}=-3,x=\frac{-13-7}{2}=-10$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8961396813392639, "perplexity": 2595.7085909834304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542060.60/warc/CC-MAIN-20161202170902-00217-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.zora.uzh.ch/id/eprint/180078/ | # Search for single production of vector-like quarks decaying to a top quark and a W boson in proton-proton collisions at $\sqrt {s} = 13$ TeV
CMS Collaboration; Canelli, Florencia; Kilminster, Benjamin; Aarrestad, Thea; Brzhechko, Danyyl; Caminada, Lea; de Cosa, Annapaoloa; Del Burgo, Riccardo; Donato, Silvio; Galloni, Camilla; Hreus, Tomas; Leontsinis, Stefanos; Mikuni, Vinicius Massami; Neutelings, Izaak; Rauco, Giorgia; Robmann, Peter; Salerno, Daniel; Schweiger, Korbinian; Seitz, Claudia; Takahashi, Yuta; Wertz, Sebastien; Zucchetta, Alberto; et al (2019). Search for single production of vector-like quarks decaying to a top quark and a W boson in proton-proton collisions at $\sqrt {s} = 13$ TeV. European Physical Journal C - Particles and Fields, C79:90.
## Abstract
A search is presented for the single production of vector-like quarks in proton–proton collisions at $\sqrt {s} = 13$ TeV. The data, corresponding to an integrated luminosity of 35.9 $fb^{−1}$, were recorded with the CMS experiment at the LHC. The analysis focuses on the vector-like quark decay into a top quark and a W boson, with one muon or electron in the final state. The mass of the vector-like quark candidate is reconstructed from hadronic jets, the lepton, and the missing transverse momentum. Methods for the identification of b quarks and of highly Lorentz boosted hadronically decaying top quarks and W bosons are exploited in this search. No significant deviation from the standard model background expectation is observed. Exclusion limits at 95% confidence level are set on the product of the production cross section and branching fraction as a function of the vector-like quark mass, which range from 0.3 to 0.03pb for vector-like quark masses of 700 to 2000GeV. Mass exclusion limits up to 1660GeV are obtained, depending on the vector-like quark type, coupling, and decay width. These represent the most stringent exclusion limits for the single production of vector-like quarks in this channel.
## Abstract
A search is presented for the single production of vector-like quarks in proton–proton collisions at $\sqrt {s} = 13$ TeV. The data, corresponding to an integrated luminosity of 35.9 $fb^{−1}$, were recorded with the CMS experiment at the LHC. The analysis focuses on the vector-like quark decay into a top quark and a W boson, with one muon or electron in the final state. The mass of the vector-like quark candidate is reconstructed from hadronic jets, the lepton, and the missing transverse momentum. Methods for the identification of b quarks and of highly Lorentz boosted hadronically decaying top quarks and W bosons are exploited in this search. No significant deviation from the standard model background expectation is observed. Exclusion limits at 95% confidence level are set on the product of the production cross section and branching fraction as a function of the vector-like quark mass, which range from 0.3 to 0.03pb for vector-like quark masses of 700 to 2000GeV. Mass exclusion limits up to 1660GeV are obtained, depending on the vector-like quark type, coupling, and decay width. These represent the most stringent exclusion limits for the single production of vector-like quarks in this channel.
## Statistics
### Citations
Dimensions.ai Metrics
5 citations in Web of Science®
3 citations in Scopus®
### Altmetrics
Detailed statistics | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9950772523880005, "perplexity": 2750.0641239290126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370505359.23/warc/CC-MAIN-20200401003422-20200401033422-00360.warc.gz"} |
https://www.krellinst.org/csgf/conf/2006/abstracts/sallerson?width=70%25&height=70%25 | ### A Numerical Study of the Kelvin-Helmholtz Instability on Large Amplitude Internal Waves
##### Amber Sallerson, University of North Carolina - Chapel Hill
Internal gravity wave dynamics in stratified fluids are recently experiencing increased attention due in part to the role played by these waves in environmental issues such as near-coastal dynamics; however, great difficulties arise in collecting data in either field or lab experiment and thus numerical simulations can be very valuable in shedding light on observed phenomenon. A conservative projection method for the variable density Euler equations is implemented to simulate numerically the generation and propagation of internal solitary waves. By using parameters and dimensions from a set of laboratory experiments involving fresh water and brine separated by a thin diffused interface, the code is validated against experimental data as well as theoretical results for regimes that include near maximum amplitude waves. The wave-induced shear instabilities that can be observed in these regimes are captured by the numerical simulations, and are studied in detail by initializing the dynamics with traveling wave solutions computed through an iterative scheme.
Abstract Author(s): Amber Sallerson and Roberto Camassa | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8851341009140015, "perplexity": 970.0006993108035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583516117.80/warc/CC-MAIN-20181023065305-20181023090805-00241.warc.gz"} |
http://physics.stackexchange.com/questions/60197/calculating-work-done-by-friction | # calculating work done by friction
I want to calculate the work done by friction if the length $L$ of uniform rope on the table slides off. There is friction between the cord and the table with coefficient of kinetic friction $\mu_k$.
$$W = \int F \cdot d \vec{s}$$ I think it would be: $$W_{fr} = \frac M L g \int_{0}^{L} dx$$
But the solutions (which could be mistaken) say: $$dW_{fr} = \mu_k \frac M L g \, x \, dx$$ which is then integrated.
Should there be an $x$ in the integral? I don't think there should be because you are summing up over an infinitesimal displacement $dx$ and the force of friction is not proportional to the displacement at any instant (I think).
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Do you know that your expression is dimensionally incorrect? – ABC Apr 6 '13 at 3:27
Let $x$ denote the length of the rope that is on the table, then $$m(x) = \frac{M}{L}x$$ is the mass of the rope on the table. It follows that the force of friction on the rope on the table is $$f(x) = \mu_k m(x) g = \mu_k\frac{M}{L}xg$$ if the rope moves an amount $dx$ then the work done by friction is $$dW = f(x)dx = \mu_k\frac{M}{L}gx dx$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.977662205696106, "perplexity": 96.64461249698623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657137698.52/warc/CC-MAIN-20140914011217-00102-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/statug_iclifetest_examples01.htm | # The ICLIFETEST Procedure
### Example 62.1 Analyzing Data with Observations below a Limit of Detection
Data that have certain values below a limit of detection (LOD) are frequently encountered by toxicologists and environmental scientists. Such data are usually analyzed by imputing the unobserved values by LOD/2 or LOD/. This type of practice often raises the question of whether the population distributions can be estimated without bias. Gillespie et al. (2010) propose using a reverse Kaplan-Meier estimator, or equivalently, Turnbull’s method (1976) by treating the unobserved data as left-censored. When the assumption of independent censoring holds, these estimators can unbiasedly estimate the population distribution functions.
The following hypothetical data have two values, 3 and 10, that are below the limit of detection:
data temp;
input C1 C2;
datalines;
. 3
4 4
6 6
8 8
. 10
12 12
;
The following statements invoke PROC ICLIFETEST to estimate the population distribution function by using Turnbull’s method:
proc iclifetest data=temp method=turnbull plots=survival(failure)
impute(seed=1234);
time (c1,c2);
run;
Specifying the PLOTS=SURVIVAL(FAILURE) option requests a failure probability plot. Results are shown in Output 62.1.1. Note that because the first Turnbull interval is , the failure probability function is undefined within that interval.
Output 62.1.1: Failure Probability Plot for Fictitious Nondetection Data
Output 62.1.2 presents the estimated failure probability, with standard errors that are estimated by the method of multiple imputations.
Output 62.1.2: Cumulative Probability Estimates
The ICLIFETEST Procedure
Nonparametric Survival Estimates
Probability Estimate Imputation
Standard
Error
Lagrange
Multiplier
Time Interval Failure Survival
3 4 0.2083 0.7917 0.1811 0.0000
4 6 0.4167 0.5833 0.2179 0.0000
6 8 0.6250 0.3750 0.2099 0.0000
8 12 0.8333 0.1667 0.1521 0.0000
12 Inf 1.0000 0.0000 0.0000 0.0000 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9213862419128418, "perplexity": 2904.5642297981703}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527474.85/warc/CC-MAIN-20190722030952-20190722052952-00262.warc.gz"} |
https://www.physicsforums.com/threads/how-directional-is-a-single-photon.789581/ | # How directional is a single photon?
1. Dec 29, 2014
### intervoxel
A photon's momentum vector points in the direction of its propagation but interacts with particles off its axis. How this directional preference is revealed by QM? Is there an ontological picture of the photon's propagation?
2. Dec 29, 2014
### bhobba
Why do you say that?
Thanks
Bill
3. Dec 29, 2014
### naima
A photon may have a definite momentum direction (plane wave) bur when it passes thru a pinhole, the wave becomes spherical.
Then every direction have the same probability.
4. Dec 30, 2014
### intervoxel
O.k., so this is not the case since the direction is defined at the moment of the interaction with the target (or if the recoiled source hits something) and the consequent simultenous definition of the recoil direction of the source by entaglement, right? My question remains though: how reflection is possible or how hits on one side of the mirror are privileged?
Hope this reply makes sense :)
5. Dec 30, 2014
### bhobba
Why do you think you can extrapolate a direction from where it hits a target - think about the double slit - with both slits open you cant say which slit it went through or what direction it had - indeed even if it had the property of direction.
Why do you think a photon has the classical property of 'propagation'?
In QM its much better not to ascribe properties like propagation etc except when observed.
Thanks
Bill
6. Dec 30, 2014
### bhobba
See Feynman - QED Strange Theory Of Light And Matter.
Thanks
Bill
7. Feb 19, 2015
### intervoxel
So you mean that, when observed, the received photon's momentum does not necessarily has to be aligned with the photon's origin position? It could be pointing for example at an angle of 90 degrees?
I suppose that for many photons most of received momentum is aligned so light sailing is possible.
8. Feb 19, 2015
### Staff: Mentor
Did you follow through on Bhobba's suggestion that you find and read Feynman's book? If not, do so.
9. Feb 21, 2015
### bhobba
Momentum aligned with position?
They are independent variables.
Thanks
Bill
10. Feb 21, 2015
### naima
I think that his question is about the conservation of the momentum.
11. Feb 21, 2015
### bhobba
The very essence of QM is all you can predict is averages and momentum conservation is preserved on the average.
Why exactly is that an issue?
Thanks
Bill
Similar Discussions: How directional is a single photon? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.877585232257843, "perplexity": 1885.3188602819373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104634.14/warc/CC-MAIN-20170818102246-20170818122246-00589.warc.gz"} |
http://quant.stackexchange.com/questions/7808/convexity-adjustment-for-a-forward-swap-rate?answertab=oldest | # Convexity adjustment for a forward swap rate
I recently heard that for a forward swap rate (for example, the fixed rate of a swap that will start in one year and end in five years), I need to do a convexity adjustment in order to get the right number. Is it true, and if so why?
-
The question in this form is incomplete. The swap rate alone does not need any convexity adjustment. You have to specify how this rate is paid. – Christian Fries Apr 26 '13 at 18:16
Yes, an adjustment has to be made and the reason is that a forward curve now will evolve and not be the same as the future spot curve. For example, a one year forward today is not equal to your spot rate a year hence. So spot curve discount factors have to be adjusted or directly replaced through the forward DFs.
-
A forward starting swap needs adjustment, yes, but in discounting terms rather than convexity. IRSs are linear AFAIK; if expected Libors rose 1bp, that would raise the par fixed rate 1bp. I'm assuming no optionality in the swap. – Phil H Apr 26 '13 at 9:02
Sorry, I was very vague and it was probably misleading. Convexity adjustments still have to be made but its more during the libor forward curve built-out (off the euro$futures). Thanks for pointing that out. Will edit my answer. – Matt Wolf Apr 26 '13 at 10:22 add comment First of all it is not clear what exactly you mean by right number, you definitely do not adjust forward swap rate. You probably mean adjusting euro dollar futures contract rates so that you can later use these values to fit the swap/forward libor curve. Reason for adjustment is simple. If you are short ED futures and rates go higher futures price drops and you make money. Clearing house of exchange reimburses you excess margin and you can reinvest them at higher rate. If rates go lower you have to put extra cash into your margin account, you can borrow this money at lower rate. Assuming some distribution and forward rate model(and here things vary from simple vasicek model to extremely complicated example Peterbarg) one can compute how much this advantage is in dollars and convert it to basis points. Pure intuition tells that longer is the expiry higher is the adjustment also higher is the vol higher is the adjustment. So the end result is that being short gives you advantage and market participants are aware of it and penalize short side by that amount. - The convexity adjustment needed for futures comes from the margining applied to the (undiscounted) future price. In contrast, swaps are collateralized by discounted value, such that a future-like convexity adjustment does not apply. However, if a forward swap rate is paid in an unnatural way (like in a CMS), a convexity adjustment applies. – Christian Fries Apr 26 '13 at 18:45 add comment For a vanilla forward-start swap, I would agree with imachabeli; convexity is an adjustment for the non-linearity of the quoted fixed rate dependence on the floating note. If expected Libors rise 1bp, the fixed leg can be increased 1bp to compensate. Convexity adjustments are made as standard to interest rate futures (i.e. 3m); with the next futures date (Jun13) 2 months away, the front contract convexity adjustment is less than .1bp, so it makes no real difference. By contrast at 5y (Jun18), the 21st contract convexity is around 15bp. It is possible the question is about swap futures, which deal over the futures dates, and which are therefore forward starting. As these are also futures, and deliver margin payments, there is a convexity adjustment to be made as per 3m futures. - good point made at the end, indeed the question may actually be about swap futures. – Matt Wolf Apr 26 '13 at 10:31 OK. If this is about swap futures, then he should be more precise... – Christian Fries Apr 26 '13 at 18:46 add comment Given an index$t \mapsto S(t)$(this may be a forward swap rate) and some value process$t \mapsto A(t)$(this may be a swap annuity) we assume that$S/A$is a traded product (which is true if$S$is the forward swap rate and A is the corresponding (!) swap annuity. Then the future payoff$S(T) \cdot A(T)$can be values as$S(t) \cdot A(t)$(since$S$is a martingale under the measure$Q^A$). Now, if we consider the payoff$S(T) \cdot P(T)$(say for example if$P$is the zero coupon bond with maturity$T$) then the value can be expressed as$S'(t) \cdot P(t)$where$S'(t)$is the so called convexity adjusted rate, that is$S'(t) = E(S(T) \cdot \frac{P(T)/P(t)}{A(T)/A(t)})$(with expectation under$Q^A\$). The convexity adjustment is a correlation term coming from the correlation of the index to the change of the payment.
That said: If you need a convexity adjustment depends on how the index is paid.
-
Most likely the question is about CMS rate convexity adjustment. i.e. today value of a swap rate that fixes at some future time T.
Mathematically, the adjustment arises from different measures (annuity versus forward measure).
This is a good reference http://www.math.nyu.edu/~alberts/spring07/Lecture4.pdf
As a rule of thumb, the size of the adjustment depends on
1. CMS maturity
2. Level of vol
3. skew
For the latter the classical reference is http://www.gorillasci.com/documents/convexity.pdf
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9458487629890442, "perplexity": 2036.7634387132014}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00293-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://winnerscience.com/2012/02/22/gauss-law-electrostatics-derivation/ | Gauss law for electrostatics derivation
Do you know that if we know the charge distribution, then we can calculate the electric field due to this charge distribution? This is based on the gauss law of electrostatics.
Statement. Total electric flux through any closed surface, is equal to 1/ε times the total charge enclosed by the surface.
ФE=∫E.dS=q/ε
Where ε is the permittivity of the medium (for free space ε=ε0),
So ФE=∫E.dS=q/ε0
Where ∫E.dS is surface integral over the closed surface and q is the charge present in the closed surface S
The imaginary closed surface is called Gaussian Surface. If the surface encloses a continuous charge distribution then q is replaced by the intergal ∫ρdV,where ρ is the volume charge density.
Proof of gauss law of electrostatics:
Consider a source producing the electric field E is a point charge +q situated at a point O inside a volume enclosed by an arbitrary closed surface S. let us consider a small area element ds around a point P on the surface where the electric field produced by the charge +q is E. if E is along OP and area vector dS is along the outward drawn normal to the area element dS, (Try to make the diagram yourself),
Then the electric flux dФ through the area element dS is given by
dФ=E.dS=EdS cosθ
Where θ is the angle between E and dS and it is zero degree, therefore
dФ = EdS (1)
Since the source producing E at dS is a point charge +q at O, therefore
E=1/4πε0 q/(OP)2=1/4πε0q/r2 (OP=r)
Substituting this value of E in equation (1),we get
dФ=E.dS=q/4πε0dS/r2 (2)
Hence , total electric flux Ф through the entire closed surface S would be
∫dФ=q/4πε0 r2∫dS (3)
But ∫dS = 4πr2
Therefore, equation(3) becomes
Ф=∫E.dS= q4πε0 r2 /4πε0 r2
Or Ф=∫E.dS= q/ε0 (4)
Equation (4) represents Gauss law for electrostatics for a single point charge. If the source producing the electric field has more than one point charges such as +q1,+q2,+q3,-q4.-q5,-q6………..etc,then the total flux due to all of them would be the algebric sum of all the fluxes as,
Ф=∫E.dS=1/ε0(q1+q2+q3-q4-q5-q6…….)
Or Ф=∫E.dS=∑q/ε0
Hence proved a charge outside the Gaussian surface would contribute nothing to the electric flux. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9906821846961975, "perplexity": 738.5770636072241}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400244231.61/warc/CC-MAIN-20200926134026-20200926164026-00172.warc.gz"} |
http://mathhelpforum.com/calculus/70628-proving-limits-help-please.html | 1. ## Proving limits help please???
Hi, can anyone give me a hand with this question, i honestly dont know what to do!!
Prove that lim (n tends to infinity) 1/an = 0 when an is a sequence with an > 0 for n being natural, and lim (n tends to infinity) an = infinity.
2. Originally Posted by Bexii
Hi, can anyone give me a hand with this question, i honestly dont know what to do!!
Prove that lim (n tends to infinity) 1/an = 0 when an is a sequence with an > 0 for n being natural, and lim (n tends to infinity) an = infinity.
1 divided by a very big number is a very small number.. so if you divide something by infinity, you will get infinite small number = 0.
3. i was told i had to use definitions in my answer...
4. $\forall \,\epsilon >0,\,\exists \,k\in \mathbb{N}\mid n\ge k\implies \left| \frac1n\right|<\epsilon .$
But this is quite straightforward, so we get that $\frac1n<\epsilon$ which means that $n>\frac1\epsilon,$ so for any $k\in\mathbb N$ being greater than $\frac1\epsilon,$ the above condition is fulfilled. For example, it's enough to take $k=\left\lfloor \frac{1}{\epsilon} \right\rfloor +1.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8585698008537292, "perplexity": 346.2429418852579}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718423.28/warc/CC-MAIN-20161020183838-00209-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/621224/question-arising-from-quantum-mechanics-concerning-groups-and-symmetries?answertab=active | # Question arising from quantum mechanics concerning groups and symmetries
I'm trying to understand a calculation my professor did in my quantum mechanics script. Here it is:
Each rotation $R \in O(3)$ induces a unitary transformation in $L^2(R^3)$, i.e. the space of square integrable real functions. $U( R ): \psi(\vec{x}) \rightarrow \psi(R^{-1} \vec{x})$.
The professor goes on to show that $R \rightarrow U( R)$ is a unitary representation of O(3). He does that by showing $U( 1 ) = id$ and $U(R_2)U(R_1) = U(R_2 R_1)$. The first one is trivial while the second one seems to be wrong for the definition of $U(R )$ above.
$U(R_2)U(R_1) \psi(\vec{x}) = U(R_2)\psi(R_1^{-1} \vec{x}) = \psi(R_2^{-1} R_1^{-1} \vec{x}) = \psi((R_1 R_2)^{-1}\vec{x})$. Which isn't the same as $U(R_2 R_1)$.
Does anybody know where I went wrong?
Second question is:
It says in the script "$U(\lambda) = U(R(\lambda))$ produces a 1 parameter unitary group. Its generating function A is: $(A\psi)(\vec{x}) = i\hbar \frac{d}{d \lambda} \psi(R(-\lambda)\vec{x})$". Without really explaining where this comes from. What definition do I need to understand that equation? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9526199698448181, "perplexity": 223.7513781281899}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500833525.81/warc/CC-MAIN-20140820021353-00063-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://mathalino.com/forum/calculus/family-plane-curves | # Family of Plane Curves
Find the differential equation of parabolas with vertex and focus on the x-axis.
Rate:
### Since the vertex and focus
Since the vertex and focus are on the $x-axis$, then their coordinates are in the form $(a,0)$ and $(b,0)$ respectively. The general equation of a parabola with axis of symmetry at the $x-axis$ is $(x-a)^2 = 4ky$. There will be two arbitrary constants since there is no fixed vertex/focus and length of latus rectum.
$$\begin{eqnarray} (x-a)^2 &=& 4ky\\ (x-a)^2 y^(-1) &=& 4k\\ 2(x-a)y^(-1) - (x-a)^2 y^(-2)y'&=& 0\\ 2y - (x-a)y' &=& 0\\ x - \dfrac{2y}{y'} &=& a\\ 1 - \dfrac{2y' y' - 2yy''}{(y')^2} &=& 0\\ (y')^2 - (2(y')^2 - 2yy'') &=& 0\\ -(y')^2 + 2yy'' &=& 0\\ (y')^2 - 2yy'' &=& 0 \rightarrow Answer \end{eqnarray}$$
Rate: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8074474334716797, "perplexity": 269.6390500432413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703529128.47/warc/CC-MAIN-20210122051338-20210122081338-00444.warc.gz"} |
https://en.famagusta.news/news/ikonomia/ta-agatha-me-ti-megalyteri-afxisi-timon-ton-septemvrio/ | # The goods with the biggest price increase in September
Inflation in September 2022 increased at a rate of 8,7%
The Consumer Price Index in September 2022 decreased by 0,22 points to 113,12 points compared to 113,34 points in August 2022. Inflation in September 2022 increased at a rate of 8,7%, according to the Statistics Office.
For the period January – September 2022, the CPI recorded an increase of 8,4% compared to the corresponding period last year.
The biggest changes in the economic categories compared to September 2021 were recorded in Electricity with a percentage of 59,9% and in Petroleum products with a percentage of 23,4%. In relation to the previous month, the biggest negative change was seen in Petroleum products with a rate of 6,3%.
Analysis of percentage changes
Compared to September 2021, the biggest changes were observed in the Housing, Water, Electricity and LPG (27,3%) and Transport (13,8%) categories.
In relation to August 2022, the biggest change occurred in the Clothing and Footwear category (7,7%).
For the period January – September 2022 compared to the corresponding period last year, the largest changes were presented in the categories Housing, Water Supply, Electricity and LPG (21,9%) and Transportation (18,3%).
Impact analysis in units
The categories Housing, Water Supply, Electricity and LPG (2022) and Transport (2021) had the greatest impact on the change in the CPI of September 3,48 compared to September 2,19.
The biggest negative impact on the CPI change compared to the previous month was the category Transportation (-1,03).
Electricity (2022) and Petroleum (2021) had the greatest impact on the September 2,28 CPI change compared to the September 2,10 index.
Finally, Oil Products (-0,75) had the greatest negative effect on the change in the September 2022 CPI compared to the corresponding index of the previous month.
Source: economytoday | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8955100178718567, "perplexity": 2967.034465311389}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499819.32/warc/CC-MAIN-20230130133622-20230130163622-00366.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/CLONE-afaf42be-9820-4186-8d76-e738423175bc/chapter-13-for-thought-and-discussion-page-245/2 | Chapter 13 - For Thought and Discussion - Page 245: 2
Work Step by Step
When the spring constant is doubled, the frequency goes up by a factor of $\sqrt 2$ since frequency is directly related to the square root of the spring constant. Since the frequency is inversely related to the square root of the mass, doubling the mass will cause the frequency to decrease by a factor of $\sqrt 2$.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.940523087978363, "perplexity": 281.3973089352374}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986670928.29/warc/CC-MAIN-20191016213112-20191017000612-00231.warc.gz"} |
https://www.arxiv-vanity.com/papers/1409.0826/ | # Collective excitations in deformed sd-shell nuclei from realistic interactions
Bastian Erler Robert Roth Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
July 14, 2022
###### Abstract
Background
Collective excitations of nuclei and their theoretical descriptions provide an insight into the structure of nuclei. Replacing traditional phenomenological interactions with unitarily transformed realistic nucleon-nucleon interactions increases the predictive power of the theoretical calculations for exotic or deformed nuclei.
Purpose
Extend the application of realistic interactions to deformed nuclei and compare the performance of different interactions, including phenomenological interactions, for collective excitations in the sd-shell.
Method
Ground-state energies and charge radii of Ne, Si and S are calculated with the Hartree-Fock method. Transition strengths and transition densities are obtained in the Random Phase Approximation with explicit angular-momentum projection.
Results
Strength distributions for monopole, dipole and quadrupole excitations are analyzed and compared to experimental data. Transition densities give insight into the structure of collective excitations in deformed nuclei.
Conclusions
Unitarily transformed realistic interactions are able to describe the collective response in deformed sd-shell nuclei in good agreement with experimental data and as good or better than purely phenomenological interactions. Explicit angular momentum projection can have a significant impact on the response.
###### pacs:
21.60.Jz,24.30.Cz,27.30+t,21.30.Fe
## I Introduction
Excited states are one of the main sources of information on the structure of atomic nuclei and are the subject of constant research in theory and experiment. Collective excitations constitute a specific class of excitations, which probe the global structure of the nucleus and allow for a geometric interpretation in terms of oscillations of intrinsic nuclear shapes. A well tested approach to describe collective excitations is the Random Phase Approximation (RPA) Rowe70 , where excited states are described by coherent particle-hole excitations starting from a mean-field-type ground state usually obtained within the Hartree-Fock (HF) approximation.
Traditionally, HF and RPA calculations are being performed with phenomenological interactions. The form of such interactions is guided by symmetry considerations and computational simplicity, and their parameters are typically fitted to binding energies and radii of a series of nuclei within the chosen approximation for treating the many-body problem, e.g., the HF approximation. Popular phenomenological interactions are the various Skyrme forces, e.g. Skyrme58 ; Chabanat98 ; Tondeur00 , and the Gogny D1S interaction DG80 . These interactions allow for efficient HF calculations of nuclear binding energies and other ground-state properties and typically yield good agreement with experiment. However, when applying these interactions in another many-body scheme, like RPA, or to other observables, like the collective response, their performance might deteriorate.
An alternative approach, which we follow in this work, uses unitarily transformed realistic nucleon-nucleon interactions, like the phenomenological Argonne V18 (AV18) Wiringa95 , CD Bonn Machleidt01 or Nijmegen Stoks94 interactions, or interactions derived from chiral effective field theory (EFT) Epelbaum02 ; Entem03 ; Epelbaum05 ; Machleidt10 . These interactions are not tuned to specific properties of finite nuclei and are not determined within a specific approximation scheme, but rather are fit to two-nucleon phase-shifts and deuteron properties in exact calculations. Thus, these interactions are universal and can be employed in different many-body approaches to describe different states and observables on equal footing. The unitary transformations help to improve the convergence of the many-body calculations with respect to the many-body model space. At the same time, the quality of simple approximations, like the HF approximation, is improved Roth10 ; Roth06 . The interactions used in this work are based on the AV18 potential, transformed either with the Unitary Correlation Operator Method (UCOM) or the Similarity Renormalization Group (SRG) method Roth10 ; Roth08 .
During the past decade, routine calculations for deformed nuclei within the HF and RPA framework have become possible. So far, these calculations have only been carried out with phenomenological interactions Peru08 ; Arteaga08 ; Losa10 . This work is the first application of the HF-RPA treatment with unitarily transformed realistic interactions to deformed nuclei. The extension to intrinsically deformed nuclei opens a new domain of application away from semi-magic nuclei and allows for a detailed study of the impact of deformation on collective modes. Due to the symmetry breaking in the mean-field density, the HF state is no longer an eigenstate of the angular-momentum operator and the proper symmetry has to be restored by an explicit angular-momentum projection.
In this work, we study nuclei in the sd-shell. These nuclei already exhibit strong deformations with only a small number of nucleons. We focus on three even-even self-conjugate nuclei, the prolate nuclei Ne and S, and the oblate nucleus Si. Ne and Si show a purely axial deformation, while S also has a small triaxial component. The deformation in these nuclei is driven by -cluster correlations in the ground states, which makes them a particularly interesting candidate for the study of deformation effects on the collective response. At the same time the presence of -clustering suppresses the pairing correlations in these open-shell nuclei, so that a simple HF approach without explicit pairing is applicable. For these first calculations we restrict ourselves to axial deformations, which simplifies the explicit angular-momentum projection significantly.
## Ii Hartree Fock method
### ii.1 Formalism
To obtain a nuclear ground-state, the starting point for the RPA, we employ the HF method. We use -coupled spherical harmonic oscillator (HO) states as the computational basis. An HO state is fully determined by the quantum numbers , , , and . In a spherical HF implementation, only states with different but equal , and can mix. Deformed HF states can be obtained, if the HF single-particle states are also allowed to contain contributions of different total and orbital angular-momentum , and
|αmt⟩=∑aCαmta|amt⟩. (1)
The HO quantum numbers , , and are combined in the index . The HF state index covers the same range as the HO index , but does not have the same physical meaning. If a nucleus attains an axially symmetric deformation, the HF single-particle states are a superposition of HO states with different , , . The single-particle projection quantum numbers remain good quantum numbers, and their sum in the HF state, denoted by , defines the angular-momentum projection onto the symmetry axis of the intrinsic frame, which is the only remaining good quantum number in the intrinsic frame (except for isospin).
To obtain quantities in the lab-frame, where the ground state is an eigenstate of the total angular-momentum operator , angular-momentum projection has to be employed Loewdin55-1 ; Brink66 ; Ring80 . The angular-momentum projected energy of an axially symmetric nucleus is
(2)
The projection operator for axial-symmetry is given by
^PJMK =2J+12∫1−1dJMK(β)eiβ^Jyd(cosβ), (3)
where denotes the reduced Wigner-Functions.
The ground state is obtained by minimizing the projected ground-state energy in a so called variation-after-projection approach. This procedure is approximated by carrying out a number of constrained HF calculations with the modified Hamiltonian
^H′=^H−λ^Q. (4)
Among these solutions, the one with the lowest projected ground-state energy is selected. The quadrupole operator is a natural choice for the constraint, as this is the dominant collective degree of freedom for axially deformed nuclei. We refer to this treatment as approximate variation-after-projection.
### ii.2 Calculation details
The HO basis used for our calculations is truncated with respect to the principal oscillator quantum number , with an additional truncation for the orbital angular-momentum . Unless stated otherwise, we use and . The ground-state energies for are converged to within less than . The optimal harmonic oscillator lengths are determined by a minimization of the ground-state energy over a set of discrete oscillator lengths, the values used throughout this work are summarized in Tab. 1.
We use a total of four interactions in this work. Three are based on unitary transformations of the AV18 potential. UCOM(VAR) is a pure two-body interaction, transformed with the UCOM, where the correlation operators are determined from a variational approach. It was first published in 2005 Roth05 and has since been used in a number of calculations Roth06 ; Paar06 ; Papakonstantinou07 ; Papakonstantinou10 ; Papakonstantinou11 and is also described in Roth10 . Since the UCOM(VAR) interaction does not reproduce the correct charge radii (cf. section II.3), other interactions have been developed, which go beyond a pure two-body interaction, e.g., the SSand S-UCOM(SRG) interactions introduced in Gunther10 . The SSinteraction is transformed via an SRG flow-evolution. For the S-UCOM(SRG) interaction, the solution of an SRG flow-evolution is used to determine the correlation operators for the UCOM transformation. In both interactions, the unitary transformation only acts on partial waves containing relative S waves, i.e., the and the coupled - partial waves. To include effects of three-body forces, these interactions also contain a simple phenomenological three-body contact force . The strength is adjusted to reproduce the charge radius systematics for doubly-magic nuclei from to , details can be found in Ref. Gunther10 . First applications of these interactions in the RPA framework for collective excitations in spherical nuclei are discussed in Ref. Gunther14 . The contact interaction is used as a computationally efficient substitute for realistic three-body forces. In mean-field calculations, contact interactions of arbitrary particle rank can be reduced to lower-order interactions which only depend on integer powers of the ground-state density. The inclusion of realistic three-body interactions, e.g. the ones from chiral EFT, is still extremely challenging for deformed nuclei. As an example of a popular phenomenological interaction, we also included the Gogny D1S interaction DG80 into our studies.
### ii.3 Results
We start by discussing the HF results for the ground states of Ne, Si and S with and without angular-momentum projection. Figure 1 shows the dependence of the ground-state energy on the deformation parameter. Here, we used a truncation of to reduce the computational cost. At this level, the ground-state energy is converged within , which is sufficient for the presentation in Fig. 1. The ground-state energy of Ne varies continuously with the deformation, with a minimum at a strong prolate deformation. For Si and S, we see some irregularities in the curves, which are an artifact of the constrained minimization procedure. From a practical point of view this does not pose a problem, as long as the discontinuities are sufficiently far away from the energy minimum, which is always the case. Si shows two practically degenerate minima, one for oblate and one for prolate deformation. For the pure two-body interaction UCOM(VAR) and the Gogny D1S interaction Peru08 , the oblate minimum has a slightly lower energy and both minima are connected through triaxial deformation. However, this is not the case for the SSand S-UCOM(SRG) interactions. Since the oblate shape of Si is well established, we focus on this solution for our RPA calculations. S shows three local minima, one at oblate deformation, one at a moderate prolate deformation and one at a strong prolate deformation. For all interactions, the minimum with the moderate prolate deformation is the absolute minimum and it is well separated from the other minima. It should be noted that the general dependence of the ground-state energy on the deformation is very robust under changes of the underlying interaction.
The intrinsic density distributions for the ground states of the relevant minima are shown in Fig. 2. Obviously, the intrinsic shapes are more complicated than the simple ellipsoids corresponding to pure quadrupole deformations. The interactions induce -cluster correlations, which generate deformations of higher multipole orders as can be seen in the intrinsic densities.
Figure 3 shows the systematics of the ground-state energy per nucleon and the nuclear radius. The most prominent feature of the ground-state energy per nucleon is the difference of about between the measured energies and the values obtained with realistic interactions. This shift is due to correlations that are not described by a single Slater determinant and cannot be recovered by angular momentum projection. Since the unitary transformations only account for the short-range correlations, these missing correlations are driven by intermediate-range contributions in the interaction. We have shown in several previous publications that these missing correlations can be described by many-body perturbation theory and that the inclusion of low-order perturbative corrections to the energy leads to a good systematic agreement of the ground-state energies with experiment Gunther10 ; Roth10 ; Roth05 . We have also shown that the RPA does describe these ground-state correlations very well and that the inclusion of the RPA correlation energy (ring summation) leads to a good agreement with the experimental ground-state energies for closed-shell nuclei Barbieri06 . The contributions to the correlation energy resulting from long-range correlations related to deformation are recovered by the angular-momentum projection and are significantly smaller than the intermediate-range correlations.
In case of the charge-radii, the missing correlations only play a minor role and the difference between the intrinsic and projected radii is negligible. For spherical nuclei, the two- plus three-body interactions SSand S-UCOM(SRG) are in good agreement with experiment (see also Gunther10 ). While the radii of Ne and S are still described rather well, the results for Si are about 10% above the measured radii.
Figure 3 also contains data obtained with the Gogny D1S interaction. Since the interaction is fitted to binding energies, these are reproduced very accurately. The radii of deformed nuclei are comparable to those obtained with the SSand S-UCOM(SRG) interactions.
## Iii Random phase approximation for deformed nuclei
### iii.1 Formalism
In the standard RPA, excitations are described as one-particle one-hole and one-hole one-particle excitations. The excitation operator is given by
^Q†ω=∑miXωma^a†m^aa−∑maYωma^a†a^am, (5)
where indices starting with denote states above the Fermi level and those starting with denote states below. The RPA ground-state is defined by the relation and excited states are given by .
The summation in (5) runs over all possible particle-hole (ph) pairs defined with respect to the HF ground state. In the case of deformed ground-states, the number of ph-pairs cannot be reduced by angular momentum coupling rules, as ceases to be a good quantum number. However, for axial deformations, the number of ph-pairs can still be reduced by considering the projection quantum number and parity. In a spherically symmetric basis spanning 15 major HO shells, calculating electric monopole excitations requires ph pairs for O and for Ca. If the basis is extended to allow axially symmetric deformations, these numbers increase to and , respectively.
The amplitudes and are obtained by the equations-of-motion method and the quasi-boson approximation Ring80 , which result in the RPA matrix equation
(ABB∗A∗)(XωYω)=Eω(100−1)(XωYω), (6)
with
Amanb=⟨HF|[^a†a^am,[^H,^a†n^ab]]|HF⟩=(εm−εa)δmnδab+⟨m,b|^H|a,n⟩,Bmanb=−⟨HF|[^a†a^am,[^H,^a†b^an]]|HF⟩=⟨m,n|^H|a,b⟩. (7)
Here, a Hamiltonian with only one- and two-body terms is assumed. If the Hamiltonian also includes a three-body interaction , it has to be separated and the following terms have to be added to Eq. (7)
A(3)manb=∑k⟨m,b,k|^V3|a,n,k⟩,B(3)manb=∑k⟨m,n,k|^V3|a,b,k⟩. (8)
### iii.2 Projected transition matrix element
The reduced transition probability for an electric transition operator from an initial state to the final state is defined as
B(Eλ,J0→Jω)=12J0+1|(Φ0∥^Tλ∥Φω)|2=12J0+1∑μ|⟨Φ0|^Tλμ|Φω⟩|2. (9)
We use the shorthand notation for and . Intrinsic transition amplitudes between the RPA ground-state and an excited state are calculated with
(10)
Transition amplitudes between angular-momentum projected RPA states, denoted by their total angular-momentum values and , are given by the following formula Ring80
(J0∥^Tλ∥Jω)=(2J0+1)N0Nω∑K0Kωμg(0)K0g(ω)Kω=×(−1)λ+Jω+K0(J0Jωλ−K0K0−μμ)=×⟨RPA|^Tλμ^PJωK0−μ,Kω^Q†ω|RPA⟩, (11)
with the normalization factors and . Similar to Eq. (10), we express this in terms of the HF ground-state and the - and -amplitudes
(J0∥^Tλ∥Jω)≈(2J0+1)N0Nω(−1)J0−KHF=×∑maμ(Xωma+(−1)KmaYωma)(J0λJω−KHFμKHF−μ)=×⟨HF|^Tλμ^PJωKHF−μ,KHF+Kma^a†m^aa|HF⟩. (12)
Notation and further details are discussed in Appendix A. We like to point out that Eq. (12) is used directly in the calculations. We do not use any further approximations for either the overlaps or the integration involved in the angular momentum projection (the numerical integration is performed using 2048 points).
### iii.3 Transition operators
We use the standard form of the electric transition operators in the long wavelength limit given by Ring80
^Tλμ=A∑iei^rλiYλμ(^Ωi). (13)
As ususal, the electric transitions are decomposed into a sum of an isoscalar and an isovector part
^Tλμ =12(^TISλμ+^TIVλμ) (14) ^TISλμ =eZ∑i^rλiYλμ(^Ωi)+eN∑i^rλiYλμ(^Ωi) (15) ^TIVλμ =eZ∑i^rλiYλμ(^Ωi)−eN∑i^rλiYλμ(^Ωi). (16)
For the electric monopole transitions the generic first-order transition operator (13) is a constant and thus cannot induce transitions. Instead the second-order term is generally used
^T00=A∑iei^r2iY00(^Ωi). (17)
Since the electric dipole operator is potentially contaminated by spurious center-of-mass contributions, corrected transition operators are used VanGiai81 ; Harakeh01
^TIS1μ =eA∑i(^r3i−53Rms^ri)Y1μ(^Ωi), (18) ^TIV1μ =eNAZ∑i^riY1μ(^Ωi)−eZAN∑i^riY1μ(^Ωi), (19)
with the mean-square radius of the nucleus.
In principle, the unitary transformation used for the interactions also has to be applied to the transition operators, but considering missing higher-order correlations in the RPA and the long-range and low-momentum character of , it is justified to neglect this transformation. For the UCOM(VAR) interaction, it was shown that the correction due to transformed transition operators is indeed small and not relevant in most cases Paar06 .
## Iv RPA results
### iv.1 Convergence and sensitivity
We start the presentation of the RPA results with a discussion of the model-space convergence and the sensitivity to the input interaction. Figure 4 shows the convergence with respect to oscillator length and basis truncation for the example of the isoscalar monopole response of Si. For ease of presentation, the discrete RPA transition strengths in this and the following figures are folded with Lorentzians of width . For energies below , all curves lie on top of each other. In the giant resonance region at , the finer details of the response differ, but the centroid of the resonance is well converged. Above , the differences increase, but the very prominent peak above is present in all calculations. We conclude that the standard basis size of warrants a sufficient degree of convergence for the following discussions. The picture is similar for other response functions and nuclei.
As a second aspect we study the sensitivity of the response to the input interaction. Figure 5 again shows the isoscalar monopole response of Si for all interactions used in this work. The pure two-body interaction UCOM(VAR) yields a rather different response than both two- plus three-body interactions, SSand S-UCOM(SRG). It only shares the general features, i.e., the multi-peak structure, followed by a smaller resonance, followed by a high-energy peak, but on a stretched energy scale. This stretching can be understood in connection to the ground-state radii, which are underestimated by this interaction. In a simple mean-field picture, too small radii entail larger spacings of the single-particle levels and thus a shift of the unperturbed response to higher excitation energies Paar06 . The three-body part of the SSand S-UCOM(SRG) interactions corrects for the description of the ground-state radii and leads to smaller energy spacings of the HF energy-levels near the Fermi level, which in turn lowers the excitation energies of the collective peaks. Although the two three-body interactions SSand S-UCOM(SRG) yield different HF ground-state energies (cf. Sec. II), the response functions are very similar and only differ in details. This shows that the HF ground-state energy has little impact on the response functions. In the following, we limit the presentation to the S-UCOM(SRG) interaction. The results obtained with the Gogny D1S interaction are quite different from the other three, but since it constitutes a completely different approach, this is not unexpected.
### iv.2 Structure of the collective response
In a next step we survey the response for the standard collective modes and discuss the role of deformation in more detail. Figure 6 shows the RPA response of the isoscalar monopole (ISM), isoscalar dipole (ISD), isovector dipole (IVD) and isoscalar quadrupole (ISQ) transition operators for Ne, Si and S, calculated with the S-UCOM(SRG) interaction. The discrete response in Fig. 6 is color-coded to identify the different -components. Since all states with are twofold degenerate, the corresponding lines are doubled in height. The continuous curves again result from folding the discrete strengths with Lorentzians of width . Note that this width might be very different from the actual escape width, which cannot be described in our RPA approach.
In axial symmetric nuclei, different oscillation modes can be classified by the projection quantum number of the total angular-momentum and the parity . Oscillations with are along the symmetry axis and preserve the axial symmetry. For the cases studied in this work, they appear as ISM breathing modes, ISQ -vibrations or as a mixture of both. Modes with and only appear for dipole transitions. As the mode, the mode is along the symmetry axis and preserves axial symmetry. However, due to the negative parity, the density increases in one half of the nucleus, while it decreases in the other. The mode shows a similar oscillation pattern, but is directed perpendicular to the symmetry axis. Therefore, it breaks the axial symmetry. Spurious center-of-mass motion can appear in both types of dipole oscillations. Together with the already mentioned mode, the and the modes make up ISQ transitions. The modes are -vibrations and the modes are rotational. As can be seen in Fig. 6, rotational modes do not contribute much to the ISQ response, but as we will discuss later, a spurious mode can appear.
In prolate nuclei, like Ne and S, the symmetry axis of the nucleus is also the longest axis. Therefore, in the mean-field picture, oscillations see a shallow potential and appear at lower energy than modes with higher . This behavior is most pronounced for IVD and ISQ transitions. IVD transitions below are dominated by modes, while those above are dominated by modes. The ISQ giant resonance consists almost exclusively of modes, and, therefore, can be found at a rather high energy of .
In oblate nuclei, like Si, the symmetry axis of the nucleus is the shortest axis. Therefore, the situation is the exact opposite to the one found in prolate nuclei. The low-lying IVD strength is made up exclusively by modes, while modes can only be found at very high energies. The ISQ giant resonance is found at a significantly lower energy of .
### iv.3 Comparison to experiment and Gogny D1S interaction
We can now compare the RPA response to experimental data. In Fig. 7, the isovector dipole response is compared to data from photonuclear experiments NSR1981AL05 ; NSR1983PY01 ; NSR1978VA15 and the isoscalar response is compared to data from -scattering Youngblood02 . Since the cross-section for dipole transitions is proportional to , the response is multiplied with the energy to ease comparison. The measurements from Youngblood02 are given as the fraction of the energy-weighted sum-rule, so the same scaling applies also for isoscalar transitions. As we do not consider absolute values of the transition strengths, all strength plots are normalized to the range from to .
Since the SSand S-UCOM(SRG) interactions yield almost identical results, we only show data for the UCOM(VAR) and S-UCOM(SRG) interactions. For both interactions, the qualitative features of the response generally agree well with measurement, however, the exact energy of the peaks is not reproduced.
For ISM transitions, Fig. 7a, -scattering shows a strong peak between and . While the UCOM(VAR) interaction reproduces the position of this peak very well, the S-UCOM(SRG) interaction yields a peak at . At energies from , the UCOM(VAR) and S-UCOM(SRG) interactions show strength, structured into multiple peaks. In this area, experiment shows a shallow and broad structure without any distinct peaks. As we can see in the figure, monopole strength at high energies is linked to angular-momentum projection. Without the angular momentum-projection, there would be no strength above . The projected response function obtained with the Gogny D1S interaction shows almost no low-lying strength, but a very pronounced peak at . In contrast, the intrinsic response reproduces the measured peak very well.
The measured ISD response, Fig. 7b, shows a few narrow peaks around , and a broad structure from to , with significant peaks around . The UCOM(VAR) and S-UCOM(SRG) interactions reproduce this structure, especially for the S-UCOM(SRG) interaction, the agreement with experiment is remarkably good. Results from calculations done with the Gogny interaction do not reproduce the measured results.
For the ISQ response, Fig. 7c, the measurement shows three peaks with increasing height in the range between and , followed by some strength up to . All interactions, including Gogny D1S, reproduce this shape. Depending on the interaction, it is found lower than the measured peak (S-UCOM(SRG) and Gogny D1S) or at higher energy (UCOM(VAR)).
In the case of IVD transitions in Ne, Fig. 7d, data is only available in an energy window from to . Measurement shows three narrow peaks at and below , followed by a continuum up to the highest measured energy. Our calculations for the UCOM(VAR) and S-UCOM(SRG) interactions show strength distributed from to above , with more distinct peaks around . This is in general agreement with the measured data. Results for the Gogny D1S interaction look a little different, but also agree with the measured data. In the case of Si, Fig. 7e, we get very similar curves for both, measurement and calculation. Here, the UCOM(VAR) and S-UCOM(SRG) interactions show a double-peak in the region of , followed by strength up to . Experiment shows a double-peak with an energy between the calculated values of the UCOM interactions. The Gogny D1S interaction predicts a peak exactly at the measured energy, however, it is much too narrow. For S, Fig. 7f, experiment shows a broad peak from to about . This is reproduced by all interactions. We do not find any sizable effects of the axial-symmetric approximation for S.
In conclusion, the S-UCOM(SRG) interaction, including a phenomenological three-body interaction yields a good overall agreement with the experimental response. The agreement is at the same level or sometimes better than for the purely phenomenological Gogny D1S interaction. Considering that the RPA is only a first-order approximation, the agreement with experiment is remarkable.
Motivated by this observation, we will present a detailed comparison of deformed RPA calculations for the S-UCOM(SRG) interaction with new high-resolution experiments for the IVD response in a joint publication with the experimental groups Fearick14 . There we analyze, in particular, the fine structure of the giant dipole resonance and elucidate the role of deformation driven by -clustering through the confrontation of high-resolution data with our calculations.
### iv.4 Transition densities
Going beyond the response, we can compute the transition densities for various discrete RPA states in order to get an intuitive geometrical understanding of the dominant excitation modes. In Fig. 8, we show the intrinsic transition densities for a few selected transitions of the ISM (a and b) and ISQ response (c and d). The figures on the left show the angular-momentum projected and the intrinsic response. For the ISM response, we see a large effect of the angular-momentum projection, while the effect on the ISQ response is rather limited.
The strong effect on the ISM response is the consequence of a mixing of breathing oscillations and -vibrations through the angular-momentum projection. The projection generates a superposition of all possible rotations of the system, weighted by the angle-dependent Wigner functions to construct a predefined angular momentum. In the case of monopole transitions, the Wigner function is a constant. Therefore, an intrinsic -vibration, like the one at , Fig. 8b, (and to some extent the state at , Fig. 8a), is converted to a monopole-type breathing oscillation by the angular-momentum projection. This results in a redistribution of strength from the ISQ to the ISM channel for these states, which can best be seen for the state at . In this case, the intrinsic ISM transition strength practically vanishes, whereas the angular-momentum projected strength provides the second strongest peak in the whole response.
At this point, a comment is in order on the so-called needle approximation for the angular-momentum projection, which is used, e.g., in Refs. Arteaga08 ; Peru08 . For monopole transitions, the needle approximation simply reproduces the intrinsic response and, thus, misses some major effects of the projection, as can be seen in Fig. 8.
Another interesting state is the ISQ state at , Fig. 8c, which has . This state appears for all deformed nuclei in the ISQ response, but is not shown in Fig. 6 and 7. The angular-momentum projection strongly reduces the strength of this state. Further investigation shows, that the state has very large -amplitudes—about the same order of magnitude as the -amplitudes. This suggest a spurious rotational state, as is expected for deformed nuclei, which is confirmed by the transition density. For this state, Fig. 8c shows not the transition density, but the total density . The shape of the nucleus remains unchanged and it is only rotated around the -axis. As this spurious mode is found at an energy significantly above zero, it can contaminate other, non-spurious states with . However, since these modes do not contribute significantly to the ISQ response, this does not pose a problem for the current studies. Spurious center-of-mass modes are also found for the and modes, but lie at zero energy.
As was already seen in Fig. 6, the ISQ response is dominated by transitions, which correspond to -vibrations. Figure 8d shows the transition density of the strongest ISQ state, which is indeed a perfect -vibration.
## V Conclusions
In this paper, HF and RPA calculations with unitarily transformed realistic interactions for axially-symmetric deformed nuclei have been carried out for the first time. To obtain ground-state energies and response functions in the lab-frame, an explicit angular-momentum projection was employed.
For all studied nuclei, we find a much stronger fragmentation of the resonances than in spherical nuclei. Due to the angular-momentum projection, the ISM response extends to energies as high as . For the IVD response, we find the expected dipole splitting. In prolate nuclei, oscillations along the symmetry axis are found at lower energies, while those perpendicular to the symmetry axis are found at higher energies. In oblate nuclei, the situation is reversed. In case of the ISD response, we find multiple peaks at high and low energies. The ISQ response is clearly dominated by -vibrations. The geometry of the individual oscillation modes was studied via transition densities, which confirmed spurious rotational states and the effect of the angular-momentum projection on breathing oscillations and -vibrations.
In comparison to experiment, the unitarily transformed interactions, in particular the S-UCOM(SRG) and SSinteractions, which also yield the correct radius systematics, provide a good overall description of the major collective modes in our deformed and angular-momentum projected RPA calculations. The quality of the agreement is comparable to, and sometimes better than results obtained with phenomenological interactions, such as the Gogny D1S interaction. This already shows that a good reproduction of the ground-state energies at the HF level is neither a necessary nor a sufficient criterion for a good description of the collective response.
This study opens multiple lines of research for the future. Motivated by the good agreement with experiment and the fact that significant fragmentation is already present in the RPA response, we will analyze the fine structure of the giant dipole resonance and compare to new high-resolution data for the nuclei discussed Fearick14 . This will shed light on the role of deformation and -clustering on the fragmentation and fine structure of giant resonances. Another obvious extension of the present work is the use of two- plus three-nucleon interactions from chiral EFT. First studies along these lines with a spherical formulation of the RPA are well advanced. However, present chiral interactions, even after the inclusion of the chiral three-nucleon terms, still underestimate the radii of medium-mass nuclei Binder14 . It remains to be seen how consistent next-generation chiral Hamiltonians will behave in this respect. Finally, the extension from first- to second-order RPA will be a target of future studies. As a first step, second-order RPA calculations for spherical nuclei including realistic 3N interactions are already under way Trippel14 .
## Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft through SFB 634, by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the state of Hesse, and the German Federal Ministry of Education and Research (BMBF 06DA9040I, 06DA7047I).
## Appendix A Angular-momentum projected transition amplitudes in the RPA framework
The unprojected transition amplitudes to the RPA ground state are obtained by applying the quasi-boson approximation Rowe70 , followed by straightforward calculation
⟨RPA|^Tλμ^Q†ω|RPA⟩≈⟨HF|[^Tλμ,^Q†ω]|HF⟩=∑ma(Xωma⟨a|^Tλμ|m⟩+Yωma⟨m|^Tλμ|a⟩). (20)
It would be desirable to derive the projected RPA transition amplitudes in a similar manner, directly from the equation for projected transition amplitudes Ring80
(J0∥^Tλ∥Jω)=(2J0+1)N0Nω∑K0Kωμg(0)K0g(ω)Kω×(−1)λ+Jω+K0(J0Jωλ−K0K0−μμ)×⟨RPA|^Tλμ^PJωK0−μ,Kω^Q†ω|RPA⟩. (21)
However, this is not possible in a consistent and unambiguous way. The canonical way of the RPA is to replace pairs of operators with their commutators and the RPA ground-state with the HF ground-state. This treatment is not possible because of the projection operator. Since the projection operator projects a fixed set of quantum numbers onto another fixed set, these quantum numbers would have to change according to the order of the operators and —otherwise the projection operator would annihilate the states. One could relax the requirement of a commutator and allow anything to be added that vanishes for the true RPA states, but still gives a -amplitude contribution in the QBA. Then, the quantum numbers of the projection operator could be changed to match the other operators. However, this scheme also allows the introduction of an arbitrary phase—and this freedom has to be exploited if one wants to reproduce the unprojected results for spherical nuclei. But since this treatment is ambiguous and leaves much to be desired in terms of simplicity, we opt for a different, less ambiguous approach.
To calculate the projected transition amplitudes, we again consider the unprojected intrinsic transition amplitudes. The complete transition amplitude of multipolarity including normalization factors is given by
⟨RPA|^Tλ|ω⟩√⟨RPA|RPA⟩⟨ω|ω⟩=∑μ⟨RPA|^Tλμ|ω⟩√⟨RPA|RPA⟩⟨ω|ω⟩=∑μ⟨RPA|^Tλμ^Q†ω|RPA⟩√⟨RPA|RPA⟩⟨RPA|^Qω^Q†ω|RPA⟩. (22)
The numerator is given by Eq. (10)
∑μ⟨HF|[^Tλμ,^Q†ω]|HF⟩=∑μ,ma(Xωma⟨a|^Tλμ|m⟩+Yωma⟨m|^Tλμ|a⟩). (23)
Assuming real matrix elements, we write the solution in a form that is more suitable for use in the projected formalism
∑μ⟨HF|[^Tλμ,^Q†ω]|HF⟩=∑μ,ma(Xωma⟨HF|^Tλμ^a†m^aa|HF⟩+Yωma(−1)μ⟨HF|^Tλ−μ^a†m^aa|HF⟩)=∑ma(Xωma+(−1)KmaYωma)∑μ⟨HF|^Tλμ^a†m^aa|HF⟩. (24)
We renamed to in the -amplitude part and used that is a well defined quantum number in axially symmetric nuclei. denotes the quantum number of the state .
To get the projected transition amplitudes, we simply include the -factor in the formula for the projected transition amplitudes (21)
(RPA∥^Tλ∥ω)=(2J0+1)N0Nω(−1)J0−K0×∑ma(Xωma+(−1)KmaYωma)∑μ(J0λJω−KHFμKHF−μ)×⟨HF|^Tλμ^PJωKHF−μ,KHF+Kma^a†m^aa|HF⟩. (25)
Simplifications arising from axial symmetry have already been applied. We treat the normalization factors accordingly. The normalization factor from the RPA ground state is given by
N0=√⟨HF|^PJ0KHF,KHF|HF⟩−1. (26)
The unprojected normalization factor for the excited state is given by
⟨RPA|^Qω^Q†ω|RPA⟩≈⟨HF|[^Qω,^Q†ω]|HF⟩=∑ma(XωmaXωma−YωmaYωma), (27)
which evaluates to unity due to the orthonormality of the RPA amplitudes. We, therefore, use the following projected normalization factor
N−2ω=∑ma,nb(XωmaXωnb−YωmaYωnb)×⟨HF|^a†m^aa^PJωKHF+kam,KHF+knb^a†n^ab|HF⟩. (28) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9156007766723633, "perplexity": 1154.8413703459105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335326.48/warc/CC-MAIN-20220929065206-20220929095206-00600.warc.gz"} |
https://math.stackexchange.com/questions/1303799/special-omegan-sequence | # Special $\omega(n)$-sequence
Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$.
The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with $0\le j\le k-1$. In other words, a sequence of $k$ numbers, such that the number of distinct prime factors increases from $1$ to $k$.
The least examples for some $k$ :
k n
2 5
3 64
4 1867
5 491851
6 17681491
Questions :
• Does a number $n$ exists for each $k$ ?
• Is there a good estimate for the magnitude of the least example ?
• Can anyone extend the table ?
• If there is an example for $k=7$ it must be $>10^{10}$. – Julián Aguirre Jun 3 '15 at 15:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8136057257652283, "perplexity": 164.88234622319766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578641278.80/warc/CC-MAIN-20190424114453-20190424140453-00199.warc.gz"} |
http://www.geoengineer.org/education/web-based-class-projects/geoenvironmental-remediation-technologies/thermal-desorption?showall=&start=12 | The International Information Center for Geotechnical Engineers
# Thermal Desorption
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https://www.texdev.net/tag/miktex/ | ## TeX on Windows: TeX Live versus MiKTeX revisited
On Windows, users have two main choices of TeX system to install: TeX Live or MiKTeX. I’ve looked at this before a couple of times: first in 2009 then again in 2011. Over the past few years both systems have developed, so it seems like a good time to revisit this. (I know from my logs that this is one of the most popular topics I’ve covered!)
The first thing to say is that for almost all ‘end users’ (with a TeX system on their own PC just for them to use), both options are fine: they’ll probably notice no difference between the two in use. It’s also worth noting that there is a third option: W32TeX. I’ve mentioned this before: it’s popular in the far East and is where the Windows binaries for TeX Live come from. (There’s a close relationship between W32TeX and TeX Live, with W32TeX more ‘focussed’ and expecting more user decisions in installing.)
Assuming you are going for one of the ‘big two’, what is there to think about? For most people, it’s simply:
• Both MiKTeX and TeX Live include a ‘full’ set of TeX-related binaries, including the engines pdfTeX, XeTeX, LuaTeX and support programs such as BibTeX, Biber, MakeIndex and Xindy.
• The standard installer for MiKTeX installs ‘just the basics’ and uses on-the-fly installation for anything else you need; the standard install for TeX Live is ‘everything’ (about 4.5 Gb!). Which is right for you will depend on how much space you have: you can of course customise the installation of either system to include more or less of the ‘complete’ set up.
• MiKTeX has a slightly more flexibly approach to licensing than TeX Live does: there are a small number of LaTeX packages that MiKTeX includes that TeX Live does not. (Probably the most odious example is thesis.)
• TeX Live has a Unix background so the management GUI looks slightly less ‘standard’ than the MiKTeX one.
• TeX Live has a strict once-a-year freeze,which means that to update you have to do a fresh install once a year. On the other hand, MiKTeX versions change only when there is a significant change and otherwise ‘roll onward’.
So the decision is likely to come down to whether you want auto-installation of packages. (If you do go for MiKTeX on a one-user PC, choose the ‘Just for me’ installation option: it makes life a lot simpler!)
For more advanced users there are a few more factors you probably want to consider
• TeX Live was originally developed on Unix and so is available for Linux and on the Mac (and other systems) as well as Windows; MiKTeX is a Windows system so is (more-or-less) Windows-only. So if you want exactly the same set up on Windows and other operating systems, this of course means you need to use TeX Live.
• Both systems have graphical management tools as well as command line interfaces. They have a lot in common, but they are not identical (in particular, MiKTeX tends to emulate TeX Live command line interfaces, but the reverse is not true).
• The engine binaries in TeX Live are (almost) never updated other than in the yearly freeze period, meaning that for a given release you know which version of pdfTeX, etc., you’ll have: MiKTeX is more flexible with such updates. (At different times, one or other of the systems can be more ‘up to date’: this is not necessarily predictable! The W32TeX system often has very up-to-date testing binaries.)
• The two systems differ slightly in handling how local trees are managed (places to add TeX files that are not controlled by the TeX system itself). TeX Live automatically expects <installation root>/texmf-local to hold system-wide ‘local’ additions and <user root>/texmf to hold per-user additions, whereas MiKTeX has no out-of-the box locations, but does make it easier to add and remove them from the command line. MiKTeX also makes it easy to add multiple per-user trees, whereas for TeX Live there’s more of an assumption that all user additions will be added in one place. (This makes it easier in MiKTeX to add/remove local additions by altering a setting in the TeX system rather than deleting files.)
• TeX Live has a team doing the work; MiKTeX is a one-man project. This cuts both ways: you know exactly who is doing everything in MiKTeX (Christian Schenk), and he’s very fast, but there is more ‘spread’ in TeX Live for the work.
• For people wanting to step quickly between different versions of TeX system, the fact that TeX Live freezes once a year makes life convenient (I have TeX Live 2009,2010, 2011, 2012, 2013, 2014, 2015 and 2016 installed at present, plus MiKTeX 2.9 of course!) You can switch installations by adjusting the PATH or by choosing the appropriate version from your editor, so have a ‘fall back’ if there is an issue when you update.
• TeX Live has build-in package backup during maintenance updates.
## TeX on Windows: MiKTeX or TeX Live
Around two years ago, I wrote a short post comparing MiKTeX and TeX Live for Windows-based TeX users. Looking at my log files, this topic is perhaps the most common search term that brings people here. As such, I think it’s time to revisit the question and bring what I said before up to date.
On Windows, there are two actively-developed TeX systems with similar coverage: MiKTeX and TeX Live. Before I look at the comparison, a reminder that they are not the only choices. W32TeX is popular in the far east, and as well as being a TeX system in its own right is the source of the Windows binaries for TeX Live. There are also the commercial BaKoMa and VTeX systems (although whether anyone can get hold of the supplier of the latter is another question). However, for most users it comes down to a choice between the ‘big two’.
The good news is that there is a lot of similarity between the two systems, so for most users both systems are equally usable. However, there are differences and depending on what you need these might be important.
• The standard settings install everything for TeX Live, but only a minimal set of packages for MiKTeX. MiKTeX will then install extra packages ‘on the fly’, while TeX Live does not (there is a package to do that in TeX Live, but it’s aimed at Linux). Install-on-the-fly is useful if space is limited, but is more problematic on server set ups. So this is very much a feature who’s usefulness depends on your circumstances. Of course, there is nothing to stop you using MiKTeX and installing everything.
• The xindy program is only available in TeX Live. For those of you not familiar with it, xindy is an index-processor, and is much more capable of dealing with multi-lingual situations than MakeIndex. If you need xindy, TeX Live really is the way to go.
• MiKTeX is very much a Windows tool set, while TeX Live comes from a Unix background. This shows up from time to time in the way TeX Live is administered, and the fact that the TeX Live GUI is written based on Perl rather than as a ‘native’ Windows application.
• As TeX Live is the basis of MacTeX, and is the TeX system for Unix, if you work cross-platform and want an identical system on all of your machines, then TeX Live is the way to go.
## Spaces in file names
Spaces in file names are a constant issue for LaTeX users. As many people will know, TeX is not really very happy with spaces, as they are used to delimit the end of input in a lot of low-level macros. This shows up particularly in two areas: graphics and shell escape. For graphics, the excellent grffile package will deal with many of the issues. When using shell escape, the issue is usually that \jobname may be slightly odd. For TeX Live users, that is not so much of a problem as the name is automatically quoted to protect the space. However, MiKTeX does things a bit differently, and uses a star in place of a space. So you end up with
\edef\example{\jobname}
\show\example
> \example=macro:
->test*file.
l.2 \show\example
which is not exactly helpful. However, it is possible to deal with this, as recently mentioned on TeX.SX. As * cannot normally appear in file names, and \jobname makes all characters have category code 12, a simple approach is to do a quick replacement
\edef\Jobname{\jobname}
\catcode\*=\active
\def*{ }
\edef\Jobname{"\scantokens\expandafter{\Jobname\noexpand}"}
\catcode\*=12 %
\show\Jobname
If you want to deal with both TeX Live and MiKTeX, you’d of course need first to test which system is in use (for example using \pdftexbanner).
## Updating LaTeX3 support in MiKTeX
The LaTeX3 Project have recently updated the organisation of the various LaTeX3-based packages on CTAN. This means that the older expl3 and xpackages need to be replaced by l3kernel and l3packages. Unfortunately, this seems to confuse MiKTeX, which does not pick up the need to install the new material. So MiKTeX users will need to do this by hand in the MiKTeX Package Manager. This should be a passing problem, but does seem to be causing some confusion for MiKTeX users.
## TeX and security
Security in computer programs is always an issue, with the balance between ease of use and security never being a simple black and white line. There’s a very interesting paper, being presented at an upcoming conference, about TeX security issues. This is particularly significant to MiKTeX users, as it’s led to a change in how MiKTeX implements certain features.
One of the well-known security questions with TeX is whether to enable \write18, and as a result this is off by default in TeX Live and MiKTeX. Another area that is of obvious concern is the \openout primitive, which allows writing a new file and could therefore be used for undesirable purposes. Of course, this functionality is also important: writing to files is how LaTeX manages a whole range of automated cross-referencing. So there is a balance to be struck: we need \openout, but not at any cost.
The TeX Live team have taken the attitude that \openout should be able to write within the current directory structure but not outside of it. This can be seen with a couple of very similar plain TeX test files. If you try
\newwrite\mywrite
\immediate\openout\mywrite test/test.xxx
\bye
then everything will be fine and the test file will be created. On the other hand
\newwrite\mywrite
\immediate\openout\mywrite ../test.xxx
\bye
will raise an error. The behaviour with MiKTeX was to allow both (and also absolute paths, etc.). That has now been altered, so that MiKTeX behaves in the same way as TeX Live (at least, that’s what it looks like in my tests).
Reading the MiKTeX lists, the new behaviour is causing issues because LaTeX’s \include relies on \openout. Quite a lot of MiKTeX users have been doing things like:
\include{C:/Users/<user>/My Documents/Chapters/chapter1.tex}
or
\include{../Chapters/chapter1.tex}
which used to work and now does not. There is a setting which enables the old behaviour, but it’s not really to be recommended, I think. So users will have to rearrange their input a bit to reflect the new more secure approach.
There are some other interesting points in the paper on TeX security. One is that making a truly secure LaTeX implementation (to use as a web service) is basically impossible. The MathTran site gets mentioned as the most secure TeX web service: it uses a specially hardened version of plain TeX, with no access to things like \csname, \catcode and so on to make it secure. For LaTeX, that is probably not possible (at least with LaTeX2e). Worth reading, but for those of us who just use TeX on our own computers not quite so immediately relevant.
## Windows TeX Users: MiKTeX or TeX Live
I was talking to someone at work recently, and the topic of whether to choose MiKTeX or TeX Live on Windows came up. With MiKTeX 2.8 released and TeX Live 2009 due out any day, I thought I’d make a few comments.
First, both systems are very capable, so there is not really a “wrong” decision. However, when installing you do have to pick one. In the past, MiKTeX was the best choice for Windows by a distance, but recent work on TeX Live has altered this. So I’d say there are some factors to balance against each other.
• If you work on Windows and on Unix, then TeX Live is the system to favour. It preforms essentially in the same way across platforms, whereas you’ll get some minor differences if you use MiKTeX on Windows and TeX Live on Unix.
• If you want to only install what you use, go for MiKTeX. TeX Live doesn’t have anything to match the auto-installation system in MiKTeX.
• On the other hand, if your happiest installing everything in one go, go for TeX Live. It does this by default, and includes any new packages when you do an update. For MiKTeX, a full installation is something you have to do deliberately.
As you’ll see, there is not much in it! I’m mainly using TeX Live, but still have MiKTeX around as well.
## EPS graphics with PDF(La)TeX
One issue a lot of people find confusing with (La)TeX is the rules about which types of graphic files work with which engines. EPS files are fine when going via the DVI route, but do not work with direct PDF creation. The solution is to turn the EPS files in PDFs, and the problem goes away. However, there is then the question of how to do the conversion.
For most documents, having to convert every file by hand is not a sensible choice. The next nearest thing is the epstopdf package, which will do the same thing but from within a LaTeX run. However, it needs \write18 enabled, and this is not always desirable. More importantly, a lot of people who struggle with the graphics problem do not know how to turn on \write18 anyway. A good way around has been added to the latest version of TeX Live, which is currently in the final testing stages. TeX Live 2009 has some restricted \write18 functions enabled as standard, and also has a version of epstopdf “built in”. The result is that EPS files are automatically converted to PDF files, in a transparent manner. Of course, this only happens if the PDF does not also exist! At the moment, this feature is not in MiKTeX 2.8, so it is one reason to favour TeX Live 2009 even on Windows.
There are places where epstopdf will not help: for example, when using psfrag or pstricks. There, the best solution will either be auto-pst-pdf or pstool. Both are written by Will Robertson, and both need \write18 enabled to work. pstool is more efficient (it only re-creates graphics as needed), but for some cases on auto-pst-pdt will work. Will has documented both packages very well, so the best way to learn about them is to have a read of the documentation.
## MiKTeX 2.8 released
A quick note to say that MiKTeX 2.8 was released on Tuesday. As I’ve already posted, the two points to note in the new version are the inclusion of TeXworks and better support for multi-user systems. There seem to be one or two teething issues, but I’m sure they’ll soon be solved.
## TeX Live on Windows
As I posted earlier, the upcoming releases of both MiKTeX and TeX Live have very similar sets of features on Windows. I’ve just stumbled upon something that points up the slight differences that exist, even though this one is a bit complicated.
To detect what system is being used, for things like shell escape tricks, there is a LaTeX package called ifplatform. However, this only works on Windows if MiKTeX is being used. The reason is that while TeX Live aims to be as similar as possible across platforms, MiKTeX can adopt a different approach and stick to Windows conventions. Most of the time, this is transparent but it shows up if you use the -shell-escape option for either system and try to do some testing. Inside ifplatform, you’ll find the lines:
\edef\ip@sig{write18-test-\the\year\the\month\the\day\the\time}
\edef\ip@win{'\ip@sig'}
\immediate\write18{echo \ip@win >"\ip@file"}
The idea is that the text written to the temporary file will be different on Windows to on a Unix-like system. MiKTeX will retain the single quotes around the test data:
'write18-test-20098231074'
whereas Unix-like systems will not:
write18-test-20098231074
But try using ifplatform with TeX Live on Windows and the test fails. First, no test file gets written at all: a bit of hacking leads to the change of the write line to
\immediate\write18{echo \ip@win > \ip@file}
and then at least the first step works. However, the test file now looks like a Unix one (with no quote marks), and ifplatform gets things wrong. So for the moment the only thing to do is create a stub package file and use it, something like:
\ProvidesPackage{ifplatform}[2007/11/18 v0.2 Testing for the operating system]
\newif\ifshellescape\shellescapetrue
\newif\ifwindows\windowstrue
\newif\ifmacosx
\newif\iflinux
I’ve reported the problem to Will Robertson, and hopefully a solution which really tests the OS rather than the TeX system can be found. However, it is a reminder that even with very general feature sets, the two major TeX distributions still act differently in some respects when used on Windows.
## Testing MiKTeX 2.8 and TeX Live 2009
Both MiKTeX and TeX Live have new versions in the offing. I’ve been testing out both MiKTeX 2.8 and TeX Live 2009, to keep up to date with what is happening. In the past, I’ve tended to stick with MiKTeX as it is designed for Windows, and so can make some platform-specific decisions and be more focussed. However, the TeX Live team have done a lot of work to make TeX Live usable across platforms, and there are advantages to that approach.
Looking through the feature lists, a lot of the new features are common to the two systems, for example:
• TeXworks installed as a distribution-maintained editor.
• XeTeX version 0.9995 (which includes the new primitives that the LaTeX3 team asked for).
• Some \write18 functions enabled without turning on full \write18 support: this is used to allow “safe” functions.
There are, of course, also differences. For example, only TeX Live includes LuaTeX at present. I also notice that MiKTeX 2.8 is adding the full path of files to the log, whereas in the past you got the relative path. I’m not so sure this is a good idea: it makes things rather wordy, and also the log will vary between systems: not so great. On the other hand, MiKTeX 2.8 does provide user-specific texmf directories. For multi-user systems, this is a real bonus: you can use the auto-install system without needing to be the Administrator.
As I said, I’ve tended to use MiKTeX to date as it’s been the best “fit” on Windows. The latest version of TeX Live makes this a pretty tight call, I think. If you are happy installing a full TeX system (which I do), then there is very little in it. MiKTeX still has the edge for small installations, as the auto-install system really pays off there. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9946247935295105, "perplexity": 1758.7793694520226}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818693940.84/warc/CC-MAIN-20170926013935-20170926033935-00023.warc.gz"} |
https://www.science.gov/topicpages/q/quantum+master+equations.html | #### Sample records for quantum master equations
1. Master Equation for a Quantum Particle in a Gas
SciTech Connect
Hornberger, Klaus
2006-08-11
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
2. Master equation for a quantum particle in a gas.
PubMed
Hornberger, Klaus
2006-08-11
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
3. Quantum Markovian master equation for scattering from surfaces.
PubMed
Li, Haifeng; Shao, Jiushu; Azuri, Asaf; Pollak, Eli; Alicki, Robert
2014-01-01
We propose a semi-phenomenological Markovian Master equation for describing the quantum dynamics of atom-surface scattering. It embodies the Lindblad-like structure and can describe both damping and pumping of energy between the system and the bath. It preserves positivity and correctly accounts for the vanishing of the interaction of the particle with the surface when the particle is distant from the surface. As a numerical test, we apply it to a model of an Ar atom scattered from a LiF surface, allowing for interaction only in the vertical direction. At low temperatures, we find that the quantum mechanical average energy loss is smaller than the classical energy loss. The numerical results obtained from the space dependent friction master equation are compared with numerical simulations for a discretized bath, using the multi-configurational time dependent Hartree methodology. The agreement between the two simulations is quantitative. PMID:24410218
4. Post-Markovian quantum master equations from classical environment fluctuations.
PubMed
2014-01-01
In this paper we demonstrate that two commonly used phenomenological post-Markovian quantum master equations can be derived without using any perturbative approximation. A system coupled to an environment characterized by self-classical configurational fluctuations, the latter obeying a Markovian dynamics, defines the underlying physical model. Both Shabani-Lidar equation [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and its associated approximated integrodifferential kernel master equation are obtained by tracing out two different bipartite Markovian Lindblad dynamics where the environment fluctuations are taken into account by an ancilla system. Furthermore, conditions under which the non-Markovian system dynamics can be unraveled in terms of an ensemble of measurement trajectories are found. In addition, a non-Markovian quantum jump approach is formulated. Contrary to recent analysis [L. Mazzola, E. M. Laine, H. P. Breuer, S. Maniscalco, and J. Piilo, Phys. Rev. A 81, 062120 (2010)], we also demonstrate that these master equations, even with exponential memory functions, may lead to non-Markovian effects such as an environment-to-system backflow of information if the Hamiltonian system does not commutate with the dissipative dynamics. PMID:24580212
5. Transport in molecular states language: Generalized quantum master equation approach
Esposito, Massimiliano; Galperin, Michael
2009-05-01
A simple scheme, capable of treating transport in molecular junctions in the language of many-body states, is presented. By introducing an ansatz in Liouville space, similar to the generalized Kadanoff-Baym approximation, a quantum master equation (QME)-like expression is derived starting from the exact equation of motion for Hubbard operators. Using an effective Liouville space propagation, a dressing similar to the standard diagrammatic one is proposed. The scheme is compared to the standard QME approach and its applicability to transport calculations is discussed.
6. Dynamics of open quantum spin systems: An assessment of the quantum master equation approach
Zhao, P.; De Raedt, H.; Miyashita, S.; Jin, F.; Michielsen, K.
2016-08-01
Data of the numerical solution of the time-dependent Schrödinger equation of a system containing one spin-1/2 particle interacting with a bath of up to 32 spin-1/2 particles is used to construct a Markovian quantum master equation describing the dynamics of the system spin. The procedure of obtaining this quantum master equation, which takes the form of a Bloch equation with time-independent coefficients, accounts for all non-Markovian effects inasmuch the general structure of the quantum master equation allows. Our simulation results show that, with a few rather exotic exceptions, the Bloch-type equation with time-independent coefficients provides a simple and accurate description of the dynamics of a spin-1/2 particle in contact with a thermal bath. A calculation of the coefficients that appear in the Redfield master equation in the Markovian limit shows that this perturbatively derived equation quantitatively differs from the numerically estimated Markovian master equation, the results of which agree very well with the solution of the time-dependent Schrödinger equation.
7. Dynamics of open quantum spin systems: An assessment of the quantum master equation approach.
PubMed
Zhao, P; De Raedt, H; Miyashita, S; Jin, F; Michielsen, K
2016-08-01
Data of the numerical solution of the time-dependent Schrödinger equation of a system containing one spin-1/2 particle interacting with a bath of up to 32 spin-1/2 particles is used to construct a Markovian quantum master equation describing the dynamics of the system spin. The procedure of obtaining this quantum master equation, which takes the form of a Bloch equation with time-independent coefficients, accounts for all non-Markovian effects inasmuch the general structure of the quantum master equation allows. Our simulation results show that, with a few rather exotic exceptions, the Bloch-type equation with time-independent coefficients provides a simple and accurate description of the dynamics of a spin-1/2 particle in contact with a thermal bath. A calculation of the coefficients that appear in the Redfield master equation in the Markovian limit shows that this perturbatively derived equation quantitatively differs from the numerically estimated Markovian master equation, the results of which agree very well with the solution of the time-dependent Schrödinger equation. PMID:27627265
8. Number-resolved master equation approach to quantum measurement and quantum transport
Li, Xin-Qi
2016-08-01
In addition to the well-known Landauer-Büttiker scattering theory and the nonequilibrium Green's function technique for mesoscopic transports, an alternative (and very useful) scheme is quantum master equation approach. In this article, we review the particle-number ( n)-resolved master equation ( n-ME) approach and its systematic applications in quantum measurement and quantum transport problems. The n-ME contains rich dynamical information, allowing efficient study of topics such as shot noise and full counting statistics analysis. Moreover, we also review a newly developed master equation approach (and its n-resolved version) under self-consistent Born approximation. The application potential of this new approach is critically examined via its ability to recover the exact results for noninteracting systems under arbitrary voltage and in presence of strong quantum interference, and the challenging non-equilibrium Kondo effect.
9. Convolutionless Non-Markovian master equations and quantum trajectories: Brownian motion
SciTech Connect
Strunz, Walter T.; Yu Ting
2004-05-01
Stochastic Schroedinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schroedinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most employed exactly soluble models of open system dynamics. We show explicitly how to establish the direct connection between the exact convolutionless master equation of QBM and the corresponding convolutionless exact stochastic Schroedinger equation.
10. Quantum master equation for dephasing of a two-level system with an initial correlation
SciTech Connect
Ban, Masashi
2009-12-15
Exact quantum master equation is derived for dephasing of a two-level system, which is an homogeneous time-convolutionless equation even though there is an initial correlation between a two-level system and a thermal reservoir. The result is compared with the quantum master equation derived by means of the projection operator method. Furthermore, the effects of the initial correlation on the dephasing process are examined.
11. Some implications of superconducting quantum interference to the application of master equations in engineering quantum technologies
Duffus, S. N. A.; Bjergstrom, K. N.; Dwyer, V. M.; Samson, J. H.; Spiller, T. P.; Zagoskin, A. M.; Munro, W. J.; Nemoto, Kae; Everitt, M. J.
2016-08-01
In this paper we consider the modeling and simulation of open quantum systems from a device engineering perspective. We derive master equations at different levels of approximation for a superconducting quantum interference device (SQUID) ring coupled to an ohmic bath. We demonstrate that the master equations we consider produce decoherences that are qualitatively and quantitatively dependent on both the level of approximation and the ring's external flux bias. We discuss the issues raised when seeking to obtain Lindbladian dissipation and show, in this case, that the external flux (which may be considered to be a control variable in some applications) is not confined to the Hamiltonian, as often assumed in quantum control, but also appears in the Lindblad terms.
12. Exact non-Markovian master equations for multiple qubit systems: Quantum-trajectory approach
Chen, Yusui; You, J. Q.; Yu, Ting
2014-11-01
A wide class of exact master equations for a multiple qubit system can be explicitly constructed by using the corresponding exact non-Markovian quantum-state diffusion equations. These exact master equations arise naturally from the quantum decoherence dynamics of qubit system as a quantum memory coupled to a collective colored noisy source. The exact master equations are also important in optimal quantum control, quantum dissipation, and quantum thermodynamics. In this paper, we show that the exact non-Markovian master equation for a dissipative N -qubit system can be derived explicitly from the statistical average of the corresponding non-Markovian quantum trajectories. We illustrated our general formulation by an explicit construction of a three-qubit system coupled to a non-Markovian bosonic environment. This multiple qubit master equation offers an accurate time evolution of quantum systems in various domains, and paves the way to investigate the memory effect of an open system in a non-Markovian regime without any approximation.
13. Non-equilibrium effects upon the non-Markovian Caldeira-Leggett quantum master equation
SciTech Connect
Bolivar, A.O.
2011-05-15
Highlights: > Classical Brownian motion described by a non-Markovian Fokker-Planck equation. > Quantization process. > Quantum Brownian motion described by a non-Markovian Caldeira-Leggett equation. > A non-equilibrium quantum thermal force is predicted. - Abstract: We obtain a non-Markovian quantum master equation directly from the quantization of a non-Markovian Fokker-Planck equation describing the Brownian motion of a particle immersed in a generic environment (e.g. a non-thermal fluid). As far as the especial case of a heat bath comprising of quantum harmonic oscillators is concerned, we derive a non-Markovian Caldeira-Leggett master equation on the basis of which we work out the concept of non-equilibrium quantum thermal force exerted by the harmonic heat bath upon the Brownian motion of a free particle. The classical limit (or dequantization process) of this sort of non-equilibrium quantum effect is scrutinized, as well.
14. Calculating work in weakly driven quantum master equations: Backward and forward equations.
PubMed
Liu, Fei
2016-01-01
I present a technical report indicating that the two methods used for calculating characteristic functions for the work distribution in weakly driven quantum master equations are equivalent. One involves applying the notion of quantum jump trajectory [Phys. Rev. E 89, 042122 (2014)PLEEE81539-375510.1103/PhysRevE.89.042122], while the other is based on two energy measurements on the combined system and reservoir [Silaev et al., Phys. Rev. E 90, 022103 (2014)PLEEE81539-375510.1103/PhysRevE.90.022103]. These represent backward and forward methods, respectively, which adopt a very similar approach to that of the Kolmogorov backward and forward equations used in classical stochastic theory. The microscopic basis for the former method is also clarified. In addition, a previously unnoticed equality related to the heat is also revealed. PMID:26871044
15. Calculating work in weakly driven quantum master equations: Backward and forward equations
Liu, Fei
2016-01-01
I present a technical report indicating that the two methods used for calculating characteristic functions for the work distribution in weakly driven quantum master equations are equivalent. One involves applying the notion of quantum jump trajectory [Phys. Rev. E 89, 042122 (2014), 10.1103/PhysRevE.89.042122], while the other is based on two energy measurements on the combined system and reservoir [Silaev et al., Phys. Rev. E 90, 022103 (2014), 10.1103/PhysRevE.90.022103]. These represent backward and forward methods, respectively, which adopt a very similar approach to that of the Kolmogorov backward and forward equations used in classical stochastic theory. The microscopic basis for the former method is also clarified. In addition, a previously unnoticed equality related to the heat is also revealed.
16. Nonadiabatic Dynamics in Atomistic Environments: Harnessing Quantum-Classical Theory with Generalized Quantum Master Equations.
PubMed
Pfalzgraff, William C; Kelly, Aaron; Markland, Thomas E
2015-12-01
The development of methods that can efficiently and accurately treat nonadiabatic dynamics in quantum systems coupled to arbitrary atomistic environments remains a significant challenge in problems ranging from exciton transport in photovoltaic materials to electron and proton transfer in catalysis. Here we show that our recently introduced MF-GQME approach, which combines Ehrenfest mean field theory with the generalized quantum master equation framework, is able to yield quantitative accuracy over a wide range of charge-transfer regimes in fully atomistic environments. This is accompanied by computational speed-ups of up to 3 orders of magnitude over a direct application of Ehrenfest theory. This development offers the opportunity to efficiently investigate the atomistic details of nonadiabatic quantum relaxation processes in regimes where obtaining accurate results has previously been elusive.
17. Calculating work in adiabatic two-level quantum Markovian master equations: a characteristic function method.
PubMed
Liu, Fei
2014-09-01
We present a characteristic function method to calculate the probability density functions of the inclusive work in adiabatic two-level quantum Markovian master equations. These systems are steered by some slowly varying parameters and the dissipations may depend on time. Our theory is based on the interpretation of the quantum jump for the master equations. In addition to the calculation, we also find that the fluctuation properties of the work can be described by the symmetry of the characteristic functions, which is exactly the same as in the case of isolated systems. A periodically driven two-level model is used to demonstrate the method. PMID:25314409
18. Closed description of arbitrariness in resolving quantum master equation
Batalin, Igor A.; Lavrov, Peter M.
2016-07-01
In the most general case of the Delta exact operator valued generators constructed of an arbitrary Fermion operator, we present a closed solution for the transformed master action in terms of the original master action in the closed form of the corresponding path integral. We show in detail how that path integral reduces to the known result in the case of being the Delta exact generators constructed of an arbitrary Fermion function.
19. Exact analytical solutions to the master equation of quantum Brownian motion for a general environment
SciTech Connect
Fleming, C.H.; Roura, Albert; Hu, B.L.
2011-05-15
Research Highlights: > We study the model of a quantum oscillator linearly coupled to a bath of oscillators. > We derive the master equation and solutions for general spectra and temperatures. > We generalize to cases with an external force and arbitrary number of oscillators. > Other derivations have incorrect diffusion and force response for nonlocal damping. > We give exact results for ohmic, sub-ohmic and supra-ohmic environments. - Abstract: We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.
20. Solutions to Master equations of quantum Brownian motion in a general environment with external force
SciTech Connect
Roura, Albert; Fleming, C H; Hu, B L
2008-01-01
We revisit the model of a system made up of a Brownian quantum oscillator linearly coupled to an environment made up of many quantum oscillators at finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] has incorrectly specified coefficients for the case of nonlocal dissipation. We rederive the QBM master equation, correctly specifying all coefficients, and determine the position uncertainty to be free of excessive cutoff sensitivity. Our coefficients and solutions are reduced entirely to contour integration for analytic spectra at arbitrary temperature, coupling strength, and cut-off. As an illustration we calculate the master equation coefficients and solve the master equation for ohmic coupling (with finite cutoff) and example supra-ohmic and sub-ohmic spectral densities. We determine the effect of an external force on the quantum oscillator and also show that our representation of the master equation and solutions naturally extends to a system of multiple oscillators bilinearly coupled to themselves and the bath in arbitrary fashion. This produces a formula for investigating the standard quantum limit which is central to addressing many theoretical issues in macroscopic quantum phenomena and experimental concerns related to low temperature precision measurements. We find that in a dissipative environment, all initial states settle down to a Gaussian density matrix whose covariance is determined by the thermal reservoir and whose mean is determined by the external force. We specify the thermal covariance for the spectral densities we explore.
1. Biorthonormal eigenbasis of a Markovian master equation for the quantum Brownian motion
SciTech Connect
Tay, B. A.; Petrosky, T.
2008-11-15
The solution to a quantum Markovian master equation of a harmonic oscillator weakly coupled to a thermal reservoir is investigated as a non-Hermitian eigenvalue problem in space coordinates. In terms of a pair of quantum action-angle variables, the equation becomes separable and a complete set of biorthogonal eigenfunctions can be constructed. Properties of quantum states, such as the change in the quantum coherence length, damping in the motion, and disappearance of the spatial interference pattern, can then be described as the decay of the nonequilibrium modes in the eigenbasis expansion. It is found that the process of gaining quantum coherence from the environment takes a longer time than the opposite process of losing quantum coherence to the environment. An estimate of the time scales of these processes is obtained.
2. Continuous Time Open Quantum Random Walks and Non-Markovian Lindblad Master Equations
Pellegrini, Clément
2014-02-01
A new type of quantum random walks, called Open Quantum Random Walks, has been developed and studied in Attal et al. (Open quantum random walks, preprint) and (Central limit theorems for open quantum random walks, preprint). In this article we present a natural continuous time extension of these Open Quantum Random Walks. This continuous time version is obtained by taking a continuous time limit of the discrete time Open Quantum Random Walks. This approximation procedure is based on some adaptation of Repeated Quantum Interactions Theory (Attal and Pautrat in Annales Henri Poincaré Physique Théorique 7:59-104, 2006) coupled with the use of correlated projectors (Breuer in Phys Rev A 75:022103, 2007). The limit evolutions obtained this way give rise to a particular type of quantum master equations. These equations appeared originally in the non-Markovian generalization of the Lindblad theory (Breuer in Phys Rev A 75:022103, 2007). We also investigate the continuous time limits of the quantum trajectories associated with Open Quantum Random Walks. We show that the limit evolutions in this context are described by jump stochastic differential equations. Finally we present a physical example which can be described in terms of Open Quantum Random Walks and their associated continuous time limits.
3. Capture process in nuclear reactions with a quantum master equation
SciTech Connect
Sargsyan, V. V.; Kanokov, Z.; Adamian, G. G.; Antonenko, N. V.; Scheid, W.
2009-09-15
Projectile-nucleus capture by a target nucleus at bombarding energies in the vicinity of the Coulomb barrier is treated with the reduced-density-matrix formalism. The effects of dissipation and fluctuations on the capture process are taken self-consistently into account within the quantum model suggested. The excitation functions for the capture in the reactions {sup 16}O, {sup 19}F, {sup 26}Mg, {sup 28}Si, {sup 32,34,36,38}S, {sup 40,48}Ca, {sup 50}Ti, {sup 52}Cr+{sup 208}Pb with spherical nuclei are calculated and compared with the experimental data. At bombarding energies about (15-25) MeV above the Coulomb barrier the maximum of capture cross section is revealed for the {sup 58}Ni+{sup 208}Pb reaction.
4. Improved master equation approach to quantum transport: From Born to self-consistent Born approximation
SciTech Connect
Jin, Jinshuang; Li, Jun; Liu, Yu; Li, Xin-Qi; Yan, YiJing
2014-06-28
Beyond the second-order Born approximation, we propose an improved master equation approach to quantum transport under self-consistent Born approximation. The basic idea is to replace the free Green's function in the tunneling self-energy diagram by an effective reduced propagator under the Born approximation. This simple modification has remarkable consequences. It not only recovers the exact results for quantum transport through noninteracting systems under arbitrary voltages, but also predicts the challenging nonequilibrium Kondo effect. Compared to the nonequilibrium Green's function technique that formulates the calculation of specific correlation functions, the master equation approach contains richer dynamical information to allow more efficient studies for such as the shot noise and full counting statistics.
5. Partial secular Bloch-Redfield master equation for incoherent excitation of multilevel quantum systems
SciTech Connect
Tscherbul, Timur V. Brumer, Paul
2015-03-14
We present an efficient theoretical method for calculating the time evolution of the density matrix of a multilevel quantum system weakly interacting with incoherent light. The method combines the Bloch-Redfield theory with a partial secular approximation for one-photon coherences, resulting in a master equation that explicitly exposes the reliance on transition rates and the angles between transition dipole moments in the energy basis. The partial secular Bloch-Redfield master equation allows an unambiguous distinction between the regimes of quantum coherent vs. incoherent energy transfer under incoherent light illumination. The fully incoherent regime is characterized by orthogonal transition dipole moments in the energy basis, leading to a dynamical evolution governed by a coherence-free Pauli-type master equation. The coherent regime requires non-orthogonal transition dipole moments in the energy basis and leads to the generation of noise-induced quantum coherences and population-to-coherence couplings. As a first application, we consider the dynamics of excited state coherences arising under incoherent light excitation from a single ground state and observe population-to-coherence transfer and the formation of non-equilibrium quasisteady states in the regime of small excited state splitting. Analytical expressions derived earlier for the V-type system [T. V. Tscherbul and P. Brumer, Phys. Rev. Lett. 113, 113601 (2014)] are found to provide a nearly quantitative description of multilevel excited-state populations and coherences in both the small- and large-molecule limits.
6. On the accuracy of the Padé-resummed master equation approach to dissipative quantum dynamics.
PubMed
Chen, Hsing-Ta; Berkelbach, Timothy C; Reichman, David R
2016-04-21
Well-defined criteria are proposed for assessing the accuracy of quantum master equations whose memory functions are approximated by Padé resummation of the first two moments in the electronic coupling. These criteria partition the parameter space into distinct levels of expected accuracy, ranging from quantitatively accurate regimes to regions of parameter space where the approach is not expected to be applicable. Extensive comparison of Padé-resummed master equations with numerically exact results in the context of the spin-boson model demonstrates that the proposed criteria correctly demarcate the regions of parameter space where the Padé approximation is reliable. The applicability analysis we present is not confined to the specifics of the Hamiltonian under consideration and should provide guidelines for other classes of resummation techniques. PMID:27389208
7. Polaronic quantum master equation theory of inelastic and coherent resonance energy transfer for soft systems.
PubMed
Yang, Lei; Devi, Murali; Jang, Seogjoo
2012-07-14
This work extends the theory of coherent resonance energy transfer [S. Jang, J. Chem. Phys. 131, 164101 (2009)] by including quantum mechanical inelastic effects due to modulation of donor-acceptor electronic coupling. Within the approach of the second order time local quantum master equation (QME) in the polaron picture and under the assumption that the bath degrees of freedom modulating the electronic coupling are independent of other modes, a general time evolution equation for the reduced system density operator is derived. Detailed expressions for the relaxation operators and inhomogeneous terms of the QME are then derived for three specific models of modulation in distance, axial angle, and dihedral angle, which are all approximated by harmonic oscillators. Numerical tests are conducted for a set of model parameters. Model calculation shows that the torsional modulation can make significant contribution to the relaxation and dephasing mechanisms.
8. Density-matrix operatorial solution of the non-Markovian master equation for quantum Brownian motion
SciTech Connect
Intravaia, F.; Maniscalco, S.; Messina, A.
2003-04-01
An original method to exactly solve the non-Markovian master equation describing the interaction of a single harmonic oscillator with a quantum environment in the weak-coupling limit is reported. By using a superoperatorial approach, we succeed in deriving the operatorial solution for the density matrix of the system. Our method is independent of the physical properties of the environment. We show the usefulness of our solution deriving explicit expressions for the dissipative time evolution of some observables of physical interest for the system, such as, for example, its mean energy.
9. On the quantum master equation for Bardeen-Cooper-Schrieffer pairing models
Huang, C. F.; Huang, K.-N.
2007-03-01
A master equation symmetric with respect to particles and holes has been introduced for systems composed of non-interacting identical fermions. [C. F. Huang and K. -N. Huang Chinese J. Phys. 42, 221 (2004); R. Gebauer R and R. Car R Phys. Rev. B 70, 125324 (2004).] Extensions to such an equation, in fact, can be obtained by incorporating two anti-hermitian terms for the lifetimes of particles and holes to construct the quantum relaxation term. In this poster, we focus on the extended equation for the interacting Fermi systems modeled by Bardeen- Cooper-Schrieffer (BCS) pairing theory. A constraint on the relaxation term is taken into account to preserve the pairing relation. Such a constraint, in fact, is also important when the coupling between quasiparticles and quasiholes is introduced to unify the BCS and antiferromagnetic/ferromagnetic models.
10. Application of quantum master equation for long-term prognosis of asset-prices
Khrennikova, Polina
2016-05-01
This study combines the disciplines of behavioral finance and an extension of econophysics, namely the concepts and mathematical structure of quantum physics. We apply the formalism of quantum theory to model the dynamics of some correlated financial assets, where the proposed model can be potentially applied for developing a long-term prognosis of asset price formation. At the informational level, the asset price states interact with each other by the means of a "financial bath". The latter is composed of agents' expectations about the future developments of asset prices on the finance market, as well as financially important information from mass-media, society, and politicians. One of the essential behavioral factors leading to the quantum-like dynamics of asset prices is the irrationality of agents' expectations operating on the finance market. These expectations lead to a deeper type of uncertainty concerning the future price dynamics of the assets, than given by a classical probability theory, e.g., in the framework of the classical financial mathematics, which is based on the theory of stochastic processes. The quantum dimension of the uncertainty in price dynamics is expressed in the form of the price-states superposition and entanglement between the prices of the different financial assets. In our model, the resolution of this deep quantum uncertainty is mathematically captured with the aid of the quantum master equation (its quantum Markov approximation). We illustrate our model of preparation of a future asset price prognosis by a numerical simulation, involving two correlated assets. Their returns interact more intensively, than understood by a classical statistical correlation. The model predictions can be extended to more complex models to obtain price configuration for multiple assets and portfolios.
11. Criteria for the accuracy of small polaron quantum master equation in simulating excitation energy transfer dynamics
SciTech Connect
Chang, Hung-Tzu; Cheng, Yuan-Chung; Zhang, Pan-Pan
2013-12-14
The small polaron quantum master equation (SPQME) proposed by Jang et al. [J. Chem. Phys. 129, 101104 (2008)] is a promising approach to describe coherent excitation energy transfer dynamics in complex molecular systems. To determine the applicable regime of the SPQME approach, we perform a comprehensive investigation of its accuracy by comparing its simulated population dynamics with numerically exact quasi-adiabatic path integral calculations. We demonstrate that the SPQME method yields accurate dynamics in a wide parameter range. Furthermore, our results show that the accuracy of polaron theory depends strongly upon the degree of exciton delocalization and timescale of polaron formation. Finally, we propose a simple criterion to assess the applicability of the SPQME theory that ensures the reliability of practical simulations of energy transfer dynamics with SPQME in light-harvesting systems.
12. Generalized quantum master equations in and out of equilibrium: When can one win?
PubMed
Kelly, Aaron; Montoya-Castillo, Andrés; Wang, Lu; Markland, Thomas E
2016-05-14
Generalized quantum master equations (GQMEs) are an important tool in modeling chemical and physical processes. For a large number of problems, it has been shown that exact and approximate quantum dynamics methods can be made dramatically more efficient, and in the latter case more accurate, by proceeding via the GQME formalism. However, there are many situations where utilizing the GQME approach with an approximate method has been observed to return the same dynamics as using that method directly. Here, for systems both in and out of equilibrium, we provide a more detailed understanding of the conditions under which using an approximate method can yield benefits when combined with the GQME formalism. In particular, we demonstrate the necessary manipulations, which are satisfied by exact quantum dynamics, that are required to recast the memory kernel in a form that can be analytically shown to yield the same result as a direct application of the dynamics regardless of the approximation used. By considering the connections between these forms of the kernel, we derive the conditions that approximate methods must satisfy if they are to offer different results when used in conjunction with the GQME formalism. These analytical results thus provide new insights as to when proceeding via the GQME approach can be used to improve the accuracy of simulations.
13. Generalized quantum master equations in and out of equilibrium: When can one win?
Kelly, Aaron; Montoya-Castillo, Andrés; Wang, Lu; Markland, Thomas E.
2016-05-01
Generalized quantum master equations (GQMEs) are an important tool in modeling chemical and physical processes. For a large number of problems, it has been shown that exact and approximate quantum dynamics methods can be made dramatically more efficient, and in the latter case more accurate, by proceeding via the GQME formalism. However, there are many situations where utilizing the GQME approach with an approximate method has been observed to return the same dynamics as using that method directly. Here, for systems both in and out of equilibrium, we provide a more detailed understanding of the conditions under which using an approximate method can yield benefits when combined with the GQME formalism. In particular, we demonstrate the necessary manipulations, which are satisfied by exact quantum dynamics, that are required to recast the memory kernel in a form that can be analytically shown to yield the same result as a direct application of the dynamics regardless of the approximation used. By considering the connections between these forms of the kernel, we derive the conditions that approximate methods must satisfy if they are to offer different results when used in conjunction with the GQME formalism. These analytical results thus provide new insights as to when proceeding via the GQME approach can be used to improve the accuracy of simulations.
14. Retrodictive derivation of the radical-ion-pair master equation and Monte Carlo simulation with single-molecule quantum trajectories.
PubMed
Kritsotakis, M; Kominis, I K
2014-10-01
Radical-ion-pair reactions, central in photosynthesis and the avian magnetic compass mechanism, have been recently shown to be a paradigm system for applying quantum information science in a biochemical setting. The fundamental quantum master equation describing radical-ion-pair reactions is still under debate. Here we use quantum retrodiction to formally refine the theory put forward in the paper by Kominis [I. K. Kominis, Phys. Rev. E 83, 056118 (2011)]. We also provide a rigorous analysis of the measure of singlet-triplet coherence required for deriving the radical-pair master equation. A Monte Carlo simulation with single-molecule quantum trajectories supports the self-consistency of our approach. PMID:25375535
15. Retrodictive derivation of the radical-ion-pair master equation and Monte Carlo simulation with single-molecule quantum trajectories
Kritsotakis, M.; Kominis, I. K.
2014-10-01
Radical-ion-pair reactions, central in photosynthesis and the avian magnetic compass mechanism, have been recently shown to be a paradigm system for applying quantum information science in a biochemical setting. The fundamental quantum master equation describing radical-ion-pair reactions is still under debate. Here we use quantum retrodiction to formally refine the theory put forward in the paper by Kominis [I. K. Kominis, Phys. Rev. E 83, 056118 (2011), 10.1103/PhysRevE.83.056118]. We also provide a rigorous analysis of the measure of singlet-triplet coherence required for deriving the radical-pair master equation. A Monte Carlo simulation with single-molecule quantum trajectories supports the self-consistency of our approach.
16. Out-of-equilibrium open quantum systems: A comparison of approximate quantum master equation approaches with exact results
Purkayastha, Archak; Dhar, Abhishek; Kulkarni, Manas
2016-06-01
We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of N sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the N =2 case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.
17. Accurate nonadiabatic quantum dynamics on the cheap: Making the most of mean field theory with master equations
SciTech Connect
Kelly, Aaron; Markland, Thomas E.; Brackbill, Nora
2015-03-07
In this article, we show how Ehrenfest mean field theory can be made both a more accurate and efficient method to treat nonadiabatic quantum dynamics by combining it with the generalized quantum master equation framework. The resulting mean field generalized quantum master equation (MF-GQME) approach is a non-perturbative and non-Markovian theory to treat open quantum systems without any restrictions on the form of the Hamiltonian that it can be applied to. By studying relaxation dynamics in a wide range of dynamical regimes, typical of charge and energy transfer, we show that MF-GQME provides a much higher accuracy than a direct application of mean field theory. In addition, these increases in accuracy are accompanied by computational speed-ups of between one and two orders of magnitude that become larger as the system becomes more nonadiabatic. This combination of quantum-classical theory and master equation techniques thus makes it possible to obtain the accuracy of much more computationally expensive approaches at a cost lower than even mean field dynamics, providing the ability to treat the quantum dynamics of atomistic condensed phase systems for long times.
18. Accurate nonadiabatic quantum dynamics on the cheap: making the most of mean field theory with master equations.
PubMed
Kelly, Aaron; Brackbill, Nora; Markland, Thomas E
2015-03-01
In this article, we show how Ehrenfest mean field theory can be made both a more accurate and efficient method to treat nonadiabatic quantum dynamics by combining it with the generalized quantum master equation framework. The resulting mean field generalized quantum master equation (MF-GQME) approach is a non-perturbative and non-Markovian theory to treat open quantum systems without any restrictions on the form of the Hamiltonian that it can be applied to. By studying relaxation dynamics in a wide range of dynamical regimes, typical of charge and energy transfer, we show that MF-GQME provides a much higher accuracy than a direct application of mean field theory. In addition, these increases in accuracy are accompanied by computational speed-ups of between one and two orders of magnitude that become larger as the system becomes more nonadiabatic. This combination of quantum-classical theory and master equation techniques thus makes it possible to obtain the accuracy of much more computationally expensive approaches at a cost lower than even mean field dynamics, providing the ability to treat the quantum dynamics of atomistic condensed phase systems for long times.
19. Thermal Decomposition of NCN: Shock-Tube Study, Quantum Chemical Calculations, and Master-Equation Modeling.
PubMed
Busch, Anna; González-García, Núria; Lendvay, György; Olzmann, Matthias
2015-07-16
The thermal decomposition of cyanonitrene, NCN, was studied behind reflected shock waves in the temperature range 1790-2960 K at pressures near 1 and 4 bar. Highly diluted mixtures of NCN3 in argon were shock-heated to produce NCN, and concentration-time profiles of C atoms as reaction product were monitored with atomic resonance absorption spectroscopy at 156.1 nm. Calibration was performed with methane pyrolysis experiments. Rate coefficients for the reaction (3)NCN + M → (3)C + N2 + M (R1) were determined from the initial slopes of the C atom concentration-time profiles. Reaction R1 was found to be in the low-pressure regime at the conditions of the experiments. The temperature dependence of the bimolecular rate coefficient can be expressed with the following Arrhenius equation: k1(bim) = (4.2 ± 2.1) × 10(14) exp[-242.3 kJ mol(-1)/(RT)] cm(3) mol(-1) s(-1). The rate coefficients were analyzed by using a master equation with specific rate coefficients from RRKM theory. The necessary molecular data and energies were calculated with quantum chemical methods up to the CCSD(T)/CBS//CCSD/cc-pVTZ level of theory. From the topography of the potential energy surface, it follows that reaction R1 proceeds via isomerization of NCN to CNN and subsequent C-N bond fission along a collinear reaction coordinate without a tight transition state. The calculations reproduce the magnitude and temperature dependence of the rate coefficient and confirm that reaction R1 is in the low-pressure regime under our experimental conditions.
20. Sharp peaks in the conductance of a double quantum dot and a quantum-dot spin valve at high temperatures: A hierarchical quantum master equation approach
Wenderoth, S.; Bätge, J.; Härtle, R.
2016-09-01
We study sharp peaks in the conductance-voltage characteristics of a double quantum dot and a quantum dot spin valve that are located around zero bias. The peaks share similarities with a Kondo peak but can be clearly distinguished, in particular as they occur at high temperatures. The underlying physical mechanism is a strong current suppression that is quenched in bias-voltage dependent ways by exchange interactions. Our theoretical results are based on the quantum master equation methodology, including the Born-Markov approximation and a numerically exact, hierarchical scheme, which we extend here to the spin-valve case. The comparison of exact and approximate results allows us to reveal the underlying physical mechanisms, the role of first-, second- and beyond-second-order processes and the robustness of the effect.
1. Using non-Markovian measures to evaluate quantum master equations for photosynthesis
Chen, Hong-Bin; Lambert, Neill; Cheng, Yuan-Chung; Chen, Yueh-Nan; Nori, Franco
2015-08-01
When dealing with system-reservoir interactions in an open quantum system, such as a photosynthetic light-harvesting complex, approximations are usually made to obtain the dynamics of the system. One question immediately arises: how good are these approximations, and in what ways can we evaluate them? Here, we propose to use entanglement and a measure of non-Markovianity as benchmarks for the deviation of approximate methods from exact results. We apply two frequently-used perturbative but non-Markovian approximations to a photosynthetic dimer model and compare their results with that of the numerically-exact hierarchy equation of motion (HEOM). This enables us to explore both entanglement and non-Markovianity measures as means to reveal how the approximations either overestimate or underestimate memory effects and quantum coherence. In addition, we show that both the approximate and exact results suggest that non-Markonivity can, counter-intuitively, increase with temperature, and with the coupling to the environment.
2. Stochastically averaged master equation for a quantum-dynamic system interacting with a thermal bath
Petrov, E. G.; Teslenko, V. I.; Goychuk, I. A.
1994-05-01
The methods of nonequilibrium density-matrix and coarse-temporal conception are used to obtain the kinetic equation for the parameters γnm(t)=Sp[ρ^(t)||n>quantum-dynamic system (QDS) interacting with a thermal bath and external stochastic field. It is important that the stochastic field is taken exactly into consideration. For diagonal QDS parameters γnn(t) this equation is reduced to the generalized Pauli equation (GPE) with stochastic time-dependent coefficients wnm(t). Special attention is given to the procedure of averaging over stochastic processes. It is shown that after averaging over energy fluctuations affected by the stochastic field, in the first cumulant approximation in terms of stochastic processes wnm(t), the GPE is transformed to the Pauli equation for the QDS state population Pn(t)=<γnn(t)>f. As an example, the relaxation behavior of a two-level system interacting with a dichotomous field (dichotomous Markovian process of kangaroo type) and a harmonic oscillator coupled with a thermal bath is considered. It is shown that the probability of relaxation transitions between energy levels may be changed by several orders of magnitude under the influence of the dichotomous field.
3. Unraveling of a detailed-balance-preserved quantum master equation and continuous feedback control of a measured qubit
Luo, JunYan; Jin, Jinshuang; Wang, Shi-Kuan; Hu, Jing; Huang, Yixiao; He, Xiao-Ling
2016-03-01
We present a generic unraveling scheme for a detailed-balance-preserved quantum master equation applicable for stochastic point processes in mesoscopic transport. It enables us to investigate continuous measurement of a qubit on the level of single quantum trajectories, where essential correlations between the inherent dynamics of the qubit and detector current fluctuations are revealed. Based on this unraveling scheme, feedback control of the charge qubit is implemented to achieve a desired pure state in the presence of the detailed-balance condition. With sufficient feedback strength, coherent oscillations of the measured qubit can be maintained for arbitrary qubit-detector coupling. Competition between the loss and restoration of coherence entailed, respectively, by measurement back action and feedback control is reflected in the noise power spectrum of the detector's output. It is demonstrated unambiguously that the signal-to-noise ratio is significantly enhanced with increasing feedback strength and could even exceed the well-known Korotkov-Averin bound in quantum measurement. The proposed unraveling and feedback scheme offers a transparent and straightforward approach to effectively sustaining ideal coherent oscillations of a charge qubit in the field of quantum computation.
4. Nonequilibrium dynamical mean-field theory: an auxiliary quantum master equation approach.
PubMed
Arrigoni, Enrico; Knap, Michael; von der Linden, Wolfgang
2013-02-22
We introduce a versatile method to compute electronic steady-state properties of strongly correlated extended quantum systems out of equilibrium. The approach is based on dynamical mean-field theory (DMFT), in which the original system is mapped onto an auxiliary nonequilibrium impurity problem imbedded in a Markovian environment. The steady-state Green's function of the auxiliary system is solved by full diagonalization of the corresponding Lindblad equation. The approach can be regarded as the nontrivial extension of the exact-diagonalization-based DMFT to the nonequilibrium case. As a first application, we consider an interacting Hubbard layer attached to two metallic leads and present results for the steady-state current and the nonequilibrium density of states.
5. Dynamical suppression of decoherence by phase kicks: Master equation approach
SciTech Connect
Ban, Masashi; Kitajima, Sachiko; Shibata, Fumiaki
2007-08-15
The irreversible time evolution of a quantum system interacting with a large environmental system can be described by a quantum master equation. When an external field is applied to a quantum system, a non-Markovian mater equation is derived in a rigorous way, where the relaxation terms in the quantum master equation include the effects of the external field. It is shown that, when the external field is a sequence of phase-modulation pulses, the decoherence of the quantum system can be suppressed under certain conditions. To see the effects of phase-modulation pulses, the irreversible time evolutions of qubit and photon systems are investigated in detail.
6. Coherent one-photon phase control in closed and open quantum systems: a general master equation approach.
PubMed
Pachón, Leonardo A; Yu, Li; Brumer, Paul
2013-01-01
The underlying mechanisms for one photon phase control are revealed through a master equation approach. Specifically, two mechanisms are identified, one operating on the laser time scale and the other on the time scale of the system-bath interaction. The effects of the secular and non-secular Markovian approximations are carefully examined.
7. Master equation based steady-state cluster perturbation theory
Nuss, Martin; Dorn, Gerhard; Dorda, Antonius; von der Linden, Wolfgang; Arrigoni, Enrico
2015-09-01
A simple and efficient approximation scheme to study electronic transport characteristics of strongly correlated nanodevices, molecular junctions, or heterostructures out of equilibrium is provided by steady-state cluster perturbation theory. In this work, we improve the starting point of this perturbative, nonequilibrium Green's function based method. Specifically, we employ an improved unperturbed (so-called reference) state ρ̂S, constructed as the steady state of a quantum master equation within the Born-Markov approximation. This resulting hybrid method inherits beneficial aspects of both the quantum master equation as well as the nonequilibrium Green's function technique. We benchmark this scheme on two experimentally relevant systems in the single-electron transistor regime: an electron-electron interaction based quantum diode and a triple quantum dot ring junction, which both feature negative differential conductance. The results of this method improve significantly with respect to the plain quantum master equation treatment at modest additional computational cost.
8. Theoretical study on exciton recurrence motion in anthracene dimer using the Ab initio MO-CI based quantum master equation approach.
PubMed
Kishi, Ryohei; Nakano, Masayoshi; Minami, Takuya; Fukui, Hitoshi; Nagai, Hiroshi; Yoneda, Kyohei; Takahashi, Hideaki
2009-05-01
We apply the ab initio molecular orbital (MO)-configuration interaction (CI) based quantum master equation (MOQME) method to the investigation of ultrafast exciton dynamics in an anthracene dimer modeled after anthracenophane, which is experimentally found to exhibit an oscillatory signal of fluorescence anisotropy decay. Two low-lying near-degenerate one-photon allowed excited states with a slight energy difference (42 cm(-1)) are obtained at the CIS/6-31G** level of approximation using full valence pi-orbitals. The time evolution of reduced exciton density matrices is performed by numerically solving the quantum master equation. After the creation of a superposition state of these near-degenerate states by irradiating a near-resonant laser field, we observe two kinds of oscillatory behaviors of polarizations: field-induced polarizations with faster periods, and amplitude oscillations of x- and z-polarizations, P(x) and P(z), with a slower period, in which the amplitudes of P(x) and P(z) attain maximum alternately. The latter behavior turns out to be associated with an oscillatory exciton motion between the two monomers, i.e., exciton recurrence motion, using the dynamic exciton expression based on the polarization density. From the analysis of contribution to the exciton distributions, such exciton recurrence motion is found to be characterized by both the difference in eigenfrequencies between the two near-degenerate states excited by the laser field and the relative phases among the frontier MOs primarily contributing to the near-degenerate states.
9. A closure scheme for chemical master equations.
PubMed
2013-08-27
Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higher-order moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kinetic Monte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.
10. Approximate probability distributions of the master equation
Thomas, Philipp; Grima, Ramon
2015-07-01
Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.
11. Ab initio molecular orbital-configuration interaction based quantum master equation (MOQME) approach to the dynamic first hyperpolarizabilities of asymmetric π-conjugated systems
SciTech Connect
Kishi, Ryohei; Fujii, Hiroaki; Minami, Takuya; Shigeta, Yasuteru; Nakano, Masayoshi
2015-01-22
In this study, we apply the ab initio molecular orbital - configuration interaction based quantum master equation (MOQME) approach to the calculation and analysis of the dynamic first hyperpolarizabilities (β) of asymmetric π-conjugated molecules. In this approach, we construct the excited state models by the ab initio configuration interaction singles method. Then, time evolutions of system reduced density matrix ρ(t) and system polarization p(t) are calculated by the QME approach. Dynamic β in the second harmonic generation is calculated based on the nonperturbative definition of nonlinear optical susceptibility, using the frequency domain system polarization p(ω). Spatial contributions of electrons to β are analyzed based on the dynamic hyperpolarizability density map, which visualizes the second-order response of charge density oscillating with a frequency of 2ω. We apply the present method to the calculation of the dynamic β of a series of donor/acceptor substituted polyene oligomers, and then discuss the applicability of the MOQME method to the calculation and analysis of dynamic NLO properties of molecular systems.
12. Master equation as a radial constraint
Hussain, Uzair; Booth, Ivan; Kunduri, Hari K.
2016-06-01
We revisit the problem of perturbations of Schwarzschild-AdS4 black holes by using a combination of the Martel-Poisson formalism for perturbations of four-dimensional spherically symmetric spacetimes [K. Martel and E. Poisson, Phys. Rev. D 71, 104003 (2005).] and the Kodama-Ishibashi formalism [H. Kodama and A. Ishibashi, Prog. Theor. Phys. 110, 701 (2003).]. We clarify the relationship between both formalisms and express the Brown-York-Balasubramanian-Krauss boundary stress-energy tensor, T¯μ ν, on a finite-r surface purely in terms of the even and odd master functions. Then, on these surfaces we find that the spacelike components of the conservation equation D¯μT¯μ ν=0 are equivalent to the wave equations for the master functions. The renormalized stress-energy tensor at the boundary r/L lim r →∞ T¯μ ν is calculated directly in terms of the master functions.
13. Exact master equation for a noncommutative Brownian particle
SciTech Connect
Costa Dias, Nuno Nuno Prata, Joao
2009-01-15
We derive the Hu-Paz-Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators on the plane with spatial noncommutativity. The results obtained are exact to all orders in the noncommutative parameter. As a by-product we derive some miscellaneous results such as the equilibrium Wigner distribution for the reservoir of noncommutative oscillators, the weak coupling limit of the master equation and a set of sufficient conditions for strict purity decrease of the Brownian particle. Finally, we consider a high-temperature Ohmic model and obtain an estimate for the time scale of the transition from noncommutative to ordinary quantum mechanics. This scale is considerably smaller than the decoherence scale.
14. A master equation for a two-sided optical cavity
PubMed Central
Barlow, Thomas M.; Bennett, Robert; Beige, Almut
2015-01-01
Quantum optical systems, like trapped ions, are routinely described by master equations. The purpose of this paper is to introduce a master equation for two-sided optical cavities with spontaneous photon emission. To do so, we use the same notion of photons as in linear optics scattering theory and consider a continuum of travelling-wave cavity photon modes. Our model predicts the same stationary state photon emission rates for the different sides of a laser-driven optical cavity as classical theories. Moreover, it predicts the same time evolution of the total cavity photon number as the standard standing-wave description in experiments with resonant and near-resonant laser driving. The proposed resonator Hamiltonian can be used, for example, to analyse coherent cavity-fiber networks [E. Kyoseva et al., New J. Phys. 14, 023023 (2012)].
15. Master equation for an oscillator coupled to the electromagnetic field
SciTech Connect
Ford, G.W. |; Lewis, J.T.; OConnell, R.F. |
1996-12-01
The macroscopic description of a quantum oscillator with linear passive dissipation is formulated in terms of a master equation for the reduced density matrix. The procedure used is based on the asymptotic methods of nonlinear dynamics, which enables one to obtain an expression for the general term in the weak coupling expansion. For the special example of a charged oscillator interacting with the electromagnetic field, an explicit form of the master equation through third order in this expansion is obtained. This form differs from that generally obtained using the rotating wave approximation in that there is no electromagnetic (Lamb) shift and that an explicit expression is given for the decay rate. Copyright {copyright} 1996 Academic Press, Inc.
16. Sufficient conditions for a memory-kernel master equation
Chruściński, Dariusz; Kossakowski, Andrzej
2016-08-01
We derive sufficient conditions for the memory-kernel governing nonlocal master equation which guarantee a legitimate (completely positive and trace-preserving) dynamical map. It turns out that these conditions provide natural parametrizations of the dynamical map being a generalization of the Markovian semigroup. This parametrization is defined by the so-called legitimate pair—monotonic quantum operation and completely positive map—and it is shown that such a class of maps covers almost all known examples from the Markovian semigroup, the semi-Markov evolution, up to collision models and their generalization.
17. Epidemics in networks: a master equation approach
Cotacallapa, M.; Hase, M. O.
2016-02-01
A problem closely related to epidemiology, where a subgraph of ‘infected’ links is defined inside a larger network, is investigated. This subgraph is generated from the underlying network by a random variable, which decides whether a link is able to propagate a disease/information. The relaxation timescale of this random variable is examined in both annealed and quenched limits, and the effectiveness of propagation of disease/information is analyzed. The dynamics of the model is governed by a master equation and two types of underlying network are considered: one is scale-free and the other has exponential degree distribution. We have shown that the relaxation timescale of the contagion variable has a major influence on the topology of the subgraph of infected links, which determines the efficiency of spreading of disease/information over the network.
18. Thermodynamics of the polaron master equation at finite bias
SciTech Connect
Krause, Thilo Brandes, Tobias; Schaller, Gernot; Esposito, Massimiliano
2015-04-07
We study coherent transport through a double quantum dot. Its two electronic leads induce electronic matter and energy transport and a phonon reservoir contributes further energy exchanges. By treating the system-lead couplings perturbatively, whereas the coupling to vibrations is treated non-perturbatively in a polaron-transformed frame, we derive a thermodynamic consistent low-dimensional master equation. When the number of phonon modes is finite, a Markovian description is only possible when these couple symmetrically to both quantum dots. For a continuum of phonon modes however, also asymmetric couplings can be described with a Markovian master equation. We compute the electronic current and dephasing rate. The electronic current enables transport spectroscopy of the phonon frequency and displays signatures of Franck-Condon blockade. For infinite external bias but finite tunneling bandwidths, we find oscillations in the current as a function of the internal bias due to the electron-phonon coupling. Furthermore, we derive the full fluctuation theorem and show its identity to the entropy production in the system.
19. Class of exact memory-kernel master equations
Lorenzo, Salvatore; Ciccarello, Francesco; Palma, G. Massimo
2016-05-01
A well-known situation in which a non-Markovian dynamics of an open quantum system S arises is when this is coherently coupled to an auxiliary system M in contact with a Markovian bath. In such cases, while the joint dynamics of S -M is Markovian and obeys a standard (bipartite) Lindblad-type master equation (ME), this is in general not true for the reduced dynamics of S . Furthermore, there are several instances (e.g., the dissipative Jaynes-Cummings model) in which a closed ME for the S 's state cannot even be worked out. Here, we find a class of bipartite Lindblad-type MEs such that the reduced ME of S can be derived exactly and in a closed form for any initial product state of S -M . We provide a detailed microscopic derivation of our result in terms of a mapping between two collision models.
20. A master functional for quantum field theory
Anselmi, Damiano
2013-04-01
We study a new generating functional of one-particle irreducible diagrams in quantum field theory, called master functional, which is invariant under the most general perturbative changes of field variables. The usual functional Γ does not behave as a scalar under the transformation law inherited from its very definition as the Legendre transform of W=ln Z, although it does behave as a scalar under an unusual transformation law. The master functional, on the other hand, is the Legendre transform of an improved functional W with respect to the sources coupled to both elementary and composite fields. The inclusion of certain improvement terms in W and Z is necessary to make this new Legendre transform well defined. The master functional behaves as a scalar under the transformation law inherited from its very definition. Moreover, it admits a proper formulation, obtained extending the set of integrated fields to so-called proper fields, which allows us to work without passing through Z, W or Γ. In the proper formulation the classical action coincides with the classical limit of the master functional, and correlation functions and renormalization are calculated applying the usual diagrammatic rules to the proper fields. Finally, the most general change of field variables, including the map relating bare and renormalized fields, is a linear redefinition of the proper fields.
1. Generalized master equation via aging continuous-time random walks.
PubMed
Allegrini, Paolo; Aquino, Gerardo; Grigolini, Paolo; Palatella, Luigi; Rosa, Angelo
2003-11-01
We discuss the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME). The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density psi(t) that is assumed to be an inverse power law with the power index micro. We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case where psi(t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai [E. Barkai, Phys. Rev. Lett. 90, 104101 (2003)], is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure we create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations. PMID:14682862
2. Generalized master equation via aging continuous-time random walks.
PubMed
Allegrini, Paolo; Aquino, Gerardo; Grigolini, Paolo; Palatella, Luigi; Rosa, Angelo
2003-11-01
We discuss the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME). The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density psi(t) that is assumed to be an inverse power law with the power index micro. We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case where psi(t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai [E. Barkai, Phys. Rev. Lett. 90, 104101 (2003)], is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure we create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations.
3. Master integrals for splitting functions from differential equations in QCD
Gituliar, Oleksandr
2016-02-01
A method for calculating phase-space master integrals for the decay process 1 → n masslesspartonsinQCDusingintegration-by-partsanddifferentialequationstechniques is discussed. The method is based on the appropriate choice of the basis for master integrals which leads to significant simplification of differential equations. We describe an algorithm how to construct the desirable basis, so that the resulting system of differential equations can be recursively solved in terms of (G) HPLs as a series in the dimensional regulator ɛ to any order. We demonstrate its power by calculating master integrals for the NLO time-like splitting functions and discuss future applications of the proposed method at the NNLO precision.
4. Number-conserving master equation theory for a dilute Bose-Einstein condensate
SciTech Connect
Schelle, Alexej; Wellens, Thomas; Buchleitner, Andreas; Delande, Dominique
2011-01-15
We describe the transition of N weakly interacting atoms into a Bose-Einstein condensate within a number-conserving quantum master equation theory. Based on the separation of time scales for condensate formation and noncondensate thermalization, we derive a master equation for the condensate subsystem in the presence of the noncondensate environment under the inclusion of all two-body interaction processes. We numerically monitor the condensate particle number distribution during condensate formation, and derive a condition under which the unique equilibrium steady state of a dilute, weakly interacting Bose-Einstein condensate is given by a Gibbs-Boltzmann thermal state of N noninteracting atoms.
5. The Approach to Equilibrium: Detailed Balance and the Master Equation
ERIC Educational Resources Information Center
Alexander, Millard H.; Hall, Gregory E.; Dagdigian, Paul J.
2011-01-01
The approach to the equilibrium (Boltzmann) distribution of populations of internal states of a molecule is governed by inelastic collisions in the gas phase and with surfaces. The set of differential equations governing the time evolution of the internal state populations is commonly called the master equation. An analytic solution to the master…
6. Operator Approach to the Master Equation for the One-Step Process
Hnatič, M.; Eferina, E. G.; Korolkova, A. V.; Kulyabov, D. S.; Sevastyanov, L. A.
2016-02-01
Background. Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The kinetics of the interaction has recently attracted attention because it often occurs in the physical, chemical, technical, biological, environmental, economic, and sociological systems. However, there are no general methods for the direct study of this equation. The expansion of the equation in a formal Taylor series (the so called Kramers-Moyal's expansion) is used in the procedure of stochastization of one-step processes. Purpose. However, this does not eliminate the need for the study of the master equation. Method. It is proposed to use quantum field perturbation theory for the statistical systems (the so-called Doi method). Results: This work is a methodological material that describes the principles of master equation solution based on quantum field perturbation theory methods. The characteristic property of the work is that it is intelligible for non-specialists in quantum field theory. Conclusions: We show the full equivalence of the operator and combinatorial methods of obtaining and study of the one-step process master equation.
7. IKT for quantum hydrodynamic equations
Tessarotto, Massimo; Ellero, Marco; Nicolini, Piero
2007-11-01
A striking feature of standard quantum mechanics (SQM) is its analogy with classical fluid dynamics. In fact, it is well-known that the Schr"odinger equation is equivalent to a closed set of partial differential equations for suitable real-valued functions of position and time (denoted as quantum fluid fields) [Madelung, 1928]. In particular, the corresponding quantum hydrodynamic equations (QHE) can be viewed as the equations of a classical compressible and non-viscous fluid, endowed with potential velocity and quantized velocity circulation. In this reference, an interesting theoretical problem, in its own right, is the construction of an inverse kinetic theory (IKT) for such a type of fluids. In this note we intend to investigate consequences of the IKT recently formulated for QHE [M.Tessarotto et al., Phys. Rev. A 75, 012105 (2007)]. In particular a basic issue is related to the definition of the quantum fluid fields.
8. A derivation of the master equation from path entropy maximization
PubMed Central
Lee, Julian; Pressé, Steve
2012-01-01
The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles. PMID:22920099
9. Scattering Theory for Lindblad Master Equations
Falconi, Marco; Faupin, Jérémy; Fröhlich, Jürg; Schubnel, Baptiste
2016-08-01
We study scattering theory for a quantum-mechanical system consisting of a particle scattered off a dynamical target that occupies a compact region in position space. After taking a trace over the degrees of freedom of the target, the dynamics of the particle is generated by a Lindbladian acting on the space of trace-class operators. We study scattering theory for a general class of Lindbladians with bounded interaction terms. First, we consider models where a particle approaching the target is always re-emitted by the target. Then we study models where the particle may be captured by the target. An important ingredient of our analysis is a scattering theory for dissipative operators on Hilbert space.
10. Kraus operator solutions to a fermionic master equation describing a thermal bath and their matrix representation
Xiang-Guo, Meng; Ji-Suo, Wang; Hong-Yi, Fan; Cheng-Wei, Xia
2016-04-01
We solve the fermionic master equation for a thermal bath to obtain its explicit Kraus operator solutions via the fermionic state approach. The normalization condition of the Kraus operators is proved. The matrix representation for these solutions is obtained, which is incongruous with the result in the book completed by Nielsen and Chuang [Quantum Computation and Quantum Information, Cambridge University Press, 2000]. As especial cases, we also present the Kraus operator solutions to master equations for describing the amplitude-decay model and the diffusion process at finite temperature. Project supported by the National Natural Science Foundation of China (Grant No. 11347026), the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2013AM012 and ZR2012AM004), and the Research Fund for the Doctoral Program and Scientific Research Project of Liaocheng University, Shandong Province, China.
11. Chemical master equation closure for computer-aided synthetic biology.
PubMed
2015-01-01
With inexpensive DNA synthesis technologies, we can now construct biological systems by quickly piecing together DNA sequences. Synthetic biology is the promising discipline that focuses on the construction of these new biological systems. Synthetic biology is an engineering discipline, and as such, it can benefit from mathematical modeling. This chapter focuses on mathematical models of biological systems. These models take the form of chemical reaction networks. The importance of stochasticity is discussed and methods to simulate stochastic reaction networks are reviewed. A closure scheme solution is also presented for the master equation of chemical reaction networks. The master equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks for over 70 years. With the first complete solution of chemical master equations, a wide range of experimental observations of biomolecular interactions may be mathematically conceptualized. We anticipate that models based on the closure scheme described herein may assist in rationally designing synthetic biological systems.
12. Maxwell boundary conditions imply non-Lindblad master equation
2016-09-01
From the Hamiltonian connecting the inside and outside of a Fabry-Pérot cavity, which is derived from the Maxwell boundary conditions at a mirror of the cavity, a master equation of a non-Lindblad form is derived when the cavity embeds matters, although we can transform it to the Lindblad form by performing the rotating-wave approximation to the connecting Hamiltonian. We calculate absorption spectra by these Lindblad and non-Lindblad master equations and also by the Maxwell boundary conditions in the framework of the classical electrodynamics, which we consider the most reliable approach. We found that, compared to the Lindblad master equation, the absorption spectra by the non-Lindblad one agree better with those by the Maxwell boundary conditions. Although the discrepancy is highlighted only in the ultrastrong light-matter interaction regime with a relatively large broadening, the master equation of the non-Lindblad form is preferable rather than of the Lindblad one for pursuing the consistency with the classical electrodynamics.
13. Limitations on the utility of exact master equations
SciTech Connect
Ford, G.W.; O'Connell, R.F. . E-mail: [email protected]
2005-10-01
The low temperature solution of the exact master equation for an oscillator coupled to a linear passive heat bath is known to give rise to serious divergences. We now show that, even in the high temperature regime, problems also exist, notably the fact that the density matrix is not necessarily positive.
14. Chemical Master Equation Closure for Computer-Aided Synthetic Biology
PubMed Central
2016-01-01
SUMMARY With inexpensive DNA synthesis technologies, we can now construct biological systems by quickly piecing together DNA sequences. Synthetic biology is the promising discipline that focuses on the construction of these new biological systems. Synthetic biology is an engineering discipline, and as such, it can benefit from mathematical modeling. This chapter focuses on mathematical models of biological systems. These models take the form of chemical reaction networks. The importance of stochasticity is discussed and methods to simulate stochastic reaction networks are reviewed. A closure scheme solution is also presented for the master equation of chemical reaction networks. The master equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks for over seventy years. With the first complete solution of chemical master equations, a wide range of experimental observations of biomolecular interactions may be mathematically conceptualized. We anticipate that models based on the closure scheme described herein may assist in rationally designing synthetic biological systems. PMID:25487098
15. Monitoring derivation of the quantum linear Boltzmann equation
SciTech Connect
Hornberger, Klaus; Vacchini, Bassano
2008-02-15
We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced by Hornberger [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a nonperturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear Boltzmann equation once the state is diagonal in momentum.
16. Stochastic simulation algorithm for the quantum linear Boltzmann equation.
PubMed
Busse, Marc; Pietrulewicz, Piotr; Breuer, Heinz-Peter; Hornberger, Klaus
2010-08-01
We develop a Monte Carlo wave function algorithm for the quantum linear Boltzmann equation, a Markovian master equation describing the quantum motion of a test particle interacting with the particles of an environmental background gas. The algorithm leads to a numerically efficient stochastic simulation procedure for the most general form of this integrodifferential equation, which involves a five-dimensional integral over microscopically defined scattering amplitudes that account for the gas interactions in a nonperturbative fashion. The simulation technique is used to assess various limiting forms of the quantum linear Boltzmann equation, such as the limits of pure collisional decoherence and quantum Brownian motion, the Born approximation, and the classical limit. Moreover, we extend the method to allow for the simulation of the dissipative and decohering dynamics of superpositions of spatially localized wave packets, which enables the study of many physically relevant quantum phenomena, occurring e.g., in the interferometry of massive particles.
17. Dependence of kinetic friction on velocity: master equation approach.
PubMed
Braun, O M; Peyrard, M
2011-04-01
We investigate the velocity dependence of kinetic friction with a model that makes minimal assumptions on the actual mechanism of friction so that it can be applied at many scales, provided the system involves multicontact friction. Using a recently developed master equation approach, we investigate the influence of two concurrent processes. First, at a nonzero temperature, thermal fluctuations allow an activated breaking of contacts that are still below the threshold. As a result, the friction force monotonically increases with velocity. Second, the aging of contacts leads to a decrease of the friction force with velocity. Aging effects include two aspects: the delay in contact formation and aging of a contact itself, i.e., the change of its characteristics with the duration of stationary contact. All these processes are considered simultaneously with the master equation approach, giving a complete dependence of the kinetic friction force on the driving velocity and system temperature, provided the interface parameters are known.
18. Solving Chemical Master Equations by an Adaptive Wavelet Method
SciTech Connect
Jahnke, Tobias; Galan, Steffen
2008-09-01
Solving chemical master equations is notoriously difficult due to the tremendous number of degrees of freedom. We present a new numerical method which efficiently reduces the size of the problem in an adaptive way. The method is based on a sparse wavelet representation and an algorithm which, in each time step, detects the essential degrees of freedom required to approximate the solution up to the desired accuracy.
19. Resummed memory kernels in generalized system-bath master equations
SciTech Connect
Mavros, Michael G.; Van Voorhis, Troy
2014-08-07
Generalized master equations provide a concise formalism for studying reduced population dynamics. Usually, these master equations require a perturbative expansion of the memory kernels governing the dynamics; in order to prevent divergences, these expansions must be resummed. Resummation techniques of perturbation series are ubiquitous in physics, but they have not been readily studied for the time-dependent memory kernels used in generalized master equations. In this paper, we present a comparison of different resummation techniques for such memory kernels up to fourth order. We study specifically the spin-boson Hamiltonian as a model system bath Hamiltonian, treating the diabatic coupling between the two states as a perturbation. A novel derivation of the fourth-order memory kernel for the spin-boson problem is presented; then, the second- and fourth-order kernels are evaluated numerically for a variety of spin-boson parameter regimes. We find that resumming the kernels through fourth order using a Padé approximant results in divergent populations in the strong electronic coupling regime due to a singularity introduced by the nature of the resummation, and thus recommend a non-divergent exponential resummation (the “Landau-Zener resummation” of previous work). The inclusion of fourth-order effects in a Landau-Zener-resummed kernel is shown to improve both the dephasing rate and the obedience of detailed balance over simpler prescriptions like the non-interacting blip approximation, showing a relatively quick convergence on the exact answer. The results suggest that including higher-order contributions to the memory kernel of a generalized master equation and performing an appropriate resummation can provide a numerically-exact solution to system-bath dynamics for a general spectral density, opening the way to a new class of methods for treating system-bath dynamics.
20. Effective equations for the quantum pendulum from momentous quantum mechanics
SciTech Connect
Hernandez, Hector H.; Chacon-Acosta, Guillermo
2012-08-24
In this work we study the quantum pendulum within the framework of momentous quantum mechanics. This description replaces the Schroedinger equation for the quantum evolution of the system with an infinite set of classical equations for expectation values of configuration variables, and quantum dispersions. We solve numerically the effective equations up to the second order, and describe its evolution.
1. Reaction rates for a generalized reaction-diffusion master equation
PubMed Central
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model, and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is on the order of the reaction radius of a reacting pair of molecules. PMID:26871190
2. Reaction rates for a generalized reaction-diffusion master equation
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.
3. Reaction rates for a generalized reaction-diffusion master equation.
PubMed
Hellander, Stefan; Petzold, Linda
2016-01-01
It has been established that there is an inherent limit to the accuracy of the reaction-diffusion master equation. Specifically, there exists a fundamental lower bound on the mesh size, below which the accuracy deteriorates as the mesh is refined further. In this paper we extend the standard reaction-diffusion master equation to allow molecules occupying neighboring voxels to react, in contrast to the traditional approach, in which molecules react only when occupying the same voxel. We derive reaction rates, in two dimensions as well as three dimensions, to obtain an optimal match to the more fine-grained Smoluchowski model and show in two numerical examples that the extended algorithm is accurate for a wide range of mesh sizes, allowing us to simulate systems that are intractable with the standard reaction-diffusion master equation. In addition, we show that for mesh sizes above the fundamental lower limit of the standard algorithm, the generalized algorithm reduces to the standard algorithm. We derive a lower limit for the generalized algorithm which, in both two dimensions and three dimensions, is of the order of the reaction radius of a reacting pair of molecules.
4. Integration of quantum hydrodynamical equation
Ulyanova, Vera G.; Sanin, Andrey L.
2007-04-01
Quantum hydrodynamics equations describing the dynamics of quantum fluid are a subject of this report (QFD).These equations can be used to decide the wide class of problem. But there are the calculated difficulties for the equations, which take place for nonlinear hyperbolic systems. In this connection, It is necessary to impose the additional restrictions which assure the existence and unique of solutions. As test sample, we use the free wave packet and study its behavior at the different initial and boundary conditions. The calculations of wave packet propagation cause in numerical algorithm the division. In numerical algorithm at the calculations of wave packet propagation, there arises the problem of division by zero. To overcome this problem we have to sew together discrete numerical and analytical continuous solutions on the boundary. We demonstrate here for the free wave packet that the numerical solution corresponds to the analytical solution.
5. Evolution equation for quantum coherence
Hu, Ming-Liang; Fan, Heng
2016-07-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures.
6. Evolution equation for quantum coherence
PubMed Central
Hu, Ming-Liang; Fan, Heng
2016-01-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933
7. Evolution equation for quantum coherence.
PubMed
Hu, Ming-Liang; Fan, Heng
2016-01-01
The estimation of the decoherence process of an open quantum system is of both theoretical significance and experimental appealing. Practically, the decoherence can be easily estimated if the coherence evolution satisfies some simple relations. We introduce a framework for studying evolution equation of coherence. Based on this framework, we prove a simple factorization relation (FR) for the l1 norm of coherence, and identified the sets of quantum channels for which this FR holds. By using this FR, we further determine condition on the transformation matrix of the quantum channel which can support permanently freezing of the l1 norm of coherence. We finally reveal the universality of this FR by showing that it holds for many other related coherence and quantum correlation measures. PMID:27382933
8. Multiple re-encounter approach to radical pair reactions and the role of nonlinear master equations
Clausen, Jens; Guerreschi, Gian Giacomo; Tiersch, Markus; Briegel, Hans J.
2014-08-01
We formulate a multiple-encounter model of the radical pair mechanism that is based on a random coupling of the radical pair to a minimal model environment. These occasional pulse-like couplings correspond to the radical encounters and give rise to both dephasing and recombination. While this is in agreement with the original model of Haberkorn and its extensions that assume additional dephasing, we show how a nonlinear master equation may be constructed to describe the conditional evolution of the radical pairs prior to the detection of their recombination. We propose a nonlinear master equation for the evolution of an ensemble of independently evolving radical pairs whose nonlinearity depends on the record of the fluorescence signal. We also reformulate Haberkorn's original argument on the physicality of reaction operators using the terminology of quantum optics/open quantum systems. Our model allows one to describe multiple encounters within the exponential model and connects this with the master equation approach. We include hitherto neglected effects of the encounters, such as a separate dephasing in the triplet subspace, and predict potential new effects, such as Grover reflections of radical spins, that may be observed if the strength and time of the encounters can be experimentally controlled.
9. Friedmann equation with quantum potential
SciTech Connect
Siong, Ch'ng Han; Radiman, Shahidan; Nikouravan, Bijan
2013-11-27
Friedmann equations are used to describe the evolution of the universe. Solving Friedmann equations for the scale factor indicates that the universe starts from an initial singularity where all the physical laws break down. However, the Friedmann equations are well describing the late-time or large scale universe. Hence now, many physicists try to find an alternative theory to avoid this initial singularity. In this paper, we generate a version of first Friedmann equation which is added with an additional term. This additional term contains the quantum potential energy which is believed to play an important role at small scale. However, it will gradually become negligible when the universe evolves to large scale.
10. Adaptive aggregation method for the Chemical Master Equation.
PubMed
Zhang, Jingwei; Watson, Layne T; Cao, Yang
2009-01-01
One important aspect of biological systems such as gene regulatory networks and protein-protein interaction networks is the stochastic nature of interactions between chemical species. Such stochastic behaviour can be accurately modelled by the Chemical Master Equation (CME). However, the CME usually imposes intensive computational requirements when used to characterise molecular biological systems. The major challenge comes from the curse of dimensionality, which has been tackled by a few research papers. The essential goal is to aggregate the system efficiently with limited approximation errors. This paper presents an adaptive way to implement the aggregation process using information collected from Monte Carlo simulations. Numerical results show the effectiveness of the proposed algorithm.
11. Extended master equation models for molecular communication networks.
PubMed
Chou, Chun Tung
2013-06-01
12. Non-Markovian master equation for a system of Fermions interacting with an electromagnetic field
SciTech Connect
Stefanescu, Eliade Scheid, Werner; Sandulescu, Aurel
2008-05-15
For a system of charged Fermions interacting with an electromagnetic field, we derive a non-Markovian master equation in the second-order approximation of the weak dissipative coupling. A complex dissipative environment including Fermions, Bosons and the free electromagnetic field is taken into account. Besides the well-known Markovian term of Lindblad's form, that describes the decay of the system by correlated transitions of the system and environment particles, this equation includes new Markovian and non-Markovian terms proceeding from the fluctuations of the self-consistent field of the environment. These terms describe fluctuations of the energy levels, transitions among these levels stimulated by the fluctuations of the self-consistent field of the environment, and the influence of the time-evolution of the environment on the system dynamics. We derive a complementary master equation describing the environment dynamics correlated with the dynamics of the system. As an application, we obtain non-Markovian Maxwell-Bloch equations and calculate the absorption spectrum of a field propagation mode transversing an array of two-level quantum dots.
13. Generalized master equations for non-Poisson dynamics on networks.
PubMed
Hoffmann, Till; Porter, Mason A; Lambiotte, Renaud
2012-10-01
The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network. This implicitly assumes that the edges are governed by Poisson processes, which is not typically the case in empirical temporal networks. Accordingly, we examine the effects of non-Poisson inter-event statistics on the dynamics of edges, and we apply the concept of a generalized master equation to the study of continuous-time random walks on networks. We show that this equation reduces to the standard rate equations when the underlying process is Poissonian and that its stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate. We conduct numerical simulations and also derive analytical results for the stationary solution under the assumption that all edges have the same waiting-time distribution. We discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature.
14. Generalized master equations for non-Poisson dynamics on networks
Hoffmann, Till; Porter, Mason A.; Lambiotte, Renaud
2012-10-01
The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network. This implicitly assumes that the edges are governed by Poisson processes, which is not typically the case in empirical temporal networks. Accordingly, we examine the effects of non-Poisson inter-event statistics on the dynamics of edges, and we apply the concept of a generalized master equation to the study of continuous-time random walks on networks. We show that this equation reduces to the standard rate equations when the underlying process is Poissonian and that its stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate. We conduct numerical simulations and also derive analytical results for the stationary solution under the assumption that all edges have the same waiting-time distribution. We discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature.
15. Construction and accuracy of partial differential equation approximations to the chemical master equation.
PubMed
Grima, Ramon
2011-11-01
The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.
16. Reaction-diffusion master equation in the microscopic limit
Hellander, Stefan; Hellander, Andreas; Petzold, Linda
2012-04-01
Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice reaction-diffusion master equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. Here we give a general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model and lets us quantify this limit in two and three dimensions. In this light we review and discuss recent work in which the RDME has been modified in different ways in order to better agree with the microscale model for very small voxel sizes.
17. Catchment residence and travel time distributions: The master equation
Botter, Gianluca; Bertuzzo, Enrico; Rinaldo, Andrea
2011-06-01
The probability density functions (pdf's) of travel and residence times are key descriptors of the mechanisms through which catchments retain and release old and event water, transporting solutes to receiving water bodies. In this paper we analyze theoretically such pdf's, whose proper characterization reveals important conceptual and practical differences. A general stochastic framework applicable to arbitrary catchment control volumes is adopted, where time-variable precipitation, evapotranspiration and discharge are assumed to be the major hydrological drivers. The master equation for the residence time pdf is derived and solved analytically, providing expressions for travel and residence time pdf's as a function of input/output fluxes and of the relevant mixing. Our solutions suggest intrinsically time-variant travel and residence time pdf's through a direct dependence on hydrological forcings and soil-vegetation dynamics. The proposed framework integrates age-dating and tracer hydrology techniques, and provides a coherent framework for catchment transport models based on travel times.
18. Fast adaptive uniformisation of the chemical master equation.
PubMed
Mateescu, M; Wolf, V; Didier, F; Henzinger, T A
2010-11-01
Within systems biology there is an increasing interest in the stochastic behaviour of biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous-time Markov chain (CTMC). The uniformisation technique is an efficient method to compute probability distributions of a CTMC if the number of states is manageable. However, the size of a CTMC that represents a biochemical reaction network is usually far beyond what is feasible. In this study, the authors present an on-the-fly variant of uniformisation, where they improve the original algorithm at the cost of a small approximation error. By means of several examples, the authors show that their approach is particularly well-suited for biochemical reaction networks.
19. Force spectroscopy of single multidomain biopolymers: A master equation approach
Braun, O.; Seifert, U.
2005-09-01
Experiments using atomic force microscopy for unfolding single multidomain biopolymers cover a broad range of time scales from equilibrium to non-equilibrium. A master equation approach allows to identify and treat coherently three dynamical regimes for increasing linear ramp velocity: i) an equilibrium regime, ii) a transient regime where refolding events still occur, and iii) a saw-tooth regime without any refolding events. For each regime, analytical approximations are derived and compared to numerically investigated examples. We analyze in the framework of this model also a periodic experimental protocol instead of a linear ramp. In this case, a major simplification arises if the dynamics can be restricted to an effectively two-dimensional subspace. For transitions with an intermediate meta-stable state, like Immunoglobulin27, a refined model allows to extract previously unknown molecular parameters related to this meta-stable state.
20. Order Reduction of the Chemical Master Equation via Balanced Realisation
PubMed Central
López-Caamal, Fernando; Marquez-Lago, Tatiana T.
2014-01-01
We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism), and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator. PMID:25121581
1. Order reduction of the chemical master equation via balanced realisation.
PubMed
López-Caamal, Fernando; Marquez-Lago, Tatiana T
2014-01-01
We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism), and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator. PMID:25121581
2. Order reduction of the chemical master equation via balanced realisation.
PubMed
López-Caamal, Fernando; Marquez-Lago, Tatiana T
2014-01-01
We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism), and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator.
3. The Master Equation for Two-Level Accelerated Systems at Finite Temperature
Tomazelli, J. L.; Cunha, R. O.
2016-07-01
In this work, we study the behaviour of two weakly coupled quantum systems, described by a separable density operator; one of them is a single oscillator, representing a microscopic system, while the other is a set of oscillators which perform the role of a reservoir in thermal equilibrium. From the Liouville-Von Neumann equation for the reduced density operator, we devise the master equation that governs the evolution of the microscopic system, incorporating the effects of temperature via Thermofield Dynamics formalism by suitably redefining the vacuum of the macroscopic system. As applications, we initially investigate the behaviour of a Fermi oscillator in the presence of a heat bath consisting of a set of Fermi oscillators and that of an atomic two-level system interacting with a scalar radiation field, considered as a reservoir, by constructing the corresponding master equation which governs the time evolution of both sub-systems at finite temperature. Finally, we calculate the energy variation rates for the atom and the field, as well as the atomic population levels, both in the inertial case and at constant proper acceleration, considering the two-level system as a prototype of an Unruh detector, for admissible couplings of the radiation field.
4. The Master Equation for Two-Level Accelerated Systems at Finite Temperature
Tomazelli, J. L.; Cunha, R. O.
2016-10-01
In this work, we study the behaviour of two weakly coupled quantum systems, described by a separable density operator; one of them is a single oscillator, representing a microscopic system, while the other is a set of oscillators which perform the role of a reservoir in thermal equilibrium. From the Liouville-Von Neumann equation for the reduced density operator, we devise the master equation that governs the evolution of the microscopic system, incorporating the effects of temperature via Thermofield Dynamics formalism by suitably redefining the vacuum of the macroscopic system. As applications, we initially investigate the behaviour of a Fermi oscillator in the presence of a heat bath consisting of a set of Fermi oscillators and that of an atomic two-level system interacting with a scalar radiation field, considered as a reservoir, by constructing the corresponding master equation which governs the time evolution of both sub-systems at finite temperature. Finally, we calculate the energy variation rates for the atom and the field, as well as the atomic population levels, both in the inertial case and at constant proper acceleration, considering the two-level system as a prototype of an Unruh detector, for admissible couplings of the radiation field.
5. Quantum oblivion: A master key for many quantum riddles
Elitzur, Avshalom C.; Cohen, Eliahu
2014-02-01
A simple quantum interaction is analyzed, where the paths of two superposed particles asymmetrically cross, while a detector set to detect an interaction between them remains silent. Despite this negative result, the particles' states leave no doubt that a peculiar interaction has occurred: One particle's momentum is changed while the other's remains unaffected, in apparent violation of momentum conservation. Revisiting the foundations of the standard quantum measurement process offers the resolution. Prior to the macroscopic recording of no interaction, a brief critical interval (CI) prevails, during which the particles and the detector's pointer form a subtle entanglement which immediately dissolves. It is this self-cancellation, henceforth "quantum oblivion (QO)," that lies at the basis of some well-known intriguing quantum effects. Such is interaction-free measurement (IFM)1 and its more paradoxical variants like Hardy's Paradox2 and the quantum liar paradox.3 Even the Aharonov-Bohm (AB) effect4 and weak measurement (WM)5 turn out to belong to this group. We next study interventions within the CI that produce some other peculiar effects. Finally, we discuss some of the conceptual issues involved. Under a greater time-resolution of the CI, some non-local phenomena turn out to be local. Momentum is conserved due to the quantum uncertainties inflicted by the particle-pointer interaction, which sets the experiment's final boundary condition.
6. Maxwell's equations, quantum physics and the quantum graviton
Gersten, Alexander; Moalem, Amnon
2011-12-01
Quantum wave equations for massless particles and arbitrary spin are derived by factorizing the d'Alembertian operator. The procedure is extensively applied to the spin one photon equation which is related to Maxwell's equations via the proportionality of the photon wavefunction Ψ to the sum E + iB of the electric and magnetic fields. Thus Maxwell's equations can be considered as the first quantized one-photon equation. The photon wave equation is written in two forms, one with additional explicit subsidiary conditions and second with the subsidiary conditions implicitly included in the main equation. The second equation was obtained by factorizing the d'Alembertian with 4×4 matrix representation of "relativistic quaternions". Furthermore, scalar Lagrangian formalism, consistent with quantization requirements is developed using derived conserved current of probability and normalization condition for the wavefunction. Lessons learned from the derivation of the photon equation are used in the derivation of the spin two quantum equation, which we call the quantum graviton. Quantum wave equation with implicit subsidiary conditions, which factorizes the d'Alembertian with 8×8 matrix representation of relativistic quaternions, is derived. Scalar Lagrangian is formulated and conserved probability current and wavefunction normalization are found, both consistent with the definitions of quantum operators and their expectation values. We are showing that the derived equations are the first quantized equations of the photon and the graviton.
7. Bistability in the chemical master equation for dual phosphorylation cycles.
PubMed
Bazzani, Armando; Castellani, Gastone C; Giampieri, Enrico; Remondini, Daniel; Cooper, Leon N
2012-06-21
Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.
8. Bistability in the chemical master equation for dual phosphorylation cycles
Bazzani, Armando; Castellani, Gastone C.; Giampieri, Enrico; Remondini, Daniel; Cooper, Leon N.
2012-06-01
Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.
9. Symmetric and antisymmetric forms of the Pauli master equation
Klimenko, A. Y.
2016-07-01
When applied to matter and antimatter states, the Pauli master equation (PME) may have two forms: time-symmetric, which is conventional, and time-antisymmetric, which is suggested in the present work. The symmetric and antisymmetric forms correspond to symmetric and antisymmetric extensions of thermodynamics from matter to antimatter — this is demonstrated by proving the corresponding H-theorem. The two forms are based on the thermodynamic similarity of matter and antimatter and differ only in the directions of thermodynamic time for matter and antimatter (the same in the time-symmetric case and the opposite in the time-antisymmetric case). We demonstrate that, while the symmetric form of PME predicts an equibalance between matter and antimatter, the antisymmetric form of PME favours full conversion of antimatter into matter. At this stage, it is impossible to make an experimentally justified choice in favour of the symmetric or antisymmetric versions of thermodynamics since we have no experience of thermodynamic properties of macroscopic objects made of antimatter, but experiments of this kind may become possible in the future.
10. Symmetric and antisymmetric forms of the Pauli master equation
PubMed Central
Klimenko, A. Y.
2016-01-01
When applied to matter and antimatter states, the Pauli master equation (PME) may have two forms: time-symmetric, which is conventional, and time-antisymmetric, which is suggested in the present work. The symmetric and antisymmetric forms correspond to symmetric and antisymmetric extensions of thermodynamics from matter to antimatter — this is demonstrated by proving the corresponding H-theorem. The two forms are based on the thermodynamic similarity of matter and antimatter and differ only in the directions of thermodynamic time for matter and antimatter (the same in the time-symmetric case and the opposite in the time-antisymmetric case). We demonstrate that, while the symmetric form of PME predicts an equibalance between matter and antimatter, the antisymmetric form of PME favours full conversion of antimatter into matter. At this stage, it is impossible to make an experimentally justified choice in favour of the symmetric or antisymmetric versions of thermodynamics since we have no experience of thermodynamic properties of macroscopic objects made of antimatter, but experiments of this kind may become possible in the future. PMID:27440454
11. Multi-time equations, classical and quantum
PubMed Central
Petrat, Sören; Tumulka, Roderich
2014-01-01
Multi-time equations are evolution equations involving several time variables, one for each particle. Such equations have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and significance in classical and quantum physics. PMID:24711721
12. Quantum simulation of the Dirac equation.
PubMed
Gerritsma, R; Kirchmair, G; Zähringer, F; Solano, E; Blatt, R; Roos, C F
2010-01-01
The Dirac equation successfully merges quantum mechanics with special relativity. It provides a natural description of the electron spin, predicts the existence of antimatter and is able to reproduce accurately the spectrum of the hydrogen atom. The realm of the Dirac equation-relativistic quantum mechanics-is considered to be the natural transition to quantum field theory. However, the Dirac equation also predicts some peculiar effects, such as Klein's paradox and 'Zitterbewegung', an unexpected quivering motion of a free relativistic quantum particle. These and other predicted phenomena are key fundamental examples for understanding relativistic quantum effects, but are difficult to observe in real particles. In recent years, there has been increased interest in simulations of relativistic quantum effects using different physical set-ups, in which parameter tunability allows access to different physical regimes. Here we perform a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a single trapped ion set to behave as a free relativistic quantum particle. We measure the particle position as a function of time and study Zitterbewegung for different initial superpositions of positive- and negative-energy spinor states, as well as the crossover from relativistic to non-relativistic dynamics. The high level of control of trapped-ion experimental parameters makes it possible to simulate textbook examples of relativistic quantum physics. PMID:20054392
13. ACCURATE CHEMICAL MASTER EQUATION SOLUTION USING MULTI-FINITE BUFFERS
PubMed Central
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-01-01
The discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multi-scale nature of many networks where reaction rates have large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the Accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multi-finite buffers for reducing the state space by O(n!), exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes, and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be pre-computed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multi-scale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks. PMID:27761104
14. Exact calculation of the time convolutionless master equation generator: Application to the nonequilibrium resonant level model.
PubMed
Kidon, Lyran; Wilner, Eli Y; Rabani, Eran
2015-12-21
The generalized quantum master equation provides a powerful tool to describe the dynamics in quantum impurity models driven away from equilibrium. Two complementary approaches, one based on Nakajima-Zwanzig-Mori time-convolution (TC) and the other on the Tokuyama-Mori time-convolutionless (TCL) formulations provide a starting point to describe the time-evolution of the reduced density matrix. A key in both approaches is to obtain the so called "memory kernel" or "generator," going beyond second or fourth order perturbation techniques. While numerically converged techniques are available for the TC memory kernel, the canonical approach to obtain the TCL generator is based on inverting a super-operator in the full Hilbert space, which is difficult to perform and thus, nearly all applications of the TCL approach rely on a perturbative scheme of some sort. Here, the TCL generator is expressed using a reduced system propagator which can be obtained from system observables alone and requires the calculation of super-operators and their inverse in the reduced Hilbert space rather than the full one. This makes the formulation amenable to quantum impurity solvers or to diagrammatic techniques, such as the nonequilibrium Green's function. We implement the TCL approach for the resonant level model driven away from equilibrium and compare the time scales for the decay of the generator with that of the memory kernel in the TC approach. Furthermore, the effects of temperature, source-drain bias, and gate potential on the TCL/TC generators are discussed.
15. Zakharov equations in quantum dusty plasmas
SciTech Connect
Sayed, F.; Vladimirov, S. V.; Ishihara, O.
2015-08-15
By generalizing the formalism of modulational interactions in quantum dusty plasmas, we derive the kinetic quantum Zakharov equations in dusty plasmas that describe nonlinear coupling of high frequency Langmuir waves to low frequency plasma density variations, for cases of non-degenerate and degenerate plasma electrons.
16. Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium.
PubMed
Horowitz, Jordan M
2015-07-28
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.
17. Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium
SciTech Connect
Horowitz, Jordan M.
2015-07-28
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.
18. Similarity reduction, nonlocal and master symmetries of sixth order Korteweg-deVries equation
2009-05-01
A systematic investigation to derive the Lie point symmetries, nonlocal and master symmetries of sixth order Korteweg-de Vries equation (KdV6) is presented. Using the obtained point symmetries, similarity reductions are derived and constructed their particular solutions wherever possible. It is shown that KdV6 admits infinitely many nonlocal and master symmetries. The existence of infinitely many master symmetries ensures that KdV6 is completely integrable.
19. Master equation solutions in the linear regime of characteristic formulation of general relativity
Cedeño M., C. E.; de Araujo, J. C. N.
2015-12-01
From the field equations in the linear regime of the characteristic formulation of general relativity, Bishop, for a Schwarzschild's background, and Mädler, for a Minkowski's background, were able to show that it is possible to derive a fourth order ordinary differential equation, called master equation, for the J metric variable of the Bondi-Sachs metric. Once β , another Bondi-Sachs potential, is obtained from the field equations, and J is obtained from the master equation, the other metric variables are solved integrating directly the rest of the field equations. In the past, the master equation was solved for the first multipolar terms, for both the Minkowski's and Schwarzschild's backgrounds. Also, Mädler recently reported a generalisation of the exact solutions to the linearised field equations when a Minkowski's background is considered, expressing the master equation family of solutions for the vacuum in terms of Bessel's functions of the first and the second kind. Here, we report new solutions to the master equation for any multipolar moment l , with and without matter sources in terms only of the first kind Bessel's functions for the Minkowski, and in terms of the Confluent Heun's functions (Generalised Hypergeometric) for radiative (nonradiative) case in the Schwarzschild's background. We particularize our families of solutions for the known cases for l =2 reported previously in the literature and find complete agreement, showing the robustness of our results.
20. Fourth-order master equation for a charged harmonic oscillator coupled to an electromagnetic field
Kurt, Arzu; Eryigit, Resul
Using Krylov averaging method, we have derived a fourth-order master equation for a charged harmonic oscillator weakly coupled to an electromagnetic field. Interaction is assumed to be of velocity coupling type which also takes into account the diagmagnetic term. Exact analytical expressions have been obtained for the second, the third and the fourth-order corrections to the diffusion and the drift terms of the master equation. We examined the validity range of the second order master equation in terms of the coupling constant and the bath cutoff frequency and found that for the most values of those parameters, the contribution from the third and the fourth order terms have opposite signs and cancel each other. Inclusion of the third and the fourth-order terms is found to not change the structure of the master equation. Bolu, Turkey.
1. Statistically testing the validity of analytical and computational approximations to the chemical master equation.
PubMed
Jenkinson, Garrett; Goutsias, John
2013-05-28
The master equation is used extensively to model chemical reaction systems with stochastic dynamics. However, and despite its phenomenological simplicity, it is not in general possible to compute the solution of this equation. Drawing exact samples from the master equation is possible, but can be computationally demanding, especially when estimating high-order statistical summaries or joint probability distributions. As a consequence, one often relies on analytical approximations to the solution of the master equation or on computational techniques that draw approximative samples from this equation. Unfortunately, it is not in general possible to check whether a particular approximation scheme is valid. The main objective of this paper is to develop an effective methodology to address this problem based on statistical hypothesis testing. By drawing a moderate number of samples from the master equation, the proposed techniques use the well-known Kolmogorov-Smirnov statistic to reject the validity of a given approximation method or accept it with a certain level of confidence. Our approach is general enough to deal with any master equation and can be used to test the validity of any analytical approximation method or any approximative sampling technique of interest. A number of examples, based on the Schlögl model of chemistry and the SIR model of epidemiology, clearly illustrate the effectiveness and potential of the proposed statistical framework.
2. Coarse-grained kinetic equations for quantum systems
Petrov, E. G.
2013-01-01
The nonequilibrium density matrix method is employed to derive a master equation for the averaged state populations of an open quantum system subjected to an external high frequency stochastic field. It is shown that if the characteristic time τstoch of the stochastic process is much lower than the characteristic time τsteady of the establishment of the system steady state populations, then on the time scale Δ t ˜ τsteady, the evolution of the system populations can be described by the coarse-grained kinetic equations with the averaged transition rates. As an example, the exact averaging is carried out for the dichotomous Markov process of the kangaroo type.
3. Exact calculation of the time convolutionless master equation generator: Application to the nonequilibrium resonant level model
SciTech Connect
Kidon, Lyran; Wilner, Eli Y.; Rabani, Eran
2015-12-21
The generalized quantum master equation provides a powerful tool to describe the dynamics in quantum impurity models driven away from equilibrium. Two complementary approaches, one based on Nakajima–Zwanzig–Mori time-convolution (TC) and the other on the Tokuyama–Mori time-convolutionless (TCL) formulations provide a starting point to describe the time-evolution of the reduced density matrix. A key in both approaches is to obtain the so called “memory kernel” or “generator,” going beyond second or fourth order perturbation techniques. While numerically converged techniques are available for the TC memory kernel, the canonical approach to obtain the TCL generator is based on inverting a super-operator in the full Hilbert space, which is difficult to perform and thus, nearly all applications of the TCL approach rely on a perturbative scheme of some sort. Here, the TCL generator is expressed using a reduced system propagator which can be obtained from system observables alone and requires the calculation of super-operators and their inverse in the reduced Hilbert space rather than the full one. This makes the formulation amenable to quantum impurity solvers or to diagrammatic techniques, such as the nonequilibrium Green’s function. We implement the TCL approach for the resonant level model driven away from equilibrium and compare the time scales for the decay of the generator with that of the memory kernel in the TC approach. Furthermore, the effects of temperature, source-drain bias, and gate potential on the TCL/TC generators are discussed.
4. Master equation for a chemical wave front with perturbation of local equilibrium.
PubMed
Dziekan, P; Lemarchand, A; Nowakowski, B
2011-08-28
In order to develop a stochastic description of gaseous reaction-diffusion systems, which includes a reaction-induced departure from local equilibrium, we derive a modified expression of the master equation from analytical calculations based on the Boltzmann equation. We apply the method to a chemical wave front of Fisher-Kolmogorov-Petrovsky-Piskunov type, whose propagation speed is known to be sensitive to small perturbations. The results of the modified master equation are compared successfully with microscopic simulations of the particle dynamics using the direct simulation Monte Carlo method. The modified master equation constitutes an efficient tool at the mesoscopic scale, which incorporates the nonequilibrium effect without need of determining the particle velocity distribution function.
5. Dimensional reduction of the master equation for stochastic chemical networks: The reduced-multiplane method.
PubMed
Barzel, Baruch; Biham, Ofer; Kupferman, Raz; Lipshtat, Azi; Zait, Amir
2010-08-01
Chemical reaction networks which exhibit strong fluctuations are common in microscopic systems in which reactants appear in low copy numbers. The analysis of these networks requires stochastic methods, which come in two forms: direct integration of the master equation and Monte Carlo simulations. The master equation becomes infeasible for large networks because the number of equations increases exponentially with the number of reactive species. Monte Carlo methods, which are more efficient in integrating over the exponentially large phase space, also become impractical due to the large amounts of noisy data that need to be stored and analyzed. The recently introduced multiplane method [A. Lipshtat and O. Biham, Phys. Rev. Lett. 93, 170601 (2004)] is an efficient framework for the stochastic analysis of large reaction networks. It is a dimensional reduction method, based on the master equation, which provides a dramatic reduction in the number of equations without compromising the accuracy of the results. The reduction is achieved by breaking the network into a set of maximal fully connected subnetworks (maximal cliques). A separate master equation is written for the reduced probability distribution associated with each clique, with suitable coupling terms between them. This method is highly efficient in the case of sparse networks, in which the maximal cliques tend to be small. However, in dense networks some of the cliques may be rather large and the dimensional reduction is not as effective. Furthermore, the derivation of the multiplane equations from the master equation is tedious and difficult. Here we present the reduced-multiplane method in which the maximal cliques are broken down to the fundamental two-vertex cliques. The number of equations is further reduced, making the method highly efficient even for dense networks. Moreover, the equations take a simpler form, which can be easily constructed using a diagrammatic procedure, for any desired network
6. Relaxation process of quantum system: Stochastic Liouville equation and initial correlation
SciTech Connect
Ban, Masashi; Kitajima, Sachiko; Shibata, Fumiaki
2010-08-15
Time evolution of a quantum system which is influenced by a stochastically fluctuating environment is studied by means of the stochastic Liouville equation. The two different types of the stochastic Liouville equation and their relation are discussed. The stochastic Liouville equation is shown to be derived from the quantum master equation of the Lindblad under certain conditions. Relaxation processes of single and bipartite quantum systems which are initially correlated with a stochastic environment are investigated. It is shown the possibility that the stochastic fluctuation can create coherence and entanglement of a quantum system with the assistance of the initial correlation. The results are examined in the pure dephasing processes of qubits, which are caused by the nonstationary Gauss-Markov process and two-state jump Markov process.
7. Vibrational energy flow in the villin headpiece subdomain: Master equation simulations
SciTech Connect
Leitner, David M. E-mail: [email protected]; Buchenberg, Sebastian; Brettel, Paul; Stock, Gerhard E-mail: [email protected]
2015-02-21
We examine vibrational energy flow in dehydrated and hydrated villin headpiece subdomain HP36 by master equation simulations. Transition rates used in the simulations are obtained from communication maps calculated for HP36. In addition to energy flow along the main chain, we identify pathways for energy transport in HP36 via hydrogen bonding between residues quite far in sequence space. The results of the master equation simulations compare well with all-atom non-equilibrium simulations to about 1 ps following initial excitation of the protein, and quite well at long times, though for some residues we observe deviations between the master equation and all-atom simulations at intermediate times from about 1–10 ps. Those deviations are less noticeable for hydrated than dehydrated HP36 due to energy flow into the water.
8. Master equation for a bistable chemical system with perturbed particle velocity distribution function.
PubMed
Dziekan, P; Lemarchand, A; Nowakowski, B
2012-02-01
We present a modified master equation for a homogeneous gaseous reactive system which includes nonequilibrium corrections due to the reaction-induced perturbation of the particle velocity distribution function. For the Schlögl model, the modified stochastic approach predicts nonequilibrium-induced transitions between different dynamical regimes, including the transformation of a monostable system into a bistable one, and vice versa. These predictions are confirmed by the comparison with microscopic simulations using the direct simulation Monte Carlo method. Compared to microscopic simulations of the particle dynamics, the modified master equation approach proves to be much more efficient.
9. Nonlinear quantum equations: Classical field theory
SciTech Connect
Rego-Monteiro, M. A.; Nobre, F. D.
2013-10-15
An exact classical field theory for nonlinear quantum equations is presented herein. It has been applied recently to a nonlinear Schrödinger equation, and it is shown herein to hold also for a nonlinear generalization of the Klein-Gordon equation. These generalizations were carried by introducing nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard, linear equations, are recovered in the limit q→ 1. The main characteristic of this field theory consists on the fact that besides the usual Ψ(x(vector sign),t), a new field Φ(x(vector sign),t) needs to be introduced in the Lagrangian, as well. The field Φ(x(vector sign),t), which is defined by means of an additional equation, becomes Ψ{sup *}(x(vector sign),t) only when q→ 1. The solutions for the fields Ψ(x(vector sign),t) and Φ(x(vector sign),t) are found herein, being expressed in terms of a q-plane wave; moreover, both field equations lead to the relation E{sup 2}=p{sup 2}c{sup 2}+m{sup 2}c{sup 4}, for all values of q. The fact that such a classical field theory works well for two very distinct nonlinear quantum equations, namely, the Schrödinger and Klein-Gordon ones, suggests that this procedure should be appropriate for a wider class nonlinear equations. It is shown that the standard global gauge invariance is broken as a consequence of the nonlinearity.
10. Direct solution of the Chemical Master Equation using quantized tensor trains.
PubMed
Kazeev, Vladimir; Khammash, Mustafa; Nip, Michael; Schwab, Christoph
2014-03-01
The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to "lift" this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging hp-discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the "basis" of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage
11. Computational Cellular Dynamics Based on the Chemical Master Equation: A Challenge for Understanding Complexity.
PubMed
Liang, Jie; Qian, Hong
2010-01-01
Modern molecular biology has always been a great source of inspiration for computational science. Half a century ago, the challenge from understanding macromolecular dynamics has led the way for computations to be part of the tool set to study molecular biology. Twenty-five years ago, the demand from genome science has inspired an entire generation of computer scientists with an interest in discrete mathematics to join the field that is now called bioinformatics. In this paper, we shall lay out a new mathematical theory for dynamics of biochemical reaction systems in a small volume (i.e., mesoscopic) in terms of a stochastic, discrete-state continuous-time formulation, called the chemical master equation (CME). Similar to the wavefunction in quantum mechanics, the dynamically changing probability landscape associated with the state space provides a fundamental characterization of the biochemical reaction system. The stochastic trajectories of the dynamics are best known through the simulations using the Gillespie algorithm. In contrast to the Metropolis algorithm, this Monte Carlo sampling technique does not follow a process with detailed balance. We shall show several examples how CMEs are used to model cellular biochemical systems. We shall also illustrate the computational challenges involved: multiscale phenomena, the interplay between stochasticity and nonlinearity, and how macroscopic determinism arises from mesoscopic dynamics. We point out recent advances in computing solutions to the CME, including exact solution of the steady state landscape and stochastic differential equations that offer alternatives to the Gilespie algorithm. We argue that the CME is an ideal system from which one can learn to understand "complex behavior" and complexity theory, and from which important biological insight can be gained.
12. Direct solution of the Chemical Master Equation using quantized tensor trains.
PubMed
Kazeev, Vladimir; Khammash, Mustafa; Nip, Michael; Schwab, Christoph
2014-03-01
The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to "lift" this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging hp-discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the "basis" of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage
13. Kinetic Approach for Quantum Hydrodynamic Equations
Tessarotto, M.; Ellero, M.; Nicolini, P.
2008-12-01
A striking feature of standard quantum mechanics (SQM) is its analogy with classical fluid dynamics. In particular it is well known the Schrödinger equation can be viewed as describing a classical compressible and non-viscous fluid, described by two (quantum) fluid fields {ρ,V}, to be identified with the quantum probability density and velocity field. This feature has suggested the construction of a phase-space hidden-variable description based on a suitable inverse kinetic theory (IKT; Tessarotto et al., 2007). The discovery of this approach has potentially important consequences since it permits to identify the classical dynamical system which advances in time the quantum fluid fields. This type of approach, however requires the identification of additional fluid fields. These can be generally identified with suitable directional fluid temperatures TQM,i (for i = 1,2,3), to be related to the expectation values of momentum fluctuations appearing in the Heisenberg inequalities. Nevertheless the definition given previously for them (Tessarotto et al., 2007) is non-unique. In this paper we intend to propose a criterion, based on the validity of a constant H-theorem, which provides an unique definition for the quantum temperatures.
14. General transient solution of the one-step master equation in one dimension
Smith, Stephen; Shahrezaei, Vahid
2015-06-01
Exact analytical solutions of the master equation are limited to special cases and exact numerical methods are inefficient. Even the generic one-dimensional, one-step master equation has evaded exact solution, aside from the steady-state case. This type of master equation describes the dynamics of a continuous-time Markov process whose range consists of positive integers and whose transitions are allowed only between adjacent sites. The solution of any master equation can be written as the exponential of a (typically huge) matrix, which requires the calculation of the eigenvalues and eigenvectors of the matrix. Here we propose a linear algebraic method for simplifying this exponential for the general one-dimensional, one-step process. In particular, we prove that the calculation of the eigenvectors is actually not necessary for the computation of exponential, thereby we dramatically cut the time of this calculation. We apply our new methodology to examples from birth-death processes and biochemical networks. We show that the computational time is significantly reduced compared to existing methods.
15. General transient solution of the one-step master equation in one dimension.
PubMed
Smith, Stephen; Shahrezaei, Vahid
2015-06-01
Exact analytical solutions of the master equation are limited to special cases and exact numerical methods are inefficient. Even the generic one-dimensional, one-step master equation has evaded exact solution, aside from the steady-state case. This type of master equation describes the dynamics of a continuous-time Markov process whose range consists of positive integers and whose transitions are allowed only between adjacent sites. The solution of any master equation can be written as the exponential of a (typically huge) matrix, which requires the calculation of the eigenvalues and eigenvectors of the matrix. Here we propose a linear algebraic method for simplifying this exponential for the general one-dimensional, one-step process. In particular, we prove that the calculation of the eigenvectors is actually not necessary for the computation of exponential, thereby we dramatically cut the time of this calculation. We apply our new methodology to examples from birth-death processes and biochemical networks. We show that the computational time is significantly reduced compared to existing methods.
16. Control of Stochastic Master Equation Models of Genetic Regulatory Networks by Approximating Their Average Behavior
2010-10-01
The central dogma of molecular biology states that information cannot be transferred back from protein to either protein or nucleic acid.'' However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of data in the cellular level and probabilistic nature of interactions. Probabilistic models like Stochastic Master Equation (SME) or deterministic models like differential equations (DE) can be used to analyze these types of interactions. SME models based on chemical master equation (CME) can provide detailed representation of genetic regulatory system, but their use is restricted by the large data requirements and computational costs of calculations. The differential equations models on the other hand, have low calculation costs and much more adequate to generate control procedures on the system; but they are not adequate to investigate the probabilistic nature of interactions. In this work the success of the mapping between SME and DE is analyzed, and the success of a control policy generated by DE model with respect to SME model is examined. Index Terms--- Stochastic Master Equation models, Differential Equation Models, Control Policy Design, Systems biology
17. Dimensional reduction of the master equation for stochastic chemical networks: The reduced-multiplane method
Barzel, Baruch; Biham, Ofer; Kupferman, Raz; Lipshtat, Azi; Zait, Amir
2010-08-01
Chemical reaction networks which exhibit strong fluctuations are common in microscopic systems in which reactants appear in low copy numbers. The analysis of these networks requires stochastic methods, which come in two forms: direct integration of the master equation and Monte Carlo simulations. The master equation becomes infeasible for large networks because the number of equations increases exponentially with the number of reactive species. Monte Carlo methods, which are more efficient in integrating over the exponentially large phase space, also become impractical due to the large amounts of noisy data that need to be stored and analyzed. The recently introduced multiplane method [A. Lipshtat and O. Biham, Phys. Rev. Lett. 93, 170601 (2004)10.1103/PhysRevLett.93.170601] is an efficient framework for the stochastic analysis of large reaction networks. It is a dimensional reduction method, based on the master equation, which provides a dramatic reduction in the number of equations without compromising the accuracy of the results. The reduction is achieved by breaking the network into a set of maximal fully connected subnetworks (maximal cliques). A separate master equation is written for the reduced probability distribution associated with each clique, with suitable coupling terms between them. This method is highly efficient in the case of sparse networks, in which the maximal cliques tend to be small. However, in dense networks some of the cliques may be rather large and the dimensional reduction is not as effective. Furthermore, the derivation of the multiplane equations from the master equation is tedious and difficult. Here we present the reduced-multiplane method in which the maximal cliques are broken down to the fundamental two-vertex cliques. The number of equations is further reduced, making the method highly efficient even for dense networks. Moreover, the equations take a simpler form, which can be easily constructed using a diagrammatic procedure
18. Efficient parametric analysis of the chemical master equation through model order reduction
PubMed Central
2012-01-01
Background Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation. Results In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. Conclusions The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis. PMID:22748204
19. Dynamics of the chemical master equation, a strip of chains of equations in d-dimensional space.
PubMed
Galstyan, Vahe; Saakian, David B
2012-07-01
We investigate the multichain version of the chemical master equation, when there are transitions between different states inside the long chains, as well as transitions between (a few) different chains. In the discrete version, such a model can describe the connected diffusion processes with jumps between different types. We apply the Hamilton-Jacobi equation to solve some aspects of the model. We derive exact (in the limit of infinite number of particles) results for the dynamic of the maximum of the distribution and the variance of distribution.
20. Dynamics of the chemical master equation, a strip of chains of equations in d-dimensional space.
PubMed
Galstyan, Vahe; Saakian, David B
2012-07-01
We investigate the multichain version of the chemical master equation, when there are transitions between different states inside the long chains, as well as transitions between (a few) different chains. In the discrete version, such a model can describe the connected diffusion processes with jumps between different types. We apply the Hamilton-Jacobi equation to solve some aspects of the model. We derive exact (in the limit of infinite number of particles) results for the dynamic of the maximum of the distribution and the variance of distribution. PMID:23005386
1. Calogero-Sutherland model and Quantum Benjamin-Ono Equation
Abanov, Alexander G.; Wiegmann, Paul B.
2005-03-01
Collective field theory for Calogero-Sutherland model represents particles with fractional statistics in terms of holomorphic bosonic field made out of the density and velocity fields. We identify an operator equation of motion for this bosonic field with a quantum deformation of a known classical Benjamin-Ono equation. The latter equation is integrable and the same is true for its quantum version. The inverse scattering transform for the classical Benjamin-Ono equation can be extended to its quantum analog. Soliton solutions of quantum Benjamin- Ono equation correspond to particle and hole excitations of Calogero- Sutherland model.
2. Exact semiclassical wave equation for stochastic quantum optics
Diósi, Lajos
1996-02-01
Semiclassical (stochastic) wave equations are proposed for the coupled dynamics of atomic quantum states and semiclassical radiation field. All relevant predictions of standard unitary quantum dynamics are exactly reproducible in the framework of the stochastic wave equation model. We stress in such a way that the concept of stochastic wave equations is not to be restricted to the widely used Markovian approximation.
3. A Master Equation Approach to Modeling Short-term Behaviors of the Stock Market
Zhao, Conan; Yang, Xiaoxiang; Mazilu, Irina
2015-03-01
Short term fluctuations in stock prices are highly random, due to the multitude of external factors acting on the price determination process. While long-term economic factors such as inflation and revenue growth rate affect short-term price fluctuation, it is difficult to obtain the complete set of information and uncertainties associated with a given period of time. Instead, we propose a simpler short-term model based on only prior price averages and extrema. In this paper, we take a master equation under the random walk hypothesis and fit parameters based on AAPL stock price data over the past ten years. We report results for small system sizes and for the short term average price. These results may lead to a general closed-form solution to this particular master equation.
4. A Steady-State Approximation to the Two-Dimensional Master Equation for Chemical Kinetics Calculations.
PubMed
Nguyen, Thanh Lam; Stanton, John F
2015-07-16
In the field of chemical kinetics, the solution of a two-dimensional master equation that depends explicitly on both total internal energy (E) and total angular momentum (J) is a challenging problem. In this work, a weak-E/fixed-J collisional model (i.e., weak-collisional internal energy relaxation/free-collisional angular momentum relaxation) is used along with the steady-state approach to solve the resulting (simplified) two-dimensional (E,J)-grained master equation. The corresponding solutions give thermal rate constants and product branching ratios as functions of both temperature and pressure. We also have developed a program that can be used to predict and analyze experimental chemical kinetics results. This expedient technique, when combined with highly accurate potential energy surfaces, is cable of providing results that may be meaningfully compared to experiments. The reaction of singlet oxygen with methane proceeding through vibrationally excited methanol is used as an illustrative example.
5. Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model
Gelß, Patrick; Matera, Sebastian; Schütte, Christof
2016-06-01
In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.
6. Qubit-mediated energy transfer between thermal reservoirs: Beyond the Markovian master equation
Segal, Dvira
2013-05-01
We study qubit-mediated energy transfer between two electron reservoirs by adopting a numerically exact influence functional path-integral method. This nonperturbative technique allows us to study the system's dynamics beyond the weak coupling limit. Our simulations for the energy current indicate that perturbative-Markovian master equation predictions significantly deviate from exact numerical results already at intermediate coupling πραj,j'≳0.4, where ρ is the metal (Fermi sea) density of states, taken as a constant, and αj,j' is the scattering potential energy of electrons, between the j and j' states. Perturbative Markovian master equation techniques should be therefore used with caution beyond the strictly weak subsystem-bath coupling limit, especially when a quantitative knowledge of transport characteristics is desired.
7. Recent applications of the Boltzmann master equation to heavy ion precompound decay phenomena
SciTech Connect
Blann, M.; Remington, B.A.
1988-06-01
The Boltzmann master equation (BME) is described and used as a tool to interpret preequilibrium neutron emission from heavy ion collisions gated on evaporation residue or fission fragments. The same approach is used to interpret neutron spectra gated on deep inelastic and quasi-elastic heavy ion collisions. Less successful applications of BME to proton inclusive data with 40 MeV/u incident /sup 12/C ions are presented, and improvements required in the exciton injection term are discussed.
8. Boundary transfer matrices and boundary quantum KZ equations
SciTech Connect
Vlaar, Bart
2015-07-15
A simple relation between inhomogeneous transfer matrices and boundary quantum Knizhnik-Zamolodchikov (KZ) equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus, the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin’s boundary transfer matrices by merely imposing appropriate reflection equations, in particular without using the conditions of crossing symmetry and unitarity of the R-matrix.
9. Breakdown of the reaction-diffusion master equation with nonelementary rates.
PubMed
Smith, Stephen; Grima, Ramon
2016-05-01
The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: Indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to a master equation but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class, then the RDME converges to the CME of the same system, whereas if the RDME has propensities not in this class, then convergence is not guaranteed. These are revealed to be elementary and nonelementary propensities, respectively. We also show that independent of the type of propensity, the RDME converges to the CME in the simultaneous limit of fast diffusion and large volumes. We illustrate our results with some simple example systems and argue that the RDME cannot generally be an accurate description of systems with nonelementary rates.
10. Breakdown of the reaction-diffusion master equation with nonelementary rates
Smith, Stephen; Grima, Ramon
2016-05-01
The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: Indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to a master equation but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class, then the RDME converges to the CME of the same system, whereas if the RDME has propensities not in this class, then convergence is not guaranteed. These are revealed to be elementary and nonelementary propensities, respectively. We also show that independent of the type of propensity, the RDME converges to the CME in the simultaneous limit of fast diffusion and large volumes. We illustrate our results with some simple example systems and argue that the RDME cannot generally be an accurate description of systems with nonelementary rates.
11. Sqeezing generated by a nonlinear master equation and by amplifying-dissipative Hamiltonians
NASA Technical Reports Server (NTRS)
Dodonov, V. V.; Marchiolli, M. A.; Mizrahi, Solomon S.; Moussa, M. H. Y.
1994-01-01
In the first part of this contribution we show that the master equation derived from the generalized version of the nonlinear Doebner-Goldin equation leads to the squeezing of one of the quadratures. In the second part we consider two familiar Hamiltonians, the Bateman- Caldirola-Kanai and the optical parametric oscillator; going back to their classical Lagrangian form we introduce a stochastic force and a dissipative factor. From this new Lagrangian we obtain a modified Hamiltonian that treats adequately the simultaneous amplification and dissipation phenomena, presenting squeezing, too.
12. Testing the master constraint programme for loop quantum gravity: V. Interacting field theories
Dittrich, B.; Thiemann, T.
2006-02-01
This is the fifth and final paper in our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. Here we consider interacting quantum field theories, specifically we consider the non-Abelian Gauss constraints of Einstein Yang Mills theory and 2 + 1 gravity. Interestingly, while Yang Mills theory in 4D is not yet rigorously defined as an ordinary (Wightman) quantum field theory on Minkowski space, in background-independent quantum field theories such as loop quantum gravity (LQG) this might become possible by working in a new, background-independent representation. While for the Gauss constraint the master constraint can be solved explicitly, for the 2 + 1 theory we are only able to rigorously define the master constraint operator. We show that the, by other methods known, physical Hilbert is contained in the kernel of the master constraint, however, to systematically derive it by only using spectral methods is as complicated as for 3 + 1 gravity and we therefore leave the complete analysis for 3 + 1 gravity.
13. Emergence of wave equations from quantum geometry
SciTech Connect
Majid, Shahn
2012-09-24
We argue that classical geometry should be viewed as a special limit of noncommutative geometry in which aspects which are inter-constrained decouple and appear arbitrary in the classical limit. In particular, the wave equation is really a partial derivative in a unified extra-dimensional noncommutative geometry and arises out of the greater rigidity of the noncommutative world not visible in the classical limit. We provide an introduction to this 'wave operator' approach to noncommutative geometry as recently used[27] to quantize any static spacetime metric admitting a spatial conformal Killing vector field, and in particular to construct the quantum Schwarzschild black hole. We also give an introduction to our related result that every classical Riemannian manifold is a shadow of a slightly noncommutative one wherein the meaning of the classical Ricci tensor becomes very natural as the square of a generalised braiding.
14. Neutrino quantum kinetic equations: The collision term
Blaschke, Daniel N.; Cirigliano, Vincenzo
2016-08-01
We derive the collision term relevant for neutrino quantum kinetic equations in the early universe and compact astrophysical objects, displaying its full matrix structure in both flavor and spin degrees of freedom. We include in our analysis neutrino-neutrino processes, scattering and annihilation with electrons and positrons, and neutrino scattering off nucleons (the latter in the low-density limit). After presenting the general structure of the collision terms, we take two instructive limiting cases. The one-flavor limit highlights the structure in helicity space and allows for a straightforward interpretation of the off-diagonal entries in terms of the product of scattering amplitudes of the two helicity states. The isotropic limit is relevant for studies of the early universe: in this case the terms involving spin coherence vanish and the collision term can be expressed in terms of two-dimensional integrals, suitable for computational implementation.
15. Neutrino quantum kinetic equations: The collision term
DOE PAGES
Blaschke, Daniel N.; Cirigliano, Vincenzo
2016-08-25
We derive the collision term relevant for neutrino quantum kinetic equations in the early universe and compact astrophysical objects, displaying its full matrix structure in both flavor and spin degrees of freedom. We include in our analysis neutrino-neutrino processes, scattering and annihilation with electrons and positrons, and neutrino scattering off nucleons (the latter in the low-density limit). After presenting the general structure of the collision terms, we take two instructive limiting cases. The one-flavor limit highlights the structure in helicity space and allows for a straightforward interpretation of the off-diagonal entries in terms of the product of scattering amplitudes ofmore » the two helicity states. As a result, the isotropic limit is relevant for studies of the early universe: in this case the terms involving spin coherence vanish and the collision term can be expressed in terms of two-dimensional integrals, suitable for computational implementation.« less
16. Unification of the general non-linear sigma model and the Virasoro master equation
SciTech Connect
Boer, J. de; Halpern, M.B. |
1997-06-01
The Virasoro master equation describes a large set of conformal field theories known as the affine-Virasoro constructions, in the operator algebra (affinie Lie algebra) of the WZW model, while the einstein equations of the general non-linear sigma model describe another large set of conformal field theories. This talk summarizes recent work which unifies these two sets of conformal field theories, together with a presumable large class of new conformal field theories. The basic idea is to consider spin-two operators of the form L{sub ij}{partial_derivative}x{sup i}{partial_derivative}x{sup j} in the background of a general sigma model. The requirement that these operators satisfy the Virasoro algebra leads to a set of equations called the unified Einstein-Virasoro master equation, in which the spin-two spacetime field L{sub ij} cuples to the usual spacetime fields of the sigma model. The one-loop form of this unified system is presented, and some of its algebraic and geometric properties are discussed.
17. Stochastic quasi-steady state approximations for asymptotic solutions of the chemical master equation.
PubMed
Alarcón, Tomás
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.
18. Stochastic quasi-steady state approximations for asymptotic solutions of the chemical master equation.
PubMed
Alarcón, Tomás
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm. PMID:24832255
19. Stochastic quasi-steady state approximations for asymptotic solutions of the chemical master equation
SciTech Connect
Alarcón, Tomás
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.
20. Proton-pumping mechanism of cytochrome c oxidase: a kinetic master-equation approach.
PubMed
Kim, Young C; Hummer, Gerhard
2012-04-01
Cytochrome c oxidase is an efficient energy transducer that reduces oxygen to water and converts the released chemical energy into an electrochemical membrane potential. As a true proton pump, cytochrome c oxidase translocates protons across the membrane against this potential. Based on a wealth of experiments and calculations, an increasingly detailed picture of the reaction intermediates in the redox cycle has emerged. However, the fundamental mechanism of proton pumping coupled to redox chemistry remains largely unresolved. Here we examine and extend a kinetic master-equation approach to gain insight into redox-coupled proton pumping in cytochrome c oxidase. Basic principles of the cytochrome c oxidase proton pump emerge from an analysis of the simplest kinetic models that retain essential elements of the experimentally determined structure, energetics, and kinetics, and that satisfy fundamental physical principles. The master-equation models allow us to address the question of how pumping can be achieved in a system in which all reaction steps are reversible. Whereas proton pumping does not require the direct modulation of microscopic reaction barriers, such kinetic gating greatly increases the pumping efficiency. Further efficiency gains can be achieved by partially decoupling the proton uptake pathway from the active-site region. Such a mechanism is consistent with the proposed Glu valve, in which the side chain of a key glutamic acid shuttles between the D channel and the active-site region. We also show that the models predict only small proton leaks even in the absence of turnover. The design principles identified here for cytochrome c oxidase provide a blueprint for novel biology-inspired fuel cells, and the master-equation formulation should prove useful also for other molecular machines. . PMID:21946020
1. Computational study of p53 regulation via the chemical master equation
Vo, Huy D.; Sidje, Roger B.
2016-06-01
A stochastic model of cellular p53 regulation was established in Leenders, and Tuszynski (2013 Front. Oncol. 3 1-16) to study the interactions of p53 with MDM2 proteins, where the stochastic analysis was done using a Monte Carlo approach. We revisit that model here using an alternative scheme, which is to directly solve the chemical master equation (CME) by an adaptive Krylov-based finite state projection method that combines the stochastic simulation algorithm with other computational strategies, namely Krylov approximation techniques to the matrix exponential, divide and conquer, and aggregation. We report numerical results that demonstrate the extend of tackling the CME with this combination of tools.
2. Understanding the finite state projection and related methods for solving the chemical master equation.
PubMed
Dinh, Khanh N; Sidje, Roger B
2016-01-01
The finite state projection (FSP) method has enabled us to solve the chemical master equation of some biological models that were considered out of reach not long ago. Since the original FSP method, much effort has gone into transforming it into an adaptive time-stepping algorithm as well as studying its accuracy. Some of the improvements include the multiple time interval FSP, the sliding windows, and most notably the Krylov-FSP approach. Our goal in this tutorial is to give the reader an overview of the current methods that build on the FSP. PMID:27176781
3. Understanding the finite state projection and related methods for solving the chemical master equation
Dinh, Khanh N.; Sidje, Roger B.
2016-06-01
The finite state projection (FSP) method has enabled us to solve the chemical master equation of some biological models that were considered out of reach not long ago. Since the original FSP method, much effort has gone into transforming it into an adaptive time-stepping algorithm as well as studying its accuracy. Some of the improvements include the multiple time interval FSP, the sliding windows, and most notably the Krylov-FSP approach. Our goal in this tutorial is to give the reader an overview of the current methods that build on the FSP.
4. Understanding the finite state projection and related methods for solving the chemical master equation.
PubMed
Dinh, Khanh N; Sidje, Roger B
2016-05-13
The finite state projection (FSP) method has enabled us to solve the chemical master equation of some biological models that were considered out of reach not long ago. Since the original FSP method, much effort has gone into transforming it into an adaptive time-stepping algorithm as well as studying its accuracy. Some of the improvements include the multiple time interval FSP, the sliding windows, and most notably the Krylov-FSP approach. Our goal in this tutorial is to give the reader an overview of the current methods that build on the FSP.
5. Computational study of p53 regulation via the chemical master equation.
PubMed
Vo, Huy D; Sidje, Roger B
2016-04-29
A stochastic model of cellular p53 regulation was established in Leenders, and Tuszynski (2013 Front. Oncol. 3 1-16) to study the interactions of p53 with MDM2 proteins, where the stochastic analysis was done using a Monte Carlo approach. We revisit that model here using an alternative scheme, which is to directly solve the chemical master equation (CME) by an adaptive Krylov-based finite state projection method that combines the stochastic simulation algorithm with other computational strategies, namely Krylov approximation techniques to the matrix exponential, divide and conquer, and aggregation. We report numerical results that demonstrate the extend of tackling the CME with this combination of tools.
6. Computational study of p53 regulation via the chemical master equation
Vo, Huy D.; Sidje, Roger B.
2016-06-01
A stochastic model of cellular p53 regulation was established in Leenders, and Tuszynski (2013 Front. Oncol. 3 1–16) to study the interactions of p53 with MDM2 proteins, where the stochastic analysis was done using a Monte Carlo approach. We revisit that model here using an alternative scheme, which is to directly solve the chemical master equation (CME) by an adaptive Krylov-based finite state projection method that combines the stochastic simulation algorithm with other computational strategies, namely Krylov approximation techniques to the matrix exponential, divide and conquer, and aggregation. We report numerical results that demonstrate the extend of tackling the CME with this combination of tools.
7. Master equation for a kinetic model of a trading market and its analytic solution.
PubMed
Chatterjee, Arnab; Chakrabarti, Bikas K; Stinchcombe, Robin B
2005-08-01
We analyze an ideal-gas-like model of a trading market with quenched random saving factors for its agents and show that the steady state income (m) distribution P(m) in the model has a power law tail with Pareto index nu exactly equal to unity, confirming the earlier numerical studies on this model. The analysis starts with the development of a master equation for the time development of P(m) . Precise solutions are then obtained in some special cases. PMID:16196663
8. Solving the quantum brachistochrone equation through differential geometry
You, Chenglong; Wilde, Mark; Dowling, Jonathan; Wang, Xiaoting
2016-05-01
The ability of generating a particular quantum state, or model a physical quantum device by exploring quantum state transfer, is important in many applications such as quantum chemistry, quantum information processing, quantum metrology and cooling. Due to the environmental noise, a quantum device suffers from decoherence causing information loss. Hence, completing the state-generation task in a time-optimal way can be considered as a straightforward method to reduce decoherence. For a quantum system whose Hamiltonian has a fixed type and a finite energy bandwidth, it has been found that the time-optimal quantum evolution can be characterized by the quantum brachistochrone equation. In addition, the brachistochrone curve is found to have a geometric interpretation: it is the limit of a one-parameter family of geodesics on a sub-Riemannian model. Such geodesic-brachistochrone connection provides an efficient numerical method to solve the quantum brachistochrone equation. In this work, we will demonstrate this numerical method by studying the time-optimal state-generating problem on a given quantum spin system. We also find that the Pareto weighted-sum optimization turns out to be a simple but efficient method in solving the quantum time-optimal problems. We would like to acknowledge support from NSF under Award No. CCF-1350397.
9. Master equation with quantized atomic motion including dipole-dipole interactions
Damanet, François; Braun, Daniel; Martin, John
2016-05-01
We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion. Our equation provides a unifying picture of the consequences of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and is relevant for experiments with ultracold trapped atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find analytical formulas for a number of relevant states (Gaussian states, Fock states and thermal states). In particular, we show that the dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion. The effects predicted should be experimentally observable with Rydberg atoms. FD would like to thank the F.R.S.-FNRS for financial support. FD is a FRIA Grant holder of the Fonds de la Recherche Scientifique-FNRS.
10. Reduction and solution of the chemical master equation using time scale separation and finite state projection.
PubMed
Peles, Slaven; Munsky, Brian; Khammash, Mustafa
2006-11-28
The dynamics of chemical reaction networks often takes place on widely differing time scales--from the order of nanoseconds to the order of several days. This is particularly true for gene regulatory networks, which are modeled by chemical kinetics. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a model reduction method for study of stochastic chemical kinetic systems that takes advantage of multiple time scales. The method applies to finite projections of the chemical master equation and allows for effective time scale separation of the system dynamics. We implement this method in a novel numerical algorithm that exploits the time scale separation to achieve model order reductions while enabling error checking and control. We illustrate the efficiency of our method in several examples motivated by recent developments in gene regulatory networks.
11. A dynamical low-rank approach to the chemical master equation.
PubMed
Jahnke, Tobias; Huisinga, Wilhelm
2008-11-01
Stochastic reaction kinetics have increasingly been used to study cellular systems, with applications ranging from viral replication to gene regulatory networks and to signaling pathways. The underlying evolution equation, known as the chemical master equation (CME), can rarely be solved with traditional methods due to the huge number of degrees of freedom. We present a new approach to directly solve the CME by a dynamical low-rank approximation based on the Dirac-Frenkel-McLachlan variational principle. The new approach has the capability to substantially reduce the number of degrees of freedom, and to turn the CME into a computationally tractable problem. We illustrate the accuracy and efficiency of our methods in application to two examples of biological interest.
12. Influence of high-order nonlinear fluctuations in the multivariate susceptible-infectious-recovered master equation
Bayati, Basil S.; Eckhoff, Philip A.
2012-12-01
We perform a high-order analytical expansion of the epidemiological susceptible-infectious-recovered multivariate master equation and include terms up to and beyond single-particle fluctuations. It is shown that higher order approximations yield qualitatively different results than low-order approximations, which is incident to the influence of additional nonlinear fluctuations. The fluctuations can be related to a meaningful physical parameter, the basic reproductive number, which is shown to dictate the rate of divergence in absolute terms from the ordinary differential equations more so than the total number of persons in the system. In epidemiological terms, the effect of single-particle fluctuations ought to be taken into account as the reproductive number approaches unity.
13. General features and master equations for structurization in complex dusty plasmas
SciTech Connect
Tsytovich, V. N.; Morfill, G. E.
2012-02-15
Dust structurization is considered to be typical for complex plasmas. Homogeneous dusty plasmas are shown to be universally unstable. The dusty plasma structurization instability is similar to the gravitational instability and can results in creation of different compact dust structures. A general approach for investigation of the nonlinear stage of structurization in dusty plasmas is proposed and master equations for the description of self-organized structures are formulated in the general form that can be used for any nonlinear model of dust screening. New effects due to the scattering of ions on the nonlinearly screened grains are calculated: nonlinear ion dust drag force and nonlinear ion diffusion. The physics of confinement of dust and plasma components in the equilibria of compact dust structures is presented and is supported by numerical calculations of master equations. The necessary conditions for the existence of equilibrium structures are found for an arbitrary nonlinearity in dust screening. Features of compact dust structures observed in recent experiments agree with the numerically calculated ones. Some proposals for future experiments in spherical chamber are given.
14. Stochastic coalescence in finite systems: an algorithm for the numerical solution of the multivariate master equation.
Alfonso, Lester; Zamora, Jose; Cruz, Pedro
2015-04-01
The stochastic approach to coagulation considers the coalescence process going in a system of a finite number of particles enclosed in a finite volume. Within this approach, the full description of the system can be obtained from the solution of the multivariate master equation, which models the evolution of the probability distribution of the state vector for the number of particles of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain type of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels and initial conditions is introduced. The performance of the method was checked by comparing the numerically calculated particle mass spectrum with analytical solutions obtained for the constant and sum kernels, with an excellent correspondence between the analytical and numerical solutions. In order to increase the speedup of the algorithm, software parallelization techniques with OpenMP standard were used, along with an implementation in order to take advantage of new accelerator technologies. Simulations results show an important speedup of the parallelized algorithms. This study was funded by a grant from Consejo Nacional de Ciencia y Tecnologia de Mexico SEP-CONACYT CB-131879. The authors also thanks LUFAC® Computacion SA de CV for CPU time and all the support provided.
15. Rate equation modelling and investigation of quantum cascade detector characteristics
Saha, Sumit; Kumar, Jitendra
2016-10-01
A simple precise transport model has been proposed using rate equation approach for the characterization of a quantum cascade detector. The resonant tunneling transport is incorporated in the rate equation model through a resonant tunneling current density term. All the major scattering processes are included in the rate equation model. The effect of temperature on the quantum cascade detector characteristics has been examined considering the temperature dependent band parameters and the carrier scattering processes. Incorporation of the resonant tunneling process in the rate equation model improves the detector performance appreciably and reproduces the detector characteristics within experimental accuracy.
16. The Spatial Chemical Langevin and Reaction Diffusion Master Equations: Moments and Qualitative Solutions
Ghosh, Atiyo; Leier, Andre; Marquez-Lago, Tatiana
2014-03-01
Spatial stochastic effects are prevalent in many biological systems spanning a variety of scales, from intracellular (e.g. gene expression) to ecological (plankton aggregation). The most common ways of simulating such systems involve drawing sample paths from either the Reaction Diffusion Master Equation (RDME) or the Smoluchowski Equation, using methods such as Gillespie's Simulation Algorithm, Green's Function Reaction Dynamics and Single Particle Tracking. The simulation times of such techniques scale with the number of simulated particles, leading to much computational expense when considering large systems. The Spatial Chemical Langevin Equation (SCLE) can be simulated with fixed time intervals, independent of the number of particles, and can thus provide significant computational savings. However, very little work has been done to investigate the behavior of the SCLE. In this talk we summarize our findings on comparing the SCLE to the well-studied RDME. We use both analytical and numerical procedures to show when one should expect the moments of the SCLE to be close to the RDME, and also when they should differ.
17. Thermal fluctuation statistics in a molecular motor described by a multidimensional master equation
Challis, K. J.; Jack, M. W.
2013-12-01
We present a theoretical investigation of thermal fluctuation statistics in a molecular motor. Energy transfer in the motor is described using a multidimensional discrete master equation with nearest-neighbor hopping. In this theory, energy transfer leads to statistical correlations between thermal fluctuations in different degrees of freedom. For long times, the energy transfer is a multivariate diffusion process with constant drift and diffusion. The fluctuations and drift align in the strong-coupling limit enabling a one-dimensional description along the coupled coordinate. We derive formal expressions for the probability distribution and simulate single trajectories of the system in the near- and far-from-equilibrium limits both for strong and weak coupling. Our results show that the hopping statistics provide an opportunity to distinguish different operating regimes.
18. A master equation for the probability distribution functions of forces in soft particle packings.
PubMed
Saitoh, Kuniyasu; Magnanimo, Vanessa; Luding, Stefan
2015-02-01
We study the microscopic response of force-chain networks in jammed soft particles to quasi-static isotropic (de)compressions by molecular dynamics simulations. We show that not only contacts but also interparticle gaps between the nearest neighbors must be considered for the stochastic evolution of the probability distribution functions (PDFs) of forces, where the mutual exchange of contacts and interparticle gaps, i.e. opening and closing contacts, are also crucial to the incremental system behavior. By numerically determining the transition rates for all changes of contacts and gaps, we formulate a Master equation for the PDFs of forces, where the insight one gets from the transition rates is striking: the mean change of forces reflects non-affine system responses, while their fluctuations obey uncorrelated Gaussian statistics. In contrast, interparticle gaps react mostly affine in average, but imply multi-scale correlations according to a much wider stable distribution function.
19. Reformulation and solution of the master equation for multiple-well chemical reactions.
PubMed
Georgievskii, Yuri; Miller, James A; Burke, Michael P; Klippenstein, Stephen J
2013-11-21
We consider an alternative formulation of the master equation for complex-forming chemical reactions with multiple wells and bimolecular products. Within this formulation the dynamical phase space consists of only the microscopic populations of the various isomers making up the reactive complex, while the bimolecular reactants and products are treated equally as sources and sinks. This reformulation yields compact expressions for the phenomenological rate coefficients describing all chemical processes, i.e., internal isomerization reactions, bimolecular-to-bimolecular reactions, isomer-to-bimolecular reactions, and bimolecular-to-isomer reactions. The applicability of the detailed balance condition is discussed and confirmed. We also consider the situation where some of the chemical eigenvalues approach the energy relaxation time scale and show how to modify the phenomenological rate coefficients so that they retain their validity.
20. Surface electromagnetic wave equations in a warm magnetized quantum plasma
SciTech Connect
Li, Chunhua; Yang, Weihong; Wu, Zhengwei; Chu, Paul K.
2014-07-15
Based on the single-fluid plasma model, a theoretical investigation of surface electromagnetic waves in a warm quantum magnetized inhomogeneous plasma is presented. The surface electromagnetic waves are assumed to propagate on the plane between a vacuum and a warm quantum magnetized plasma. The quantum magnetohydrodynamic model includes quantum diffraction effect (Bohm potential), and quantum statistical pressure is used to derive the new dispersion relation of surface electromagnetic waves. And the general dispersion relation is analyzed in some special cases of interest. It is shown that surface plasma oscillations can be propagated due to quantum effects, and the propagation velocity is enhanced. Furthermore, the external magnetic field has a significant effect on surface wave's dispersion equation. Our work should be of a useful tool for investigating the physical characteristic of surface waves and physical properties of the bounded quantum plasmas.
1. Master equation approach to charge injection and transport in organic insulators
Freire, José A.; Voss, Grasiela
2005-03-01
We develop a master equation model of a disordered organic insulator sandwiched between metallic electrodes by treating as rate processes both the injection and the internal transport. We show how the master equation model allows for the inclusion of crucial correlation effects in the charge transport, particularly of the Pauli exclusion principle and of space-charge effects, besides, being dependent on just the microscopic form of the transfer rate between the localized electronic states, it allows for the investigation of different microscopic scenarios in the organic, such as polaronic hopping, correlated energy levels, interaction with image charge, etc. The model allows for a separate analysis of the injection and the recombination currents. We find that the disorder, besides increasing the injection current, eliminates the possibility of observation of a Fowler-Nordheim injection current at zero temperature, and that it does not alter the Schottky barrier size of the zero-field thermionic injection current from the value based on the energy difference between the electrode Fermi level and the highest occupied molecular orbital/lowest unoccupied molecular orbital levels in the organic, but it makes the Arrhenius temperature dependence appear at larger temperatures. We investigate how the I(V ) characteristics of a device is affected by the presence of correlations in the site energy distribution and by the form of the internal hopping rate, specifically the Miller-Abrahams rate and the Marcus or small-polaron rate. We show that the disorder does not modify significantly the eβ√E field dependence of the net current due to the Schottky barrier lowering caused by the attraction between the charge and its image in the electrode.
2. Solution of the master equation for Wigner's quasiprobability distribution in phase space for the Brownian motion of a particle in a double well potential
SciTech Connect
Coffey, William T.; Kalmykov, Yuri P.; Titov, Serguey V.
2007-08-21
Quantum effects in the Brownian motion of a particle in the symmetric double well potential V(x)=ax{sup 2}/2+bx{sup 4}/4 are treated using the semiclassical master equation for the time evolution of the Wigner distribution function W(x,p,t) in phase space (x,p). The equilibrium position autocorrelation function, dynamic susceptibility, and escape rate are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded has a quantum correction depending strongly on the barrier height and is compared with that given analytically by the quantum mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution. Moreover, the low-frequency part of the spectrum associated with noise assisted Kramers transitions across the potential barrier may be accurately described by a single Lorentzian with characteristic frequency given by the quantum mechanical reaction rate.
3. The Schroedinger equation with friction from the quantum trajectory perspective
SciTech Connect
Garashchuk, Sophya; Dixit, Vaibhav; Gu Bing; Mazzuca, James
2013-02-07
Similarity of equations of motion for the classical and quantum trajectories is used to introduce a friction term dependent on the wavefunction phase into the time-dependent Schroedinger equation. The term describes irreversible energy loss by the quantum system. The force of friction is proportional to the velocity of a quantum trajectory. The resulting Schroedinger equation is nonlinear, conserves wavefunction normalization, and evolves an arbitrary wavefunction into the ground state of the system (of appropriate symmetry if applicable). Decrease in energy is proportional to the average kinetic energy of the quantum trajectory ensemble. Dynamics in the high friction regime is suitable for simple models of reactions proceeding with energy transfer from the system to the environment. Examples of dynamics are given for single and symmetric and asymmetric double well potentials.
4. Quantum Hydrodynamics, the Quantum Benjamin-Ono Equation, and the Calogero Model
Abanov, Alexander G.; Wiegmann, Paul B.
2005-08-01
Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes—density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field—the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.
5. Degeneration of Trigonometric Dynamical Difference Equations for Quantum Loop Algebras to Trigonometric Casimir Equations for Yangians
Balagović, Martina
2015-03-01
We show that, under Drinfeld's degeneration (Proceedings of the International Congress of Mathematicians. American Mathematical Society, Providence, pp 798-820, 1987) of quantum loop algebras to Yangians, the trigonometric dynamical difference equations [Etingof and Varchenko (Adv Math 167:74-127, 2002)] for the quantum affine algebra degenerate to the trigonometric Casimir differential equations [Toledano Laredo (J Algebra 329:286-327, 2011)] for Yangians.
6. Equations of motion in general relativity and quantum mechanics
O'Hara, Paul
2011-12-01
In a previous article a relationship was established between the linearized metrics of General Relativity associated with geodesics and the Dirac Equation of quantum mechanics. In this paper the extension of that result to arbitrary curves is investigated. A generalized Dirac equation is derived and shown to be related to the Lie derivative of the momentum along the curve. In addition,the equations of motion are derived from the Hamilton-Jacobi equation associated with the metric and the wave equation associated with the Hamiltonian is then shown not to commute with the Dirac operator. Finally, the Maxwell-Boltzmann distribution is shown to be a consequence of geodesic motion.
7. Bimolecular recombination reactions: K-adiabatic and K-active forms of the bimolecular master equations and analytic solutions
2016-03-01
Expressions for a K-adiabatic master equation for a bimolecular recombination rate constant krec are derived for a bimolecular reaction forming a complex with a single well or complexes with multiple well, where K is the component of the total angular momentum along the axis of least moment of inertia of the recombination product. The K-active master equation is also considered. The exact analytic solutions, i.e., the K-adiabatic and K-active steady-state population distribution function of reactive complexes, g(EJK) and g(EJ), respectively, are derived for the K-adiabatic and K-active master equation cases using properties of inhomogeneous integral equations (Fredholm type). The solutions accommodate arbitrary intermolecular energy transfer models, e.g., the single exponential, double exponential, Gaussian, step-ladder, and near-singularity models. At the high pressure limit, the krec for both the K-adiabatic and K-active master equations reduce, respectively, to the K-adiabatic and K-active bimolecular Rice-Ramsperger-Kassel-Marcus theory (high pressure limit expressions). Ozone and its formation from O + O2 are known to exhibit an adiabatic K. The ratio of the K-adiabatic to the K-active recombination rate constants for ozone formation at the high pressure limit is calculated to be ˜0.9 at 300 K. Results on the temperature and pressure dependence of the recombination rate constants and populations of O3 will be presented elsewhere.
8. Bimolecular recombination reactions: K-adiabatic and K-active forms of the bimolecular master equations and analytic solutions.
PubMed
2016-03-28
Expressions for a K-adiabatic master equation for a bimolecular recombination rate constant krec are derived for a bimolecular reaction forming a complex with a single well or complexes with multiple well, where K is the component of the total angular momentum along the axis of least moment of inertia of the recombination product. The K-active master equation is also considered. The exact analytic solutions, i.e., the K-adiabatic and K-active steady-state population distribution function of reactive complexes, g(EJK) and g(EJ), respectively, are derived for the K-adiabatic and K-active master equation cases using properties of inhomogeneous integral equations (Fredholm type). The solutions accommodate arbitrary intermolecular energy transfer models, e.g., the single exponential, double exponential, Gaussian, step-ladder, and near-singularity models. At the high pressure limit, the krec for both the K-adiabatic and K-active master equations reduce, respectively, to the K-adiabatic and K-active bimolecular Rice-Ramsperger-Kassel-Marcus theory (high pressure limit expressions). Ozone and its formation from O + O2 are known to exhibit an adiabatic K. The ratio of the K-adiabatic to the K-active recombination rate constants for ozone formation at the high pressure limit is calculated to be ∼0.9 at 300 K. Results on the temperature and pressure dependence of the recombination rate constants and populations of O3 will be presented elsewhere. PMID:27036434
9. Quantum Fokker-Planck-Kramers equation and entropy production.
PubMed
de Oliveira, Mário J
2016-07-01
We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production that properly describes quantum systems in a nonequilibrium stationary state is also provided. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance. PMID:27575097
10. Pattern dynamics and spatiotemporal chaos in the quantum Zakharov equations
SciTech Connect
Misra, A. P.; Shukla, P. K.
2009-05-15
The dynamical behavior of the nonlinear interaction of quantum Langmuir waves (QLWs) and quantum ion-acoustic waves (QIAWs) is studied in the one-dimensional quantum Zakharov equations. Numerical simulations of coupled QLWs and QIAWs reveal that many coherent solitary patterns can be excited and saturated via the modulational instability of unstable harmonic modes excited by a modulation wave number of monoenergetic QLWs. The evolution of such solitary patterns may undergo the states of spatially partial coherence (SPC), coexistence of temporal chaos and spatiotemporal chaos (STC), as well as STC. The SPC state is essentially due to ion-acoustic wave emission and due to quantum diffraction, while the STC is caused by the combined effects of SPC and quantum diffraction, as well as by collisions and fusions among patterns in stochastic motion. The energy in the system is strongly redistributed, which may switch on the onset of weak turbulence in dense quantum plasmas.
11. A wave equation interpolating between classical and quantum mechanics
Schleich, W. P.; Greenberger, D. M.; Kobe, D. H.; Scully, M. O.
2015-10-01
We derive a ‘master’ wave equation for a family of complex-valued waves {{Φ }}\\equiv R{exp}[{{{i}}S}({cl)}/{{\\hbar }}] whose phase dynamics is dictated by the Hamilton-Jacobi equation for the classical action {S}({cl)}. For a special choice of the dynamics of the amplitude R which eliminates all remnants of classical mechanics associated with {S}({cl)} our wave equation reduces to the Schrödinger equation. In this case the amplitude satisfies a Schrödinger equation analogous to that of a charged particle in an electromagnetic field where the roles of the scalar and the vector potentials are played by the classical energy and the momentum, respectively. In general this amplitude is complex and thereby creates in addition to the classical phase {S}({cl)}/{{\\hbar }} a quantum phase. Classical statistical mechanics, as described by a classical matter wave, follows from our wave equation when we choose the dynamics of the amplitude such that it remains real for all times. Our analysis shows that classical and quantum matter waves are distinguished by two different choices of the dynamics of their amplitudes rather than two values of Planck’s constant. We dedicate this paper to the memory of Richard Lewis Arnowitt—a pioneer of many-body theory, a path finder at the interface of gravity and quantum mechanics, and a true leader in non-relativistic and relativistic quantum field theory.
12. Experimental quantum computing to solve systems of linear equations.
PubMed
Cai, X-D; Weedbrook, C; Su, Z-E; Chen, M-C; Gu, Mile; Zhu, M-J; Li, Li; Liu, Nai-Le; Lu, Chao-Yang; Pan, Jian-Wei
2013-06-01
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2×2 linear equations for various input vectors on a quantum computer. We use four quantum bits and four controlled logic gates to implement every subroutine required, demonstrating the working principle of this algorithm.
13. Schroedinger-equation formalism for a dissipative quantum system
SciTech Connect
Anisimovas, E.; Matulis, A.
2007-02-15
We consider a model dissipative quantum-mechanical system realized by coupling a quantum oscillator to a semi-infinite classical string which serves as a means of energy transfer from the oscillator to the infinity and thus plays the role of a dissipative element. The coupling between the two--quantum and classical--parts of the compound system is treated in the spirit of the mean-field approximation and justification of the validity of such an approach is given. The equations of motion of the classical subsystem are solved explicitly and an effective dissipative Schroedinger equation for the quantum subsystem is obtained. The proposed formalism is illustrated by its application to two basic problems: the decay of the quasistationary state and the calculation of the nonlinear resonance line shape.
14. Loop equations and KDV hierarchy in 2-D quantum gravity
SciTech Connect
Fucito, F. ); Martellini, M. )
1992-04-20
In this paper a derivation of the loop equation for two-dimensional quantum gravity from the KdV equations and the string equation of the one-matrix model is given. The loop equation was found to be equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the virasoro algebra. Starting form the equations expressing these constraints. The authors are able to rederive the equations of the KdV hierarchy using the vertex operator construction of the A{sup (I)}{sub I} infinite dimensional twisted Kac-Moody algebra. From these considerations it follows that the solutions of the string equation of the one-matrix model are given by a subset of the solutions of the KdV hierarchy.
15. High-power arrays of quantum cascade laser master-oscillator power-amplifiers.
PubMed
Rauter, Patrick; Menzel, Stefan; Goyal, Anish K; Wang, Christine A; Sanchez, Antonio; Turner, George; Capasso, Federico
2013-02-25
We report on multi-wavelength arrays of master-oscillator power-amplifier quantum cascade lasers operating at wavelengths between 9.2 and 9.8 μm. All elements of the high-performance array feature longitudinal (spectral) as well as transverse single-mode emission at peak powers between 2.7 and 10 W at room temperature. The performance of two arrays that are based on different seed-section designs is thoroughly studied and compared. High output power and excellent beam quality render the arrays highly suitable for stand-off spectroscopy applications.
16. Scalar field equations from quantum gravity during inflation
SciTech Connect
Kahya, E. O.; Woodard, R. P.
2008-04-15
We exploit a previous computation of the self-mass-squared from quantum gravity to include quantum corrections to the scalar evolution equation. The plane wave mode functions are shown to receive no significant one loop corrections at late times. This result probably applies as well to the inflaton of scalar-driven inflation. If so, there is no significant correction to the {phi}{phi} correlator that plays a crucial role in computations of the power spectrum.
17. Variational Equation for Quantum Number Projection at Finite Temperature
2008-04-01
To describe phase transitions in a finite system at finite temperature, we develop a formalism of the variation-after-projection (VAP) of quantum numbers based on the thermofield dynamics (TFD). We derive a new Bardeen-Cooper-Schrieffer (BCS)-type equation by variating the free energy with approximate entropy without violating Peierls inequality. The solution to the new BCS equation describes the S-shape in the specific heat curve and the superfluid-to-normal phase transition caused by the temperature effect. It simulates the exact quantum Monte Carlo results well.
18. Derivation of the Drude conductivity from quantum kinetic equations
Kitamura, Hikaru
2015-11-01
The Drude formula of ac (frequency-dependent) electric conductivity has been established as a simple and practically useful model to understand the electromagnetic response of simple free-electron-like metals. In most textbooks of solid-state physics, the Drude formula is derived from either a classical equation of motion or the semiclassical Boltzmann transport equation. On the other hand, quantum-mechanical derivation of the Drude conductivity, which requires an appropriate treatment of phonon-assisted intraband transitions with small momentum transfer, has not been well documented except for the zero- or high-frequency case. Here, a lucid derivation of the Drude conductivity that covers the entire frequency range is presented by means of quantum kinetic equations in the density-matrix formalism. The derivation is straightforward so that advanced undergraduate students or early-year graduate students will be able to gain insight into the link between the microscopic Schrödinger equation and macroscopic transport.
19. Quantum N-body problem: Matrix structures and equations
Yakovlev, S. L.
2014-10-01
We consider matrix structures in the quantum N-body problem that generalize the Faddeev components for resolvents, T-matrices, and eigenfunctions of the continuous spectrum. We write matrix equations for the introduced components of T-matrices and resolvents and use these equations to obtain matrix operators generalizing the matrix three-particle Faddeev operators to the case of arbitrarily many particles. We determine the eigenfunctions of the continuous spectrum of these matrix operators.
20. Absorbing boundary conditions for relativistic quantum mechanics equations
SciTech Connect
Antoine, X.; Sater, J.; Fillion-Gourdeau, F.; Bandrauk, A.D.
2014-11-15
This paper is devoted to the derivation of absorbing boundary conditions for the Klein–Gordon and Dirac equations modeling quantum and relativistic particles subject to classical electromagnetic fields. Microlocal analysis is the main ingredient in the derivation of these boundary conditions, which are obtained in the form of pseudo-differential equations. Basic numerical schemes are derived and analyzed to illustrate the accuracy of the derived boundary conditions.
1. Local error estimates for adaptive simulation of the Reaction–Diffusion Master Equation via operator splitting
PubMed Central
Hellander, Andreas; Lawson, Michael J; Drawert, Brian; Petzold, Linda
2015-01-01
The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps are adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the Diffusive Finite-State Projection (DFSP) method, to incorporate temporal adaptivity. PMID:26865735
2. Local error estimates for adaptive simulation of the reaction-diffusion master equation via operator splitting
Hellander, Andreas; Lawson, Michael J.; Drawert, Brian; Petzold, Linda
2014-06-01
The efficiency of exact simulation methods for the reaction-diffusion master equation (RDME) is severely limited by the large number of diffusion events if the mesh is fine or if diffusion constants are large. Furthermore, inherent properties of exact kinetic-Monte Carlo simulation methods limit the efficiency of parallel implementations. Several approximate and hybrid methods have appeared that enable more efficient simulation of the RDME. A common feature to most of them is that they rely on splitting the system into its reaction and diffusion parts and updating them sequentially over a discrete timestep. This use of operator splitting enables more efficient simulation but it comes at the price of a temporal discretization error that depends on the size of the timestep. So far, existing methods have not attempted to estimate or control this error in a systematic manner. This makes the solvers hard to use for practitioners since they must guess an appropriate timestep. It also makes the solvers potentially less efficient than if the timesteps were adapted to control the error. Here, we derive estimates of the local error and propose a strategy to adaptively select the timestep when the RDME is simulated via a first order operator splitting. While the strategy is general and applicable to a wide range of approximate and hybrid methods, we exemplify it here by extending a previously published approximate method, the diffusive finite-state projection (DFSP) method, to incorporate temporal adaptivity.
3. Finite state projection based bounds to compare chemical master equation models using single-cell data
Fox, Zachary; Neuert, Gregor; Munsky, Brian
2016-08-01
Emerging techniques now allow for precise quantification of distributions of biological molecules in single cells. These rapidly advancing experimental methods have created a need for more rigorous and efficient modeling tools. Here, we derive new bounds on the likelihood that observations of single-cell, single-molecule responses come from a discrete stochastic model, posed in the form of the chemical master equation. These strict upper and lower bounds are based on a finite state projection approach, and they converge monotonically to the exact likelihood value. These bounds allow one to discriminate rigorously between models and with a minimum level of computational effort. In practice, these bounds can be incorporated into stochastic model identification and parameter inference routines, which improve the accuracy and efficiency of endeavors to analyze and predict single-cell behavior. We demonstrate the applicability of our approach using simulated data for three example models as well as for experimental measurements of a time-varying stochastic transcriptional response in yeast.
4. Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions.
PubMed
Ammar, Amine; Cueto, Elías; Chinesta, Francisco
2012-09-01
The numerical solution of the chemical master equation (CME) governing gene regulatory networks and cell signaling processes remains a challenging task owing to its complexity, exponentially growing with the number of species involved. Although most of the existing techniques rely on the use of Monte Carlo-like techniques, we present here a new technique based on the approximation of the unknown variable (the probability of having a particular chemical state) in terms of a finite sum of separable functions. In this framework, the complexity of the CME grows only linearly with the number of state space dimensions. This technique generalizes the so-called Hartree approximation, by using terms as needed in the finite sums decomposition for ensuring convergence. But noteworthy, the ease of the approximation allows for an easy treatment of unknown parameters (as is frequently the case when modeling gene regulatory networks, for instance). These unknown parameters can be considered as new space dimensions. In this way, the proposed method provides solutions for any value of the unknown parameters (within some interval of arbitrary size) in one execution of the program.
5. Finite state projection based bounds to compare chemical master equation models using single-cell data.
PubMed
Fox, Zachary; Neuert, Gregor; Munsky, Brian
2016-08-21
Emerging techniques now allow for precise quantification of distributions of biological molecules in single cells. These rapidly advancing experimental methods have created a need for more rigorous and efficient modeling tools. Here, we derive new bounds on the likelihood that observations of single-cell, single-molecule responses come from a discrete stochastic model, posed in the form of the chemical master equation. These strict upper and lower bounds are based on a finite state projection approach, and they converge monotonically to the exact likelihood value. These bounds allow one to discriminate rigorously between models and with a minimum level of computational effort. In practice, these bounds can be incorporated into stochastic model identification and parameter inference routines, which improve the accuracy and efficiency of endeavors to analyze and predict single-cell behavior. We demonstrate the applicability of our approach using simulated data for three example models as well as for experimental measurements of a time-varying stochastic transcriptional response in yeast.
6. A graph-based approach for the approximate solution of the chemical master equation.
PubMed
Basile, Raffaele; Grima, Ramon; Popović, Nikola
2013-10-01
The chemical master equation (CME) represents the accepted stochastic description of chemical reaction kinetics in mesoscopic systems. As its exact solution—which gives the corresponding probability density function—is possible only in very simple cases; there is a clear need for approximation techniques. Here, we propose a novel perturbative three-step approach, which draws heavily on graph theory: (i) we expand the eigenvalues of the transition state matrix in the CME as a series in a nondimensional parameter that depends on the reaction rates and the reaction volume; (ii) we derive an analogous series for the corresponding eigenvectors via a graph-based algorithm; (iii) we combine the resulting expansions into an approximate solution to the CME. We illustrate our approach by applying it to a reversible dimerization reaction; then we formulate a set of conditions, which ensure its applicability to more general reaction networks, and we verify those conditions for two common catalytic mechanisms. Comparing our results with the linear-noise approximation (LNA), we find that our methodology is consistently more accurate for sufficiently small values of the nondimensional parameter. This superior accuracy is particularly evident in scenarios characterized by small molecule numbers, which are typical of conditions inside biological cells.
7. Finite state projection based bounds to compare chemical master equation models using single-cell data.
PubMed
Fox, Zachary; Neuert, Gregor; Munsky, Brian
2016-08-21
Emerging techniques now allow for precise quantification of distributions of biological molecules in single cells. These rapidly advancing experimental methods have created a need for more rigorous and efficient modeling tools. Here, we derive new bounds on the likelihood that observations of single-cell, single-molecule responses come from a discrete stochastic model, posed in the form of the chemical master equation. These strict upper and lower bounds are based on a finite state projection approach, and they converge monotonically to the exact likelihood value. These bounds allow one to discriminate rigorously between models and with a minimum level of computational effort. In practice, these bounds can be incorporated into stochastic model identification and parameter inference routines, which improve the accuracy and efficiency of endeavors to analyze and predict single-cell behavior. We demonstrate the applicability of our approach using simulated data for three example models as well as for experimental measurements of a time-varying stochastic transcriptional response in yeast. PMID:27544081
8. Non-Markovian reduced propagator, multiple-time correlation functions, and master equations with general initial conditions in the weak-coupling limit
SciTech Connect
Vega, Ines de; Alonso, Daniel
2006-02-15
In this paper we derive the evolution equation for the reduced propagator, an object that evolves vectors of the Hilbert space of a system S interacting with an environment B in a non-Markovian way. This evolution is conditioned to certain initial and final states of the environment. Once an average over these environmental states is made, reduced propagators permit the evaluation of multiple-time correlation functions of system observables. When this average is done stochastically the reduced propagator evolves according to a stochastic Schroedinger equation. In addition, it is possible to obtain the evolution equations of the multiple-time correlation functions which generalize the well-known quantum regression theorem to the non-Markovian case. Here, both methods, stochastic and evolution equations, are described by assuming a weak coupling between system and environment. Finally, we show that reduced propagators can be used to obtain a master equation with general initial conditions, and not necessarily an initial vacuum state for the environment. We illustrate the theory with several examples.
9. Mapping quantum-classical Liouville equation: projectors and trajectories.
PubMed
Kelly, Aaron; van Zon, Ramses; Schofield, Jeremy; Kapral, Raymond
2012-02-28
The evolution of a mixed quantum-classical system is expressed in the mapping formalism where discrete quantum states are mapped onto oscillator states, resulting in a phase space description of the quantum degrees of freedom. By defining projection operators onto the mapping states corresponding to the physical quantum states, it is shown that the mapping quantum-classical Liouville operator commutes with the projection operator so that the dynamics is confined to the physical space. It is also shown that a trajectory-based solution of this equation can be constructed that requires the simulation of an ensemble of entangled trajectories. An approximation to this evolution equation which retains only the Poisson bracket contribution to the evolution operator does admit a solution in an ensemble of independent trajectories but it is shown that this operator does not commute with the projection operators and the dynamics may take the system outside the physical space. The dynamical instabilities, utility, and domain of validity of this approximate dynamics are discussed. The effects are illustrated by simulations on several quantum systems.
10. Quantum-mechanical transport equation for atomic systems.
NASA Technical Reports Server (NTRS)
Berman, P. R.
1972-01-01
A quantum-mechanical transport equation (QMTE) is derived which should be applicable to a wide range of problems involving the interaction of radiation with atoms or molecules which are also subject to collisions with perturber atoms. The equation follows the time evolution of the macroscopic atomic density matrix elements of atoms located at classical position R and moving with classical velocity v. It is quantum mechanical in the sense that all collision kernels or rates which appear have been obtained from a quantum-mechanical theory and, as such, properly take into account the energy-level variations and velocity changes of the active (emitting or absorbing) atom produced in collisions with perturber atoms. The present formulation is better suited to problems involving high-intensity external fields, such as those encountered in laser physics.
11. High resolution finite volume scheme for the quantum hydrodynamic equations
Lin, Chin-Tien; Yeh, Jia-Yi; Chen, Jiun-Yeu
2009-03-01
The theory of quantum fluid dynamics (QFD) helps nanotechnology engineers to understand the physical effect of quantum forces. Although the governing equations of quantum fluid dynamics and classical fluid mechanics have the same form, there are two numerical simulation problems must be solved in QFD. The first is that the quantum potential term becomes singular and causes a divergence in the numerical simulation when the probability density is very small and close to zero. The second is that the unitarity in the time evolution of the quantum wave packet is significant. Accurate numerical evaluations are critical to the simulations of the flow fields that are generated by various quantum fluid systems. A finite volume scheme is developed herein to solve the quantum hydrodynamic equations of motion, which significantly improve the accuracy and stability of this method. The QFD equation is numerically implemented within the Eulerian method. A third-order modified Osher-Chakravarthy (MOC) upwind-centered finite volume scheme was constructed for conservation law to evaluate the convective terms, and a second-order central finite volume scheme was used to map the quantum potential field. An explicit Runge-Kutta method is used to perform the time integration to achieve fast convergence of the proposed scheme. In order to meet the numerical result can conform to the physical phenomenon and avoid numerical divergence happening due to extremely low probability density, the minimum value setting of probability density must exceed zero and smaller than certain value. The optimal value was found in the proposed numerical approach to maintain a converging numerical simulation when the minimum probability density is 10 -5 to 10 -12. The normalization of the wave packet remains close to unity through a long numerical simulation and the deviations from 1.0 is about 10 -4. To check the QFD finite difference numerical computations, one- and two-dimensional particle motions were
12. High resolution finite volume scheme for the quantum hydrodynamic equations
SciTech Connect
Lin, C.-T. Yeh, J.-Y. Chen, J.-Y.
2009-03-20
The theory of quantum fluid dynamics (QFD) helps nanotechnology engineers to understand the physical effect of quantum forces. Although the governing equations of quantum fluid dynamics and classical fluid mechanics have the same form, there are two numerical simulation problems must be solved in QFD. The first is that the quantum potential term becomes singular and causes a divergence in the numerical simulation when the probability density is very small and close to zero. The second is that the unitarity in the time evolution of the quantum wave packet is significant. Accurate numerical evaluations are critical to the simulations of the flow fields that are generated by various quantum fluid systems. A finite volume scheme is developed herein to solve the quantum hydrodynamic equations of motion, which significantly improve the accuracy and stability of this method. The QFD equation is numerically implemented within the Eulerian method. A third-order modified Osher-Chakravarthy (MOC) upwind-centered finite volume scheme was constructed for conservation law to evaluate the convective terms, and a second-order central finite volume scheme was used to map the quantum potential field. An explicit Runge-Kutta method is used to perform the time integration to achieve fast convergence of the proposed scheme. In order to meet the numerical result can conform to the physical phenomenon and avoid numerical divergence happening due to extremely low probability density, the minimum value setting of probability density must exceed zero and smaller than certain value. The optimal value was found in the proposed numerical approach to maintain a converging numerical simulation when the minimum probability density is 10{sup -5} to 10{sup -12}. The normalization of the wave packet remains close to unity through a long numerical simulation and the deviations from 1.0 is about 10{sup -4}. To check the QFD finite difference numerical computations, one- and two-dimensional particle
13. Derivation of the Schrodinger Equation from the Hamilton-Jacobi Equation in Feynman's Path Integral Formulation of Quantum Mechanics
ERIC Educational Resources Information Center
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
14. Kraus Operator-Sum Solution to the Master Equation Describing the Single-Mode Cavity Driven by an Oscillating External Field in the Heat Reservoir
Meng, Xiang-Guo; Wang, Ji-Suo; Gao, Hua-Chao
2016-08-01
Exploiting the thermo entangled state approach, we successfully solve the master equation for describing the single-mode cavity driven by an oscillating external field in the heat reservoir and then get the analytical time-evolution rule for the density operator in the infinitive Kraus operator-sum representation. It is worth noting that the Kraus operator M l, m is proved to be a trace-preserving quantum operation. As an application, the time-evolution for an initial coherent state ρ | β> = | β>< β| in such an environment is investigated, which shows that the initial coherent state decays to a new mixed state as a result of thermal noise, however the coherence can still be reserved for amplitude damping.
15. Distributions for negative-feedback-regulated stochastic gene expression: dimension reduction and numerical solution of the chemical master equation.
PubMed
Zeron, Eduardo S; Santillán, Moisés
2010-05-21
In this work we introduce a novel approach to study biochemical noise. It comprises a simplification of the master equation of complex reaction schemes (via an adiabatic approximation) and the numerical solution of the reduced master equation. The accuracy of this procedure is tested by comparing its results with analytic solutions (when available) and with Gillespie stochastic simulations. We further employ our approach to study the stochastic expression of a simple gene network, which is subject to negative feedback regulation at the transcriptional level. Special attention is paid to the influence of negative feedback on the amplitude of intrinsic noise, as well as on the relaxation rate of the system probability distribution function to the steady solution. Our results suggest the existence of an optimal feedback strength that maximizes this relaxation rate.
16. Resolution of the vibrational energy distribution function using a direct simulation Monte Carlo-master equation approach
Boyd, Iain D.; Josyula, Eswar
2016-01-01
The direct simulation Monte Carlo (DSMC) method is the primary numerical technique for analysis of rarefied gas flows. While recent progress in computational chemistry is beginning to provide vibrationally resolved transition and reaction cross sections that can be employed in DSMC calculations, the particle nature of the standard DSMC method makes it difficult to use this information in a statistically significant way. The current study introduces a new technique that makes it possible to resolve all of the vibrational energy levels by using a master equation approach along with temperature-dependent transition rates. The new method is compared to the standard DSMC technique for several heat bath and shock wave conditions and demonstrates the ability to resolve the full vibrational manifold at the expected overall rates of relaxation. The ability of the new master equation approach to the DSMC method for resolving, in particular, the high-energy states addresses a well-known, longstanding deficiency of the standard DSMC method.
17. HO + CO reaction rates and H/D kinetic isotope effects: master equation models with ab initio SCTST rate constants.
PubMed
Weston, Ralph E; Nguyen, Thanh Lam; Stanton, John F; Barker, John R
2013-02-01
Ab initio microcanonical rate constants were computed using Semi-Classical Transition State Theory (SCTST) and used in two master equation formulations (1D, depending on active energy with centrifugal corrections, and 2D, depending on total energy and angular momentum) to compute temperature-dependent rate constants for the title reactions using a potential energy surface obtained by sophisticated ab initio calculations. The 2D master equation was used at the P = 0 and P = ∞ limits, while the 1D master equation with centrifugal corrections and an empirical energy transfer parameter could be used over the entire pressure range. Rate constants were computed for 75 K ≤ T ≤ 2500 K and 0 ≤ [He] ≤ 10(23) cm(-3). For all temperatures and pressures important for combustion and for the terrestrial atmosphere, the agreement with the experimental rate constants is very good, but at very high pressures and T ≤ 200 K, the theoretical rate constants are significantly smaller than the experimental values. This effect is possibly due to the presence in the experiments of dimers and prereactive complexes, which were not included in the model calculations. The computed H/D kinetic isotope effects are in acceptable agreement with experimental data, which show considerable scatter. Overall, the agreement between experimental and theoretical H/D kinetic isotope effects is much better than in previous work, and an assumption of non-RRKM behavior does not appear to be needed to reproduce experimental observations.
18. Uniqueness of the equation for quantum state vector collapse.
PubMed
Bassi, Angelo; Dürr, Detlef; Hinrichs, Günter
2013-11-22
The linearity of quantum mechanics leads, under the assumption that the wave function offers a complete description of reality, to grotesque situations famously known as Schrödinger's cat. Ways out are either adding elements of reality or replacing the linear evolution by a nonlinear one. Models of spontaneous wave function collapses took the latter path. The way such models are constructed leaves the question of whether such models are in some sense unique, i.e., whether the nonlinear equations replacing Schrödinger's equation are uniquely determined as collapse equations. Various people worked on identifying the class of nonlinear modifications of the Schrödinger equation, compatible with general physical requirements. Here we identify the most general class of continuous wave function evolutions under the assumption of no-faster-than-light signaling.
19. A Matter of Principle: The Principles of Quantum Theory, Dirac's Equation, and Quantum Information
2015-10-01
This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be addressed as well, in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac's work, which, in particular Dirac's derivation of his relativistic equation of the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall also consider Heisenberg's earlier work leading him to the discovery of quantum mechanics, which inspired Dirac's work. I argue that Heisenberg's and Dirac's work was guided by their adherence to and their confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by D'Ariano and coworkers on the principles of quantum information theory, which extend quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac's equations from these principles alone, without using the principles of relativity.
20. State space truncation with quantified errors for accurate solutions to discrete Chemical Master Equation
PubMed Central
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-01-01
The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in mesoscopic reaction networks. Since its direct solution rapidly becomes intractable due to the increasing size of the state space, truncation of the state space is necessary for solving most dCMEs. It is therefore important to assess the consequences of state space truncations so errors can be quantified and minimized. Here we describe a novel method for state space truncation. By partitioning a reaction network into multiple molecular equivalence groups (MEG), we truncate the state space by limiting the total molecular copy numbers in each MEG. We further describe a theoretical framework for analysis of the truncation error in the steady state probability landscape using reflecting boundaries. By aggregating the state space based on the usage of a MEG and constructing an aggregated Markov process, we show that the truncation error of a MEG can be asymptotically bounded by the probability of states on the reflecting boundary of the MEG. Furthermore, truncating states of an arbitrary MEG will not undermine the estimated error of truncating any other MEGs. We then provide an overall error estimate for networks with multiple MEGs. To rapidly determine the appropriate size of an arbitrary MEG, we also introduce an a priori method to estimate the upper bound of its truncation error. This a priori estimate can be rapidly computed from reaction rates of the network, without the need of costly trial solutions of the dCME. As examples, we show results of applying our methods to the four stochastic networks of 1) the birth and death model, 2) the single gene expression model, 3) the genetic toggle switch model, and 4) the phage lambda bistable epigenetic switch model. We demonstrate how truncation errors and steady state probability landscapes can be computed using different sizes of the MEG(s) and how the results validate out theories. Overall, the novel state space
1. State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation.
PubMed
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-04-01
The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in mesoscopic reaction networks. Since its direct solution rapidly becomes intractable due to the increasing size of the state space, truncation of the state space is necessary for solving most dCMEs. It is therefore important to assess the consequences of state space truncations so errors can be quantified and minimized. Here we describe a novel method for state space truncation. By partitioning a reaction network into multiple molecular equivalence groups (MEGs), we truncate the state space by limiting the total molecular copy numbers in each MEG. We further describe a theoretical framework for analysis of the truncation error in the steady-state probability landscape using reflecting boundaries. By aggregating the state space based on the usage of a MEG and constructing an aggregated Markov process, we show that the truncation error of a MEG can be asymptotically bounded by the probability of states on the reflecting boundary of the MEG. Furthermore, truncating states of an arbitrary MEG will not undermine the estimated error of truncating any other MEGs. We then provide an overall error estimate for networks with multiple MEGs. To rapidly determine the appropriate size of an arbitrary MEG, we also introduce an a priori method to estimate the upper bound of its truncation error. This a priori estimate can be rapidly computed from reaction rates of the network, without the need of costly trial solutions of the dCME. As examples, we show results of applying our methods to the four stochastic networks of (1) the birth and death model, (2) the single gene expression model, (3) the genetic toggle switch model, and (4) the phage lambda bistable epigenetic switch model. We demonstrate how truncation errors and steady-state probability landscapes can be computed using different sizes of the MEG(s) and how the results validate our theories. Overall, the novel state space
2. Description of quantum noise by a Langevin equation
NASA Technical Reports Server (NTRS)
Metiu, H.; Schon, G.
1984-01-01
General features of the quantum noise problem expressed as the equations of motion for a particle coupled to a set of oscillators are investigated analytically. Account is taken of the properties of the companion oscillators by formulating quantum statistical correlation Langevin equations (QSLE). The frequency of the oscillators is then retained as a natural cut-off for the quantum noise. The QSLE is further extended to encompass the particle trajectory and is bounded by initial and final states of the oscillator. The states are expressed as the probability of existence at the moment of particle collision that takes the oscillator into a final state. Two noise sources then exist: a statistical uncertainty of the initial state and the quantum dynamical uncertainty associated with a transition from the initial to final state. Feynman's path-integral formulation is used to characterize the functional of the particle trajectory, which slows the particle. It is shown that the energy loss may be attributed to friction, which satisfies energy conservation laws.
3. Dirac equation with an ultraviolet cutoff and a quantum walk
SciTech Connect
Sato, Fumihito; Katori, Makoto
2010-01-15
The weak convergence theorems of the one- and two-dimensional simple quantum walks, SQW{sup (d)},d=1,2, show a striking contrast to the classical counterparts, the simple random walks, SRW{sup (d)}. In the SRW{sup (d)}, the distribution of position X(t) of the particle starting from the origin converges to the Gaussian distribution in the diffusion scaling limit, in which the time scale T and spatial scale L both go to infinity as the ratio L/sq root(T) is kept finite. On the other hand, in the SQW{sup (d)}, the ratio L/T is kept to define the pseudovelocity V(t)=X(t)/t, and then all joint moments of the components V{sub j}(t),1<=j<=d, of V(t) converge in the T=L->infinity limit. The limit distributions have novel structures such that they are inverted-bell shaped and their supports are bounded. In the present paper we claim that these properties of the SQW{sup (d)} can be explained by the theory of relativistic quantum mechanics. We show that the Dirac equation with a proper ultraviolet cutoff can provide a quantum walk model in three dimensions, where the walker has a four-component qubit. We clarify that the pseudovelocity V(t) of the quantum walker, which solves the Dirac equation, is identified with the relativistic velocity. Since the quantum walker should be a tardyon, not a tachyon, |V(t)|quantum walk models. By reducing the number of components of momentum in the Dirac equation, we obtain the limit distributions of pseudovelocities for the lower dimensional quantum walks. We show that the obtained limit distributions for the one- and two-dimensional systems have common features with those of SQW{sup (1)} and SQW{sup (2)}.
4. Exact results in the large system size limit for the dynamics of the chemical master equation, a one dimensional chain of equations.
PubMed
Martirosyan, A; Saakian, David B
2011-08-01
We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the chemical master equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of the Van Kampen method from the HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ordinary differential equations (ODEs) to define the variance in a two-dimensional case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bistability in the case of stationary noise in one-dimensional CMEs.
5. Exact results in the large system size limit for the dynamics of the chemical master equation, a one dimensional chain of equations.
PubMed
Martirosyan, A; Saakian, David B
2011-08-01
We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the chemical master equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of the Van Kampen method from the HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ordinary differential equations (ODEs) to define the variance in a two-dimensional case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bistability in the case of stationary noise in one-dimensional CMEs. PMID:21928964
6. Quantum theory of rotational isomerism and Hill equation
SciTech Connect
Ugulava, A.; Toklikishvili, Z.; Chkhaidze, S.; Abramishvili, R.; Chotorlishvili, L.
2012-06-15
The process of rotational isomerism of linear triatomic molecules is described by the potential with two different-depth minima and one barrier between them. The corresponding quantum-mechanical equation is represented in the form that is a special case of the Hill equation. It is shown that the Hill-Schroedinger equation has a Klein's quadratic group symmetry which, in its turn, contains three invariant subgroups. The presence of these subgroups makes it possible to create a picture of energy spectrum which depends on a parameter and has many merging and branch points. The parameter-dependent energy spectrum of the Hill-Schroedinger equation, like Mathieu-characteristics, contains branch points from the left and from the right of the demarcation line. However, compared to the Mathieu-characteristics, in the Hill-Schroedinger equation spectrum the 'right' points are moved away even further for some distance that is the bigger, the bigger is the less deep well. The asymptotic wave functions of the Hill-Schroedinger equation for the energy values near the potential minimum contain two isolated sharp peaks indicating a possibility of the presence of two stable isomers. At high energy values near the potential maximum, the height of two peaks decreases, and between them there appear chaotic oscillations. This form of the wave functions corresponds to the process of isomerization.
7. Derivation of a time dependent Schrödinger equation as the quantum mechanical Landau-Lifshitz-Bloch equation.
PubMed
Wieser, R
2016-10-01
The derivation of the time dependent Schrödinger equation with transversal and longitudinal relaxation, as the quantum mechanical analog of the classical Landau-Lifshitz-Bloch equation, has been described. Starting from the classical Landau-Lifshitz-Bloch equation the transition to quantum mechanics has been performed and the corresponding von-Neumann equation deduced. In a second step the time Schrödinger equation has been derived. Analytical proofs and computer simulations show the correctness and applicability of the derived Schrödinger equation.
8. Derivation of a time dependent Schrödinger equation as the quantum mechanical Landau-Lifshitz-Bloch equation.
PubMed
Wieser, R
2016-10-01
The derivation of the time dependent Schrödinger equation with transversal and longitudinal relaxation, as the quantum mechanical analog of the classical Landau-Lifshitz-Bloch equation, has been described. Starting from the classical Landau-Lifshitz-Bloch equation the transition to quantum mechanics has been performed and the corresponding von-Neumann equation deduced. In a second step the time Schrödinger equation has been derived. Analytical proofs and computer simulations show the correctness and applicability of the derived Schrödinger equation. PMID:27494599
9. Derivation of a time dependent Schrödinger equation as the quantum mechanical Landau-Lifshitz-Bloch equation
Wieser, R.
2016-10-01
The derivation of the time dependent Schrödinger equation with transversal and longitudinal relaxation, as the quantum mechanical analog of the classical Landau-Lifshitz-Bloch equation, has been described. Starting from the classical Landau-Lifshitz-Bloch equation the transition to quantum mechanics has been performed and the corresponding von-Neumann equation deduced. In a second step the time Schrödinger equation has been derived. Analytical proofs and computer simulations show the correctness and applicability of the derived Schrödinger equation.
10. A novel quantum-mechanical interpretation of the Dirac equation
K-H Kiessling, M.; Tahvildar-Zadeh, A. S.
2016-04-01
A novel interpretation is given of Dirac’s ‘wave equation for the relativistic electron’ as a quantum-mechanical one-particle equation. In this interpretation the electron and the positron are merely the two different ‘topological spin’ states of a single more fundamental particle, not distinct particles in their own right. The new interpretation is backed up by the existence of such ‘bi-particle’ structures in general relativity, in particular the ring singularity present in any spacelike section of the spacetime singularity of the maximal-analytically extended, topologically non-trivial, electromagnetic Kerr-Newman (KN)spacetime in the zero-gravity limit (here, ‘zero-gravity’ means the limit G\\to 0, where G is Newton’s constant of universal gravitation). This novel interpretation resolves the dilemma that Dirac’s wave equation seems to be capable of describing both the electron and the positron in ‘external’ fields in many relevant situations, while the bi-spinorial wave function has only a single position variable in its argument, not two—as it should if it were a quantum-mechanical two-particle wave equation. A Dirac equation is formulated for such a ring-like bi-particle which interacts with a static point charge located elsewhere in the topologically non-trivial physical space associated with the moving ring particle, the motion being governed by a de Broglie-Bohm type law extracted from the Dirac equation. As an application, the pertinent general-relativistic zero-gravity hydrogen problem is studied in the usual Born-Oppenheimer approximation. Its spectral results suggest that the zero-G KN magnetic moment be identified with the so-called ‘anomalous magnetic moment of the physical electron,’ not with the Bohr magneton, so that the ring radius is only a tiny fraction of the electron’s reduced Compton wavelength.
11. Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom
SciTech Connect
Yang, C.-D. . E-mail: [email protected]
2006-12-15
This paper gives a thorough investigation on formulating and solving quantum problems by extended analytical mechanics that extends canonical variables to complex domain. With this complex extension, we show that quantum mechanics becomes a part of analytical mechanics and hence can be treated integrally with classical mechanics. Complex canonical variables are governed by Hamilton equations of motion, which can be derived naturally from Schroedinger equation. Using complex canonical variables, a formal proof of the quantization axiom p {sup {yields}} p = -ih{nabla}, which is the kernel in constructing quantum-mechanical systems, becomes a one-line corollary of Hamilton mechanics. The derivation of quantum operators from Hamilton mechanics is coordinate independent and thus allows us to derive quantum operators directly under any coordinate system without transforming back to Cartesian coordinates. Besides deriving quantum operators, we also show that the various prominent quantum effects, such as quantization, tunneling, atomic shell structure, Aharonov-Bohm effect, and spin, all have the root in Hamilton mechanics and can be described entirely by Hamilton equations of motion.
12. Hierarchical Equation of Motion Investigation of Decoherence and Relaxation Dynamics in Nonequilibrium Transport through Interacting Quantum Dots
Hartle, Rainer; Cohen, Guy; Reichman, David R.; Millis, Andrew J.
2014-03-01
A recently developed hierarchical quantum master equation approach is used to investigate nonequilibrium electron transport through an interacting double quantum dot system in the regime where the inter-dot coupling is weaker than the coupling to the electrodes. The corresponding eigenstates provide tunneling paths that may interfere constructively or destructively, depending on the energy of the tunneling electrons. Electron-electron interactions are shown to quench these interference effects in bias-voltage dependent ways, leading, in particular, to negative differential resistance, population inversion and an enhanced broadening of resonances in the respective transport characteristics. Relaxation times are found to be very long, and to be correlated with very slow dynamics of the inter-dot coherences (off diagonal density matrix elements). The ability of the hierarchical quantum master equation approach to access very long time scales is crucial for the study of this physics. This work is supported by the National Science Foundation (NSF DMR-1006282 and NSF CHE-1213247), the Yad Hanadiv-Rothschild Foundation (via a Rothschild Fellowship for GC) and the Alexander von Humboldt Foundation (via a Feodor Lynen fellowship for RH).
13. A master equation and moment approach for biochemical systems with creation-time-dependent bimolecular rate functions.
PubMed
2014-12-01
Noise and stochasticity are fundamental to biology and derive from the very nature of biochemical reactions where thermal motion of molecules translates into randomness in the sequence and timing of reactions. This randomness leads to cell-to-cell variability even in clonal populations. Stochastic biochemical networks have been traditionally modeled as continuous-time discrete-state Markov processes whose probability density functions evolve according to a chemical master equation (CME). In diffusion reaction systems on membranes, the Markov formalism, which assumes constant reaction propensities is not directly appropriate. This is because the instantaneous propensity for a diffusion reaction to occur depends on the creation times of the molecules involved. In this work, we develop a chemical master equation for systems of this type. While this new CME is computationally intractable, we make rational dimensional reductions to form an approximate equation, whose moments are also derived and are shown to yield efficient, accurate results. This new framework forms a more general approach than the Markov CME and expands upon the realm of possible stochastic biochemical systems that can be efficiently modeled.
14. A master equation and moment approach for biochemical systems with creation-time-dependent bimolecular rate functions
SciTech Connect
2014-12-07
Noise and stochasticity are fundamental to biology and derive from the very nature of biochemical reactions where thermal motion of molecules translates into randomness in the sequence and timing of reactions. This randomness leads to cell-to-cell variability even in clonal populations. Stochastic biochemical networks have been traditionally modeled as continuous-time discrete-state Markov processes whose probability density functions evolve according to a chemical master equation (CME). In diffusion reaction systems on membranes, the Markov formalism, which assumes constant reaction propensities is not directly appropriate. This is because the instantaneous propensity for a diffusion reaction to occur depends on the creation times of the molecules involved. In this work, we develop a chemical master equation for systems of this type. While this new CME is computationally intractable, we make rational dimensional reductions to form an approximate equation, whose moments are also derived and are shown to yield efficient, accurate results. This new framework forms a more general approach than the Markov CME and expands upon the realm of possible stochastic biochemical systems that can be efficiently modeled.
15. Non-linear corrections to the time-covariance function derived from a multi-state chemical master equation.
PubMed
Scott, M
2012-08-01
The time-covariance function captures the dynamics of biochemical fluctuations and contains important information about the underlying kinetic rate parameters. Intrinsic fluctuations in biochemical reaction networks are typically modelled using a master equation formalism. In general, the equation cannot be solved exactly and approximation methods are required. For small fluctuations close to equilibrium, a linearisation of the dynamics provides a very good description of the relaxation of the time-covariance function. As the number of molecules in the system decrease, deviations from the linear theory appear. Carrying out a systematic perturbation expansion of the master equation to capture these effects results in formidable algebra; however, symbolic mathematics packages considerably expedite the computation. The authors demonstrate that non-linear effects can reveal features of the underlying dynamics, such as reaction stoichiometry, not available in linearised theory. Furthermore, in models that exhibit noise-induced oscillations, non-linear corrections result in a shift in the base frequency along with the appearance of a secondary harmonic.
16. A master equation and moment approach for biochemical systems with creation-time-dependent bimolecular rate functions
2014-12-01
Noise and stochasticity are fundamental to biology and derive from the very nature of biochemical reactions where thermal motion of molecules translates into randomness in the sequence and timing of reactions. This randomness leads to cell-to-cell variability even in clonal populations. Stochastic biochemical networks have been traditionally modeled as continuous-time discrete-state Markov processes whose probability density functions evolve according to a chemical master equation (CME). In diffusion reaction systems on membranes, the Markov formalism, which assumes constant reaction propensities is not directly appropriate. This is because the instantaneous propensity for a diffusion reaction to occur depends on the creation times of the molecules involved. In this work, we develop a chemical master equation for systems of this type. While this new CME is computationally intractable, we make rational dimensional reductions to form an approximate equation, whose moments are also derived and are shown to yield efficient, accurate results. This new framework forms a more general approach than the Markov CME and expands upon the realm of possible stochastic biochemical systems that can be efficiently modeled.
17. Dissipation equation of motion approach to open quantum systems
Yan, YiJing; Jin, Jinshuang; Xu, Rui-Xue; Zheng, Xiao
2016-08-01
This paper presents a comprehensive account of the dissipaton-equation-of-motion (DEOM) theory for open quantum systems. This newly developed theory treats not only the quantum dissipative systems of primary interest, but also the hybrid environment dynamics that are also experimentally measurable. Despite the fact that DEOM recovers the celebrated hierarchical-equations-of-motion (HEOM) formalism, these two approaches have some fundamental differences. To show these differences, we also scrutinize the HEOM construction via its root at the influence functional path integral formalism. We conclude that many unique features of DEOM are beyond the reach of the HEOM framework. The new DEOM approach renders a statistical quasi-particle picture to account for the environment, which can be either bosonic or fermionic. The review covers the DEOM construction, the physical meanings of dynamical variables, the underlying theorems and dissipaton algebra, and recent numerical advancements for efficient DEOM evaluations of various problems. We also address the issue of high-order many-dissipaton truncations with respect to the invariance principle of quantum mechanics of Schrödinger versus Heisenberg prescriptions. DEOM serves as a universal tool for characterizing of stationary and dynamic properties of system-and-bath interferences, as highlighted with its real-time evaluation of both linear and nonlinear current noise spectra of nonequilibrium electronic transport.
18. Effective equations of cosmological models in (loop) quantum gravity
Simpson, David
This dissertation focuses on the properties of several differing models within quantum cosmology. Specifically, by using the method of effective equations, we explore: a linear discrete Schrodinger model, a non-linear discrete Schrodinger model, factor ordering ambiguities in the Hamiltonian constraint (with a focus on large-volume behavior), and the use of the electric vector potential as deparameterized time. In the linear and non-linear Schrodinger models, we arrive at a new possibility for studying inhomogeneous quantum cosmology (where the non-linearities are interpreted as non-local deviations from the spatial average) that allows for a variety of dynamics and raises a number of questions for future research. We then turn our focus to the general effects of factor ordering ambiguities and their possible role in large-volume collapse of a k = 0 isotropic quantum cosmology with a free, massless scalar field. With the additional inclusion of holonomy and inverse-triad corrections, the choice in factor ordering of the Hamiltonian constraint is quite relevant; however, with our assumptions, we do not see any significant departure from classical large-volume behavior. The final model discussed is formulated with the electric vector potential as the global internal time in a Wheeler-DeWitt setting sourced by radiation. While further analysis is required to make a definitive statement on the impact that the choice of deparameterization makes, we find that the specific form of quantum state can affect early-universe dynamics and even lead to new possibilities.
19. Quantum corrections to the Mukhanov-Sasaki equations
Castelló Gomar, Laura; Mena Marugán, Guillermo A.; Martín-Benito, Mercedes
2016-05-01
Recently, a lot of attention has been paid to the modifications of the power spectrum of primordial fluctuations caused by quantum cosmology effects. The origin of these modifications is corrections to the Mukhanov-Sasaki equations that govern the propagation of the primeval cosmological perturbations. The specific form of these corrections depends on a series of details of the quantization approach and of the prescription followed to implement it. Generally, the complexity of the theoretical quantum formulation is simplified in practice appealing to a semiclassical or effective approximation in order to perform concrete numerical computations. In this work, we introduce technical tools and design a procedure to deal with these quantum corrections beyond the most direct approximations employed so far in the literature. In particular, by introducing an interaction picture, we extract the quantum dynamics of the homogeneous geometry in absence of scalar field potential and inhomogeneities, dynamics that has been intensively studied and that can be integrated. The rest of our analysis focuses on the interaction evolution, putting forward methods to cope with it. The ultimate aim is to develop treatments that increase our ability to discriminate between the predictions of different quantization proposals for cosmological perturbations.
20. Electromagnetic wave equations for relativistically degenerate quantum magnetoplasmas.
PubMed
Masood, Waqas; Eliasson, Bengt; Shukla, Padma K
2010-06-01
A generalized set of nonlinear electromagnetic quantum hydrodynamic (QHD) equations is derived for a magnetized quantum plasma, including collisional, electron spin- 1/2, and relativistically degenerate electron pressure effects that are relevant for dense astrophysical systems, such as white dwarfs. For illustrative purposes, linear dispersion relations are derived for one-dimensional magnetoacoustic waves for a collisionless nonrelativistic degenerate gas in the presence of the electron spin- 1/2 contribution and for magnetoacoustic waves in a plasma containing relativistically degenerate electrons. It is found that both the spin and relativistic degeneracy at high densities tend to slow down the magnetoacoustic wave due to the Pauli paramagnetic effect and relativistic electron mass increase. The present study outlines the theoretical framework for the investigation of linear and nonlinear behaviors of electromagnetic waves in dense astrophysical systems. The results are applied to calculate the magnetoacoustic speeds for both the nonrelativistic and relativistic electron degeneracy cases typical for white dwarf stars. PMID:20866534
1. Electromagnetic wave equations for relativistically degenerate quantum magnetoplasmas.
PubMed
Masood, Waqas; Eliasson, Bengt; Shukla, Padma K
2010-06-01
A generalized set of nonlinear electromagnetic quantum hydrodynamic (QHD) equations is derived for a magnetized quantum plasma, including collisional, electron spin- 1/2, and relativistically degenerate electron pressure effects that are relevant for dense astrophysical systems, such as white dwarfs. For illustrative purposes, linear dispersion relations are derived for one-dimensional magnetoacoustic waves for a collisionless nonrelativistic degenerate gas in the presence of the electron spin- 1/2 contribution and for magnetoacoustic waves in a plasma containing relativistically degenerate electrons. It is found that both the spin and relativistic degeneracy at high densities tend to slow down the magnetoacoustic wave due to the Pauli paramagnetic effect and relativistic electron mass increase. The present study outlines the theoretical framework for the investigation of linear and nonlinear behaviors of electromagnetic waves in dense astrophysical systems. The results are applied to calculate the magnetoacoustic speeds for both the nonrelativistic and relativistic electron degeneracy cases typical for white dwarf stars.
2. Vortex equations governing the fractional quantum Hall effect
SciTech Connect
Medina, Luciano
2015-09-15
An existence theory is established for a coupled non-linear elliptic system, known as “vortex equations,” describing the fractional quantum Hall effect in 2-dimensional double-layered electron systems. Via variational methods, we prove the existence and uniqueness of multiple vortices over a doubly periodic domain and the full plane. In the doubly periodic situation, explicit sufficient and necessary conditions are obtained that relate the size of the domain and the vortex numbers. For the full plane case, existence is established for all finite-energy solutions and exponential decay estimates are proved. Quantization phenomena of the magnetic flux are found in both cases.
3. Supersymmetric quantum mechanics and Painlevé equations
SciTech Connect
Bermudez, David; Fernández C, David J.
2014-01-08
In these lecture notes we shall study first the supersymmetric quantum mechanics (SUSY QM), specially when applied to the harmonic and radial oscillators. In addition, we will define the polynomial Heisenberg algebras (PHA), and we will study the general systems ruled by them: for zero and first order we obtain the harmonic and radial oscillators, respectively; for second and third order the potential is determined by solutions to Painlevé IV (PIV) and Painlevé V (PV) equations. Taking advantage of this connection, later on we will find solutions to PIV and PV equations expressed in terms of confluent hypergeometric functions. Furthermore, we will classify them into several solution hierarchies, according to the specific special functions they are connected with.
4. Quantum cascade laser master-oscillator power-amplifier with 1.5 W output power at 300 K.
PubMed
Menzel, Stefan; Diehl, Laurent; Pflügl, Christian; Goyal, Anish; Wang, Christine; Sanchez, Antonio; Turner, George; Capasso, Federico
2011-08-15
We report quantum cascade laser (QCL) master-oscillator power-amplifiers (MOPAs) at 300 K reaching output power of 1.5 W for tapered devices and 0.9 W for untapered devices. The devices display single-longitudinal-mode emission at λ = 7.26 µm and single-transverse-mode emission at TM(00). The maximum amplification factor is 12 dB for the tapered devices. PMID:21934985
5. Jump-diffusion unravelling of a non-Markovian generalized Lindblad master equation
SciTech Connect
Barchielli, A.; Pellegrini, C.
2010-11-15
The ''correlated-projection technique'' has been successfully applied to derive a large class of highly non-Markovian dynamics, the so called non-Markovian generalized Lindblad-type equations or Lindblad rate equations. In this article, general unravelings are presented for these equations, described in terms of jump-diffusion stochastic differential equations for wave functions. We show also that the proposed unraveling can be interpreted in terms of measurements continuous in time but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and discuss possible effects on the heterodyne spectrum of a two-level system due to a structured thermal-like bath with memory.
6. Quantum-mechanical and quantum-electrodynamic equations for spectroscopic transitions
Yearchuck, Dmitry; Yerchak, Yauhen; Dovlatova, Alla
2010-09-01
Transition operator method is developed for description of the dynamics of spectroscopic transitions. Quantum-mechanical and quantum-electrodynamic difference-differential equations in general discrete space case and differential equations in continuum limit have been derived for spectroscopic transitions in the system of periodical ferroelectrically (ferromagnetically) ordered chains, interacting with external electromagnetic field. It was shown, that given equations can be represented in the form of Landau-Lifshitz equation in continuum limit and its generalization in discrete space case. Landau-Lifshitz equation was represented in Lorentz invariant form by Hilbert space definition over the ring of quaternions. It has been shown, that spin vector can be considered to be quaternion vector of the state of the system studied. From comparison with pure optical experiments the value of spin S=1/2 for spin-Peierls solitons in carbon chains has been found and it has also been established, that given quasiparticles are dually charged. The ratio of magnetic to electric (imaginary to real) components of electromagnetic dual (complex) charge is evaluated for given centers to be ge ≈(1.1-1.3)10 in correspondence with Dirac theory of charge quantization. The given results seem to be obtained for the first time.
7. Riccati equation for simulation of leads in quantum transport
Bravi, M.; Farchioni, R.; Grosso, G.; Pastori Parravicini, G.
2014-10-01
We present a theoretical procedure with numerical demonstration of a workable and efficient method to evaluate the surface Green's function of semi-infinite leads connected to a device. Such a problem always occurs in quantum transport calculations but also in the study of surfaces and heterojunctions. We show here that these semi-infinite leads can be properly described by real-energy Green's functions obtained analytically by a smart solution of the Riccati matrix equation. The performance of our method is demonstrated in the case of a multichain two-dimensional electron-gas system, composed of a central ribbon connected to two semi-infinite leads, pierced by two opposite magnetic fields.
8. Solving the chemical master equation by a fast adaptive finite state projection based on the stochastic simulation algorithm.
PubMed
Sidje, R B; Vo, H D
2015-11-01
The mathematical framework of the chemical master equation (CME) uses a Markov chain to model the biochemical reactions that are taking place within a biological cell. Computing the transient probability distribution of this Markov chain allows us to track the composition of molecules inside the cell over time, with important practical applications in a number of areas such as molecular biology or medicine. However the CME is typically difficult to solve, since the state space involved can be very large or even countably infinite. We present a novel way of using the stochastic simulation algorithm (SSA) to reduce the size of the finite state projection (FSP) method. Numerical experiments that demonstrate the effectiveness of the reduction are included.
9. Optical manipulation of a multilevel nuclear spin in ZnO: Master equation and experiment
Buß, J. H.; Rudolph, J.; Wassner, T. A.; Eickhoff, M.; Hägele, D.
2016-04-01
We demonstrate the dynamics and optical control of a large quantum mechanical solid state spin system consisting of a donor electron spin strongly coupled to the 9/2 nuclear spin of 115In in the semiconductor ZnO. Comparison of electron spin dynamics observed by time-resolved pump-probe spectroscopy with density matrix theory reveals nuclear spin pumping via optically oriented electron spins, coherent spin-spin interaction, and quantization effects of the ten nuclear spin levels. Modulation of the optical electron spin orientation at frequencies above 1 MHz gives evidence for fast optical manipulation of the nuclear spin state.
10. Stratified quantization approach to dissipative quantum systems: Derivation of the Hamiltonian and kinetic equations for reduced density matrices
SciTech Connect
Richardson, W.H. . E-mail: [email protected]
2006-06-15
A technique for describing dissipative quantum systems that utilizes a fundamental Hamiltonian, which is composed of intrinsic operators of the system, is presented. The specific system considered is a capacitor (or free particle) that is coupled to a resistor (or dissipative medium). The microscopic mechanisms that lead to dissipation are represented by the standard Hamiltonian. Now dissipation is really a collective phenomenon of entities that comprise a macroscopic or mesoscopic object. Hence operators that describe the collective features of the dissipative medium are utilized to construct the Hamiltonian that represents the coupled resistor and capacitor. Quantization of the spatial gauge function is introduced. The magnetic energy part of the coupled Hamiltonian is written in terms of that quantized gauge function and the current density of the dissipative medium. A detailed derivation of the kinetic equation that represents the capacitor or free particle is presented. The partial spectral densities and functions related to spectral densities, which enter the kinetic equations as coefficients of commutators, are evaluated. Explicit expressions for the nonMarkoffian contribution in terms of products of spectral densities and related functions are given. The influence of all two-time correlation functions are considered. Also stated is a remainder term that is a product of partial spectral densities and which is due to higher order terms in the correlation density matrix. The Markoffian part of the kinetic equation is compared with the Master equation that is obtained using the standard generator in the axiomatic approach. A detailed derivation of the Master equation that represents the dissipative medium is also presented. The dynamical equation for the resistor depends on the spatial wavevector, and the influence of the free particle on the diagonal elements (in wavevector space) is stated.
11. Comparison of Control Approaches in Genetic Regulatory Networks by Using Stochastic Master Equation Models, Probabilistic Boolean Network Models and Differential Equation Models and Estimated Error Analyzes
2011-03-01
Central dogma of molecular biology states that information cannot be transferred back from protein to either protein or nucleic acid''. However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of cell level data and probabilistic - nonlinear nature of interactions. Several models widely used to analyze and simulate these types of nonlinear interactions. Stochastic Master Equation (SME) models give probabilistic nature of the interactions in a detailed manner, with a high calculation cost. On the other hand Probabilistic Boolean Network (PBN) models give a coarse scale picture of the stochastic processes, with a less calculation cost. Differential Equation (DE) models give the time evolution of mean values of processes in a highly cost effective way. The understanding of the relations between the predictions of these models is important to understand the reliability of the simulations of genetic regulatory networks. In this work the success of the mapping between SME, PBN and DE models is analyzed and the accuracy and affectivity of the control policies generated by using PBN and DE models is compared.
12. Dielectric function and plasmons in graphene: A self-consistent-field calculation within a Markovian master equation formalism
Karimi, F.; Davoody, A. H.; Knezevic, I.
2016-05-01
We introduce a method for calculating the dielectric function of nanostructures with an arbitrary band dispersion and Bloch wave functions. The linear response of a dissipative electronic system to an external electromagnetic field is calculated by a self-consistent-field approach within a Markovian master-equation formalism (SCF-MMEF) coupled with full-wave electromagnetic equations. The SCF-MMEF accurately accounts for several concurrent scattering mechanisms. The method captures interband electron-hole-pair generation, as well as the interband and intraband electron scattering with phonons and impurities. We employ the SCF-MMEF to calculate the dielectric function, complex conductivity, and loss function for supported graphene. From the loss-function maximum, we obtain plasmon dispersion and propagation length for different substrate types [nonpolar diamondlike carbon (DLC) and polar SiO2 and hBN], impurity densities, carrier densities, and temperatures. Plasmons on the two polar substrates are suppressed below the highest surface phonon energy, while the spectrum is broad on the nonpolar DLC. Plasmon propagation lengths are comparable on polar and nonpolar substrates and are on the order of tens of nanometers, considerably shorter than previously reported. They improve with fewer impurities, at lower temperatures, and at higher carrier densities.
13. Modern integral equation techniques for quantum reactive scattering theory
SciTech Connect
Auerbach, S.M.
1993-11-01
Rigorous calculations of cross sections and rate constants for elementary gas phase chemical reactions are performed for comparison with experiment, to ensure that our picture of the chemical reaction is complete. We focus on the H/D+H{sub 2} {yields} H{sub 2}/DH + H reaction, and use the time independent integral equation technique in quantum reactive scattering theory. We examine the sensitivity of H+H{sub 2} state resolved integral cross sections {sigma}{sub v{prime}j{prime},vj}(E) for the transitions (v = 0,j = 0) to (v{prime} = 1,j{prime} = 1,3), to the difference between the Liu-Siegbahn-Truhlar-Horowitz (LSTH) and double many body expansion (DMBE) ab initio potential energy surfaces (PES). This sensitivity analysis is performed to determine the origin of a large discrepancy between experimental cross sections with sharply peaked energy dependence and theoretical ones with smooth energy dependence. We find that the LSTH and DMBE PESs give virtually identical cross sections, which lends credence to the theoretical energy dependence.
14. Reply to the comment on "Quantum trajectory tests of radical-pair quantum dynamics in CIDNP measurements of photosynthetic reaction centers" by G. Jeschke
Kominis, I. K.
2016-03-01
We recently unraveled a major inconsistency in the traditional description of radical-pair quantum dynamics by studying single-molecule quantum trajectories and comparing their prediction with Haberkorn's master equation. A comment by Jeschke claimed that the inconsistency arises because we did not properly include quantum state projections in the traditional approach. We here show that Jeschke stipulates quantum trajectories involving unphysical quantum states with negative populations. Moreover, the author's Monte Carlo simulation and its agreement with Haberkorn's master equation is a demonstration of an algebraic tautology, establishing the consistency of an unphysical master equation with circularly defined unphysical trajectories.
15. A Hybrid of the Chemical Master Equation and the Gillespie Algorithm for Efficient Stochastic Simulations of Sub-Networks.
PubMed
Albert, Jaroslav
2016-01-01
Modeling stochastic behavior of chemical reaction networks is an important endeavor in many aspects of chemistry and systems biology. The chemical master equation (CME) and the Gillespie algorithm (GA) are the two most fundamental approaches to such modeling; however, each of them has its own limitations: the GA may require long computing times, while the CME may demand unrealistic memory storage capacity. We propose a method that combines the CME and the GA that allows one to simulate stochastically a part of a reaction network. First, a reaction network is divided into two parts. The first part is simulated via the GA, while the solution of the CME for the second part is fed into the GA in order to update its propensities. The advantage of this method is that it avoids the need to solve the CME or stochastically simulate the entire network, which makes it highly efficient. One of its drawbacks, however, is that most of the information about the second part of the network is lost in the process. Therefore, this method is most useful when only partial information about a reaction network is needed. We tested this method against the GA on two systems of interest in biology--the gene switch and the Griffith model of a genetic oscillator--and have shown it to be highly accurate. Comparing this method to four different stochastic algorithms revealed it to be at least an order of magnitude faster than the fastest among them.
16. A Hybrid of the Chemical Master Equation and the Gillespie Algorithm for Efficient Stochastic Simulations of Sub-Networks
PubMed Central
Albert, Jaroslav
2016-01-01
Modeling stochastic behavior of chemical reaction networks is an important endeavor in many aspects of chemistry and systems biology. The chemical master equation (CME) and the Gillespie algorithm (GA) are the two most fundamental approaches to such modeling; however, each of them has its own limitations: the GA may require long computing times, while the CME may demand unrealistic memory storage capacity. We propose a method that combines the CME and the GA that allows one to simulate stochastically a part of a reaction network. First, a reaction network is divided into two parts. The first part is simulated via the GA, while the solution of the CME for the second part is fed into the GA in order to update its propensities. The advantage of this method is that it avoids the need to solve the CME or stochastically simulate the entire network, which makes it highly efficient. One of its drawbacks, however, is that most of the information about the second part of the network is lost in the process. Therefore, this method is most useful when only partial information about a reaction network is needed. We tested this method against the GA on two systems of interest in biology - the gene switch and the Griffith model of a genetic oscillator—and have shown it to be highly accurate. Comparing this method to four different stochastic algorithms revealed it to be at least an order of magnitude faster than the fastest among them. PMID:26930199
17. The Born Rule and Time-Reversal Symmetry of Quantum Equations of Motion
Ilyin, Aleksey V.
2016-07-01
It was repeatedly underlined in literature that quantum mechanics cannot be considered a closed theory if the Born Rule is postulated rather than derived from the first principles. In this work the Born Rule is derived from the time-reversal symmetry of quantum equations of motion. The derivation is based on a simple functional equation that takes into account properties of probability, as well as the linearity and time-reversal symmetry of quantum equations of motion. The derivation presented in this work also allows to determine certain limits to applicability of the Born Rule.
18. Composite bound states and broken U(1) symmetry in the chemical-master-equation derivation of the Gray-Scott model.
PubMed
Cooper, Fred; Ghoshal, Gourab; Pérez-Mercader, Juan
2013-10-01
We give a first principles derivation of the stochastic partial differential equations that describe the chemical reactions of the Gray-Scott model (GS): U+2V →[λ]3V and V → [μ]P, U → [ν]Q, with a constant feed rate for U. We find that the conservation of probability ensured by the chemical master equation leads to a modification of the usual differential equations for the GS model, which now involves two composite fields and also intrinsic noise terms. One of the composites is ψ(1) = φ(v)(2), where {φ(v)}(η) =v is the concentration of the species V and the averaging is over the internal noise η(u,v,ψ(1)). The second composite field is the product of three fields χ = λφ(u)φ(v)(2) and requires a noise source to ensure probability conservation. A third composite ψ(2) = φ(u)φ(v) can also be identified from the noise-induced reactions. The Hamiltonian that governs the time evolution of the many-body wave function, associated with the master equation, has a broken U(1) symmetry related to particle number conservation. By expanding around the (broken symmetry) zero-energy solution of the Hamiltonian (by performing a Doi shift) one obtains from our path integral formulation the usual reaction diffusion equation, at the classical level. The Langevin equations that are derived from the chemical master equation have multiplicative noise sources for the density fields φ(u), φ(v),χ that induce higher-order processes such as n → n scattering for n>3. The amplitude of the noise acting on φ(v) is itself stochastic in nature.
19. Existence of mild solution of impulsive quantum stochastic differential equation with nonlocal conditions
Bishop, S. A.; Ayoola, E. O.; Oghonyon, G. J.
2016-08-01
New results on existence and uniqueness of solution of impulsive quantum stochastic differential equation with nonlocal conditions are established. The nonlocal conditions are completely continuous. The methods applied here are simple extension of the methods applied in the classical case to this noncummutative quantum setting.
20. Notes on the Riccati operator equation in open quantum systems
Gardas, Bartłomiej; Puchała, Zbigniew
2012-01-01
A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.
1. Quantum cascade laser in a master oscillator power amplifier configuration with Watt-level optical output power.
PubMed
Hinkov, Borislav; Beck, Mattias; Gini, Emilio; Faist, Jérôme
2013-08-12
We present the design and realization of short-wavelength (λ = 4.53 μm) and buried-heterostructure quantum cascade lasers in a master oscillator power amplifier configuration. Watt-level, singlemode peak optical output power is demonstrated for typical non-tapered 4 μm wide and 5.25 mm long devices. Farfield measurements prove a symmetric, single transverse-mode emission in TM(00)-mode with typical divergences of 25° and 27° in and perpendicular to growth direction, respectively. We demonstrate singlemode tuning over a range of 7.9 cm(-1) for temperatures between 263K and 313K and also singlemode emission for different driving currents. The side mode suppression ratio is measured to be higher than 20 dB. PMID:23938833
2. The role of non-equilibrium fluxes in the relaxation processes of the linear chemical master equation
SciTech Connect
Oliveira, Luciana Renata de; Bazzani, Armando; Giampieri, Enrico; Castellani, Gastone C.
2014-08-14
We propose a non-equilibrium thermodynamical description in terms of the Chemical Master Equation (CME) to characterize the dynamics of a chemical cycle chain reaction among m different species. These systems can be closed or open for energy and molecules exchange with the environment, which determines how they relax to the stationary state. Closed systems reach an equilibrium state (characterized by the detailed balance condition (D.B.)), while open systems will reach a non-equilibrium steady state (NESS). The principal difference between D.B. and NESS is due to the presence of chemical fluxes. In the D.B. condition the fluxes are absent while for the NESS case, the chemical fluxes are necessary for the state maintaining. All the biological systems are characterized by their “far from equilibrium behavior,” hence the NESS is a good candidate for a realistic description of the dynamical and thermodynamical properties of living organisms. In this work we consider a CME written in terms of a discrete Kolmogorov forward equation, which lead us to write explicitly the non-equilibrium chemical fluxes. For systems in NESS, we show that there is a non-conservative “external vector field” whose is linearly proportional to the chemical fluxes. We also demonstrate that the modulation of these external fields does not change their stationary distributions, which ensure us to study the same system and outline the differences in the system's behavior when it switches from the D.B. regime to NESS. We were interested to see how the non-equilibrium fluxes influence the relaxation process during the reaching of the stationary distribution. By performing analytical and numerical analysis, our central result is that the presence of the non-equilibrium chemical fluxes reduces the characteristic relaxation time with respect to the D.B. condition. Within a biochemical and biological perspective, this result can be related to the “plasticity property” of biological systems
3. The role of non-equilibrium fluxes in the relaxation processes of the linear chemical master equation.
PubMed
de Oliveira, Luciana Renata; Bazzani, Armando; Giampieri, Enrico; Castellani, Gastone C
2014-08-14
We propose a non-equilibrium thermodynamical description in terms of the Chemical Master Equation (CME) to characterize the dynamics of a chemical cycle chain reaction among m different species. These systems can be closed or open for energy and molecules exchange with the environment, which determines how they relax to the stationary state. Closed systems reach an equilibrium state (characterized by the detailed balance condition (D.B.)), while open systems will reach a non-equilibrium steady state (NESS). The principal difference between D.B. and NESS is due to the presence of chemical fluxes. In the D.B. condition the fluxes are absent while for the NESS case, the chemical fluxes are necessary for the state maintaining. All the biological systems are characterized by their "far from equilibrium behavior," hence the NESS is a good candidate for a realistic description of the dynamical and thermodynamical properties of living organisms. In this work we consider a CME written in terms of a discrete Kolmogorov forward equation, which lead us to write explicitly the non-equilibrium chemical fluxes. For systems in NESS, we show that there is a non-conservative "external vector field" whose is linearly proportional to the chemical fluxes. We also demonstrate that the modulation of these external fields does not change their stationary distributions, which ensure us to study the same system and outline the differences in the system's behavior when it switches from the D.B. regime to NESS. We were interested to see how the non-equilibrium fluxes influence the relaxation process during the reaching of the stationary distribution. By performing analytical and numerical analysis, our central result is that the presence of the non-equilibrium chemical fluxes reduces the characteristic relaxation time with respect to the D.B. condition. Within a biochemical and biological perspective, this result can be related to the "plasticity property" of biological systems and to their
4. The role of non-equilibrium fluxes in the relaxation processes of the linear chemical master equation
de Oliveira, Luciana Renata; Bazzani, Armando; Giampieri, Enrico; Castellani, Gastone C.
2014-08-01
We propose a non-equilibrium thermodynamical description in terms of the Chemical Master Equation (CME) to characterize the dynamics of a chemical cycle chain reaction among m different species. These systems can be closed or open for energy and molecules exchange with the environment, which determines how they relax to the stationary state. Closed systems reach an equilibrium state (characterized by the detailed balance condition (D.B.)), while open systems will reach a non-equilibrium steady state (NESS). The principal difference between D.B. and NESS is due to the presence of chemical fluxes. In the D.B. condition the fluxes are absent while for the NESS case, the chemical fluxes are necessary for the state maintaining. All the biological systems are characterized by their "far from equilibrium behavior," hence the NESS is a good candidate for a realistic description of the dynamical and thermodynamical properties of living organisms. In this work we consider a CME written in terms of a discrete Kolmogorov forward equation, which lead us to write explicitly the non-equilibrium chemical fluxes. For systems in NESS, we show that there is a non-conservative "external vector field" whose is linearly proportional to the chemical fluxes. We also demonstrate that the modulation of these external fields does not change their stationary distributions, which ensure us to study the same system and outline the differences in the system's behavior when it switches from the D.B. regime to NESS. We were interested to see how the non-equilibrium fluxes influence the relaxation process during the reaching of the stationary distribution. By performing analytical and numerical analysis, our central result is that the presence of the non-equilibrium chemical fluxes reduces the characteristic relaxation time with respect to the D.B. condition. Within a biochemical and biological perspective, this result can be related to the "plasticity property" of biological systems and to their
5. Nonlinear Riccati equations as a unifying link between linear quantum mechanics and other fields of physics
Schuch, Dieter
2014-04-01
Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.
6. Linear-noise approximation and the chemical master equation agree up to second-order moments for a class of chemical systems
Grima, Ramon
2015-10-01
It is well known that the linear-noise approximation (LNA) agrees with the chemical master equation, up to second-order moments, for chemical systems composed of zero and first-order reactions. Here we show that this is also a property of the LNA for a subset of chemical systems with second-order reactions. This agreement is independent of the number of interacting molecules.
7. Linear-noise approximation and the chemical master equation agree up to second-order moments for a class of chemical systems.
PubMed
Grima, Ramon
2015-10-01
It is well known that the linear-noise approximation (LNA) agrees with the chemical master equation, up to second-order moments, for chemical systems composed of zero and first-order reactions. Here we show that this is also a property of the LNA for a subset of chemical systems with second-order reactions. This agreement is independent of the number of interacting molecules.
8. An Analytical Framework for Studying Small-Number Effects in Catalytic Reaction Networks: A Probability Generating Function Approach to Chemical Master Equations
PubMed Central
Nakagawa, Masaki; Togashi, Yuichi
2016-01-01
Cell activities primarily depend on chemical reactions, especially those mediated by enzymes, and this has led to these activities being modeled as catalytic reaction networks. Although deterministic ordinary differential equations of concentrations (rate equations) have been widely used for modeling purposes in the field of systems biology, it has been pointed out that these catalytic reaction networks may behave in a way that is qualitatively different from such deterministic representation when the number of molecules for certain chemical species in the system is small. Apart from this, representing these phenomena by simple binary (on/off) systems that omit the quantities would also not be feasible. As recent experiments have revealed the existence of rare chemical species in cells, the importance of being able to model potential small-number phenomena is being recognized. However, most preceding studies were based on numerical simulations, and theoretical frameworks to analyze these phenomena have not been sufficiently developed. Motivated by the small-number issue, this work aimed to develop an analytical framework for the chemical master equation describing the distributional behavior of catalytic reaction networks. For simplicity, we considered networks consisting of two-body catalytic reactions. We used the probability generating function method to obtain the steady-state solutions of the chemical master equation without specifying the parameters. We obtained the time evolution equations of the first- and second-order moments of concentrations, and the steady-state analytical solution of the chemical master equation under certain conditions. These results led to the rank conservation law, the connecting state to the winner-takes-all state, and analysis of 2-molecules M-species systems. A possible interpretation of the theoretical conclusion for actual biochemical pathways is also discussed. PMID:27047384
9. An Analytical Framework for Studying Small-Number Effects in Catalytic Reaction Networks: A Probability Generating Function Approach to Chemical Master Equations.
PubMed
Nakagawa, Masaki; Togashi, Yuichi
2016-01-01
Cell activities primarily depend on chemical reactions, especially those mediated by enzymes, and this has led to these activities being modeled as catalytic reaction networks. Although deterministic ordinary differential equations of concentrations (rate equations) have been widely used for modeling purposes in the field of systems biology, it has been pointed out that these catalytic reaction networks may behave in a way that is qualitatively different from such deterministic representation when the number of molecules for certain chemical species in the system is small. Apart from this, representing these phenomena by simple binary (on/off) systems that omit the quantities would also not be feasible. As recent experiments have revealed the existence of rare chemical species in cells, the importance of being able to model potential small-number phenomena is being recognized. However, most preceding studies were based on numerical simulations, and theoretical frameworks to analyze these phenomena have not been sufficiently developed. Motivated by the small-number issue, this work aimed to develop an analytical framework for the chemical master equation describing the distributional behavior of catalytic reaction networks. For simplicity, we considered networks consisting of two-body catalytic reactions. We used the probability generating function method to obtain the steady-state solutions of the chemical master equation without specifying the parameters. We obtained the time evolution equations of the first- and second-order moments of concentrations, and the steady-state analytical solution of the chemical master equation under certain conditions. These results led to the rank conservation law, the connecting state to the winner-takes-all state, and analysis of 2-molecules M-species systems. A possible interpretation of the theoretical conclusion for actual biochemical pathways is also discussed.
10. Complex quantum Hamilton-Jacobi equation with Bohmian trajectories: Application to the photodissociation dynamics of NOCl
SciTech Connect
Chou, Chia-Chun
2014-03-14
The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics.
11. Complex quantum Hamilton-Jacobi equation with Bohmian trajectories: application to the photodissociation dynamics of NOCl.
PubMed
Chou, Chia-Chun
2014-03-14
The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics. PMID:24628169
12. Simulation of the Burgers equation by NMR quantum-information processing
SciTech Connect
Chen Zhiying; Cory, David G.; Yepez, Jeffrey
2006-10-15
We report on the implementation of Burgers equation as a type-II quantum computation on a NMR quantum-information processor. Since the flow field evolving under the Burgers equation develops sharp features over time, this is a better test of liquid-state NMR implementations of type-II quantum computers than the previous examples using the diffusion equation. In particular, we show that Fourier approximations used in the encoding step are not the dominant error. Small systematic errors in the collision operator accumulate and swamp all other errors. We propose, and demonstrate, that the accumulation of this error can be avoided to a large extent by replacing the single collision operator with a set of operators with random errors and similar fidelities. Experiments have been implemented on 16 two-qubit sites for eight successive time steps for the Burgers equation.
13. Reply to "Comment on 'Fractional quantum mechanics' and 'Fractional Schrödinger equation' ".
PubMed
2016-06-01
The fractional uncertainty relation is a mathematical formulation of Heisenberg's uncertainty principle in the framework of fractional quantum mechanics. Two mistaken statements presented in the Comment have been revealed. The origin of each mistaken statement has been clarified and corrected statements have been made. A map between standard quantum mechanics and fractional quantum mechanics has been presented to emphasize the features of fractional quantum mechanics and to avoid misinterpretations of the fractional uncertainty relation. It has been shown that the fractional probability current equation is correct in the area of its applicability. Further studies have to be done to find meaningful quantum physics problems with involvement of the fractional probability current density vector and the extra term emerging in the framework of fractional quantum mechanics.
14. Reply to "Comment on 'Fractional quantum mechanics' and 'Fractional Schrödinger equation' ".
PubMed
2016-06-01
The fractional uncertainty relation is a mathematical formulation of Heisenberg's uncertainty principle in the framework of fractional quantum mechanics. Two mistaken statements presented in the Comment have been revealed. The origin of each mistaken statement has been clarified and corrected statements have been made. A map between standard quantum mechanics and fractional quantum mechanics has been presented to emphasize the features of fractional quantum mechanics and to avoid misinterpretations of the fractional uncertainty relation. It has been shown that the fractional probability current equation is correct in the area of its applicability. Further studies have to be done to find meaningful quantum physics problems with involvement of the fractional probability current density vector and the extra term emerging in the framework of fractional quantum mechanics. PMID:27415398
15. Reply to "Comment on Fractional quantum mechanics' and Fractional Schrödinger equation' "
2016-06-01
The fractional uncertainty relation is a mathematical formulation of Heisenberg's uncertainty principle in the framework of fractional quantum mechanics. Two mistaken statements presented in the Comment have been revealed. The origin of each mistaken statement has been clarified and corrected statements have been made. A map between standard quantum mechanics and fractional quantum mechanics has been presented to emphasize the features of fractional quantum mechanics and to avoid misinterpretations of the fractional uncertainty relation. It has been shown that the fractional probability current equation is correct in the area of its applicability. Further studies have to be done to find meaningful quantum physics problems with involvement of the fractional probability current density vector and the extra term emerging in the framework of fractional quantum mechanics.
16. Quantum harmonic oscillator in a thermal bath
NASA Technical Reports Server (NTRS)
Zhang, Yuhong
1993-01-01
The influence functional path-integral treatment of quantum Brownian motion is briefly reviewed. A newly derived exact master equation of a quantum harmonic oscillator coupled to a general environment at arbitrary temperature is discussed. It is applied to the problem of loss of quantum coherence.
17. Nucleated polymerization with secondary pathways II. Determination of self-consistent solutions to growth processes described by non-linear master equations
PubMed Central
Cohen, Samuel I. A.; Vendruscolo, Michele; Dobson, Christopher M.; Knowles, Tuomas P. J.
2016-01-01
Nucleated polymerisation processes are involved in many growth phenomena in nature, including the formation of cytoskeletal filaments and the assembly of sickle hemoglobin and amyloid fibrils. Closed form rate equations have, however, been challenging to derive for these growth phenomena in cases where secondary nucleation processes are active, a difficulty exemplified by the highly non-linear nature of the equation systems that describe monomer dependent secondary nucleation pathways. We explore here the use of fixed point analysis to provide self-consistent solutions to such growth problems. We present iterative solutions and discuss their convergence behaviour. We establish a range of closed form results for linear growth processes, including the scaling behaviours of the maximum growth rate and of the reaction end-point. We further show that a self-consistent approach applied to the master equation of filamentous growth allows the determination of the evolution of the shape of the length distribution including the mean, the standard deviation and the mode. Our results demonstrate the power of fixed-point approaches in finding closed form self-consistent solutions to growth problems characterised by highly non-linear master equations. PMID:21842955
18. Integrating the quantum Hamilton-Jacobi equations by wavefront expansion and phase space analysis
Bittner, Eric R.; Wyatt, Robert E.
2000-11-01
In this paper we report upon our computational methodology for numerically integrating the quantum Hamilton-Jacobi equations using hydrodynamic trajectories. Our method builds upon the moving least squares method developed by Lopreore and Wyatt [Phys. Rev. Lett. 82, 5190 (1999)] in which Lagrangian fluid elements representing probability volume elements of the wave function evolve under Newtonian equations of motion which include a nonlocal quantum force. This quantum force, which depends upon the third derivative of the quantum density, ρ, can vary rapidly in x and become singular in the presence of nodal points. Here, we present a new approach for performing quantum trajectory calculations which does not involve calculating the quantum force directly, but uses the wavefront to calculate the velocity field using mv=∇S, where S/ℏ is the argument of the wave function ψ. Additional numerical stability is gained by performing local gauge transformations to remove oscillatory components of the wave function. Finally, we use a dynamical Rayleigh-Ritz approach to derive ancillary equations-of-motion for the spatial derivatives of ρ, S, and v. The methodologies described herein dramatically improve the long time stability and accuracy of the quantum trajectory approach even in the presence of nodes. The method is applied to both barrier crossing and tunneling systems. We also compare our results to semiclassical based descriptions of barrier tunneling.
19. Generalized guidance equation for peaked quantum solitons and effective gravity
Durt, Thomas
2016-04-01
Bouncing oil droplets have been shown to follow de Broglie-Bohm–like trajectories and at the same time they exhibit attractive and repulsive pseudo-gravitation. We propose a model aimed at rendering account of these phenomenological observations. It inspires, in a more speculative approach, a toy model for quantum gravity.
20. Quantum structure emerging from the transformation design of the Dirac equation
SciTech Connect
Lin, De-Hone
2014-06-15
It is shown that a quantum structure can be created by a set of chosen constraint conditions that emerge from the transformation design of the Dirac equation in general relativity. As an explanation, the constraints that cause novel bound states with the quantization rule of a 2D Coulomb system are presented. The discussion in this paper provides a systematic way to look for constraints that generate a required quantization rule. -- Highlights: •We perform the transformation design of space and time for spin-1/2 matter waves. •A quantum rule could naturally emerge as constraints imposed on the Dirac equation itself. •New fermion states share the quantum spectrum of a 2D Coulomb system. •Transformation design uncovers a new exact solvable model. •A quantum spectrum can be created by a geometric structure.
1. A two-qubit photonic quantum processor and its application to solving systems of linear equations.
PubMed
Barz, Stefanie; Kassal, Ivan; Ringbauer, Martin; Lipp, Yannick Ole; Dakić, Borivoje; Aspuru-Guzik, Alán; Walther, Philip
2014-01-01
Large-scale quantum computers will require the ability to apply long sequences of entangling gates to many qubits. In a photonic architecture, where single-qubit gates can be performed easily and precisely, the application of consecutive two-qubit entangling gates has been a significant obstacle. Here, we demonstrate a two-qubit photonic quantum processor that implements two consecutive CNOT gates on the same pair of polarisation-encoded qubits. To demonstrate the flexibility of our system, we implement various instances of the quantum algorithm for solving of systems of linear equations.
2. A two-qubit photonic quantum processor and its application to solving systems of linear equations.
PubMed
Barz, Stefanie; Kassal, Ivan; Ringbauer, Martin; Lipp, Yannick Ole; Dakić, Borivoje; Aspuru-Guzik, Alán; Walther, Philip
2014-01-01
Large-scale quantum computers will require the ability to apply long sequences of entangling gates to many qubits. In a photonic architecture, where single-qubit gates can be performed easily and precisely, the application of consecutive two-qubit entangling gates has been a significant obstacle. Here, we demonstrate a two-qubit photonic quantum processor that implements two consecutive CNOT gates on the same pair of polarisation-encoded qubits. To demonstrate the flexibility of our system, we implement various instances of the quantum algorithm for solving of systems of linear equations. PMID:25135432
3. Quantum-mechanical derivation of the Davydov equations for multi-quanta states
SciTech Connect
Kerr, W.C.; Lomdahl, P.S.
1989-01-01
Our purpose here is to present a derivation of the Davydov equations which employs only quantum-mechanical techniques. The derivation here is more general than our previous treatment of this problem because we use an Ansatz which has present several quanta of the high frequency oscillator system rather than just one quantum. Since some steps of the calculation are the same as those in our paper which treats the single quantum case, reference will be made to that paper for some of the those details. 9 refs.
4. Time-evolution of quantum systems via a complex nonlinear Riccati equation. II. Dissipative systems
Cruz, Hans; Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar
2016-10-01
In our former contribution (Cruz et al., 2015), we have shown the sensitivity to the choice of initial conditions in the evolution of Gaussian wave packets via the nonlinear Riccati equation. The formalism developed in the previous work is extended to effective approaches for the description of dissipative quantum systems. By means of simple examples we show the effects of the environment on the quantum uncertainties, correlation function, quantum energy contribution and tunnelling currents. We prove that the environmental parameter γ is strongly related with the sensitivity to the choice of initial conditions.
5. Analysis of the forward-backward trajectory solution for the mixed quantum-classical Liouville equation.
PubMed
Hsieh, Chang-Yu; Kapral, Raymond
2013-04-01
Mixed quantum-classical methods provide powerful algorithms for the simulation of quantum processes in large and complex systems. The forward-backward trajectory solution of the mixed quantum-classical Liouville equation in the mapping basis [C.-Y. Hsieh and R. Kapral, J. Chem. Phys. 137, 22A507 (2012)] is one such scheme. It simulates the dynamics via the propagation of forward and backward trajectories of quantum coherent state variables, and the propagation of bath trajectories on a mean-field potential determined jointly by the forward and backward trajectories. An analysis of the properties of this solution, numerical tests of its validity and an investigation of its utility for the study of nonadiabtic quantum processes are given. In addition, we present an extension of this approximate solution that allows one to systematically improve the results. This extension, termed the jump forward-backward trajectory solution, is analyzed and tested in detail and its various implementations are discussed. PMID:23574211
6. Lessons from the quantum control landscape: Robust optimal control of quantum systems and optimal control of nonlinear Schrodinger equations
Hocker, David Lance
The control of quantum systems occurs across a broad range of length and energy scales in modern science, and efforts have demonstrated that locating suitable controls to perform a range of objectives has been widely successful. The justification for this success arises from a favorable topology of a quantum control landscape, defined as a mapping of the controls to a cost function measuring the success of the operation. This is summarized in the landscape principle that no suboptimal extrema exist on the landscape for well-suited control problems, explaining a trend of successful optimizations in both theory and experiment. This dissertation explores what additional lessons may be gleaned from the quantum control landscape through numerical and theoretical studies. The first topic examines the experimentally relevant problem of assessing and reducing disturbances due to noise. The local curvature of the landscape is found to play an important role on noise effects in the control of targeted quantum unitary operations, and provides a conceptual framework for assessing robustness to noise. Software for assessing noise effects in quantum computing architectures was also developed and applied to survey the performance of current quantum control techniques for quantum computing. A lack of competition between robustness and perfect unitary control operation was discovered to fundamentally limit noise effects, and highlights a renewed focus upon system engineering for reducing noise. This convergent behavior generally arises for any secondary objective in the situation of high primary objective fidelity. The other dissertation topic examines the utility of quantum control for a class of nonlinear Hamiltonians not previously considered under the landscape principle. Nonlinear Schrodinger equations are commonly used to model the dynamics of Bose-Einstein condensates (BECs), one of the largest known quantum objects. Optimizations of BEC dynamics were performed in which the
7. Logical inference approach to relativistic quantum mechanics: Derivation of the Klein-Gordon equation
Donker, H. C.; Katsnelson, M. I.; De Raedt, H.; Michielsen, K.
2016-09-01
The logical inference approach to quantum theory, proposed earlier De Raedt et al. (2014), is considered in a relativistic setting. It is shown that the Klein-Gordon equation for a massive, charged, and spinless particle derives from the combination of the requirements that the space-time data collected by probing the particle is obtained from the most robust experiment and that on average, the classical relativistic equation of motion of a particle holds.
8. Curvatures and discrete Gauss-Codazzi equation in (2 + 1)-dimensional loop quantum gravity
Ariwahjoedi, Seramika; Kosasih, Jusak Sali; Rovelli, Carlo; Zen, Freddy P.
2015-07-01
We derive the Gauss-Codazzi equation in the holonomy and plane-angle representations and we use the result to write a Gauss-Codazzi equation for a discrete (2 + 1)-dimensional manifold, triangulated by isosceles tetrahedra. This allows us to write operators acting on spin network states in (2 + 1)-dimensional loop quantum gravity, representing the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvatures.
9. A parametric approach to supersymmetric quantum mechanics in the solution of Schrödinger equation
SciTech Connect
Tezcan, Cevdet; Sever, Ramazan
2014-03-15
We study exact solutions of the Schrödinger equation for some potentials. We introduce a parametric approach to supersymmetric quantum mechanics to calculate energy eigenvalues and corresponding wave functions exactly. As an application we solve Schrödinger equation for the generalized Morse potential, modified Hulthen potential, deformed Rosen-Morse potential and Poschl-Teller potential. The method is simple and effective to get the results.
10. Digital quantum simulation of Dirac equation with a trapped ion
Shen, Yangchao; Zhang, Xiang; Zhang, Junhua; Casanova, Jorge; Lamata, Lucas; Solano, Enrique; Yung, Man-Hong; Zhang, Jingning; Kim, Kihwan; Department Of Physical Chemistry Collaboration
2014-05-01
Recently there has been growing interest in simulating relativistic effects in controllable physical system. We digitally simulate the Dirac equation in 3 +1 dimensions with a single trapped ion. We map four internal levels of 171Yb+ ion to the Dirac bispinor. The time evolution of the Dirac equation is implemented by trotter expansion. In the 3 +1 dimension, we can observe a helicoidal motion of a free Dirac particle which reduces to Zitterbewegung in 1 +1 dimension. This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540. KK acknowledge the support from the recruitment program of global youth experts.
11. The stochastic radiative transfer equation: quantum damping, Kirchoff's law and NLTE
SciTech Connect
Graziani, F R
2005-01-24
A method is presented based on the theory of quantum damping, for deriving a self consistent but approximate form of the quantum transport for photons interacting with fully ionized electron plasma. Specifically, we propose in this paper a technique of approximately including the effects of background plasma on a photon distribution function without directly solving any kinetic equations for the plasma itself. The result is a quantum Langevin equation for the photon number operator; the quantum radiative transfer equation. A dissipation term appears which is the imaginary part of the dielectric function for an electron gas with photon mediated electron-electron interactions due to absorption and re-emission. It depends only on the initial state of the plasma. A quantum noise operator also appears as a result of spontaneous emission of photons from the electron plasma. The thermal expectation value of this noise operator yields the emissivity which is exactly of the form of the Kirchoff-Planck relation. This non-zero thermal expectation value is a direct consequence of a fluctuation-dissipation relation (FDR).
12. The Schr{umlt o}dinger and Dirac free particle equations without quantum mechanics
SciTech Connect
Ord, G.N.
1996-08-01
Einstein{close_quote}s theory of Brownian Movement has provided a well accepted microscopic model of diffusion for many years. Until recently the relationship between this model and Quantum Mechanics has been completely formal. Brownian motion provides a microscopic model for diffusion, but quantum mechanics and diffusion are related by a formal analytic continuation, so the relationship between Brownian motion and Quantum Mechanics has been correspondingly vague. Some recent work has changed this picture somewhat and here we show that a random walk model of Brownian motion produces the diffusion equation or the telegraph equations as a descriptions of particle densities, while at the same time the correlations in the space-time geometry of these same Brownian particles obey the Schr{umlt o}dinger and Dirac equations respectively. This is of interest because the equations of Quantum Mechanics appear here naturally in a classical context without the problems of interpretation they have in the usual context. {copyright} 1996 Academic Press, Inc.
13. The equation of motion of an electron : a debate in classical and quantum physics.
SciTech Connect
Kim, K.-J.
1999-01-27
The current status of understanding of the equation of motion of an electron is summarized. Classically, a consistent, linearized theory exists for an electron of finite extent, as long as the size of the electron is larger than the classical electron radius. Nonrelativistic quantum mechanics seems to offer a tine theory even in the point-particle limit.
14. Scaling of magneto-quantum-radiative hydrodynamic equations: from laser-produced plasmas to astrophysics
SciTech Connect
Cross, J. E.; Gregori, G.; Reville, B.
2014-11-01
We introduce the equations of magneto-quantum-radiative hydrodynamics. By rewriting them in a dimensionless form, we obtain a set of parameters that describe scale-dependent ratios of characteristic hydrodynamic quantities. We discuss how these dimensionless parameters relate to the scaling between astrophysical observations and laboratory experiments.
15. Theta function solutions of the quantum Knizhnik-Zamolodchikov-Bernard equation for a face model
Finch, Peter E.; Weston, Robert; Zinn-Justin, Paul
2016-02-01
We consider the quantum Knizhnik-Zamolodchikov-Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent to the three-colour model, these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions.
16. Quantum flywheel
Levy, Amikam; Diósi, Lajos; Kosloff, Ronnie
2016-05-01
In this work we present the concept of a quantum flywheel coupled to a quantum heat engine. The flywheel stores useful work in its energy levels, while additional power is extracted continuously from the device. Generally, the energy exchange between a quantum engine and a quantized work repository is accompanied by heat, which degrades the charging efficiency. Specifically when the quantum harmonic oscillator acts as a work repository, quantum and thermal fluctuations dominate the dynamics. Quantum monitoring and feedback control are applied to the flywheel in order to reach steady state and regulate its operation. To maximize the charging efficiency one needs a balance between the information gained by measuring the system and the information fed back to the system. The dynamics of the flywheel are described by a stochastic master equation that accounts for the engine, the external driving, the measurement, and the feedback operations.
17. An iterative finite difference method for solving the quantum hydrodynamic equations of motion
SciTech Connect
Kendrick, Brian K
2010-01-01
The quantum hydrodynamic equations of motion associated with the de Broglie-Bohm description of quantum mechanics describe a time evolving probability density whose 'fluid' elements evolve as a correlated ensemble of particle trajectories. These equations are intuitively appealing due to their similarities with classical fluid dynamics and the appearance of a generalized Newton's equation of motion in which the total force contains both a classical and quantum contribution. However, the direct numerical solution of the quantum hydrodynamic equations (QHE) is fraught with challenges: the probability 'fluid' is highly-compressible, it has zero viscosity, the quantum potential ('pressure') is non-linear, and if that weren't enough the quantum potential can also become singular during the course of the calculations. Collectively these properties are responsible for the notorious numerical instabilities associated with a direct numerical solution of the QHE. The most successful and stable numerical approach that has been used to date is based on the Moving Least Squares (MLS) algorithm. The improved stability of this approach is due to the repeated local least squares fitting which effectively filters or reduces the numerical noise which tends to accumulate with time. However, this method is also subject to instabilities if it is pushed too hard. In addition, the stability of the MLS approach often comes at the expense of reduced resolution or fidelity of the calculation (i.e., the MLS filtering eliminates the higher-frequency components of the solution which may be of interest). Recently, a promising new solution method has been developed which is based on an iterative solution of the QHE using finite differences. This method (referred to as the Iterative Finite Difference Method or IFDM) is straightforward to implement, computationally efficient, stable, and its accuracy and convergence properties are well understood. A brief overview of the IFDM will be presented
18. On a derivation of the Boltzmann equation in Quantum Field Theory
Leiler, Gregor
The Boltzmann equation (BE) is a commonly used tool for the study of non-equilibrium many particle systems. It has been introduced in 1872 by Ludwig Boltzmann and has been widely generalized throughout the years. Today it is commonly used in physical applications, from the study of ordinary fluids to problems in particle Cosmology where Quantum Field Theoretical techniques are essential. Despite its numerous experimental successes, the conceptual basis of the BE is not entirely clear. For instance, it is well known that it is not a fundamental equation of physics like, say, the Heisenberg equation (HE). A natural question then arises whether it is possible to derive the BE from physical first principles, i.e. the Heisenberg equation in Quantum Field Theory. In this work we attempted to answer this question and succeeded in deriving the BE from the HE, thus further clarifying its conceptual status. In particular, the results we have obtained are as follows. Firstly, we establish the non-perturbative validity of what we call the "pre-Boltzmann equation". The crucial point here is that this latter equation is equivalent to the Heisenberg equation. Secondly, we proceed to consider various limits of the pre-Boltzmann equation, namly the "low density" and the "weak coupling" limits, to obtain two equations that can be considered as generalizations of the BE. These limits are always taken together with the "long time" limit, which allows us to interpret the BE as an appropriate long time limit of the HE. The generalization we obtain consists in additional "correction" terms to the usual Boltzmann collision factor, and can be associated to multiple particle scattering. Unlike the pre-Boltzmann equation, these latter results are only valid pertubatively. Finally, we briefly consider the possibility to extend these results beyond said limits and outline some important aspects in this case.
19. Some Mathematical Structures Including Simplified Non-Relativistic Quantum Teleportation Equations and Special Relativity
SciTech Connect
Woesler, Richard
2007-02-21
The computations of the present text with non-relativistic quantum teleportation equations and special relativity are totally speculative, physically correct computations can be done using quantum field theory, which remain to be done in future. Proposals for what might be called statistical time loop experiments with, e.g., photon polarization states are described when assuming the simplified non-relativistic quantum teleportation equations and special relativity. However, a closed time loop would usually not occur due to phase incompatibilities of the quantum states. Histories with such phase incompatibilities are called inconsistent ones in the present text, and it is assumed that only consistent histories would occur. This is called an exclusion principle for inconsistent histories, and it would yield that probabilities for certain measurement results change. Extended multiple parallel experiments are proposed to use this statistically for transmission of classical information over distances, and regarding time. Experiments might be testable in near future. However, first a deeper analysis, including quantum field theory, remains to be done in future.
20. Nonlinear quantum-dynamical system based on the Kadomtsev-Petviashvili II equation
Zarmi, Yair
2013-06-01
The structure of soliton solutions of classical integrable nonlinear evolution equations, which can be solved through the Hirota transformation, suggests a new way for the construction of nonlinear quantum-dynamical systems that are based on the classical equations. In the new approach, the classical soliton solution is mapped into an operator, U, which is a nonlinear functional of the particle-number operators over a Fock space of quantum particles. U obeys the evolution equation; the classical soliton solutions are the eigenvalues of U in multi-particle states in the Fock space. The construction easily allows for the incorporation of particle interactions, which generate soliton effects that do not have a classical analog. In this paper, this new approach is applied to the case of the Kadomtsev-Petviashvili II equation. The nonlinear quantum-dynamical system describes a set of M = (2S + 1) particles with intrinsic spin S, which interact in clusters of 1 ≤ N ≤ (M - 1) particles.
1. APPEL Masters with Masters #3
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Masters with Masters #3 features Mike Hawes, Assoc. Administrator, Office of Independent Program Cost & Evaluation, and Lynn Cline, Deputy Assoc. Administrator for Special Operations Missions, disc...
2. Solutions to the Painlevé V equation through supersymmetric quantum mechanics
Bermudez, David; Fernández C, David J.; Negro, Javier
2016-08-01
In this paper we shall use the algebraic method known as supersymmetric quantum mechanics (SUSY QM) to obtain solutions to the Painlevé V (PV) equation, a second-order nonlinear ordinary differential equation. For this purpose, we will apply first the SUSY QM treatment to the radial oscillator. In addition, we will revisit the polynomial Heisenberg algebras (PHAs) and we will study the general systems ruled by them: for first-order PHAs we obtain the radial oscillator while for third-order PHAs the potential will be determined by solutions to the PV equation. This connection allows us to introduce a simple technique for generating solutions of the PV equation expressed in terms of confluent hypergeometric functions. Finally, we will classify them into several solution hierarchies.
3. Astrophysical Applications of Quantum Corrections to the Equation of State of a Plasma
NASA Technical Reports Server (NTRS)
Heckler, Andrew F.
1994-01-01
The quantum electrodynamic correction to the equation of state of a plasma at finite temperature is applied to the areas of solar physics and cosmology. A previously neglected, purely quantum term in the correction is found to change the equation of state in the solar core by -0.37%, which is roughly estimated to decrease the calculated high energy neutrino flux by about 2.2%. We also show that a previous calculation of the effect of this correction on big bang nucleosynthesis is incomplete, and we estimate the correction to the primordial helium abundance Y to be Delta A= 1.4 x 10(exp -4). A physical explanation for the correction is found in terms of corrections to the dispersion relation of the electron, positron, and photon.
4. Temporal dynamics in the one-dimensional quantum Zakharov equations for plasmas
SciTech Connect
Misra, A. P.; Haas, F.; Banerjee, S.; Shukla, P. K.; Assis, L. P. G.
2010-03-15
The temporal dynamics of the quantum Zakharov equations in one spatial dimension, which describes the nonlinear interaction of quantum Langmuir waves and quantum ion-acoustic waves, is revisited by considering their solution as a superposition of three interacting wave modes in Fourier space. Previous results in the literature are modified and rectified. Periodic, chaotic, and hyperchaotic behaviors of the Fourier-mode amplitudes are identified by the analysis of Lyapunov exponent spectra and the power spectrum. The periodic route to chaos is explained through a one-parameter bifurcation analysis. The system is shown to be destabilized via a supercritical Hopf-bifurcation. The adiabatic limits of the fully spatiotemporal and reduced systems are compared from the viewpoint of integrability properties.
5. Monotonically convergent algorithms for solving quantum optimal control problems described by an integrodifferential equation of motion
Ohtsuki, Yukiyoshi; Teranishi, Yoshiaki; Saalfrank, Peter; Turinici, Gabriel; Rabitz, Herschel
2007-03-01
A family of monotonically convergent algorithms is presented for solving a wide class of quantum optimal control problems satisfying an inhomogeneous integrodifferential equation of motion. The convergence behavior is examined using a four-level model system under the influence of non-Markovian relaxation. The results show that high quality solutions can be obtained over a wide range of parameters that characterize the algorithms, independent of the presence or absence of relaxation.
6. Selection rules for the Wheeler-DeWitt equation in quantum cosmology
Barvinsky, A. O.; Kamenshchik, A. Yu.
2014-02-01
The incompleteness of the Dirac quantization scheme leads to a redundant set of solutions of the Wheeler-DeWitt equation for the wave function in the superspace of quantum cosmology. The selection of physically meaningful solutions that match quantum initial data can be attained by a reduction of the theory to the sector of true physical degrees of freedom and their canonical quantization. The resulting physical wave function unitarily evolving in the time variable introduced within this reduction can then be raised to the level of the cosmological wave function in the superspace of 3-metrics to form a needed subset of all solutions of the Wheeler-DeWitt equation. We apply this technique in several simple but nonlinear minisuperspace models and discuss (at both the classical and quantum level) the physical reduction in extrinsic time—the time variable determined in terms of extrinsic curvature (or momentum conjugated to the cosmological scale factor). Only this extrinsic time gauge can be consistently used in the vicinity of turning points and bounces where the scale factor reaches extremum and cannot monotonically parametrize the evolution of the system. Since the 3-metric scale factor is canonically dual to the extrinsic time variable, the transition from the physical wave function to the wave function in superspace represents a kind of generalized Fourier transform. This transformation selects square-integrable solutions of the Wheeler-DeWitt equation, which guarantees the Hermiticity of canonical operators of the Dirac quantization scheme. This makes this scheme consistent, a property that is not guaranteed with general solutions of the Wheeler-DeWitt equation. Semiclassically this means that wave functions are represented by oscillating waves in classically allowed domains of superspace and exponentially fall off in classically forbidden (underbarrier) regions. This is explicitly demonstrated in a flat Friedmann-Robertson-Walker (FRW) model with a scalar
7. Masters Network
SciTech Connect
Myers, R.E.; Riley, S.C.
1994-12-31
A state-wide network for Mathematics And Science Teaching Excellence Through Resources, Renewal, and Services, MASTER{sup 3}S Network is a grant-supported project designed to provide a comprehensive teacher support system for enhancing MST industry representatives, and MST professional system for enhancing MST industry representatives, and MST professional societies, a unique feature in the broad alliance of organizations involved. Made up of three inter-related projects, MASTER{sup 3}S facilitates the provision of resources, services, and opportunities for renewal for the instructional expert. Project One provides support through expert assistance as a cadre of scientists and engineers volunteer to serve as assistants in the classroom. Project Two provides support in the form of math/science summer courses, plus forums throughout the school year to offer ongoing collegial support. Project Three will soon provide support through information with the development of TROL (Teaching Resources on Line), our catalog database of locally available MST resources. The guiding principle of MASTER3S is the belief that by empowering, supporting and serving the teacher, we can most effectively impact student attitudes toward and achievement in MST subjects.
8. Quantum theory as a description of robust experiments: Derivation of the Pauli equation
SciTech Connect
De Raedt, Hans; Katsnelson, Mikhail I.; Donker, Hylke C.; Michielsen, Kristel
2015-08-15
It is shown that the Pauli equation and the concept of spin naturally emerge from logical inference applied to experiments on a charged particle under the conditions that (i) space is homogeneous (ii) the observed events are logically independent, and (iii) the observed frequency distributions are robust with respect to small changes in the conditions under which the experiment is carried out. The derivation does not take recourse to concepts of quantum theory and is based on the same principles which have already been shown to lead to e.g. the Schrödinger equation and the probability distributions of pairs of particles in the singlet or triplet state. Application to Stern–Gerlach experiments with chargeless, magnetic particles, provides additional support for the thesis that quantum theory follows from logical inference applied to a well-defined class of experiments. - Highlights: • The Pauli equation is obtained through logical inference applied to robust experiments on a charged particle. • The concept of spin appears as an inference resulting from the treatment of two-valued data. • The same reasoning yields the quantum theoretical description of neutral magnetic particles. • Logical inference provides a framework to establish a bridge between objective knowledge gathered through experiments and their description in terms of concepts.
9. Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition
Jang, Seogjoo
2016-06-01
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.
10. Functional integral equation for the complete effective action in quantum field theory
Scharnhorst, K.
1997-02-01
Based on a methodological analysis of the effective action approach, certain conceptual foundations of quantum field theory are reconsidered to establish a quest for an equation for the effective action. Relying on the functional integral formulation of Lagrangian quantum field theory, we propose a functional integral equation for the complete effective action which can be understood as a certain fixed-point condition. This is motivated by a critical attitude toward the distinction, artificial from an experimental point of view, between classical and effective action. While for free field theories nothing new is accomplished, for interacting theories the concept differs from the established paradigm. The analysis of this new concept concentrates on gauge field theories, treating QED as the prototype model. An approximative approach to the functional integral equation for the complete effective action of QED is exploited to obtain certain nonperturbative information about the quadratic kernels of the action. As a particular application the approximate calculation of the QED coupling constant α is explicitly studied. It is understood as one of the characteristics of a fixed point given as a solution of the functional integral equation proposed. Finally, within the present approach the vacuum energy problem is considered, as are possible implications for the concept of induced gravity.
11. Nonlinear quantum-mechanical system associated with Sine-Gordon equation in (1 + 2) dimensions
SciTech Connect
Zarmi, Yair
2014-10-15
Despite the fact that it is not integrable, the (1 + 2)-dimensional Sine-Gordon equation has N-soliton solutions, whose velocities are lower than the speed of light (c = 1), for all N ≥ 1. Based on these solutions, a quantum-mechanical system is constructed over a Fock space of particles. The coordinate of each particle is an angle around the unit circle. U, a nonlinear functional of the particle number-operators, which obeys the Sine-Gordon equation in (1 + 2) dimensions, is constructed. Its eigenvalues on N-particle states in the Fock space are the slower-than-light, N-soliton solutions of the equation. A projection operator (a nonlinear functional of U), which vanishes on the single-particle subspace, is a mass-density generator. Its eigenvalues on multi-particle states play the role of the mass density of structures that emulate free, spatially extended, relativistic particles. The simplicity of the quantum-mechanical system allows for the incorporation of perturbations with particle interactions, which have the capacity to “annihilate” and “create” solitons – an effect that does not have an analog in perturbed classical nonlinear evolution equations.
12. Unified treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: reduced hierarchy equation approach.
PubMed
Ishizaki, Akihito; Fleming, Graham R
2009-06-21
A new quantum dynamic equation for excitation energy transfer is developed which can describe quantum coherent wavelike motion and incoherent hopping in a unified manner. The developed equation reduces to the conventional Redfield theory and Forster theory in their respective limits of validity. In the regime of coherent wavelike motion, the equation predicts several times longer lifetime of electronic coherence between chromophores than does the conventional Redfield equation. Furthermore, we show quantum coherent motion can be observed even when reorganization energy is large in comparison to intersite electronic coupling (the Forster incoherent regime). In the region of small reorganization energy, slow fluctuation sustains longer-lived coherent oscillation, whereas the Markov approximation in the Redfield framework causes infinitely fast fluctuation and then collapses the quantum coherence. In the region of large reorganization energy, sluggish dissipation of reorganization energy increases the time electronic excitation stays above an energy barrier separating chromophores and thus prolongs delocalization over the chromophores.
13. Orbital HP-Clouds for Solving Schr?dinger Equation inQuantum Mechanics
SciTech Connect
Chen, J; Hu, W; Puso, M
2006-10-19
Solving Schroedinger equation in quantum mechanics presents a challenging task in numerical methods due to the high order behavior and high dimension characteristics in the wave functions, in addition to the highly coupled nature between wave functions. This work introduces orbital and polynomial enrichment functions to the partition of unity for solution of Schroedinger equation under the framework of HP-Clouds. An intrinsic enrichment of orbital function and extrinsic enrichment of monomial functions are proposed. Due to the employment of higher order basis functions, a higher order stabilized conforming nodal integration is developed. The proposed methods are implemented using the density functional theory for solution of Schroedinger equation. Analysis of several single and multi-electron/nucleus structures demonstrates the effectiveness of the proposed method.
14. Matrix continued fraction approach to the relativistic quantum mechanical spin-zero Feshbach-Villars equations
Brown, Natalie
In this thesis we solve the Feshbach-Villars equations for spin-zero particles through use of matrix continued fractions. The Feshbach-Villars equations are derived from the Klein-Gordon equation and admit, for the Coulomb potential on an appropriate basis, a Hamiltonian form that has infinite symmetric band-matrix structure. The corresponding representation of the Green's operator of such a matrix can be given as a matrix continued fraction. Furthermore, we propose a finite dimensional representation for the potential operator such that it retains some information about the whole Hilbert space. Combining these two techniques, we are able to solve relativistic quantum mechanical problems of a spin-zero particle in a Coulomb-like potential with a high level of accuracy.
15. A new functional flow equation for Einstein-Cartan quantum gravity
Harst, U.; Reuter, M.
2015-03-01
We construct a special-purpose functional flow equation which facilitates non-perturbative renormalization group (RG) studies on theory spaces involving a large number of independent field components that are prohibitively complicated using standard methods. Its main motivation are quantum gravity theories in which the gravitational degrees of freedom are carried by a complex system of tensor fields, a prime example being Einstein-Cartan theory, possibly coupled to matter. We describe a sequence of approximation steps leading from the functional RG equation of the Effective Average Action to the new flow equation which, as a consequence, is no longer fully exact on the untruncated theory space. However, it is by far more "user friendly" when it comes to projecting the abstract equation on a concrete (truncated) theory space and computing explicit beta-functions. The necessary amount of (tensor) algebra reduces drastically, and the usually very hard problem of diagonalizing the pertinent Hessian operator is sidestepped completely. In this paper we demonstrate the reliability of the simplified equation by applying it to a truncation of the Einstein-Cartan theory space. It is parametrized by a scale dependent Holst action, depending on a O(4) spin-connection and the tetrad as the independent field variables. We compute the resulting RG flow, focusing in particular on the running of the Immirzi parameter, and compare it to the results of an earlier computation where the exact equation had been applied to the same truncation. We find consistency between the two approaches and provide further evidence for the conjectured non-perturbative renormalizability (asymptotic safety) of quantum Einstein-Cartan gravity. We also investigate a duality symmetry relating small and large values of the Immirzi parameter (γ → 1 / γ) which is displayed by the beta-functions in the absence of a cosmological constant.
16. Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian
SciTech Connect
Cruz, Hans; Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar
2015-09-15
The sensitivity of the evolution of quantum uncertainties to the choice of the initial conditions is shown via a complex nonlinear Riccati equation leading to a reformulation of quantum dynamics. This sensitivity is demonstrated for systems with exact analytic solutions with the form of Gaussian wave packets. In particular, one-dimensional conservative systems with at most quadratic Hamiltonians are studied.
17. Wave front-ray synthesis for solving the multidimensional quantum Hamilton-Jacobi equation.
PubMed
Wyatt, Robert E; Chou, Chia-Chun
2011-08-21
A Cauchy initial-value approach to the complex-valued quantum Hamilton-Jacobi equation (QHJE) is investigated for multidimensional systems. In this approach, ray segments foliate configuration space which is laminated by surfaces of constant action. The QHJE incorporates all quantum effects through a term involving the divergence of the quantum momentum function (QMF). The divergence term may be expressed as a sum of two terms, one involving displacement along the ray and the other incorporating the local curvature of the action surface. It is shown that curvature of the wave front may be computed from coefficients of the first and second fundamental forms from differential geometry that are associated with the surface. Using the expression for the divergence, the QHJE becomes a Riccati-type ordinary differential equation (ODE) for the complex-valued QMF, which is parametrized by the arc length along the ray. In order to integrate over possible singularities in the QMF, a stable and accurate Möbius propagator is introduced. This method is then used to evolve rays and wave fronts for four systems in two and three dimensions. From the QMF along each ray, the wave function can be easily computed. Computational difficulties that may arise are described and some ways to circumvent them are presented.
18. Bernoulli's formula and Poisson's equations for a confined quantum gas: Effects due to a moving piston
Nakamura, Katsuhiro; Sobirov, Zarifboy A.; Matrasulov, Davron U.; Avazbaev, Sanat K.
2012-12-01
We study a nonequilibrium equation of states of an ideal quantum gas confined in the cavity under a moving piston with a small but finite velocity in the case in which the cavity wall suddenly begins to move at the time origin. Confining ourselves to the thermally isolated process, the quantum nonadiabatic (QNA) contribution to Poisson's adiabatic equations and to Bernoulli's formula which bridges the pressure and internal energy is elucidated. We carry out a statistical mean of the nonadiabatic (time-reversal-symmetric) force operator found in our preceding paper [Nakamura , Phys. Rev. EPLEEE81539-375510.1103/PhysRevE.83.041133 83, 041133 (2011)] in both the low-temperature quantum-mechanical and high-temperature quasiclassical regimes. The QNA contribution, which is proportional to the square of the piston's velocity and to the inverse of the longitudinal size of the cavity, has a coefficient that is dependent on the temperature, gas density, and dimensionality of the cavity. The investigation is done for a unidirectionally expanding three-dimensional (3D) rectangular parallelepiped cavity as well as its 1D version. Its relevance in a realistic nanoscale heat engine is discussed.
19. Wave front-ray synthesis for solving the multidimensional quantum Hamilton-Jacobi equation.
PubMed
Wyatt, Robert E; Chou, Chia-Chun
2011-08-21
A Cauchy initial-value approach to the complex-valued quantum Hamilton-Jacobi equation (QHJE) is investigated for multidimensional systems. In this approach, ray segments foliate configuration space which is laminated by surfaces of constant action. The QHJE incorporates all quantum effects through a term involving the divergence of the quantum momentum function (QMF). The divergence term may be expressed as a sum of two terms, one involving displacement along the ray and the other incorporating the local curvature of the action surface. It is shown that curvature of the wave front may be computed from coefficients of the first and second fundamental forms from differential geometry that are associated with the surface. Using the expression for the divergence, the QHJE becomes a Riccati-type ordinary differential equation (ODE) for the complex-valued QMF, which is parametrized by the arc length along the ray. In order to integrate over possible singularities in the QMF, a stable and accurate Möbius propagator is introduced. This method is then used to evolve rays and wave fronts for four systems in two and three dimensions. From the QMF along each ray, the wave function can be easily computed. Computational difficulties that may arise are described and some ways to circumvent them are presented. PMID:21861551
20. Wave front-ray synthesis for solving the multidimensional quantum Hamilton-Jacobi equation
SciTech Connect
Wyatt, Robert E.; Chou, Chia-Chun
2011-08-21
A Cauchy initial-value approach to the complex-valued quantum Hamilton-Jacobi equation (QHJE) is investigated for multidimensional systems. In this approach, ray segments foliate configuration space which is laminated by surfaces of constant action. The QHJE incorporates all quantum effects through a term involving the divergence of the quantum momentum function (QMF). The divergence term may be expressed as a sum of two terms, one involving displacement along the ray and the other incorporating the local curvature of the action surface. It is shown that curvature of the wave front may be computed from coefficients of the first and second fundamental forms from differential geometry that are associated with the surface. Using the expression for the divergence, the QHJE becomes a Riccati-type ordinary differential equation (ODE) for the complex-valued QMF, which is parametrized by the arc length along the ray. In order to integrate over possible singularities in the QMF, a stable and accurate Moebius propagator is introduced. This method is then used to evolve rays and wave fronts for four systems in two and three dimensions. From the QMF along each ray, the wave function can be easily computed. Computational difficulties that may arise are described and some ways to circumvent them are presented.
1. Photocurrents in semiconductors and semiconductor quantum wells analyzed by k.p-based Bloch equations
Podzimski, Reinold; Duc, Huynh Thanh; Priyadarshi, Shekhar; Schmidt, Christian; Bieler, Mark; Meier, Torsten
2016-03-01
Using a microscopic theory that combines k.p band structure calculations with multisubband semiconductor Bloch equations we are capable of computing coherent optically-induced rectification, injection, and shift currents in semiconductors and semiconductor nanostructures. A 14-band k.p theory has been employed to obtain electron states in non-centrosymmetric semiconductor systems. Numerical solutions of the multisubband Bloch equations provide a detailed and transparent description of the dynamics of the material excitations in terms of interband and intersubband polarizations/coherences and occupations. Our approach allows us to calculate and analyze photocurrents in the time and the frequency domains for bulk as well as quantum well and quantum wire systems with various growth directions. As examples, we present theoretical results on the rectification and shift currents in bulk GaAs and GaAs-based quantum wells. Moreover, we compare our results with experiments on shift currents. In the experiments the terahertz radiation emitted from the transient currents is detected via electro-optic sampling. This comparison is important from two perspectives. First, it helps to validate the theoretical model. Second, it allows us to investigate the microscopic origins of interesting features observed in the experiments.
2. Open Quantum Dynamics Calculations with the Hierarchy Equations of Motion on Parallel Computers.
PubMed
Strümpfer, Johan; Schulten, Klaus
2012-08-14
Calculating the evolution of an open quantum system, i.e., a system in contact with a thermal environment, has presented a theoretical and computational challenge for many years. With the advent of supercomputers containing large amounts of memory and many processors, the computational challenge posed by the previously intractable theoretical models can now be addressed. The hierarchy equations of motion present one such model and offer a powerful method that remained under-utilized so far due to its considerable computational expense. By exploiting concurrent processing on parallel computers the hierarchy equations of motion can be applied to biological-scale systems. Herein we introduce the quantum dynamics software PHI, that solves the hierarchical equations of motion. We describe the integrator employed by PHI and demonstrate PHI's scaling and efficiency running on large parallel computers by applying the software to the calculation of inter-complex excitation transfer between the light harvesting complexes 1 and 2 of purple photosynthetic bacteria, a 50 pigment system. PMID:23105920
3. A device adaptive inflow boundary condition for Wigner equations of quantum transport
SciTech Connect
Jiang, Haiyan; Lu, Tiao; Cai, Wei
2014-02-01
In this paper, an improved inflow boundary condition is proposed for Wigner equations in simulating a resonant tunneling diode (RTD), which takes into consideration the band structure of the device. The original Frensley inflow boundary condition prescribes the Wigner distribution function at the device boundary to be the semi-classical Fermi–Dirac distribution for free electrons in the device contacts without considering the effect of the quantum interaction inside the quantum device. The proposed device adaptive inflow boundary condition includes this effect by assigning the Wigner distribution to the value obtained from the Wigner transform of wave functions inside the device at zero external bias voltage, thus including the dominant effect on the electron distribution in the contacts due to the device internal band energy profile. Numerical results on computing the electron density inside the RTD under various incident waves and non-zero bias conditions show much improvement by the new boundary condition over the traditional Frensley inflow boundary condition.
4. Projected equations of motion approach to hybrid quantum/classical dynamics in dielectric-metal composites
McMillan, Ryan J.; Stella, Lorenzo; Grüning, Myrta
2016-09-01
We introduce a hybrid method for dielectric-metal composites that describes the dynamics of the metallic system classically while retaining a quantum description of the dielectric. The time-dependent dipole moment of the classical system is mimicked by the introduction of projected equations of motion (PEOM), and the coupling between the two systems is achieved through an effective dipole-dipole interaction. To benchmark this method, we model a test system (semiconducting quantum dot-metal nanoparticle hybrid). We begin by examining the energy absorption rate, showing agreement between the PEOM method and the analytical rotating wave approximation (RWA) solution. We then investigate population inversion and show that the PEOM method provides an accurate model for the interaction under ultrashort pulse excitation where the traditional RWA breaks down.
5. A device adaptive inflow boundary condition for Wigner equations of quantum transport
Jiang, Haiyan; Lu, Tiao; Cai, Wei
2014-02-01
In this paper, an improved inflow boundary condition is proposed for Wigner equations in simulating a resonant tunneling diode (RTD), which takes into consideration the band structure of the device. The original Frensley inflow boundary condition prescribes the Wigner distribution function at the device boundary to be the semi-classical Fermi-Dirac distribution for free electrons in the device contacts without considering the effect of the quantum interaction inside the quantum device. The proposed device adaptive inflow boundary condition includes this effect by assigning the Wigner distribution to the value obtained from the Wigner transform of wave functions inside the device at zero external bias voltage, thus including the dominant effect on the electron distribution in the contacts due to the device internal band energy profile. Numerical results on computing the electron density inside the RTD under various incident waves and non-zero bias conditions show much improvement by the new boundary condition over the traditional Frensley inflow boundary condition.
6. Quantum Numbers of Eigenstates of Generalized de Broglie-Bargmann- Wigner Equations for Fermions with Partonic Substructure
Stumpf, H.
2003-01-01
Generalized de Broglie-Bargmann-Wigner (BBW) equations are relativistically invariant quantum mechanical many body equations with nontrivial interaction, selfregularization and probability interpretation. Owing to these properties these equations are a suitable means for describing relativistic bound states of fermions. In accordance with de Broglie's fusion theory and modern assumptions about the partonic substructure of elementary fermions, i.e., leptons and quarks, the three-body generalized BBW-equations are investigated. The transformation properties and quantum numbers of the three-parton equations under the relevant group actions are elaborated in detail. Section 3 deals with the action of the isospin group SU(2), a U(1) global gauge group for the fermion number, the hypercharge and charge generators. The resulting quantum numbers of the composite partonic systems can be adapted to those of the phenomenological particles to be described. The space-time transformations and in particular rotations generated by angular momentum operators are considered in Section 4. Based on the compatibility of the BBW-equations and the group theoretical constraints, in Sect. 5 integral equations are formulated in a representation with diagonal energy and total angular momentum variables. The paper provides new insight into the solution space and quantum labels of resulting integral equations for three parton states and prepares the ground for representing leptons and quarks as composite systems.
7. Production of a sterile species: Quantum kinetics
SciTech Connect
Ho, Chiu Man; Boyanovsky, D.; Ho, C.M.
2007-04-23
Production of a sterile species is studied within an effective model of active-sterile neutrino mixing in a medium in thermal equilibrium. The quantum kinetic equations for the distribution functions and coherences are obtained from two independent methods: the effective action and the quantum master equation. The decoherence time scale for active-sterile oscillations is tau(dec)=2/Gamma(aa), but the evolution of the distribution functions is determined by the two different time scales associated with the damping rates of the quasiparticle modes in the medium: Gamma(1)=Gamma(aa)cos^2theta(m); Gamma(2)=Gamma(aa)sin^2theta(m) where Gamma(aa) is the interaction rate of the active species in the absence of mixing and theta(m) the mixing angle in the medium. These two time scales are widely different away from Mikheyev-Smirnov-Wolfenstein resonances and preclude the kinetic description of active-sterile production in terms of a simple rate equation. We give the complete set of quantum kinetic equations for the active and sterile populations and coherences and discuss in detail the various approximations. A generalization of the active-sterile transition probability in a medium is provided via the quantum master equation. We derive explicitly the usual quantum kinetic equations in terms of the"polarization vector" and show their equivalence to those obtained from the quantum master equation and effective action.
8. Theoretical study on electromagnetically induced transparency in molecular aggregate models using quantum Liouville equation method
SciTech Connect
Minami, Takuya; Nakano, Masayoshi
2015-01-22
Electromagnetically induced transparency (EIT), which is known as an efficient control method of optical absorption property, is investigated using the polarizability spectra and population dynamics obtained by solving the quantum Liouville equation. In order to clarify the intermolecular interaction effect on EIT, we examine several molecular aggregate models composed of three-state monomers with the dipole-dipole coupling. On the basis of the present results, we discuss the applicability of EIT in molecular aggregate systems to a new type of optical switch.
9. Time-dependent treatment of scattering. II - Novel integral equation approach to quantum wave packets
NASA Technical Reports Server (NTRS)
Sharafeddin, Omar A.; Judson, Richard S.; Kouri, Donald J.; Hoffman, David K.
1990-01-01
The novel wave-packet propagation scheme presented is based on the time-dependent form of the Lippman-Schwinger integral equation and does not require extensive matrix inversions, thereby facilitating application to systems in which some degrees of freedom express the potential in a basis expansion. The matrix to be inverted is a function of the kinetic energy operator, and is accordingly diagonal in a Bessel function basis set. Transition amplitudes for various orbital angular momentum quantum numbers are obtainable via either Fourier transform of the amplitude density from the time to the energy domain, or the direct analysis of the scattered wave packet.
10. Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion
SciTech Connect
Lamata, L.; Leon, J.; Schaetz, T.; Solano, E.
2007-06-22
We present a method of simulating the Dirac equation in 3+1 dimensions for a free spin-1/2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1+1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters.
11. Scattering problems in the fractional quantum mechanics governed by the 2D space-fractional Schrödinger equation
SciTech Connect
Dong, Jianping
2014-03-15
The 2D space-fractional Schrödinger equation in the time-independent and time-dependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.
12. Rate equations model and optical external efficiency of optically pumped electrically driven terahertz quantum cascade lasers
Hamadou, A.; Thobel, J.-L.; Lamari, S.
2016-10-01
A four level rate equations model for a terahertz optically pumped electrically driven quantum cascade laser is here introduced and used to model the system both analytically and numerically. In the steady state, both in the presence and absence of the terahertz optical field, we solve the resulting nonlinear system of equations and obtain closed form expressions for the levels occupation, population inversion as well as the mid-infrared pump threshold intensity in terms of the device parameters. We also derive, for the first time for this system, an analytical formula for the optical external efficiency and analyze the simultaneous effects of the cavity length and pump intensity on it. At moderate to high pump intensities, we find that the optical external efficiency scales roughly as the reciprocal of the cavity length.
13. The quantum equations of state of plasma under the influence of a weak magnetic field
SciTech Connect
Hussein, N. A.; Eisa, D. A.; Eldin, M. G.
2012-05-15
The aim of this paper is to calculate the magnetic quantum equations of state of plasma, the calculation is based on the magnetic binary Slater sum in the case of low density. We consider only the thermal equilibrium plasma in the case of n{lambda}{sub ab}{sup 3} Much-Less-Than 1, where {lambda}{sub ab}{sup 2}=( Planck-Constant-Over-Two-Pi {sup 2}/m{sub ab}KT) is the thermal De Broglie wave length between two particles. The formulas contain the contributions of the magnetic field effects. Using these results we compute the magnetization and the magnetic susceptibility. Our equation of state is compared with others.
14. Partial differential equations constrained combinatorial optimization on an adiabatic quantum computer
Chandra, Rishabh
Partial differential equation-constrained combinatorial optimization (PDECCO) problems are a mixture of continuous and discrete optimization problems. PDECCO problems have discrete controls, but since the partial differential equations (PDE) are continuous, the optimization space is continuous as well. Such problems have several applications, such as gas/water network optimization, traffic optimization, micro-chip cooling optimization, etc. Currently, no efficient classical algorithm which guarantees a global minimum for PDECCO problems exists. A new mapping has been developed that transforms PDECCO problem, which only have linear PDEs as constraints, into quadratic unconstrained binary optimization (QUBO) problems that can be solved using an adiabatic quantum optimizer (AQO). The mapping is efficient, it scales polynomially with the size of the PDECCO problem, requires only one PDE solve to form the QUBO problem, and if the QUBO problem is solved correctly and efficiently on an AQO, guarantees a global optimal solution for the original PDECCO problem.
15. Orbital free DFT versus single density equation: a perspective through quantum domain behavior of a classically chaotic system.
PubMed
Chakraborty, Debdutta; Kar, Susmita; Chattaraj, Pratim Kumar
2015-12-21
The orbital free density functional theory and the single density equation approach are formally equivalent. An orbital free density based quantum dynamical strategy is used to study the quantum-classical correspondence in both weakly and strongly coupled van der Pol and Duffing oscillators in the presence of an external electric field in one dimension. The resulting quantum hydrodynamic equations of motion are solved through an implicit Euler type real space method involving a moving weighted least square technique. The Lagrangian framework used here allows the numerical grid points to follow the wave packet trajectory. The associated classical equations of motion are solved using a sixth order Runge-Kutta method and the Ehrenfest dynamics is followed through the solution of the time dependent Schrodinger equation using a time dependent Fourier Grid Hamiltonian technique. Various diagnostics reveal a close parallelism between classical regular as well as chaotic dynamics and that obtained from the Bohmian mechanics. PMID:26033095
16. Real-time and imaginary-time quantum hierarchal Fokker-Planck equations
SciTech Connect
Tanimura, Yoshitaka
2015-04-14
We consider a quantum mechanical system represented in phase space (referred to hereafter as “Wigner space”), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
17. Real-time and imaginary-time quantum hierarchal Fokker-Planck equations.
PubMed
Tanimura, Yoshitaka
2015-04-14
We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material. PMID:25877565
18. Real-time and imaginary-time quantum hierarchal Fokker-Planck equations.
PubMed
Tanimura, Yoshitaka
2015-04-14
We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
19. Real-time and imaginary-time quantum hierarchal Fokker-Planck equations
Tanimura, Yoshitaka
2015-04-01
We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
20. Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation.
PubMed
Macnamara, Shev; Bersani, Alberto M; Burrage, Kevin; Sidje, Roger B
2008-09-01
Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis-Menten enzyme kinetics, double phosphorylation, the Goldbeter-Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.
1. The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks.
PubMed
Qian, Hong; Bishop, Lisa M
2010-09-20
We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner.
2. Method of conditional moments (MCM) for the Chemical Master Equation: a unified framework for the method of moments and hybrid stochastic-deterministic models.
PubMed
Hasenauer, J; Wolf, V; Kazeroonian, A; Theis, F J
2014-09-01
The time-evolution of continuous-time discrete-state biochemical processes is governed by the Chemical Master Equation (CME), which describes the probability of the molecular counts of each chemical species. As the corresponding number of discrete states is, for most processes, large, a direct numerical simulation of the CME is in general infeasible. In this paper we introduce the method of conditional moments (MCM), a novel approximation method for the solution of the CME. The MCM employs a discrete stochastic description for low-copy number species and a moment-based description for medium/high-copy number species. The moments of the medium/high-copy number species are conditioned on the state of the low abundance species, which allows us to capture complex correlation structures arising, e.g., for multi-attractor and oscillatory systems. We prove that the MCM provides a generalization of previous approximations of the CME based on hybrid modeling and moment-based methods. Furthermore, it improves upon these existing methods, as we illustrate using a model for the dynamics of stochastic single-gene expression. This application example shows that due to the more general structure, the MCM allows for the approximation of multi-modal distributions.
3. The Chemical Master Equation Approach to Nonequilibrium Steady-State of Open Biochemical Systems: Linear Single-Molecule Enzyme Kinetics and Nonlinear Biochemical Reaction Networks
PubMed Central
Qian, Hong; Bishop, Lisa M.
2010-01-01
We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a “punctuated equilibrium” manner. PMID:20957107
4. Superlinear scaling in master-slave quantum chemical calculations using in-core storage of two-electron integrals.
PubMed
Fossgård, Eirik; Ruud, Kenneth
2006-02-01
We describe the implementation of a parallel, in-core, integral-direct Hartree-Fock and density functional theory code for the efficient calculation of Hartree-Fock wave functions and density functional theory. The algorithm is based on a parallel master-slave algorithm, and the two-electron integrals calculated by a slave are stored in available local memory. To ensure the greatest computational savings, the master node keeps track of all integral batches stored on the different slaves. The code can reuse undifferentiated two-electron integrals both in the wave function optimization and in the evaluation of second-, third-, and fourth-order molecular properties. Superlinear scaling is achieved in a series of test examples, with speedups of up to 55 achieved for calculations run on medium-sized molecules on 16 processors with respect to the time used on a single processor.
5. Investigating non-Markovian dynamics of quantum open systems
Chen, Yusui
Quantum open system coupled to a non-Markovian environment has recently attracted widespread interest for its important applications in quantum information processing and quantum dissipative systems. New phenomena induced by the non-Markovian environment have been discovered in variety of research areas ranging from quantum optics, quantum decoherence to condensed matter physics. However, the study of the non-Markovian quantum open system is known a difficult problem due to its technical complexity in deriving the fundamental equation of motion and elusive conceptual issues involving non-equilibrium dynamics for a strong coupled environment. The main purpose of this thesis is to introduce several new techniques of solving the quantum open systems including a systematic approach to dealing with non-Markovian master equations from a generic quantum-state diffusion (QSD) equation. In the first part of this thesis, we briefly introduce the non-Markovian quantum-state diffusion approach, and illustrate some pronounced non-Markovian quantum effects through numerical investigation on a cavity-QED model. Then we extend the non-Markovian QSD theory to an interesting model where the environment has a hierarchical structure, and find out the exact non-Markovian QSD equation of this model system. We observe the generation of quantum entanglement due to the interplay between the non-Markovian environment and the cavity. In the second part, we show an innovative method to obtain the exact non-Markovian master equations for a set of generic quantum open systems based on the corresponding non-Markovian QSD equations. Multiple-qubit systems and multilevel systems are discussed in details as two typical examples. Particularly, we derive the exact master equation for a model consisting of a three-level atom coupled to an optical cavity and controlled by an external laser field. Additionally, we discuss in more general context the mathematical similarity between the multiple
6. Interpretation of quantum Hall effect from angular momentum theory and Dirac equation.
Shrivastava, Keshav
2007-03-01
It is found that when suitable modifications to the g values are made, the effective charge of a particle is determined by eeff =(1/2)ge, which enters in the Dirac equation to yield the fractional charges. The calculated values of the fractional charges agree with the data on fractional charge deduced from the quantum Hall effect. Therefore, the Dirac equation can accommodate not only particles of charges 0 and ± 1 but also fractional charges such as 1/3 and 2/3. This means that spin and charge get coupled. There are two g values for two signs of the spin. Hence 4 eigen values emerge, two for positive spin and two for negative spin. Therefore a 4x4 matrix has to be added to the eigen value E in the Dirac equation. This matrix has interesting anticommuting properties. K. N. Shrivastava, Phys. Lett. A 113,435-6(1986);115, 459(1986)(E). K. N. Shrivastava, Phys. Lett. A 326, 469-472(2004) K. N. Shrivastava, Mod. Phys. Lett. B 13, 1087-1090(1999); 14, 1009-1013(2000).
7. Description of a dissipative quantum spin dynamics with a Landau-Lifshitz/Gilbert like damping and complete derivation of the classical Landau-Lifshitz equation
Wieser, Robert
2015-03-01
The classical Landau-Lifshitz equation has been derived from quantum mechanics. Starting point is the assumption of a non-Hermitian Hamilton operator to take the energy dissipation into account. The corresponding quantum mechanical spin dynamics along with the time dependent Schrödinger, Liouville and Heisenberg equation has been described and the similarities and differences between classical and quantum mechanical spin dynamics have been discussed. Furthermore, a time dependent Schrödinger equation corresponding to the classical Landau-Lifshitz-Gilbert equation and two ways to include temperature into the quantum mechanical spin dynamics have been proposed.
8. Synergies from using higher order symplectic decompositions both for ordinary differential equations and quantum Monte Carlo methods
SciTech Connect
Matuttis, Hans-Georg; Wang, Xiaoxing
2015-03-10
Decomposition methods of the Suzuki-Trotter type of various orders have been derived in different fields. Applying them both to classical ordinary differential equations (ODEs) and quantum systems allows to judge their effectiveness and gives new insights for many body quantum mechanics where reference data are scarce. Further, based on data for 6 × 6 system we conclude that sampling with sign (minus-sign problem) is probably detrimental to the accuracy of fermionic simulations with determinant algorithms.
9. Reanalysis of Rate Data for the Reaction CH3 + CH3 → C2H6 Using Revised Cross Sections and a Linearized Second-Order Master Equation.
PubMed
Blitz, M A; Green, N J B; Shannon, R J; Pilling, M J; Seakins, P W; Western, C M; Robertson, S H
2015-07-16
Rate coefficients for the CH3 + CH3 reaction, over the temperature range 300-900 K, have been corrected for errors in the absorption coefficients used in the original publication ( Slagle et al., J. Phys. Chem. 1988 , 92 , 2455 - 2462 ). These corrections necessitated the development of a detailed model of the B̃(2)A1' (3s)-X̃(2)A2″ transition in CH3 and its validation against both low temperature and high temperature experimental absorption cross sections. A master equation (ME) model was developed, using a local linearization of the second-order decay, which allows the use of standard matrix diagonalization methods for the determination of the rate coefficients for CH3 + CH3. The ME model utilized inverse Laplace transformation to link the microcanonical rate constants for dissociation of C2H6 to the limiting high pressure rate coefficient for association, k∞(T); it was used to fit the experimental rate coefficients using the Levenberg-Marquardt algorithm to minimize χ(2) calculated from the differences between experimental and calculated rate coefficients. Parameters for both k∞(T) and for energy transfer ⟨ΔE⟩down(T) were varied and optimized in the fitting procedure. A wide range of experimental data were fitted, covering the temperature range 300-2000 K. A high pressure limit of k∞(T) = 5.76 × 10(-11)(T/298 K)(-0.34) cm(3) molecule(-1) s(-1) was obtained, which agrees well with the best available theoretical expression.
10. Explicit formulas for generalized harmonic perturbations of the infinite quantum well with an application to Mathieu equations
SciTech Connect
Garcia-Ravelo, J.; Trujillo, A. L.; Schulze-Halberg, A.
2012-10-15
We obtain explicit formulas for perturbative corrections of the infinite quantum well model. The formulas we obtain are based on a class of matrix elements that we construct by means of two-parameter ladder operators associated with the infinite quantum well system. Our approach can be used to construct solutions to Schroedinger-type equations that involve generalized harmonic perturbations of their potentials, such as cosine powers, Fourier series, and more general functions. As a particular case, we obtain characteristic values for odd periodic solutions of the Mathieu equation.
11. Quantum molecular dynamics simulations of equation of state of warm dense ethane
Li, Chuan-Ying; Wang, Cong; Li, Yong-Sheng; Li, Da-Fang; Li, Zi; Zhang, Ping
2016-09-01
The equation of state of warm dense ethane is obtained using quantum molecular dynamics simulations based on finite-temperature density functional theory for densities from 0.1 g / cm 3 to 3.1 g / cm 3 and temperatures from 0.1 eV to 5.17 eV. The calculated pressure and internal energy are fitted with cubic polynomials in terms of density and temperature. Specific density-temperature-pressure tracks such as the principal and double shock Hugoniot curves along with release isentropes are predicted which are fundamental for the analysis and interpretation of high-pressure experiments. The principal and double shock Hugoniot curves are in agreement with the experimental data from the Sandia Z-Machine [Magyar et al., Phys. Rev. B 91, 134109 (2015)].
12. A non-equilibrium equation-of-motion approach to quantum transport utilizing projection operators
Ochoa, Maicol A.; Galperin, Michael; Ratner, Mark A.
2014-11-01
We consider a projection operator approach to the non-equilbrium Green function equation-of-motion (PO-NEGF EOM) method. The technique resolves problems of arbitrariness in truncation of an infinite chain of EOMs and prevents violation of symmetry relations resulting from the truncation (equivalence of left- and right-sided EOMs is shown and symmetry with respect to interchange of Fermi or Bose operators before truncation is preserved). The approach, originally developed by Tserkovnikov (1999 Theor. Math. Phys. 118 85) for equilibrium systems, is reformulated to be applicable to time-dependent non-equilibrium situations. We derive a canonical form of EOMs, thus explicitly demonstrating a proper result for the non-equilibrium atomic limit in junction problems. A simple practical scheme applicable to quantum transport simulations is formulated. We perform numerical simulations within simple models and compare results of the approach to other techniques and (where available) also to exact results.
13. Comment on "Fractional quantum mechanics" and "Fractional Schrödinger equation".
PubMed
Wei, Yuchuan
2016-06-01
In this Comment we point out some shortcomings in two papers [N. Laskin, Phys. Rev. E 62, 3135 (2000)10.1103/PhysRevE.62.3135; N. Laskin, Phys. Rev. E 66, 056108 (2002)10.1103/PhysRevE.66.056108]. We prove that the fractional uncertainty relation does not hold generally. The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.e., a particle can teleport from a place to another. Since the relativistic kinetic energy can be viewed as an approximate realization of the fractional kinetic energy, the particle teleportation should be an observable relativistic effect in quantum mechanics. With the help of this concept, superconductivity could be viewed as the teleportation of electrons from one side of a superconductor to another and superfluidity could be viewed as the teleportation of helium atoms from one end of a capillary tube to the other. We also point out how to teleport a particle to an arbitrary destination.
14. Comment on "Fractional quantum mechanics" and "Fractional Schrödinger equation".
PubMed
Wei, Yuchuan
2016-06-01
In this Comment we point out some shortcomings in two papers [N. Laskin, Phys. Rev. E 62, 3135 (2000)10.1103/PhysRevE.62.3135; N. Laskin, Phys. Rev. E 66, 056108 (2002)10.1103/PhysRevE.66.056108]. We prove that the fractional uncertainty relation does not hold generally. The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.e., a particle can teleport from a place to another. Since the relativistic kinetic energy can be viewed as an approximate realization of the fractional kinetic energy, the particle teleportation should be an observable relativistic effect in quantum mechanics. With the help of this concept, superconductivity could be viewed as the teleportation of electrons from one side of a superconductor to another and superfluidity could be viewed as the teleportation of helium atoms from one end of a capillary tube to the other. We also point out how to teleport a particle to an arbitrary destination. PMID:27415397
15. Comment on "Fractional quantum mechanics" and "Fractional Schrödinger equation"
Wei, Yuchuan
2016-06-01
In this Comment we point out some shortcomings in two papers [N. Laskin, Phys. Rev. E 62, 3135 (2000), 10.1103/PhysRevE.62.3135; N. Laskin, Phys. Rev. E 66, 056108 (2002), 10.1103/PhysRevE.66.056108]. We prove that the fractional uncertainty relation does not hold generally. The probability continuity equation in fractional quantum mechanics has a missing source term, which leads to particle teleportation, i.e., a particle can teleport from a place to another. Since the relativistic kinetic energy can be viewed as an approximate realization of the fractional kinetic energy, the particle teleportation should be an observable relativistic effect in quantum mechanics. With the help of this concept, superconductivity could be viewed as the teleportation of electrons from one side of a superconductor to another and superfluidity could be viewed as the teleportation of helium atoms from one end of a capillary tube to the other. We also point out how to teleport a particle to an arbitrary destination.
16. Perturbative approach to Markovian open quantum systems
Li, Andy C. Y.; Petruccione, F.; Koch, Jens
2014-05-01
The exact treatment of Markovian open quantum systems, when based on numerical diagonalization of the Liouville super-operator or averaging over quantum trajectories, is severely limited by Hilbert space size. Perturbation theory, standard in the investigation of closed quantum systems, has remained much less developed for open quantum systems where a direct application to the Lindblad master equation is desirable. We present such a perturbative treatment which will be useful for an analytical understanding of open quantum systems and for numerical calculation of system observables which would otherwise be impractical.
17. Decoherence and dissipation for a quantum system coupled to a local environment
NASA Technical Reports Server (NTRS)
Gallis, Michael R.
1994-01-01
Decoherence and dissipation in quantum systems has been studied extensively in the context of Quantum Brownian Motion. Effective decoherence in coarse grained quantum systems has been a central issue in recent efforts by Zurek and by Hartle and Gell-Mann to address the Quantum Measurement Problem. Although these models can yield very general classical phenomenology, they are incapable of reproducing relevant characteristics expected of a local environment on a quantum system, such as the characteristic dependence of decoherence on environment spatial correlations. I discuss the characteristics of Quantum Brownian Motion in a local environment by examining aspects of first principle calculations and by the construction of phenomenological models. Effective quantum Langevin equations and master equations are presented in a variety of representations. Comparisons are made with standard results such as the Caldeira-Leggett master equation.
18. Solution of the quantum fluid dynamical equations with radial basis function interpolation
Hu, Xu-Guang; Ho, Tak-San; Rabitz, Herschel; Askar, Attila
2000-05-01
The paper proposes a numerical technique within the Lagrangian description for propagating the quantum fluid dynamical (QFD) equations in terms of the Madelung field variables R and S, which are connected to the wave function via the transformation ψ=exp\\{(R+iS)/ħ\\}. The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density ρ=\\|ψ\\|2 and on the structure of R=ħ/2 ln ρ generally being simpler and smoother than ρ. The spatially smooth functions R and S are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme. Examples of two-dimensional model systems show that the method rivals, in both efficiency and accuracy, the split-operator and Chebychev expansion methods. The results on a three-dimensional model system indicates that the present method is superior to the existing ones, especially, for its low storage requirement and its uniform accuracy. The advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool.
19. Solution of the quantum fluid dynamical equations with radial basis function interpolation
PubMed
2000-05-01
The paper proposes a numerical technique within the Lagrangian description for propagating the quantum fluid dynamical (QFD) equations in terms of the Madelung field variables R and S, which are connected to the wave function via the transformation psi = exp(R + iS)/[symbol: see text]. The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density rho = magnitude of psi 2 and on the structure of R = [symbol: see text]/2 ln rho generally being simpler and smoother than rho. The spatially smooth functions R and S are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme. Examples of two-dimensional model systems show that the method rivals, in both efficiency and accuracy, the split-operator and Chebychev expansion methods. The results on a three-dimensional model system indicates that the present method is superior to the existing ones, especially, for its low storage requirement and its uniform accuracy. The advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool. PMID:11031661
20. Solution of the quantum fluid dynamical equations with radial basis function interpolation
SciTech Connect
Hu, Xu-Guang; Ho, Tak-San; Rabitz, Herschel; Askar, Attila
2000-05-01
The paper proposes a numerical technique within the Lagrangian description for propagating the quantum fluid dynamical (QFD) equations in terms of the Madelung field variables R and S, which are connected to the wave function via the transformation {psi}=exp{l_brace}(R+iS)/({Dirac_h}/2{pi})(right brace). The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density {rho}=|{psi}|{sup 2} and on the structure of R=({Dirac_h}/2{pi})/2 ln {rho} generally being simpler and smoother than {rho}. The spatially smooth functions R and S are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme. Examples of two-dimensional model systems show that the method rivals, in both efficiency and accuracy, the split-operator and Chebychev expansion methods. The results on a three-dimensional model system indicates that the present method is superior to the existing ones, especially, for its low storage requirement and its uniform accuracy. The advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool. (c) 2000 The American Physical Society.
1. Supersymmetric quantum mechanics and solitons of the sine-Gordon and nonlinear Schroedinger equations
SciTech Connect
Koller, Andrew; Olshanii, Maxim
2011-12-15
We present a case demonstrating the connection between supersymmetric quantum mechanics (SUSYQM), reflectionless scattering, and soliton solutions of integrable partial differential equations. We show that the members of a class of reflectionless Hamiltonians, namely, Akulin's Hamiltonians, are connected via supersymmetric chains to a potential-free Hamiltonian, explaining their reflectionless nature. While the reflectionless property in question has been mentioned in the literature for over two decades, the enabling algebraic mechanism was previously unknown. Our results indicate that the multisoliton solutions of the sine-Gordon and nonlinear Schroedinger equations can be systematically generated via the supersymmetric chains connecting Akulin's Hamiltonians. Our findings also explain a well-known but little-understood effect in laser physics: when a two-level atom, initially in the ground state, is subjected to a laser pulse of the form V(t)=(n({h_bar}/2{pi})/{tau})/cosh(t/{tau}), with n being an integer and {tau} being the pulse duration, it remains in the ground state after the pulse has been applied, for any choice of the laser detuning.
2. Disaster Master
MedlinePlus
... levels. But watch out! The wrong choice could end the game. Survive all 7 levels plus a turn in the hot seat and become a Disaster Master! Print ... 6 - Tsunami/Earthquake Level 7: Thunderstorm/Lightning ...
3. Quantum Stochastic Equations for an Opto-Mechanical Oscillator with Radiation Pressure Interaction and Non-Markovian Effects
Barchielli, Alberto
2016-06-01
The quantum stochastic Schrödinger equation or Hudson-Parthasarathy (HP) equation is a powerful tool to construct unitary dilations of quantum dynamical semigroups and to develop the theory of measurements in continuous time via the construction of output fields. An important feature of such an equation is that it allows to treat not only absorption and emission of quanta, but also scattering processes, which however had very few applications in physical modelling. Moreover, recent developments have shown that also some non-Markovian dynamics can be generated by suitable choices of the state of the quantum noises involved in the HP-equation. This paper is devoted to an application involving these two features, non-Markovianity and scattering process. We consider a micro-mirror mounted on a vibrating structure and reflecting a laser beam, a process giving rise to a radiation-pressure force on the mirror. We show that this process needs the scattering part of the HP-equation to be described. On the other side, non-Markovianity is introduced by the dissipation due to the interaction with some thermal environment which we represent by a phonon field, with a nearly arbitrary excitation spectrum, and by the introduction of phase noise in the laser beam. Finally, we study the full power spectrum of the reflected light and we show how the laser beam can be used as a temperature probe.
4. Dynamics of open bosonic quantum systems in coherent state representation
SciTech Connect
Dalvit, D. A. R.; Berman, G. P.; Vishik, M.
2006-01-15
We consider the problem of decoherence and relaxation of open bosonic quantum systems from a perspective alternative to the standard master equation or quantum trajectories approaches. Our method is based on the dynamics of expectation values of observables evaluated in a coherent state representation. We examine a model of a quantum nonlinear oscillator with a density-density interaction with a collection of environmental oscillators at finite temperature. We derive the exact solution for dynamics of observables and demonstrate a consistent perturbation approach.
5. Non-Markovian dynamics without using a quantum trajectory
SciTech Connect
Wu Chengjun; Li Yang; Zhu Mingyi; Guo Hong
2011-05-15
Open quantum systems interacting with structured environments is important and manifests non-Markovian behavior, which was conventionally studied using a quantum trajectory stochastic method. In this paper, by dividing the effects of the environment into two parts, we propose a deterministic method without using a quantum trajectory. This method is more efficient and accurate than the stochastic method in most Markovian and non-Markovian cases. We also extend this method to the generalized Lindblad master equation.
6. Schwinger-Dyson equations in large-N quantum field theories and nonlinear random processes
SciTech Connect
Buividovich, P. V.
2011-02-15
We propose a stochastic method for solving Schwinger-Dyson equations in large-N quantum field theories. Expectation values of single-trace operators are sampled by stationary probability distributions of the so-called nonlinear random processes. The set of all the histories of such processes corresponds to the set of all planar diagrams in the perturbative expansions of the expectation values of singlet operators. We illustrate the method on examples of the matrix-valued scalar field theory and the Weingarten model of random planar surfaces on the lattice. For theories with compact field variables, such as sigma models or non-Abelian lattice gauge theories, the method does not converge in the physically most interesting weak-coupling limit. In this case one can absorb the divergences into a self-consistent redefinition of expansion parameters. A stochastic solution of the self-consistency conditions can be implemented as a 'memory' of the random process, so that some parameters of the process are estimated from its previous history. We illustrate this idea on the two-dimensional O(N) sigma model. The extension to non-Abelian lattice gauge theories is discussed.
7. Equation of state and transport properties of warm dense helium via quantum molecular dynamics simulations
Li, Zhi-Guo; Cheng, Yan; Chen, Qi-Feng; Chen, Xiang-Rong
2016-05-01
The equation of state, self-diffusion, and viscosity coefficients of helium have been investigated by quantum molecular dynamics (QMD) simulations in the warm dense matter regime. Our simulations are validated through the comparison with the reliable experimental data. The calculated principal and reshock Hugoniots of liquid helium are in good agreement with the gas-gun data. On this basis, we revisit the issue for helium, i.e., the possibility of the instabilities predicted by chemical models at around 2000 GPa and 10 g/cm3 along the pressure isotherms of 6309, 15 849, and 31 623 K. Our calculations show no indications of instability in this pressure-temperature region, which reconfirm the predictions of previous QMD simulations. The self-diffusion and viscosity coefficients of warm dense helium have been systematically investigated by the QMD simulations. We carefully test the finite-size effects and convergences of statistics, and obtain numerically converged self-diffusion and viscosity coefficients by using the Kubo-Green formulas. The present results have been used to evaluate the existing one component plasma models. Finally, the validation of the Stokes-Einstein relationship for helium in the warm dense regime is discussed.
8. Gaussian quantum operator representation for bosons
SciTech Connect
Corney, Joel F.; Drummond, Peter D.
2003-12-01
We introduce a Gaussian quantum operator representation, using the most general possible multimode Gaussian operator basis. The representation unifies and substantially extends existing phase-space representations of density matrices for Bose systems and also includes generalized squeezed-state and thermal bases. It enables first-principles dynamical or equilibrium calculations in quantum many-body systems, with quantum uncertainties appearing as dynamical objects. Any quadratic Liouville equation for the density operator results in a purely deterministic time evolution. Any cubic or quartic master equation can be treated using stochastic methods.
9. Master Teacher
ERIC Educational Resources Information Center
Miranda, Maria Eugenia
2011-01-01
Dr. Carole Berotte Joseph, the new president of Bronx Community College, or BCC, has been training to lead an institution of higher education since grade school, taking on the role of master teacher since she played on her parents' stoop with the neighborhood children in Brooklyn. Growing up, she didn't play with dolls much. She played with real…
10. Role of nonlinearity in non-Hermitian quantum mechanics: Description of linear quantum electrodynamics from the nonlinear Schrödinger-Poisson equation
Reinisch, Gilbert C.; Gazeau, Maxime
2016-07-01
In this paper we consider a basic two-level nonlinear quantum model consisting in a two-particle interacting bound-state system. It is described by means of two different approaches: i) the mean-field stationary nonlinear Schrödinger-Poisson equation with classical Coulomb interaction and harmonic potential; ii) the linear quantum electrodynamics Hamiltonian of a quantized field coupled to two fixed charges. Computing numerically the ground state and the first excited state about the maximum eigenstate overlap (which is not zero because of eigenstate non-orthogonality), we numerically demonstrate that these two descriptions coincide at first order. As a consequence, a specific definition of the fine-structure constant α is provided within 99.95% accuracy by the present first-order non-relativistic and nonlinear quantum description. This result also means that the internal Coulomb interaction commutes with external particle confinement for the calculation of the ground state. Consequently peculiar nonlinear quantum properties become observable (an experiment with GaAs quantum-dot helium is suggested).
11. Quantum theory of open systems based on stochastic differential equations of generalized Langevin (non-Wiener) type
Basharov, A. M.
2012-09-01
It is shown that the effective Hamiltonian representation, as it is formulated in author's papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are "locked" inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.
12. Complex master slave interferometry.
PubMed
2016-02-01
A general theoretical model is developed to improve the novel Spectral Domain Interferometry method denoted as Master/Slave (MS) Interferometry. In this model, two functions, g and h are introduced to describe the modulation chirp of the channeled spectrum signal due to nonlinearities in the decoding process from wavenumber to time and due to dispersion in the interferometer. The utilization of these two functions brings two major improvements to previous implementations of the MS method. A first improvement consists in reducing the number of channeled spectra necessary to be collected at Master stage. In previous MSI implementation, the number of channeled spectra at the Master stage equated the number of depths where information was selected from at the Slave stage. The paper demonstrates that two experimental channeled spectra only acquired at Master stage suffice to produce A-scans from any number of resolved depths at the Slave stage. A second improvement is the utilization of complex signal processing. Previous MSI implementations discarded the phase. Complex processing of the electrical signal determined by the channeled spectrum allows phase processing that opens several novel avenues. A first consequence of such signal processing is reduction in the random component of the phase without affecting the axial resolution. In previous MSI implementations, phase instabilities were reduced by an average over the wavenumber that led to reduction in the axial resolution.
13. Teleportation-based quantum computation, extended Temperley-Lieb diagrammatical approach and Yang-Baxter equation
Zhang, Yong; Zhang, Kun; Pang, Jinglong
2016-01-01
This paper focuses on the study of topological features in teleportation-based quantum computation and aims at presenting a detailed review on teleportation-based quantum computation (Gottesman and Chuang in Nature 402: 390, 1999). In the extended Temperley-Lieb diagrammatical approach, we clearly show that such topological features bring about the fault-tolerant construction of both universal quantum gates and four-partite entangled states more intuitive and simpler. Furthermore, we describe the Yang-Baxter gate by its extended Temperley-Lieb configuration and then study teleportation-based quantum circuit models using the Yang-Baxter gate. Moreover, we discuss the relationship between the extended Temperley-Lieb diagrammatical approach and the Yang-Baxter gate approach. With these research results, we propose a worthwhile subject, the extended Temperley-Lieb diagrammatical approach, for physicists in quantum information and quantum computation.
14. The quantum-mechanical basis of an extended Landau-Lifshitz-Gilbert equation for a current-carrying ferromagnetic wire.
PubMed
Edwards, D M; Wessely, O
2009-04-01
An extended Landau-Lifshitz-Gilbert (LLG) equation is introduced to describe the dynamics of inhomogeneous magnetization in a current-carrying wire. The coefficients of all the terms in this equation are calculated quantum-mechanically for a simple model which includes impurity scattering. This is done by comparing the energies and lifetimes of a spin wave calculated from the LLG equation and from the explicit model. Two terms are of particular importance since they describe non-adiabatic spin-transfer torque and damping processes which do not rely on spin-orbit coupling. It is shown that these terms may have a significant influence on the velocity of a current-driven domain wall and they become dominant in the case of a narrow wall.
15. Toward a Solution of the Edwards Equation for the Vertex Function of Quantum Electrodynamics in the Region of Large Momenta
Agamalieva, L. A.; Gadjiev, S. A.; Jafarov, R. G.
2016-03-01
An asymptotic expression for the vertex function in the region of large momenta in quantum electrodynamics is investigated in the ladder approximation. To formulate a calculational model in the ladder approximation, an iterative scheme has been used to solve the Schwinger-Dyson equation in the formalism of a bilocal source of fields. For the chirally symmetric leading approximation, the Edwards equation for the electron-positron-photon vertex has been obtained in the case of arbitrary values of the photon momentum. Our primary task is to develop a method to solve the vertex equation in the region of large momenta. Nontrivial behavior of the vertex function in the deeply inelastic region of momenta has been revealed.
16. Bethe-Salpeter equation for quantum-well exciton states in an inhomogeneous magnetic field
Koinov, Z. G.; Nash, P.; Witzel, J.
2003-04-01
The trapping of excitons in a single quantum well due to the presence of a strong homogeneous magnetic field and a weak inhomogeneous cylindrical symmetric magnetic field, created by the deposition of a magnetized disk on top of the quantum well, both applied perpendicular to the x-y plane of confinement is studied theoretically. The numerical calculations are performed for GaAs/AlxGa1-xAs quantum wells and the formation of bound exciton states with nonzero values for the center-of-mass exciton wave function only in a small area is predicted.
17. Bethe-Salpeter equation for exciton states in quantum well in a nonhomogeneous magnetic field
Koinov, Z.; Nash, P.; Witzel, J.
2003-03-01
The trapping of excitons in a single quantum well due to the presence of an external strong constant magnetic field and a small nonhomogeneous cylindrical symmetric magnetic field, created by a magnetized disk on top of the quantum well, is studied by applying the Bethe-Salpeter formalism. The numerical calculations are performed for GaAs/AlGaAs quantum wells. We find that the nonhomogeneous magnetic field leads to the formation of bound exciton states with nonzero values for the center-of-mass exciton wave function only in a sufficiently small area.
18. Multi-band Bloch equations and gain spectra of highly excited II-VI semiconductor quantum wells
SciTech Connect
Girndt, A.; Jahnke, F.; Knorr, A.; Koch, S.W.; Chow, W.W.
1997-04-21
Quasi-equilibrium excitation dependent optical probe spectra of II-VI semiconductor quantum wells at room temperature are investigated within the framework of multi-band semiconductor Bloch equations. The calculations include correlation effects beyond the Hartree-Fock level which describe dephasing, interband Coulomb correlations and band-gap renormalization in second Born approximation. In addition to the carrier-Coulomb interaction, the influence of carrier-phonon scattering and inhomogeneous broadening is considered. The explicit calculation of single particle properties like band structure and dipole matrix elements using k {center_dot} p theory makes it possible to investigate various II-VI material combinations. Numerical results are presented for CdZnSe/ZnSe and CdZnSe/MnZnSSe semiconductor quantum-well systems.
19. Finite difference method for solving the Schroedinger equation with band nonparabolicity in mid-infrared quantum cascade lasers
SciTech Connect
Cooper, J. D.; Valavanis, A.; Ikonic, Z.; Harrison, P.; Cunningham, J. E.
2010-12-01
The nonparabolic Schroedinger equation for electrons in quantum cascade lasers (QCLs) is a cubic eigenvalue problem (EVP) which cannot be solved directly. While a method for linearizing this cubic EVP has been proposed in principle for quantum dots [Hwang et al., Math. Comput. Modell., 40, 519 (2004)] it was deemed too computationally expensive because of the three-dimensional geometry under consideration. We adapt this linearization approach to the one-dimensional geometry of QCLs, and arrive at a direct and exact solution to the cubic EVP. The method is then compared with the well established shooting method, and it is shown to be more accurate and reliable for calculating the bandstructure of mid-infrared QCLs.
20. On Improving Accuracy of Finite-Element Solutions of the Effective-Mass Schrödinger Equation for Interdiffused Quantum Wells and Quantum Wires
Topalović, D. B.; Arsoski, V. V.; Pavlović, S.; Čukarić, N. A.; Tadić, M. Ž.; Peeters, F. M.
2016-01-01
We use the Galerkin approach and the finite-element method to numerically solve the effective-mass Schrödinger equation. The accuracy of the solution is explored as it varies with the range of the numerical domain. The model potentials are those of interdiffused semiconductor quantum wells and axially symmetric quantum wires. Also, the model of a linear harmonic oscillator is considered for comparison reasons. It is demonstrated that the absolute error of the electron ground state energy level exhibits a minimum at a certain domain range, which is thus considered to be optimal. This range is found to depend on the number of mesh nodes N approximately as α0 logeα1(α2N), where the values of the constants α0, α1, and α2 are determined by fitting the numerical data. And the optimal range is found to be a weak function of the diffusion length. Moreover, it was demonstrated that a domain range adaptation to the optimal value leads to substantial improvement of accuracy of the solution of the Schrödinger equation. Supported by the Ministry of Education, Science, and Technological Development of Serbia and the Flemish fund for Scientific Research (FWO Vlaanderen)
1. Approximation solution of Schrodinger equation for Q-deformed Rosen-Morse using supersymmetry quantum mechanics (SUSY QM)
SciTech Connect
Alemgadmi, Khaled I. K. Suparmi; Cari; Deta, U. A.
2015-09-30
The approximate analytical solution of Schrodinger equation for Q-Deformed Rosen-Morse potential was investigated using Supersymmetry Quantum Mechanics (SUSY QM) method. The approximate bound state energy is given in the closed form and the corresponding approximate wave function for arbitrary l-state given for ground state wave function. The first excited state obtained using upper operator and ground state wave function. The special case is given for the ground state in various number of q. The existence of Rosen-Morse potential reduce energy spectra of system. The larger value of q, the smaller energy spectra of system.
2. Microscopic description of quantum Lorentz gas and extension of the Boltzmann equation to entire space-time scale.
PubMed
Hashimoto, K; Kanki, K; Tanaka, S; Petrosky, T
2016-02-01
Irreversible processes of weakly coupled one-dimensional quantum perfect Lorentz gas are studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouville-von Neumann operator. Without any phenomenological operations, such as a coarse-graining of space-time, or a truncation of the higher order correlation, we obtained irreversible processes in a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free length. Based on this solution, a limitation of the usual phenomenological Boltzmann equation, as well as an extension of the Boltzmann equation to entire space-time scale, is discussed.
3. Quantum-Kinetic Approach to Deriving Optical Bloch Equations for Light Emitters in a Weakly Absorbing Dielectric
2015-09-01
We obtained the system of Maxwell-Bloch equations (MB) that describe the interaction of cw laser with optically active impurity centers (particles) embedded in a dielectric material. The dielectric material is considered as a continuous medium with sufficient laser detuning from its absorption lines. The model takes into account the effects associated with both the real and the imaginary part of the dielectric constant of the material. MB equations were derived within a many-particle quantum-kinetic formalism, which is based on Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for reduced density matrices and correlation operators of material particles and the quantized radiation field modes. It is shown that this method is beneficial to describe the effects of individual and collective behavior of the light emitters and requires no phenomenological procedures. It automatically takes into account the characteristics associated with the presence of non-resonant and resonant particles filling the space between the optical centers.
4. Sinc-based method for an efficient solution in the direct space of quantum wave equations with periodic boundary conditions
SciTech Connect
Marconcini, Paolo; Logoteta, Demetrio; Macucci, Massimo
2013-11-07
The solution of differential problems, and in particular of quantum wave equations, can in general be performed both in the direct and in the reciprocal space. However, to achieve the same accuracy, direct-space finite-difference approaches usually involve handling larger algebraic problems with respect to the approaches based on the Fourier transform in reciprocal space. This is the result of the errors that direct-space discretization formulas introduce into the treatment of derivatives. Here, we propose an approach, relying on a set of sinc-based functions, that allows us to achieve an exact representation of the derivatives in the direct space and that is equivalent to the solution in the reciprocal space. We apply this method to the numerical solution of the Dirac equation in an armchair graphene nanoribbon with a potential varying only in the transverse direction.
5. Microscopic description of quantum Lorentz gas and extension of the Boltzmann equation to entire space-time scale.
PubMed
Hashimoto, K; Kanki, K; Tanaka, S; Petrosky, T
2016-02-01
Irreversible processes of weakly coupled one-dimensional quantum perfect Lorentz gas are studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouville-von Neumann operator. Without any phenomenological operations, such as a coarse-graining of space-time, or a truncation of the higher order correlation, we obtained irreversible processes in a purely dynamical basis in all space and time scale including the microscopic atomic interaction range that is much smaller than the mean-free length. Based on this solution, a limitation of the usual phenomenological Boltzmann equation, as well as an extension of the Boltzmann equation to entire space-time scale, is discussed. PMID:26986313
6. Quantum Monte Carlo Calculation for the Equation of State of MgSiO3 perovskite at high pressures
Lin, Yangzheng; Cohen, R. E.; Driver, Kevin P.; Militzer, Burkhard; Shulenburger, Luke; Kim, Jeongnim
2014-03-01
Magnesium silicate (MgSiO3) is among the most abundant minerals in the Earth's mantle. Its phase behavior under high pressure has important implications for the physical properties of deep Earth and the core-mantle boundary. A number of experiments and density functional theory calculations have studied perovskite and its transition to the post-perovskite phase. Here, we present our initial work on the equation of state of perovskite at pressures up to 200 GPa using quantum Monte Carlo (QMC), a benchmark ab initio method. Our QMC calculations optimize electron correlation by using a Slater-Jastrow type wave function with a single determinant comprised of single-particle orbitals extracted from fully converged DFT calculations. The equation of state obtained from QMC calculations agrees with experimental data. E-mail: [email protected]; This work is supported by NSF.
7. Numerical solution of the chemical master equation uniqueness and stability of the stationary distribution for chemical networks, and mRNA bursting in a gene network with negative feedback regulation.
PubMed
Zeron, E S; Santillán, M
2011-01-01
In this work, we introduce a couple of algorithms to compute the stationary probability distribution for the chemical master equation (CME) of arbitrary chemical networks. We further find the conditions guaranteeing the algorithms' convergence and the unity and stability of the stationary distribution. Next, we employ these algorithms to study the mRNA and protein probability distributions in a gene regulatory network subject to negative feedback regulation. In particular, we analyze the influence of the promoter activation/deactivation speed on the shape of such distributions. We find that a reduction of the promoter activation/deactivation speed modifies the shape of those distributions in a way consistent with the phenomenon known as mRNA (or transcription) bursting.
8. Calculation of wave-functions with frozen orbitals in mixed quantum mechanics/molecular mechanics methods. II. Application of the local basis equation.
PubMed
Ferenczy, György G
2013-04-01
The application of the local basis equation (Ferenczy and Adams, J. Chem. Phys. 2009, 130, 134108) in mixed quantum mechanics/molecular mechanics (QM/MM) and quantum mechanics/quantum mechanics (QM/QM) methods is investigated. This equation is suitable to derive local basis nonorthogonal orbitals that minimize the energy of the system and it exhibits good convergence properties in a self-consistent field solution. These features make the equation appropriate to be used in mixed QM/MM and QM/QM methods to optimize orbitals in the field of frozen localized orbitals connecting the subsystems. Calculations performed for several properties in divers systems show that the method is robust with various choices of the frozen orbitals and frontier atom properties. With appropriate basis set assignment, it gives results equivalent with those of a related approach [G. G. Ferenczy previous paper in this issue] using the Huzinaga equation. Thus, the local basis equation can be used in mixed QM/MM methods with small size quantum subsystems to calculate properties in good agreement with reference Hartree-Fock-Roothaan results. It is shown that bond charges are not necessary when the local basis equation is applied, although they are required for the self-consistent field solution of the Huzinaga equation based method. Conversely, the deformation of the wave-function near to the boundary is observed without bond charges and this has a significant effect on deprotonation energies but a less pronounced effect when the total charge of the system is conserved. The local basis equation can also be used to define a two layer quantum system with nonorthogonal localized orbitals surrounding the central delocalized quantum subsystem.
9. Exact solution of the one- and three-dimensional quantum kinetic equations with velocity-dependent collision rates: Comparative analysis
Privalov, T.; Shalagin, A.
1999-06-01
The interaction of a plane monochromatic traveling wave with two-level particles suffering collisions with buffer-gas particles is considered. Collision rates are assumed to be velocity dependent. The collision integral is obtained on the basis of the strong-collision model, generalized to the case of velocity-dependent collision rates (the so-called kangaroo'' model). We obtained the exact analytical solution of the problem for arbitrary intensity of radiation, arbitrary ratio of homogeneous and Doppler widths of the absorption line, and arbitrary mass ratio between absorbing- and buffer-gas particles. The obtained analytical solutions of the quantum kinetic equations allowed us to analyze the spectral shape of the strong-field absorption line as well as the probe-field absorption line (the nonlinear part of the work done by the probe field) and the frequency dependence of the light-induced drift (LID) velocity. A comprehensive comparative analysis for the three- and one-dimensional versions of the model is given. On the basis of this analysis, we reach the conclusion that the one-dimensional quantum kinetic equation has quite a wide range of application. We also reveal the conditions for the strongest manifestation of the velocity dependence of the collision rates, which affects most strongly the anomalous LID.
10. Quantum correlations among optical and vibrational quanta
Carlig, Sergiu; Macovei, Mihai A.
2014-05-01
We investigate the feasibility of correlating an optical cavity field and a vibrational phonon mode. A laser pumped quantum dot fixed on a nanomechanical resonator beam interacts as a whole with the optical resonator mode. When the quantum dot variables are faster than the optical and phonon ones, we obtain a final master equation describing the involved modes only. Increasing the temperature, which directly affects the vibrational degrees of freedom, one can as well influence the cavity photon intensity, i.e., the optical and phonon modes are correlated. Furthermore, the corresponding Cauchy-Schwarz inequality is violated demonstrating the quantum nature of those correlations.
11. The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy
Zabrodin, Anton
2014-01-01
Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310], we show how to construct the master T-operator for the quantum inhomogeneous GL(N) XXX spin chain with twisted boundary conditions. It satisfies the bilinear identity and Hirota equations for the classical mKP hierarchy. We also characterize the class of solutions to the mKP hierarchy that correspond to eigenvalues of the master T-operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum spin chain and the classical Ruijsenaars-Schneider system of particles.
12. Quantum dissipation in unbounded systems.
PubMed
Maddox, Jeremy B; Bittner, Eric R
2002-02-01
In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie--Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wave function of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements (Bohm particles) which evolve according to nonclassical equations of motion. Until very recently these schemes have been limited to conservative systems. In this paper, we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme that incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy, we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.
13. Non-Perturbative, Unitary Quantum-Particle Scattering Amplitudes from Three-Particle Equations
SciTech Connect
Lindesay, James V
2002-03-19
We here use our non-perturbative, cluster decomposable relativistic scattering formalism to calculate photon-spinor scattering, including the related particle-antiparticle annihilation amplitude. We start from a three-body system in which the unitary pair interactions contain the kinematic possibility of single quantum exchange and the symmetry properties needed to identify and substitute antiparticles for particles. We extract from it unitary two-particle amplitude for quantum-particle scattering. We verify that we have done this correctly by showing that our calculated photon-spinor amplitude reduces in the weak coupling limit to the usual lowest order, manifestly covariant (QED) result with the correct normalization. That we are able to successfully do this directly demonstrates that renormalizability need not be a fundamental requirement for all physically viable models.
14. Numerical solution to the Boltzmann equation for use in calculating pumping rates in a CO sub 2 discharge laser. Master's thesis
SciTech Connect
Honey, D.A.
1989-12-01
The collisional Boltzmann equation was solved numerically to obtain excitation rates for use in a CO{sub 2} laser design program. The program was written in Microsoft QuickBasic for use on the IBM Personal Computer or equivalent. Program validation involved comparisons of computed transport coefficients with experimental data and previous theoretical work. Four different numerical algorithms were evaluated in terms of accuracy and efficiency. L-U decomposition was identified as the preferred approach. The calculated transport coefficients were found to agree with empirical data within one to five percent. The program was integrated into a CO{sub 2} laser design program. Studies were then performed to evaluate the effects on predicted laser output power and energy density as parameters affecting electron kinetics were changed. Plotting routines were written for both programs.
15. Quantum Monte Carlo calculation of the equation of state of neutron matter
SciTech Connect
Gandolfi, S.; Illarionov, A. Yu.; Schmidt, K. E.; Pederiva, F.; Fantoni, S.
2009-05-15
We calculated the equation of state of neutron matter at zero temperature by means of the auxiliary field diffusion Monte Carlo (AFDMC) method combined with a fixed-phase approximation. The calculation of the energy was carried out by simulating up to 114 neutrons in a periodic box. Special attention was given to reducing finite-size effects at the energy evaluation by adding to the interaction the effect due to the truncation of the simulation box, and by performing several simulations using different numbers of neutrons. The finite-size effects due to kinetic energy were also checked by employing the twist-averaged boundary conditions. We considered a realistic nuclear Hamiltonian containing modern two- and three-body interactions of the Argonne and Urbana family. The equation of state can be used to compare and calibrate other many-body calculations and to predict properties of neutron stars.
16. Reduced quantum dynamics with arbitrary bath spectral densities: Hierarchical equations of motion based on several different bath decomposition schemes
SciTech Connect
Liu, Hao; Zhu, Lili; Bai, Shuming; Shi, Qiang
2014-04-07
We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.
17. Reduced quantum dynamics with arbitrary bath spectral densities: hierarchical equations of motion based on several different bath decomposition schemes.
PubMed
Liu, Hao; Zhu, Lili; Bai, Shuming; Shi, Qiang
2014-04-01
We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.
18. Nonperturbative confinement in quantum chromodynamics. I. Study of an approximate equation of Mandelstam
Atkinson, D.; Drohm, J. K.; Johnson, P. W.; Stam, K.
1981-11-01
An approximated form of the Dyson-Schwinger equation for the gluon propagator in quarkless QCD is subjected to nonlinear functional and numerical analysis. It is found that solutions exist, and that these have a double pole at the origin of the square of the propagator momentum, together with an accumulation of soft branch points. This analytic structure is strongly suggestive of confinement by infrared slavery.
19. Quantum algorithm for simulating the dynamics of an open quantum system
SciTech Connect
Wang Hefeng; Ashhab, S.; Nori, Franco
2011-06-15
In the study of open quantum systems, one typically obtains the decoherence dynamics by solving a master equation. The master equation is derived using knowledge of some basic properties of the system, the environment, and their interaction: One basically needs to know the operators through which the system couples to the environment and the spectral density of the environment. For a large system, it could become prohibitively difficult to even write down the appropriate master equation, let alone solve it on a classical computer. In this paper, we present a quantum algorithm for simulating the dynamics of an open quantum system. On a quantum computer, the environment can be simulated using ancilla qubits with properly chosen single-qubit frequencies and with properly designed coupling to the system qubits. The parameters used in the simulation are easily derived from the parameters of the system + environment Hamiltonian. The algorithm is designed to simulate Markovian dynamics, but it can also be used to simulate non-Markovian dynamics provided that this dynamics can be obtained by embedding the system of interest into a larger system that obeys Markovian dynamics. We estimate the resource requirements for the algorithm. In particular, we show that for sufficiently slow decoherence a single ancilla qubit could be sufficient to represent the entire environment, in principle.
20. Collision efficiency of water in the unimolecular reaction CH4 (+H2O) ⇆ CH3 + H (+H2O): one-dimensional and two-dimensional solutions of the low-pressure-limit master equation.
PubMed
Jasper, Ahren W; Miller, James A; Klippenstein, Stephen J
2013-11-27
The low-pressure-limit unimolecular decomposition of methane, CH4 (+M) ⇆ CH3 + H (+M), is characterized via low-order moments of the total energy, E, and angular momentum, J, transferred due to collisions. The low-order moments are calculated using ensembles of classical trajectories, with new direct dynamics results for M = H2O and new results for M = O2 compared with previous results for several typical atomic (M = He, Ne, Ar, Kr) and diatomic (M = H2 and N2) bath gases and one polyatomic bath gas, M = CH4. The calculated moments are used to parametrize three different models of the energy transfer function, from which low-pressure-limit rate coefficients for dissociation, k0, are calculated. Both one-dimensional and two-dimensional collisional energy transfer models are considered. The collision efficiency for M = H2O relative to the other bath gases (defined as the ratio of low-pressure limit rate coefficients) is found to depend on temperature, with, e.g., k0(H2O)/k0(Ar) = 7 at 2000 K but only 3 at 300 K. We also consider the rotational collision efficiency of the various baths. Water is the only bath gas found to fully equilibrate rotations, and only at temperatures below 1000 K. At elevated temperatures, the kinetic effect of "weak-collider-in-J" collisions is found to be small. At room temperature, however, the use of an explicitly two-dimensional master equation model that includes weak-collider-in-J effects predicts smaller rate coefficients by 50% relative to the use of a statistical model for rotations. The accuracies of several methods for predicting relative collision efficiencies that do not require solving the master equation and that are based on the calculated low-order moments are tested. Troe's weak collider efficiency, βc, includes the effect of saturation of collision outcomes above threshold and accurately predicts the relative collision efficiencies of the nine baths. Finally, a brief discussion is presented of mechanistic details of the
1. Time-dependent density functional theory of open quantum systems in the linear-response regime.
PubMed
Tempel, David G; Watson, Mark A; Olivares-Amaya, Roberto; Aspuru-Guzik, Alán
2011-02-21
Time-dependent density functional theory (TDDFT) has recently been extended to describe many-body open quantum systems evolving under nonunitary dynamics according to a quantum master equation. In the master equation approach, electronic excitation spectra are broadened and shifted due to relaxation and dephasing of the electronic degrees of freedom by the surrounding environment. In this paper, we develop a formulation of TDDFT linear-response theory (LR-TDDFT) for many-body electronic systems evolving under a master equation, yielding broadened excitation spectra. This is done by mapping an interacting open quantum system onto a noninteracting open Kohn-Sham system yielding the correct nonequilibrium density evolution. A pseudoeigenvalue equation analogous to the Casida equations of the usual LR-TDDFT is derived for the Redfield master equation, yielding complex energies and Lamb shifts. As a simple demonstration, we calculate the spectrum of a C(2 +) atom including natural linewidths, by treating the electromagnetic field vacuum as a photon bath. The performance of an adiabatic exchange-correlation kernel is analyzed and a first-order frequency-dependent correction to the bare Kohn-Sham linewidth based on the Görling-Levy perturbation theory is calculated.
2. Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconfluent Heun equation
SciTech Connect
Caruso, F.; Martins, J.; Oguri, V.
2014-08-15
The effects on the non-relativistic dynamics of a system compound by two electrons interacting by a Coulomb potential and with an external harmonic oscillator potential, confined to move in a two dimensional Euclidean space, are investigated. In particular, it is shown that it is possible to determine exactly and in a closed form a finite portion of the energy spectrum and the associated eigenfunctions for the Schrödinger equation describing the relative motion of the electrons, by putting it into the form of a biconfluent Heun equation. In the same framework, another set of solutions of this type can be straightforwardly obtained for the case when the two electrons are submitted also to an external constant magnetic field. - Highlights: • Exact solution of a quantum dot model. • Determination of the energy levels. • Investigation of how the radial wave functions depend on the external excitation frequency. • Characteristic length between the two-electrons are found to be compatible with the semiconductor lattice parameter.
3. Equation of state of dense plasmas: Orbital-free molecular dynamics as the limit of quantum molecular dynamics for high-Z elements
SciTech Connect
Danel, J.-F.; Blottiau, P.; Kazandjian, L.; Piron, R.; Torrent, M.
2014-10-15
The applicability of quantum molecular dynamics to the calculation of the equation of state of a dense plasma is limited at high temperature by computational cost. Orbital-free molecular dynamics, based on a semiclassical approximation and possibly on a gradient correction, is a simulation method available at high temperature. For a high-Z element such as lutetium, we examine how orbital-free molecular dynamics applied to the equation of state of a dense plasma can be regarded as the limit of quantum molecular dynamics at high temperature. For the normal mass density and twice the normal mass density, we show that the pressures calculated with the quantum approach converge monotonically towards those calculated with the orbital-free approach; we observe a faster convergence when the orbital-free approach includes the gradient correction. We propose a method to obtain an equation of state reproducing quantum molecular dynamics results up to high temperatures where this approach cannot be directly implemented. With the results already obtained for low-Z plasmas, the present study opens the way for reproducing the quantum molecular dynamics pressure for all elements up to high temperatures.
4. Theory of damped quantum rotation in nuclear magnetic resonance spectra. III. Nuclear permutation symmetry of the line shape equation.
PubMed
Szymański, S
2009-12-28
The damped quantum rotation (DQR) theory describes manifestations in nuclear magnetic resonance spectra of the coherent and stochastic dynamics of N-fold molecular rotors composed of indistinguishable particles. The standard jump model is only a limiting case of the DQR approach; outside this limit, the stochastic motions of such rotors have no kinematic description. In this paper, completing the previous two of this series, consequences of nuclear permutation symmetry for the properties of the DQR line shape equation are considered. The systems addressed are planar rotors, such as aromatic hydrocarbons' rings, occurring inside of molecular crystals oriented in the magnetic field. Under such conditions, oddfold rotors can have nontrivial permutation symmetries only for peculiar orientations while evenfold ones always retain their intrinsic symmetry element, which is rotation by 180 degrees about the N-fold axis; in specific orientations the latter can gain two additional symmetry elements. It is shown that the symmetry selection rules applicable to the classical rate processes in fluids, once recognized as having two diverse aspects, macroscopic and microscopic, are also rigorously valid for the DQR processes in the solid state. However, formal justification of these rules is different because the DQR equation is based on the Pauli principle, which is ignored in the jump model. For objects like the benzene ring, exploitation of these rules in simulations of spectra using the DQR equation can be of critical significance for the feasibility of the calculations. Examples of such calculations for the proton system of the benzene ring in a general orientation are provided. It is also shown that, because of the intrinsic symmetries of the evenfold rotors, many of the DQR processes, which such rotors can undergo, are unobservable in NMR spectra.
5. Mixed Quantum/Classical Theory for Molecule-Molecule Inelastic Scattering: Derivations of Equations and Application to N2 + H2 System.
PubMed
Semenov, Alexander; Babikov, Dmitri
2015-12-17
The mixed quantum classical theory, MQCT, for inelastic scattering of two molecules is developed, in which the internal (rotational, vibrational) motion of both collision partners is treated with quantum mechanics, and the molecule-molecule scattering (translational motion) is described by classical trajectories. The resultant MQCT formalism includes a system of coupled differential equations for quantum probability amplitudes, and the classical equations of motion in the mean-field potential. Numerical tests of this theory are carried out for several most important rotational state-to-state transitions in the N2 + H2 system, in a broad range of collision energies. Besides scattering resonances (at low collision energies) excellent agreement with full-quantum results is obtained, including the excitation thresholds, the maxima of cross sections, and even some smaller features, such as slight oscillations of energy dependencies. Most importantly, at higher energies the results of MQCT are nearly identical to the full quantum results, which makes this approach a good alternative to the full-quantum calculations that become computationally expensive at higher collision energies and for heavier collision partners. Extensions of this theory to include vibrational transitions or general asymmetric-top rotor (polyatomic) molecules are relatively straightforward.
6. Complex generalized minimal residual algorithm for iterative solution of quantum-mechanical reactive scattering equations
Chatfield, David C.; Reeves, Melissa S.; Truhlar, Donald G.; Duneczky, Csilla; Schwenke, David W.
1992-12-01
Complex dense matrices corresponding to the D + H2 and O + HD reactions were solved using a complex generalized minimal residual (GMRes) algorithm described by Saad and Schultz (1986) and Saad (1990). To provide a test case with a different structure, the H + H2 system was also considered. It is shown that the computational effort for solutions with the GMRes algorithm depends on the dimension of the linear system, the total energy of the scattering problem, and the accuracy criterion. In several cases with dimensions in the range 1110-5632, the GMRes algorithm outperformed the LAPACK direct solver, with speedups for the linear equation solution as large as a factor of 23.
7. Complex generalized minimal residual algorithm for iterative solution of quantum-mechanical reactive scattering equations
NASA Technical Reports Server (NTRS)
Chatfield, David C.; Reeves, Melissa S.; Truhlar, Donald G.; Duneczky, Csilla; Schwenke, David W.
1992-01-01
Complex dense matrices corresponding to the D + H2 and O + HD reactions were solved using a complex generalized minimal residual (GMRes) algorithm described by Saad and Schultz (1986) and Saad (1990). To provide a test case with a different structure, the H + H2 system was also considered. It is shown that the computational effort for solutions with the GMRes algorithm depends on the dimension of the linear system, the total energy of the scattering problem, and the accuracy criterion. In several cases with dimensions in the range 1110-5632, the GMRes algorithm outperformed the LAPACK direct solver, with speedups for the linear equation solution as large as a factor of 23.
8. Master-Iac Master-Saao OTs
Pogrosheva, T.; Vladimirov, V.; Lipunov, V.; Lopez, R. Rebolo; Buckley, D.; Gorbovskoy, E.; Tiurina, N.; Kuznetsov, A.; Balanutsa, P.; Kornilov, V.; Gorbunov, I.; Gress, O.; Shumkov, V.; Ricart, M. Serra; Israelian, G.; Potter, S.; Kniazev, A.
2016-06-01
MASTER-IAC auto-detection system( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 30L ) discovered OT source at (RA, Dec) = 20h 58m 16.98s +23d 58m 12.6s on 2016-06-09.03890 UT. The OT unfiltered magnitude is 17.5m (limit 19.4m).
9. The optomechanical instability in the quantum regime
Ludwig, Max; Kubala, Björn; Marquardt, Florian
2008-09-01
We consider a generic optomechanical system, consisting of a driven optical cavity and a movable mirror attached to a cantilever. Systems of this kind (and analogues) have been realized in many recent experiments. It is well known that these systems can exhibit an instability towards a regime where the cantilever settles into self-sustained oscillations. In this paper, we briefly review the classical theory of the optomechanical instability, and then discuss the features arising in the quantum regime. We solve numerically a full quantum master equation for the coupled system, and use it to analyze the photon number, the cantilever's mechanical energy, the phonon probability distribution and the mechanical Wigner density, as a function of experimentally accessible control parameters. When a suitable dimensionless 'quantum parameter' is sent to zero, the results of the quantum mechanical model converge towards the classical predictions. We discuss this quantum-to-classical transition in some detail.
10. Solving non-Markovian open quantum systems with multi-channel reservoir coupling
SciTech Connect
Broadbent, Curtis J.; Jing, Jun; Yu, Ting; Eberly, Joseph H.
2012-08-15
We extend the non-Markovian quantum state diffusion (QSD) equation to open quantum systems which exhibit multi-channel coupling to a harmonic oscillator reservoir. Open quantum systems which have multi-channel reservoir coupling are those in which canonical transformation of reservoir modes cannot reduce the number of reservoir operators appearing in the interaction Hamiltonian to one. We show that the non-Markovian QSD equation for multi-channel reservoir coupling can, in some cases, lead to an exact master equation which we derive. We then derive the exact master equation for the three-level system in a vee-type configuration which has multi-channel reservoir coupling and give the analytical solution. Finally, we examine the evolution of the three-level vee-type system with generalized Ornstein-Uhlenbeck reservoir correlations numerically. - Highlights: Black-Right-Pointing-Pointer The concept of multi-channel vs. single-channel reservoir coupling is rigorously defined. Black-Right-Pointing-Pointer The non-Markovian quantum state diffusion equation for arbitrary multi-channel reservoir coupling is derived. Black-Right-Pointing-Pointer An exact time-local master equation is derived under certain conditions. Black-Right-Pointing-Pointer The analytical solution to the three-level system in a vee-type configuration is found. Black-Right-Pointing-Pointer The evolution of the three-level system under generalized Ornstein-Uhlenbeck noise is plotted for many parameter regimes.
11. Quantum-Mechanical Variant of the Thouless-Anderson-Palmer Equation for Error-Correcting Codes
Inoue, J.; Saika, Y.; Okada, M.
Statistical mechanics of information has been applied to problems in various research topics of information science and technology [1],[2]. Among those research topics, error-correcting code is one of the most developed subjects. In the research field of error-correcting codes, Nicolas Sourlas showed that the so-called convolutional codes can be constructed by spin glass with infinite range p-body interactions and the decoded message should be corresponded to the ground state of the Hamiltonian [3]. Ruján pointed out that the bit error can be suppressed if one uses finite temperature equilibrium states as the decoding result, instead of the ground state [4], and the so-called Bayes-optimal decoding at some specific condition was proved by Nishimori [5] and Nishimori and Wong [6]. Kabashima and Saad succeeded in constructing more practical codes, namely low-density parity check (LDPC) codes by using the infinite range spin glass model with finite connectivities [7]. They used the so-called TAP (Thouless-Anderson-Palmer) equations to decode the original message for a given parity check.
12. The Quantum World of Ultra-Cold Atoms and Light - Book 1: Foundations of Quantum Optics
Gardiner, Crispin; Zoller, Peter
2014-03-01
Abstract The Table of Contents is as follows: * I - THE PHYSICAL BACKGROUND * 1. Controlling the Quantum World * 1.1 Quantum Optics * 1.2 Quantum Information * 2. Describing the Quantum World * 2.1 Classical Stochastic Processes * 2.2. Theoretical Quantum Optics * 2.3. Quantum Stochastic Methods * 2.4. Ultra-Cold Atoms * II - CLASSICAL STOCHASTIC METHODS * 3. Physics in a Noisy World * 3.1. Brownian Motion and the Thermal Origin of Noise * 3.2. Brownian Motion, Friction, Noise and Temperature * 3.3. Measurement in a Fluctuating System * 4. Stochastic Differential Equations * 4.1. Ito Stochastic Differential Equation * 4.2. The Fokker-Planck Equation * 4.3. The Stratonovich Stochastic Differential Equation * 4.4. Systems with Many Variables * 4.5. Numerical Simulation of Stochastic Differential Equations * 5. The Fokker-Planck Equation * 5.1. Fokker-Planck Equation in One Dimension * 5.2. Eigenfunctions of the Fokker-Planck Equation * 5.3. Many-Variable Fokker-Planck Equations * 6. Master Equations and Jump Processes * 6.1. The Master Equation * 7. Applications of Random Processes * 7.1. The Ornstein-Uhlenbeck Process * 7.2. Johnson Noise * 7.3. Complex Variable Oscillator Processes * 8. The Markov Limit * 8.1. The White Noise Limit * 8.2. Interpretation and Generalizations of the White Noise Limit * 8.3. Linear Non-Markovian Stochastic Differential Equations * 9. Adiabatic Elimination of Fast Variables * 9.1 Slow and Fast Variables * 9.2. Other Applications of the Adiabatic Elimination Method * III - FIELDS, QUANTA AND ATOMS * 10. Ideal Bose and Fermi Systems * 10.1. The Quantum Gas * 10.2. Thermal States * 10.3. Fluctuations in the Ideal Bose Gas * 10.4. Bosonic Quantum Gaussian Systems * 10.5. Coherent States * 10.6. Fluctuations in Systems of Fermions * 10.7. Two-Level Systems and Pauli Matrices * 11. Quantum Fields * 11.1 Kinds of Quantum Field * 11.2 Coherence and Correlation Functions * 12. Atoms, Light and their Interaction * 12.1. Interaction with the
13. Quantum Simulation of Dissipative Processes without Reservoir Engineering
PubMed Central
Di Candia, R.; Pedernales, J. S.; del Campo, A.; Solano, E.; Casanova, J.
2015-01-01
We present a quantum algorithm to simulate general finite dimensional Lindblad master equations without the requirement of engineering the system-environment interactions. The proposed method is able to simulate both Markovian and non-Markovian quantum dynamics. It consists in the quantum computation of the dissipative corrections to the unitary evolution of the system of interest, via the reconstruction of the response functions associated with the Lindblad operators. Our approach is equally applicable to dynamics generated by effectively non-Hermitian Hamiltonians. We confirm the quality of our method providing specific error bounds that quantify its accuracy. PMID:26024437
14. Stability of continuous-time quantum filters with measurement imperfections
Amini, H.; Pellegrini, C.; Rouchon, P.
2014-07-01
The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modelled by a left stochastic matrix.
15. Quantum simulation of dissipative processes without reservoir engineering
DOE PAGES
Di Candia, R.; Pedernales, J. S.; del Campo, A.; Solano, E.; Casanova, J.
2015-05-29
We present a quantum algorithm to simulate general finite dimensional Lindblad master equations without the requirement of engineering the system-environment interactions. The proposed method is able to simulate both Markovian and non-Markovian quantum dynamics. It consists in the quantum computation of the dissipative corrections to the unitary evolution of the system of interest, via the reconstruction of the response functions associated with the Lindblad operators. Our approach is equally applicable to dynamics generated by effectively non-Hermitian Hamiltonians. We confirm the quality of our method providing specific error bounds that quantify its accuracy.
16. Factorized three-body S-matrix restrained by the Yang–Baxter equation and quantum entanglements
SciTech Connect
Yu, Li-Wei; Zhao, Qing; Ge, Mo-Lin
2014-09-15
This paper investigates the physical effects of the Yang–Baxter equation (YBE) to quantum entanglements through the 3-body S-matrix in entangling parameter space. The explicit form of 3-body S-matrix Ř{sub 123}(θ,φ) based on the 2-body S-matrices is given due to the factorization condition of YBE. The corresponding chain Hamiltonian has been obtained and diagonalized, also the Berry phase for 3-body system is given. It turns out that by choosing different spectral parameters the Ř(θ,φ)-matrix gives GHZ and W states respectively. The extended 1-D Kitaev toy model has been derived. Examples of the role of the model in entanglement transfer are discussed. - Highlights: • We give the relation between 3-body S-matrix and 3-qubit entanglement. • The relation between 3-qubit and 2-qubit entanglements is investigated via YBE. • 1D Kitaev toy model is derived by the Type-II solution of YBE. • The condition of YBE kills the “Zero boundary mode” in our chain model.
17. Can we derive Tully's surface-hopping algorithm from the semiclassical quantum Liouville equation? Almost, but only with decoherence
SciTech Connect
Subotnik, Joseph E. Ouyang, Wenjun; Landry, Brian R.
2013-12-07
In this article, we demonstrate that Tully's fewest-switches surface hopping (FSSH) algorithm approximately obeys the mixed quantum-classical Liouville equation (QCLE), provided that several conditions are satisfied – some major conditions, and some minor. The major conditions are: (1) nuclei must be moving quickly with large momenta; (2) there cannot be explicit recoherences or interference effects between nuclear wave packets; (3) force-based decoherence must be added to the FSSH algorithm, and the trajectories can no longer rigorously be independent (though approximations for independent trajectories are possible). We furthermore expect that FSSH (with decoherence) will be most robust when nonadiabatic transitions in an adiabatic basis are dictated primarily by derivative couplings that are presumably localized to crossing regions, rather than by small but pervasive off-diagonal force matrix elements. In the end, our results emphasize the strengths of and possibilities for the FSSH algorithm when decoherence is included, while also demonstrating the limitations of the FSSH algorithm and its inherent inability to follow the QCLE exactly.
18. Contracted Schrödinger equation: Determining quantum energies and two-particle density matrices without wave functions
Mazziotti, David A.
1998-06-01
The contracted Schrödinger equation (CSE) technique through its direct determination of the two-particle reduced density matrix (2RDM) without the wave function may offer a fresh alternative to traditional many-body quantum calculations. Without additional information the CSE, also known as the density equation, cannot be solved for the 2RDM because it also requires a knowledge of the 4RDM. We provide theoretical foundations through a reconstruction theorem for recent attempts at generating higher RDMs from the 2RDM to remove the indeterminacy of the CSE. With Grassmann algebra a more concise representation for Valdemoro's reconstruction functionals [F. Colmenero, C. Perez del Valle, and C. Valdemoro, Phys. Rev. A 47, 971 (1993)] is presented. From the perspective of the particle-hole equivalence we obtain Nakatsuji and Yasuda's correction for the 4RDM formula [H. Nakatsuji and K. Yasuda, Phys. Rev. Lett. 76, 1039 (1996)] as well as a corrective approach for the 3RDM functional. A different reconstruction strategy, the ensemble representability method (ERM), is introduced to build the 3- and 4-RDMs by enforcing four-ensemble representability and contraction conditions. We derive the CSE in second quantization without Valdemoro's matrix contraction mapping and offer the first proof of Nakatsuji's theorem for the second-quantized CSE. Both the functional and ERM reconstruction strategies are employed with the CSE to solve for the energies and the 2RDMs of a quasispin model without wave functions. We elucidate the iterative solution of the CSE through an analogy with the power method for eigenvalue equations. Resulting energies of the CSE methods are comparable to single-double configuration-interaction (SDCI) energies, and the 2RDMs are more accurate by an order of magnitude than those from SDCI. While the CSE has been applied to systems with 14 electrons, we present results for as many as 40 particles. Results indicate that the 2RDM remains accurate as the number
19. Interaction times in the {sup 136}Xe+{sup 136}Xe and {sup 238}U+{sup 238}U reactions with a quantum master equation
SciTech Connect
Sargsyan, V. V.; Kanokov, Z.; Adamian, G. G.; Antonenko, N. V.; Scheid, W.
2009-10-15
Using the reduced-density-matrix formalism, the interaction time of two heavy nuclei is studied. In the reactions {sup 136}Xe+{sup 136}Xe and {sup 238}U+{sup 238}U, the mean interaction time and variance of interaction time distribution are calculated and compared with those of other treatments.
20. A quantum equation of motion for chemical reaction systems on an adiabatic double-well potential surface in solution based on the framework of mixed quantum-classical molecular dynamics.
PubMed
2008-01-28
We present a quantum equation of motion for chemical reaction systems on an adiabatic double-well potential surface in solution in the framework of mixed quantum-classical molecular dynamics, where the reactant and product states are explicitly defined by dividing the double-well potential into the reactant and product wells. The equation can describe quantum reaction processes such as tunneling and thermal excitation and relaxation assisted by the solvent. Fluctuations of the zero-point energy level, the height of the barrier, and the curvature of the well are all included in the equation. Here, the equation was combined with the surface hopping technique in order to describe the motion of the classical solvent. Applying the present method to model systems, we show two numerical examples in order to demonstrate the potential power of the present method. The first example is a proton transfer by tunneling where the high-energy product state was stabilized very rapidly by solvation. The second example shows a thermal activation mechanism, i.e., the initial vibrational excitation in the reactant well followed by the reacting transition above the barrier and the final vibrational relaxation in the product well.
1. High power diode laser Master Oscillator-Power Amplifier (MOPA)
NASA Technical Reports Server (NTRS)
Andrews, John R.; Mouroulis, P.; Wicks, G.
1994-01-01
High power multiple quantum well AlGaAs diode laser master oscillator - power amplifier (MOPA) systems were examined both experimentally and theoretically. For two pass operation, it was found that powers in excess of 0.3 W per 100 micrometers of facet length were achievable while maintaining diffraction-limited beam quality. Internal electrical-to-optical conversion efficiencies as high as 25 percent were observed at an internal amplifier gain of 9 dB. Theoretical modeling of multiple quantum well amplifiers was done using appropriate rate equations and a heuristic model of the carrier density dependent gain. The model gave a qualitative agreement with the experimental results. In addition, the model allowed exploration of a wider design space for the amplifiers. The model predicted that internal electrical-to-optical conversion efficiencies in excess of 50 percent should be achievable with careful system design. The model predicted that no global optimum design exists, but gain, efficiency, and optical confinement (coupling efficiency) can be mutually adjusted to meet a specific system requirement. A three quantum well, low optical confinement amplifier was fabricated using molecular beam epitaxial growth. Coherent beam combining of two high power amplifiers injected from a common master oscillator was also examined. Coherent beam combining with an efficiency of 93 percent resulted in a single beam having diffraction-limited characteristics. This beam combining efficiency is a world record result for such a system. Interferometric observations of the output of the amplifier indicated that spatial mode matching was a significant factor in the less than perfect beam combining. Finally, the system issues of arrays of amplifiers in a coherent beam combining system were investigated. Based upon experimentally observed parameters coherent beam combining could result in a megawatt-scale coherent beam with a 10 percent electrical-to-optical conversion efficiency.
2. One-step implementation of the 1->3 orbital state quantum cloning machine via quantum Zeno dynamics
SciTech Connect
Shao Xiaoqiang; Wang Hongfu; Zhang Shou; Chen Li; Zhao Yongfang; Yeon, Kyu-Hwang
2009-12-15
We present an approach for implementation of a 1->3 orbital state quantum cloning machine based on the quantum Zeno dynamics via manipulating three rf superconducting quantum interference device (SQUID) qubits to resonantly interact with a superconducting cavity assisted by classical fields. Through appropriate modulation of the coupling constants between rf SQUIDs and classical fields, the quantum cloning machine can be realized within one step. We also discuss the effects of decoherence such as spontaneous emission and the loss of cavity in virtue of master equation. The numerical simulation result reveals that the quantum cloning machine is especially robust against the cavity decay, since all qubits evolve in the decoherence-free subspace with respect to cavity decay due to the quantum Zeno dynamics.
3. Causal signal transmission by quantum fields. VI: The Lorentz condition and Maxwell’s equations for fluctuations of the electromagnetic field
SciTech Connect
Plimak, L.I.; Stenholm, S.
2013-11-15
The general structure of electromagnetic interactions in the so-called response representation of quantum electrodynamics (QED) is analysed. A formal solution to the general quantum problem of the electromagnetic field interacting with matter is found. Independently, a formal solution to the corresponding problem in classical stochastic electrodynamics (CSED) is constructed. CSED and QED differ only in the replacement of stochastic averages of c-number fields and currents by time-normal averages of the corresponding Heisenberg operators. All relations of QED connecting quantum field to quantum current lack Planck’s constant, and thus coincide with their counterparts in CSED. In Feynman’s terms, one encounters complete disentanglement of the potential and current operators in response picture. Based on this parallelism between QED and CSED, it is natural to expect validity of the Lorentz condition and Maxwell’s equations for the time-normal averages of the potential and current. Things however turn out to be more complicated. Maxwell’s equations under the time-normal ordering can only be demonstrated subject to cancellation of the so-called Schwinger terms by gauge-invariant regularisations. We presume this pattern to be general, formulating this as “commutativity conjecture”. Consistency of the latter with the Heisenberg uncertainty principle is discussed. -- Highlights: •The general structure of interaction in quantum electrodynamics (QED) is analysed. •A detailed parallelism between QED and classical stochastic electrodynamics is shown. •Validity of Maxwell’s equations for fluctuations of the field is discussed. •This validity turns out to be in essence a renormalisation postulate.
4. Remarks on solving the one-dimensional time-dependent Schrödinger equation on the interval ?: the case of a quantum bouncer
Dembinski, S. T.; Wolniewicz, L.
1996-01-01
It is shown that the 1D Hamiltonian, which is a sum of operators which generate a finite nilpotent Lie algebra and depends explicitly on time existing closed form solutions of the time-dependent Schrödinger equation, cannot fulfil in general boundary and normalization conditions on a positive semi-axis. An explanation of the controversy surrounding the solutions of the quantum bouncer model, which appeared recently in the literature, is given.
5. Quantum localization of classical mechanics
Batalin, Igor A.; Lavrov, Peter M.
2016-07-01
Quantum localization of classical mechanics within the BRST-BFV and BV (or field-antifield) quantization methods are studied. It is shown that a special choice of gauge fixing functions (or BRST-BFV charge) together with the unitary limit leads to Hamiltonian localization in the path integral of the BRST-BFV formalism. In turn, we find that a special choice of gauge fixing functions being proportional to extremals of an initial non-degenerate classical action together with a very special solution of the classical master equation result in Lagrangian localization in the partition function of the BV formalism.
6. A quantum mechanical alternative to the Arrhenius equation in the interpretation of proton spin-lattice relaxation data for the methyl groups in solids.
PubMed
Bernatowicz, Piotr; Shkurenko, Aleksander; Osior, Agnieszka; Kamieński, Bohdan; Szymański, Sławomir
2015-11-21
The theory of nuclear spin-lattice relaxation in methyl groups in solids has been a recurring problem in nuclear magnetic resonance (NMR) spectroscopy. The current view is that, except for extreme cases of low torsional barriers where special quantum effects are at stake, the relaxation behaviour of the nuclear spins in methyl groups is controlled by thermally activated classical jumps of the methyl group between its three orientations. The temperature effects on the relaxation rates can be modelled by Arrhenius behaviour of the correlation time of the jump process. The entire variety of relaxation effects in protonated methyl groups have recently been given a consistent quantum mechanical explanation not invoking the jump model regardless of the temperature range. It exploits the damped quantum rotation (DQR) theory originally developed to describe NMR line shape effects for hindered methyl groups. In the DQR model, the incoherent dynamics of the methyl group include two quantum rate (i.e., coherence-damping) processes. For proton relaxation only one of these processes is relevant. In this paper, temperature-dependent proton spin-lattice relaxation data for the methyl groups in polycrystalline methyltriphenyl silane and methyltriphenyl germanium, both deuterated in aromatic positions, are reported and interpreted in terms of the DQR model. A comparison with the conventional approach exploiting the phenomenological Arrhenius equation is made. The present observations provide further indications that incoherent motions of molecular moieties in the condensed phase can retain quantum character over much broader temperature range than is commonly thought. PMID:26451661
7. MASTER TELEVISION ANTENNA SYSTEM.
ERIC Educational Resources Information Center
Rhode Island State Dept. of Education, Providence.
SPECIFICATIONS FOR THE FURNISHING AND INSTALLATION OF TELEVISION MASTER ANTENNA SYSTEMS FOR SECONDARY AND ELEMENTARY SCHOOLS ARE GIVEN. CONTRACTOR REQUIREMENTS, EQUIPMENT, PERFORMANCE STANDARDS, AND FUNCTIONS ARE DESCRIBED. (MS)
8. Generalized Korteweg-de Vries equation induced from position-dependent effective mass quantum models and mass-deformed soliton solution through inverse scattering transform
SciTech Connect
Ganguly, A. E-mail: [email protected]; Das, A.
2014-11-15
We consider one-dimensional stationary position-dependent effective mass quantum model and derive a generalized Korteweg-de Vries (KdV) equation in (1+1) dimension through Lax pair formulation, one being the effective mass Schrödinger operator and the other being the time-evolution of wave functions. We obtain an infinite number of conserved quantities for the generated nonlinear equation and explicitly show that the new generalized KdV equation is an integrable system. Inverse scattering transform method is applied to obtain general solution of the nonlinear equation, and then N-soliton solution is derived for reflectionless potentials. Finally, a special choice has been made for the variable mass function to get mass-deformed soliton solution. The influence of position and time-dependence of mass and also of the different representations of kinetic energy operator on the nature of such solitons is investigated in detail. The remarkable features of such solitons are demonstrated in several interesting figures and are contrasted with the conventional KdV-soliton associated with constant-mass quantum model.
9. Interior design. Mastering the master plan.
PubMed
Mesbah, C E
1995-10-01
Reflecting on the results of the survey, this proposed interior design master planning process addresses the concerns and issues of both CEOs and facility managers in ways that focus on problem-solving strategies and methods. Use of the interior design master plan process further promotes the goals and outcomes expressed in the survey by both groups. These include enhanced facility image, the efficient selection of finishes and furnishings, continuity despite staff changes, and overall savings in both costs and time. The interior design master plan allows administrators and facility managers to anticipate changes resulting from the restructuring of health care delivery. The administrators and facility managers are then able to respond in ways that manage those changes in the flexible and cost-effective manner they are striving for. This framework permits staff members to concentrate their time and energy on the care of their patients--which is, after all, what it's all about.
10. Equations of state and stability of MgSiO3 perovskite and post-perovskite phases from quantum Monte Carlo simulations
SciTech Connect
Lin, Yangzheng; Cohen, Ronald E.; Stackhouse, Stephen; Driver, Kevin P.; Militzer, Burkhard; Shulenburger, Luke; Kim, Jeongnim
2014-11-10
In this study, we have performed quantum Monte Carlo (QMC) simulations and density functional theory calculations to study the equations of state of MgSiO3 perovskite (Pv, bridgmanite) and post-perovskite (PPv) up to the pressure and temperature conditions of the base of Earth's lower mantle. The ground-state energies were derived using QMC simulations and the temperature-dependent Helmholtz free energies were calculated within the quasiharmonic approximation and density functional perturbation theory. The equations of state for both phases of MgSiO3 agree well with experiments, and better than those from generalized gradient approximation calculations. The Pv-PPv phase boundary calculated from our QMC equations of state is also consistent with experiments, and better than previous local density approximation calculations. Lastly, we discuss the implications for double crossing of the Pv-PPv boundary in the Earth.
11. Decoherence and dissipation of a quantum harmonic oscillator coupled to two-level systems
SciTech Connect
Schlosshauer, Maximilian; Hines, A. P.; Milburn, G. J.
2008-02-15
We derive and analyze the Born-Markov master equation for a quantum harmonic oscillator interacting with a bath of independent two-level systems. This hitherto virtually unexplored model plays a fundamental role as one of the four 'canonical' system-environment models for decoherence and dissipation. To investigate the influence of further couplings of the environmental spins to a dissipative bath, we also derive the master equation for a harmonic oscillator interacting with a single spin coupled to a bosonic bath. Our models are experimentally motivated by quantum-electromechanical systems and micron-scale ion traps. Decoherence and dissipation rates are found to exhibit temperature dependencies significantly different from those in quantum Brownian motion. In particular, the systematic dissipation rate for the central oscillator decreases with increasing temperature and goes to zero at zero temperature, but there also exists a temperature-independent momentum-diffusion (heating) rate.
12. Calculation of wave-functions with frozen orbitals in mixed quantum mechanics/molecular mechanics methods. Part I. Application of the Huzinaga equation.
PubMed
Ferenczy, György G
2013-04-01
Mixed quantum mechanics/quantum mechanics (QM/QM) and quantum mechanics/molecular mechanics (QM/MM) methods make computations feasible for extended chemical systems by separating them into subsystems that are treated at different level of sophistication. In many applications, the subsystems are covalently bound and the use of frozen localized orbitals at the boundary is a possible way to separate the subsystems and to ensure a sensible description of the electronic structure near to the boundary. A complication in these methods is that orthogonality between optimized and frozen orbitals has to be warranted and this is usually achieved by an explicit orthogonalization of the basis set to the frozen orbitals. An alternative to this approach is proposed by calculating the wave-function from the Huzinaga equation that guaranties orthogonality to the frozen orbitals without basis set orthogonalization. The theoretical background and the practical aspects of the application of the Huzinaga equation in mixed methods are discussed. Forces have been derived to perform geometry optimization with wave-functions from the Huzinaga equation. Various properties have been calculated by applying the Huzinaga equation for the central QM subsystem, representing the environment by point charges and using frozen strictly localized orbitals to connect the subsystems. It is shown that a two to three bond separation of the chemical or physical event from the frozen bonds allows a very good reproduction (typically around 1 kcal/mol) of standard Hartree-Fock-Roothaan results. The proposed scheme provides an appropriate framework for mixed QM/QM and QM/MM methods.
13. Calculation of wave-functions with frozen orbitals in mixed quantum mechanics/molecular mechanics methods. Part I. Application of the Huzinaga equation.
PubMed
Ferenczy, György G
2013-04-01
Mixed quantum mechanics/quantum mechanics (QM/QM) and quantum mechanics/molecular mechanics (QM/MM) methods make computations feasible for extended chemical systems by separating them into subsystems that are treated at different level of sophistication. In many applications, the subsystems are covalently bound and the use of frozen localized orbitals at the boundary is a possible way to separate the subsystems and to ensure a sensible description of the electronic structure near to the boundary. A complication in these methods is that orthogonality between optimized and frozen orbitals has to be warranted and this is usually achieved by an explicit orthogonalization of the basis set to the frozen orbitals. An alternative to this approach is proposed by calculating the wave-function from the Huzinaga equation that guaranties orthogonality to the frozen orbitals without basis set orthogonalization. The theoretical background and the practical aspects of the application of the Huzinaga equation in mixed methods are discussed. Forces have been derived to perform geometry optimization with wave-functions from the Huzinaga equation. Various properties have been calculated by applying the Huzinaga equation for the central QM subsystem, representing the environment by point charges and using frozen strictly localized orbitals to connect the subsystems. It is shown that a two to three bond separation of the chemical or physical event from the frozen bonds allows a very good reproduction (typically around 1 kcal/mol) of standard Hartree-Fock-Roothaan results. The proposed scheme provides an appropriate framework for mixed QM/QM and QM/MM methods. PMID:23281055
14. MASTER: optical transients
Gress, O.; Balanutsa, P.; Shumkov, V.; Lipunov, V.; Buckley, D.; Rebolo, R.; Ricart, M. Serra; Podesta, R.; Levato, H.; Gorbovskoy, E.; Tiurina, N.; Kuznetsov, A.; Kornilov, V.; Pogrosheva, T.; Chazov, V.; Vladimirov, V.; Vlasenko, D.; Potter, S.; Lopez, C.; Podesta, F.; Saffe, C.
2016-10-01
MASTER-SAAO auto-detection system ( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 30L ) discovered OT source at (RA, Dec) = 14h 38m 49.60s -44d 37m 24.5s on 2016-10-01.73438 UT with unfiltered m_OT=16.4m (mlim=19.7m).
15. MASTER Net: optical transients
Gress, O.; Shumkov, V.; Pogrosheva, T.; Shurpakov, S.; Lipunov, V.; Buckley, D.; Lopez, R. Rebolo; Ricart, M. Serra; Podesta, R.; Levato, H. O.; Gorbovskoy, E.; Tiurina, N.; Balanutsa, P.; Kuznetsov, A.; Chazov, V.; Kornilov, V.; Ivanov, K.; Vladimirov, V.; Potter, S.; Lopez, C.; Podesta, F.; Saffe, C.
2016-10-01
MASTER-SAAO auto-detection system ( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 30L ) discovered OT source at (RA, Dec) = 14h 38m 49.60s -44d 37m 24.5s on 2016-10-01.73438 UT with unfiltered m_OT=16.4m (mlim=19.7m).
16. Master: 3 OT
Balanutsa, P.; Lipunov, V.; Buckley, D.; Tlatov, A.; Gorbovskoy, E.; Tyurina, N.; Kuznetsov, A.; Kornilov, V.; Kuvshinov, D.; Popova, E.; Vlasenko, D.; Shumkov, V.; Potter, S.; Kniazev, A.; Budnev, N.; Gress, O.; Ivanov, K.; Senik, V.; Dormidontov, D.; Parhomenko, A. V.
2016-08-01
MASTER-Kislovodsk auto-detection system ( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 30L ) discovered OT source at (RA, Dec) = 21h 33m 50.58s +06d 51m 22.5s on 2016-07-27.94690 UT. The OT unfiltered magnitude is 17.8m (limit 18.0m).
17. MASTER: 2 optical transients
Vladimirov, V.; Shumkov, V.; Lipunov, V.; Buckley, D.; Lopez, R. Rebolo; Serra-Ricart, M.; Gorbovskoy, E.; Tiurina, N.; Balanutsa, P.; Kornilov, V.; Kuvshinov, D.; Kuznetsov, A.; Chazov, V.
2016-08-01
MASTER-SAAO auto-detection system ( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 30L ) discovered OT source at (RA, Dec) = 15h 25m 39.43s -70d 22m 45.1s on 2016-08-16.80259 UT. The OT unfiltered magnitude is 17.0m (mlim=17.9m).
18. MASTER: optical transients
Balanutsa, P.; Gress, O.; Lipunov, V.; Buckley, D.; Lopez, R. Rebolo; Serra-Ricart, M.; Gorbovskoy, E.; Popova, E.; Kuvshinov, D.; Kuznetsov, A.; Kornilov, V.; Vlasenko, D.; Gress, O.; Shurpakov, S.; Potter, S.; Kniazev, A.
2016-07-01
MASTER-IAC auto-detection system ( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 349171 ) discovered OT source at (RA, Dec) = 19h 45m 07.47s -20d 07m 20.5s on 2016-07-11.00663 UT. The OT unfiltered magnitude is (limit 18.8m).
19. MASTER optical transients
Balanutsa, P.; Shumkov, V.; Gorbovskoy, E.; Lipunov, V.; Buckley, D.; Potter, S.; Rebolo, R.; Serra-Ricart, M.; Israelyan, G.; Tiurina, N.; Kornilov, V.; Gorbunv, I.; Popova, E.; Vladimirov, V.; Kuvshinov, D.; Budnev, N.; Gress, O.; Ivanov, K.
2016-05-01
MASTER-SAAO auto-detection system ( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 349171 ) discovered OT source at (RA, Dec) = 15h 31m 05.22s -26d 04m 37.2s on 2016-05-07.99397 UT. The OT unfiltered magnitude is 17.7m (limit 20.0m).
20. MASTER: 3 optical transients
Balanutsa, P.; Pogrosheva, T.; Lipunov, V.; Buckley, D.; Gorbovskoy, E.; Tiurina, N.; Kuznetsov, A.; Kornilov, V.; Shumkov, V.; Gorbunov, I.; Gress, O.; Kniazev, S. Potter A.
2016-04-01
MASTER-SAAO auto-detection system ( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 349171 ) discovered OT source at (RA, Dec) = 10h 34m 32.65s -38d 51m 35.3s on 2016-04-15.84756 UT. The OT unfiltered magnitude is 17.5m (limit 19.3m).
1. MASTER: 6 optical transients
Gress, O.; Balanutsa, P.; Vladimirov, V.; Lipunov, V.; Buckley, D.; Tlatov, A.; Gorbovskoy, E.; Tiurina, N.; Kuznetsov, A.; Kornilov, V.; Gorbunov, I.; Shumkov, V.; Kochutina, N.; Potter, S.; Kniazev, A.; Senik, V.; Dormidontov, D.; Parkhomenko, A.
2016-06-01
MASTER-IAC auto-detection system (Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 30L) discovered OT source at (RA, Dec) = 20h 03m 08.64s +02d 34m 10.4s on 2016-06-02.14910 UT with unfiltered m_OT=16.3m (mlim=18.8m).
2. Evolution of quantum-like modeling in decision making processes
SciTech Connect
Khrennikova, Polina
2012-12-18
The application of the mathematical formalism of quantum mechanics to model behavioral patterns in social science and economics is a novel and constantly emerging field. The aim of the so called 'quantum like' models is to model the decision making processes in a macroscopic setting, capturing the particular 'context' in which the decisions are taken. Several subsequent empirical findings proved that when making a decision people tend to violate the axioms of expected utility theory and Savage's Sure Thing principle, thus violating the law of total probability. A quantum probability formula was devised to describe more accurately the decision making processes. A next step in the development of QL-modeling in decision making was the application of Schroedinger equation to describe the evolution of people's mental states. A shortcoming of Schroedinger equation is its inability to capture dynamics of an open system; the brain of the decision maker can be regarded as such, actively interacting with the external environment. Recently the master equation, by which quantum physics describes the process of decoherence as the result of interaction of the mental state with the environmental 'bath', was introduced for modeling the human decision making. The external environment and memory can be referred to as a complex 'context' influencing the final decision outcomes. The master equation can be considered as a pioneering and promising apparatus for modeling the dynamics of decision making in different contexts.
3. Evolution of quantum-like modeling in decision making processes
Khrennikova, Polina
2012-12-01
The application of the mathematical formalism of quantum mechanics to model behavioral patterns in social science and economics is a novel and constantly emerging field. The aim of the so called 'quantum like' models is to model the decision making processes in a macroscopic setting, capturing the particular 'context' in which the decisions are taken. Several subsequent empirical findings proved that when making a decision people tend to violate the axioms of expected utility theory and Savage's Sure Thing principle, thus violating the law of total probability. A quantum probability formula was devised to describe more accurately the decision making processes. A next step in the development of QL-modeling in decision making was the application of Schrödinger equation to describe the evolution of people's mental states. A shortcoming of Schrödinger equation is its inability to capture dynamics of an open system; the brain of the decision maker can be regarded as such, actively interacting with the external environment. Recently the master equation, by which quantum physics describes the process of decoherence as the result of interaction of the mental state with the environmental 'bath', was introduced for modeling the human decision making. The external environment and memory can be referred to as a complex 'context' influencing the final decision outcomes. The master equation can be considered as a pioneering and promising apparatus for modeling the dynamics of decision making in different contexts.
4. Combined rate equation and Monte Carlo studies of electron transport in a GaAs/Al0.45Ga0.55As quantum-cascade laser
Borowik, Piotr; Thobel, Jean-Luc; Adamowicz, Leszek
2012-11-01
Comparison of the Monte Carlo and rate equation methods as applied to the study of electron transport in a mid-infrared quantum cascade laser structure initially proposed by Page et al (2001 Appl. Phys. Lett. 78 3529) is presented for a range of realistic injector doping levels. An analysis of the difference between these two methods is given. It is suggested that justified approximations of the rate equation method, originated by imposing Fermi-Dirac statistics and the same electron effective temperature for each of the energy sub-bands, can be interpreted as partial inclusion of electron-electron interactions. Results of the rate equation method may be used as good initial conditions for a more precise Monte Carlo simulation. An algorithm combining rate equation and Monte Carlo simulations is examined. A reasonable agreement between the introduced method and a fully self-consistent resolution of Monte Carlo and Schrödinger coupled with Poisson equations is demonstrated. The computation time may be reduced when the combined algorithm is used.
5. Analysis of low efficiency droop of semipolar InGaN quantum well light-emitting diodes by modified rate equation with weak phase-space filling effect
Fu, Houqiang; Lu, Zhijian; Zhao, Yuji
2016-06-01
We study the low efficiency droop characteristics of semipolar InGaN light-emitting diodes (LEDs) using modified rate equation incoporating the phase-space filling (PSF) effect where the results on c-plane LEDs are also obtained and compared. Internal quantum efficiency (IQE) of LEDs was simulated using a modified ABC model with different PSF filling (n0), Shockley-Read-Hall (A), radiative (B), Auger (C) coefficients and different active layer thickness (d), where the PSF effect showed a strong impact on the simulated LED efficiency results. A weaker PSF effect was found for low-droop semipolar LEDs possibly due to small quantum confined Stark effect, short carrier lifetime, and small average carrier density. A very good agreement between experimental data and the theoretical modeling was obtained for low-droop semipolar LEDs with weak PSF effect. These results suggest the low droop performance may be explained by different mechanisms for semipolar LEDs.
6. BOOK REVIEW Quantum Measurement and Control Quantum Measurement and Control
Kiefer, Claus
2010-12-01
In the last two decades there has been an enormous progress in the experimental investigation of single quantum systems. This progress covers fields such as quantum optics, quantum computation, quantum cryptography, and quantum metrology, which are sometimes summarized as quantum technologies'. A key issue there is entanglement, which can be considered as the characteristic feature of quantum theory. As disparate as these various fields maybe, they all have to deal with a quantum mechanical treatment of the measurement process and, in particular, the control process. Quantum control is, according to the authors, control for which the design requires knowledge of quantum mechanics'. Quantum control situations in which measurements occur at important steps are called feedback (or feedforward) control of quantum systems and play a central role here. This book presents a comprehensive and accessible treatment of the theoretical tools that are needed to cope with these situations. It also provides the reader with the necessary background information about the experimental developments. The authors are both experts in this field to which they have made significant contributions. After an introduction to quantum measurement theory and a chapter on quantum parameter estimation, the central topic of open quantum systems is treated at some length. This chapter includes a derivation of master equations, the discussion of the Lindblad form, and decoherence - the irreversible emergence of classical properties through interaction with the environment. A separate chapter is devoted to the description of open systems by the method of quantum trajectories. Two chapters then deal with the central topic of quantum feedback control, while the last chapter gives a concise introduction to one of the central applications - quantum information. All sections contain a bunch of exercises which serve as a useful tool in learning the material. Especially helpful are also various separate
7. Simple rate equation model for hypothetical doubly stimulated emission of both photons and phonons in quantum-well lasers
SciTech Connect
Kroemer, H.
1981-06-15
The dissipation processes by which electrons and holes lose energy after being trapped in quantum wells might, in a sufficiently heavily pumped quantum well laser, lead to the buildup of such a high phonon population that phonon-assisted laser action by doubly stimulated emission of photons and phonons acquires a higher gain than unassisted laser action. The resulting mode switching exhibits a pronounced hysteresis with pump rate, which should be a characteristic identifying feature of phonon-assisted laser action.
8. Phase dependent loading of Bloch bands and quantum simulation of relativistic wave equation predictions with ultracold atoms in variably shaped optical lattice potentials
Grossert, Christopher; Leder, Martin; Weitz, Martin
2016-10-01
The dispersion relation of ultracold atoms in variably shaped optical lattices can be tuned to resemble that of a relativistic particle, i.e. be linear instead of the usual nonrelativistic quadratic dispersion relation of a free atom. Cold atoms in such a lattice can be used to carry out quantum simulations of relativistic wave equation predictions. We begin this article by describing a Raman technique that allows to selectively load atoms into a desired Bloch band of the lattice near a band crossing. Subsequently, we review two recent experiments with quasirelativistic rubidium atoms in a bichromatic lattice, demonstrating the analogues of Klein tunnelling and Veselago lensing with ultracold atoms, respectively.
9. Quantum discord of the two-atom system in non-Markovian environments
Zou, Hong-Mei; Fang, Mao-Fa; Guo, You-Neng; Yang, Bai-Yuan
2015-03-01
The quantum discord of the two-atom system, which is in two independent Lorentzian reservoirs and in two independent Ohmic reservoirs with the Lorentz-Drude cutoff function, respectively, and the reservoirs are at zero temperature, is studied by applying the time-convolutionless master-equation method. We find that the quantum discord of the two-atom system is dependent on the characteristics of non-Markovian environments. The results show that the quantum discord can be effectively protected not only in Lorentzian reservoirs, but also in ohmic reservoirs with the Lorentz-Drude cutoff function. Finally, the physical interpretations for these results are given via the correlation function.
10. Quantum Fisher information flow and non-Markovian processes of open systems
SciTech Connect
Lu Xiaoming; Wang Xiaoguang; Sun, C. P.
2010-10-15
We establish an information-theoretic approach for quantitatively characterizing the non-Markovianity of open quantum processes. Here, the quantum Fisher information (QFI) flow provides a measure to statistically distinguish Markovian and non-Markovian processes. A basic relation between the QFI flow and non-Markovianity is unveiled for quantum dynamics of open systems. For a class of time-local master equations, the exactly analytic solution shows that for each fixed time the QFI flow is decomposed into additive subflows according to different dissipative channels.
11. Equations of state and stability of MgSiO3 perovskite and post-perovskite phases from quantum Monte Carlo simulations
DOE PAGES
Lin, Yangzheng; Cohen, Ronald E.; Stackhouse, Stephen; Driver, Kevin P.; Militzer, Burkhard; Shulenburger, Luke; Kim, Jeongnim
2014-11-10
In this study, we have performed quantum Monte Carlo (QMC) simulations and density functional theory calculations to study the equations of state of MgSiO3 perovskite (Pv, bridgmanite) and post-perovskite (PPv) up to the pressure and temperature conditions of the base of Earth's lower mantle. The ground-state energies were derived using QMC simulations and the temperature-dependent Helmholtz free energies were calculated within the quasiharmonic approximation and density functional perturbation theory. The equations of state for both phases of MgSiO3 agree well with experiments, and better than those from generalized gradient approximation calculations. The Pv-PPv phase boundary calculated from our QMC equationsmore » of state is also consistent with experiments, and better than previous local density approximation calculations. Lastly, we discuss the implications for double crossing of the Pv-PPv boundary in the Earth.« less
12. Surface enhanced quantum control of a two-level system
Rangan, Chitra; Mirzaee, Somayeh M. A.
2012-06-01
We demonstrate the enhanced purification of the quantum state of a two-level system subject to a near-resonant driving field when in proximity to a gold nanoparticle. The quantum dynamics of the driven two-level system in the presence of decay is modelled by the Lindblad Master equation. The electrodynamics of the gold nanoparticle illuminated by the driving field and the field radiated by the atomic dipole is solved using a finite-difference time-domain method. We discover that the presence of a proximate gold nanoparticle enhances the purity of a driven two-level system even at short times.
13. Quantum turing machine and brain model represented by Fock space
Iriyama, Satoshi; Ohya, Masanori
2016-05-01
The adaptive dynamics is known as a new mathematics to treat with a complex phenomena, for example, chaos, quantum algorithm and psychological phenomena. In this paper, we briefly review the notion of the adaptive dynamics, and explain the definition of the generalized Turing machine (GTM) and recognition process represented by the Fock space. Moreover, we show that there exists the quantum channel which is described by the GKSL master equation to achieve the Chaos Amplifier used in [M. Ohya and I. V. Volovich, J. Opt. B 5(6) (2003) 639., M. Ohya and I. V. Volovich, Rep. Math. Phys. 52(1) (2003) 25.
14. The master science teacher
Toh, Kok-Aun; Tsoi, Mun-Fie
2008-11-01
The dire need of some schools to boost the academic performance of their students inevitably rests with their ability to attract highly qualified teachers. As such, the UK has put in place the Advanced Skills Teacher (AST) scheme, while the US has set the ball rolling in laying down standards for the certification of the master science teacher, to distil the best teachers from this pool of highly qualified science teachers. This article examines some of the practices in the teaching-learning process one would associate with those of master science teachers. It argues for the master science teacher to have a well-developed pedagogical content knowledge rather than be an expert in content knowledge. It argues for the master science teacher to be someone who has moved 'from personal comprehension to preparing for the comprehension of others' (Shulman 1987 Harv. Educ. Rev. 75 1-22).
15. Acronym master list
SciTech Connect
1995-06-01
This document is a master list of acronyms and other abbreviations that are used by or could be useful to, the personnel at Los Alamos National Laboratory. Many specialized and well-known abbreviations are not included in this list.
16. Natural approach to quantum dissipation
Taj, David; Öttinger, Hans Christian
2015-12-01
The dissipative dynamics of a quantum system weakly coupled to one or several reservoirs is usually described in terms of a Lindblad generator. The popularity of this approach is certainly due to the linear character of the latter. However, while such linearity finds justification from an underlying Hamiltonian evolution in some scaling limit, it does not rely on solid physical motivations at small but finite values of the coupling constants, where the generator is typically used for applications. The Markovian quantum master equations we propose are instead supported by very natural thermodynamic arguments. They themselves arise from Markovian master equations for the system and the environment which preserve factorized states and mean energy and generate entropy at a non-negative rate. The dissipative structure is driven by an entropic map, called modular, which introduces nonlinearity. The generated modular dynamical semigroup (MDS) guarantees for the positivity of the time evolved state the correct steady state properties, the positivity of the entropy production, and a positive Onsager matrix with symmetry relations arising from Green-Kubo formulas. We show that the celebrated Davies Lindblad generator, obtained through the Born and the secular approximations, generates a MDS. In doing so we also provide a nonlinear MDS which is supported by a weak coupling argument and is free from the limitations of the Davies generator.
17. Part i: Lie-Backlund Theory and Linearization of Differential Equations. Part II: Monte Carlo Simulations of 1-D Quantum Spin Models.
Cullen, John J.
Part I begins with an account of groups of Lie -Back-lund (L-B) tangent transformations; it is then shown that L-B symmetry operators depending on integrals (nonlocal variables), such as discussed by Konopelchenko and Mokhnachev (1979), are related by change of variables to the L-B operators which involve no more than derivatives. A general method is set down for transforming a given L-B operator into a new one, by any invertible transformation depending on (. . ., D(,x)('-1) u, u, u(,x), . . .). It is shown that once a given differential equation admits a L-B operator, there is in general a very large number of related ("secondary") equations which admit the same operator. The L-B Theory involving nonlocal variables is used to characterize group theoretically the linearization both of the Burgers equation, u(,t) + uu(,x) - u(,xx) = 0, and of the o.d.e. u(,xx) + (omega)('2)(x)u + Ku('-3) = 0. Secondary equations are found to play an important role in understanding the group theoretical background to the linearization of differential equations. Part II deals with Monte Carlo simulations of the l-d quantum Heisenberg and XY-models, using an approach suggested by Suzuki (1976). The simulation is actually carried out on a 2-d, m x N, Isinglike system, equivalent to the original N-spin quantum system when m (--->) (INFIN). The results for m (LESSTHEQ) 10 and kT/(VBAR)J(VBAR) (GREATERTHEQ) .0125 are good enough to show that the method is generally applicable to quantum spin models; however some difficulties caused by singular bonding in the classical lattice (Wiesler 1982) and by the generation of unwanted states have to be taken into account in practice. The finite-size scaling method of Fisher and Ferdinard is adapted for use near T = 0 in the ferromagnetic Heisenberg model; applied to the simulation data it shows that the low temperature susceptibiltiy behaves at T('-(gamma)), where (gamma) = 1.32 (+OR-) 10%. Also, simple and potentially useful finite-size scaling
18. IDC Integrated Master Plan.
SciTech Connect
Clifford, David J.; Harris, James M.
2014-12-01
This is the IDC Re-Engineering Phase 2 project Integrated Master Plan (IMP). The IMP presents the major accomplishments planned over time to re-engineer the IDC system. The IMP and the associate Integrated Master Schedule (IMS) are used for planning, scheduling, executing, and tracking the project technical work efforts. REVISIONS Version Date Author/Team Revision Description Authorized by V1.0 12/2014 IDC Re- engineering Project Team Initial delivery M. Harris
19. Three-terminal quantum-dot refrigerators
Zhang, Yanchao; Lin, Guoxing; Chen, Jincan
2015-05-01
Based on two capacitively coupled quantum dots in the Coulomb-blockade regime, a model of three-terminal quantum-dot refrigerators is proposed. With the help of the master equation, the transport properties of steady-state charge current and energy flow between two quantum dots and thermal reservoirs are revealed. It is expounded that such a structure can be used to construct a refrigerator by controlling the voltage bias and temperature ratio. The thermodynamic performance characteristics of the refrigerator are analyzed, including the cooling power, coefficient of performance (COP), maximum cooling power, and maximum COP. Moreover, the optimal regions of main performance parameters are determined. The influence of dissipative tunnel processes on the optimal performance is discussed in detail. Finally, the performance characteristics of the refrigerators operated in two different cases are compared.
20. Equation of state of a dense plasma by orbital-free and quantum molecular dynamics: Examination of two isothermal-isobaric mixing rules
Danel, J.-F.; Kazandjian, L.
2015-01-01
We test two isothermal-isobaric mixing rules, respectively based on excess-pressure and total-pressure equilibration, applied to the equation of state of a dense plasma. While the equation of state is generally known for pure species, that of arbitrary mixtures is not available so that the validation of accurate mixing rules, that implies resorting to first-principles simulations, is very useful. Here we consider the case of a plastic with composition C2H3 and we implement two complementary ab initio approaches adapted to the dense plasma domain: quantum molecular dynamics, limited to low temperature by its computational cost, and orbital-free molecular dynamics, that can be implemented at high temperature. The temperature and density range considered is 1-10 eV and 0.6-10 g/cm 3 for quantum molecular dynamics, and 5-1000 eV and 1-10 g/cm 3 for orbital-free molecular dynamics. Simulations for the full C2H3 mixture are the benchmark against which to assess the mixing rules, and both pressure and internal energy are compared. We find that the mixing rule based on excess-pressure equilibration is overall more accurate than that based on total-pressure equilibration; except for quantum molecular dynamics and a thermodynamic domain characterized by very low or negative excess pressures, it gives pressures which are generally within statistical error or within 1% of the exact ones. Besides, its superiority is amplified in the calculation of a principal Hugoniot.
1. Non-Markovian correlation functions for open quantum systems
Jin, Jinshuang; Karlewski, Christian; Marthaler, Michael
2016-08-01
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. We approach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non-Markovian characteristics are demonstrated.
2. Non-Markovian quantum Brownian motion of a harmonic oscillator
SciTech Connect
Tang, J.
1994-02-01
We apply the density-matrix method to the study of quantum Brownian motion of a harmonic oscillator coupled to a heat bath, a system investigated previously by Caldeira and Leggett using a different method. Unlike the earlier work, in our derivation of the master equation the non-Markovian terms are maintained. Although the same model of interaction is used, discrepancy is found between their results and our equation in the Markovian limit. We also point out that the particular interaction model used by both works cannot lead to the phenomenological generalized Langevin theory of Kubo.
3. The Masters Athlete
PubMed Central
Tayrose, Gregory A.; Beutel, Bryan G.; Cardone, Dennis A.; Sherman, Orrin H.
2015-01-01
Context: With the ever-increasing number of masters athletes, it is necessary to understand how to best provide medical support to this expanding population using a multidisciplinary approach. Evidence Acquisition: Relevant articles published between 2000 and 2013 using the search terms masters athlete and aging and exercise were identified using MEDLINE. Study Design: Clinical review. Level of Evidence: Level 3. Results: Preparticipation screening should assess a variety of medical comorbidities, with emphasis on cardiovascular health in high-risk patients. The masters athlete should partake in moderate aerobic exercise and also incorporate resistance and flexibility training. A basic understanding of physiology and age-related changes in muscle composition and declines in performance are prerequisites for providing appropriate care. Osteoarthritis and joint arthroplasty are not contraindications to exercise, and analgesia has an appropriate role in the setting of acute or chronic injuries. Masters athletes should follow regular training regimens to maximize their potential while minimizing their likelihood of injuries. Conclusion: Overall, masters athletes represent a unique population and should be cared for utilizing a multidisciplinary approach. This care should be implemented not only during competitions but also between events when training and injury are more likely to occur. Strength of Recommendation Taxonomy (SORT): B. PMID:26131307
4. Inner Products of Energy Eigenstates for a 1-D Quantum Barrier
Julve, J.; Turrini, S.; de Urríes, F. J.
2013-11-01
The features of the standard inner products between all the types of real and complex-energy solutions of the Schrödinger equation for 1-dimensional cut-off quantum potentials are worked out using a Gaussian regularization. A general Master Solution is introduced which describes any of the above solutions as particular cases. From it, a Master Inner Product is obtained which yields all the particular products. We show that the Outgoing and the Incoming Boundary Conditions fully determine the location of the momenta respectively in the lower and upper half complex plane even for purely imaginary momenta (anti-bound and bound solutions).
5. Hydrodynamic simulations of relativistic heavy-ion collisions with different lattice quantum chromodynamics calculations of the equation of state
Moreland, J. Scott; Soltz, Ron A.
2016-04-01
Hydrodynamic calculations of ultrarelativistic heavy-ion collisions are performed using the iebe-vishnu 2+1-dimensional code with fluctuating initial conditions and three different parametrizations of the lattice QCD equations of state: continuum extrapolations for stout and HISQ/tree actions, as well as the s95p-v1 parametrization based upon calculations using the p4 action. All parametrizations are matched to a hadron resonance gas equation of state at T =155 MeV, at which point the calculations are continued using the urqmd hadronic cascade. Simulations of √{sN N}=200 GeV Au+Au collisions in three centrality classes are used to quantify anisotropic flow developed in the hydrodynamic phase of the collision as well as particle spectra and pion Hanbury-Brown-Twiss (HBT) radii after hadronic rescattering, which are compared with experimental data. Experimental observables for the stout and HISQ/tree equations of state are observed to differ by less than a few percent for all observables, while the s95p-v1 equation of state generates spectra and flow coefficients which differ by ˜10 -20 % . Calculations in which the HISQ/tree equation of state is sampled from the published error distribution are also observed to differ by less than a few percent.
6. A Nanosize Quantum-Dot Photoelectric Refrigerator
Li, Cong; Zhang, Yan-Chao; He, Ji-Zhou
2013-10-01
We investigate the thermodynamic performance of a nanosized photoelectric refrigerator consisting of two single energy levels embedded between two reservoirs at different temperatures. Based on the quantum master equation, expressions for the cooling power and coefficient of performance (COP) of the refrigerator are derived. The characteristic curves between the cooling power and COP are plotted. Moreover, the optimal performance parameters are analyzed by the numerical calculation and graphic method. The influence of the nonradiative processes on the performance characteristics and optimal performance parameters are discussed in detail.
7. Continuous quantum measurement in spin environments
Xie, Dong; Wang, An Min
2015-08-01
We derive a stochastic master equation (SME) which describes the decoherence dynamics of a system in spin environments conditioned on the measurement record. Markovian and non-Markovian nature of environment can be revealed by a spectroscopy method based on weak continuous quantum measurement. On account of that correlated environments can lead to a non-local open system which exhibits strong non-Markovian effects although the local dynamics are Markovian, the spectroscopy method can be used to demonstrate that there is correlation between two environments.
8. Quantum transport equation for systems with rough surfaces and its application to ultracold neutrons in a quantizing gravity field
SciTech Connect
Escobar, M.; Meyerovich, A. E.
2014-12-15
We discuss transport of particles along random rough surfaces in quantum size effect conditions. As an intriguing application, we analyze gravitationally quantized ultracold neutrons in rough waveguides in conjunction with GRANIT experiments (ILL, Grenoble). We present a theoretical description of these experiments in the biased diffusion approximation for neutron mirrors with both one- and two-dimensional (1D and 2D) roughness. All system parameters collapse into a single constant which determines the depletion times for the gravitational quantum states and the exit neutron count. This constant is determined by a complicated integral of the correlation function (CF) of surface roughness. The reliable identification of this CF is always hindered by the presence of long fluctuation-driven correlation tails in finite-size samples. We report numerical experiments relevant for the identification of roughness of a new GRANIT waveguide and make predictions for ongoing experiments. We also propose a radically new design for the rough waveguide.
9. Quantum state transfer in optomechanical arrays
de Moraes Neto, G. D.; Andrade, F. M.; Montenegro, V.; Bose, S.
2016-06-01
Quantum state transfer between distant nodes is at the heart of quantum processing and quantum networking. Stimulated by this, we propose a scheme where one can achieve quantum state transfer with a high fidelity between sites in a cavity quantum optomechanical network. In our lattice, each individual site is composed of a localized mechanical mode which interacts with a laser-driven cavity mode via radiation pressure, while photons hop between neighboring sites. After diagonalization of the Hamiltonian of each cell, we show that the system can be reduced to an effective Hamiltonian of two decoupled bosonic chains, and therefore we can apply the well-known results in quantum state transfer together with an additional condition on the transfer times. In fact, we show that our transfer protocol works for any arbitrary joint quantum state of a mechanical and an optical mode. Finally, in order to analyze a more realistic scenario we take into account the effects of independent thermal reservoirs for each site. By solving the standard master equation within the Born-Markov approximation, we reassure both the effective model and the feasibility of our protocol.
10. MASTER: dwarf nova outburst
Popova, E.; Gress, O.; Lipunov, V.; Gabovich, A.; Yurkov, V.; Sergienko, Yu.; Budnev, N.; Ivanov, K.; Gorbovskoy, E.; Tiurina, N.; Balanutsa, P.; Kuznetsov, A.; Kornilov, V.
2016-07-01
MASTER-Amur auto-detection system ( Lipunov et al., "MASTER Global Robotic Net", Advances in Astronomy, 2010, 349171 ) discovered OT source at (RA, Dec) = 05h 00m 02.2s +74d 04m 07.4s on 2016-07-06 16:47:02.264UT with unfiltered(0.2B2+0.8R2 calibrated to USNO B1) m_OT=16.9. The OT is seen on 6 images. There is no minor planet at this place.
11. Thick Photoresist Original Master:
Mizuno, Hirotaka; Sugihara, Okihiro; Kaino, Toshikuni; Ohe, Yuka; Okamoto, Naomichi; Hoshino, Masahito
A simple and low-cost fabrication method of polymeric optical waveguides with large core sizes for plastic optical fibers is presented. The waveguides are fabricated by hot embossing with a rectangular ridge ultraviolet (UV)-cured epoxy resin stamper. The stamper is fabricated by replication of a rectangular groove mold that is made from silicone rubber replicated from a rectangular ridge original master made from thick photoresist (SU-8). A rectangular ridge shape of the original photoresist master of 1 mm size was realized by using a flattening process, which involves hot embossing before the exposure process and using a UV-cut filter during the exposure process.
12. ELECTRONIC MASTER SLAVE MANIPULATOR
DOEpatents
Goertz, R.C.; Thompson, Wm.M.; Olsen, R.A.
1958-08-01
A remote control manipulator is described in which the master and slave arms are electrically connected to produce the desired motions. A response signal is provided in the master unit in order that the operator may sense a feel of the object and may not thereby exert such pressures that would ordinarily damage delicate objects. This apparatus will permit the manipulation of objects at a great distance, that may be viewed over a closed TV circuit, thereby permitting a remote operator to carry out operations in an extremely dangerous area with complete safety.
13. Generalized Quantum Theory and Mathematical Foundations of Quantum Field Theory
Maroun, Michael Anthony
This dissertation is divided into two main topics. The first is the generalization of quantum dynamics when the Schrodinger partial differential equation is not defined even in the weak mathematical sense because the potential function itself is a distribution in the spatial variable, the same variable that is used to define the kinetic energy operator, i.e. the Laplace operator. The procedure is an extension and broadening of the distributional calculus and offers spectral results as an alternative to the only other two known methods to date, namely a) the functional calculi; and b) non-standard analysis. Furthermore, the generalizations of quantum dynamics presented within give a resolution to the time asymmetry paradox created by multi-particle quantum mechanics due to the time evolution still being unitary. A consequence is the randomization of phases needed for the fundamental justification Pauli master equation. The second topic is foundations of the quantum theory of fields. The title is phrased as `foundations'' to emphasize that there is no claim of uniqueness but rather a proposal is put forth, which is markedly different than that of constructive or axiomatic field theory. In particular, the space of fields is defined as a space of generalized functions with involutive symmetry maps (the CPT invariance) that affect the topology of the field space. The space of quantum fields is then endowed the Frechet property and interactions change the topology in such a way as to cause some field spaces to be incompatible with others. This is seen in the consequences of the Haag theorem. Various examples and discussions are given that elucidate a new view of the quantum theory of fields and its (lack of) mathematical structure.
14. Efficient quantum mechanical calculation of solvation free energies based on density functional theory, numerical atomic orbitals and Poisson Boltzmann equation
Wang, Mingliang; Wong, Chung F.; Liu, Jianhong; Zhang, Peixin
2007-07-01
We have successfully coupled the Kohn-Sham with Poisson-Boltzmann equations to predict the solvation free energy, where the Kohn-Sham equations were solved by implementing the flexible pseudo atomic orbitals as in S IESTA package. It was found that the calculated solvation free energy is in good agreement with experimental results for small neutral molecules, and its standard error is 1.33 kcal/mol, the correlation coefficient is 0.97. Due to its high efficiency and accuracy, the proposed model can be a promising tool for computing solvation free energies in computer aided drug design in future.
15. Non-stochastic matrix Schrödinger equation for open systems
Joubert-Doriol, Loïc; Ryabinkin, Ilya G.; Izmaylov, Artur F.
2014-12-01
We propose an extension of the Schrödinger equation for a quantum system interacting with environment. This extension describes dynamics of a collection of auxiliary wavefunctions organized as a matrix m, from which the system density matrix can be reconstructed as hat{ρ }= {m} {m}^dagger. We formulate a compatibility condition, which ensures that the reconstructed density satisfies a given quantum master equation for the system density. The resulting non-stochastic evolution equation preserves positive-definiteness of the system density and is applicable to both Markovian and non-Markovian system-bath treatments. Our formalism also resolves a long-standing problem of energy loss in the time-dependent variational principle applied to mixed states of closed systems.
16. Non-stochastic matrix Schrödinger equation for open systems
SciTech Connect
Joubert-Doriol, Loïc; Ryabinkin, Ilya G.; Izmaylov, Artur F.
2014-12-21
We propose an extension of the Schrödinger equation for a quantum system interacting with environment. This extension describes dynamics of a collection of auxiliary wavefunctions organized as a matrix m, from which the system density matrix can be reconstructed as ρ{sup ^}=mm{sup †}. We formulate a compatibility condition, which ensures that the reconstructed density satisfies a given quantum master equation for the system density. The resulting non-stochastic evolution equation preserves positive-definiteness of the system density and is applicable to both Markovian and non-Markovian system-bath treatments. Our formalism also resolves a long-standing problem of energy loss in the time-dependent variational principle applied to mixed states of closed systems.
17. Problems with the Newton-Schrödinger equations
Anastopoulos, C.; Hu, B. L.
2014-08-01
We examine the origin of the Newton-Schrödinger equations (NSEs) that play an important role in alternative quantum theories (AQT), macroscopic quantum mechanics and gravity-induced decoherence. We show that NSEs for individual particles do not follow from general relativity (GR) plus quantum field theory (QFT). Contrary to what is commonly assumed, the NSEs are not the weak-field (WF), non-relativistic (NR) limit of the semi-classical Einstein equation (SCE) (this nomenclature is preferred over the ‘Moller-Rosenfeld equation’) based on GR+QFT. The wave-function in the NSEs makes sense only as that for a mean field describing a system of N particles as N\\to \\infty , not that of a single or finite many particles. From GR+QFT the gravitational self-interaction leads to mass renormalization, not to a non-linear term in the evolution equations of some AQTs. The WF-NR limit of the gravitational interaction in GR+QFT involves no dynamics. To see the contrast, we give a derivation of the equation (i) governing the many-body wave function from GR+QFT and (ii) for the non-relativistic limit of quantum electrodynamics. They have the same structure, being linear, and very different from NSEs. Adding to this our earlier consideration that for gravitational decoherence the master equations based on GR+QFT lead to decoherence in the energy basis and not in the position basis, despite some AQTs desiring it for the ‘collapse of the wave function’, we conclude that the origins and consequences of NSEs are very different, and should be clearly demarcated from those of the SCE equation, the only legitimate representative of semiclassical gravity, based on GR+QFT.
18. Master Watershed Stewards.
ERIC Educational Resources Information Center
Comer, Gary L.
The Master Watershed Stewards (MWS) Program is a pilot project (developed through the cooperation of the Ohio State University Extension Logan and Hardin County Offices and the Indian Lake Watershed Project) offering the opportunity for communities to get involved at the local level to protect their water quality. The program grew out of the…
19. William Brickman, Master Teacher
ERIC Educational Resources Information Center
Swing, Elizabeth Sherman
2010-01-01
In this article, the author shares her encounter and relationship with William Brickman as her master teacher. William Brickman was her professor, her dissertation advisor, her mentor, and her friend. Her pursuit of a Ph.D. in late middle age may have seemed strange to friends, family, and some of her professors, but not to Brickman. She enrolled…
20. The "Clinical" Masters Degree.
ERIC Educational Resources Information Center
Perlman, Baron; Lane, Robert
1981-01-01
Discusses issues surrounding the clinical master's degree: the belief that the only true psychologist is a PhD, public confusion between doctoral and subdoctoral psychologists, training guidelines, role responsibility, employment, licensing and competency, accreditation, and supervision. Suggests an APA sponsored conference to discuss and resolve…
1. The Change Masters.
ERIC Educational Resources Information Center
Kanter, Rosabeth Moss
1984-01-01
The change masters are identified as corporate managers who have the resources and the vision to effect an economic renaissance in the United States. Strategies for change should emphasize horizontal as well as vertical communication, and should reward enterprise and innovation at all levels. (JB)
2. The Master Science Teacher
ERIC Educational Resources Information Center
Toh, Kok-Aun; Tsoi, Mun-Fie
2008-01-01
The dire need of some schools to boost the academic performance of their students inevitably rests with their ability to attract highly qualified teachers. As such, the UK has put in place the Advanced Skills Teacher (AST) scheme, while the US has set the ball rolling in laying down standards for the certification of the master science teacher, to…
3. The Master Plan Revisited.
ERIC Educational Resources Information Center
Mulrooney, Virginia F.
The State of California, as it seeks to revise the Master Plan for Higher Education, is grappling with circumstances similar to those underpinning the revision of the United States' original Articles of Confederation. The delegates to the 1787 Constitutional Convention had to identify the task before them; determine how the country would be…
4. Sketchbook & Mastering Technical Skills
ERIC Educational Resources Information Center
Lappe, Steve
2004-01-01
A student sketchbook is not a new idea. However, the appropriate application and assessment capabilities of the sketchbook are often overlooked. Historically, artists were trained via copying the works of master painters and draftsmen. In the same tradition, the author has infused the art curriculum with sketchbook copying exercises to improve…
5. Soils. Transparency Masters.
ERIC Educational Resources Information Center
Clemson Univ., SC. Vocational Education Media Center.
This document is a collection of 43 overhead transparency masters to be used as teaching aids in a course of study involving soils such as geology, agronomy, hydrology, earth science, or land use study. Some transparencies are in color. Selected titles of transparencies may give the reader a better understanding of the graphic content. Titles are:…
6. Comment on "Surface electromagnetic wave equations in a warm magnetized quantum plasma" [Phys. Plasmas 21, 072114 (2014)
2016-07-01
In a recent article [C. Li et al., Phys. Plasmas 21, 072114 (2014)], Li et al. studied the propagation of surface waves on a magnetized quantum plasma half-space in the Voigt configuration (in this case, the magnetic field is parallel to the surface but is perpendicular to the direction of propagation). Here, we present a fresh look at the problem and obtain a new form of dispersion relation of surface waves of the system. We find that our new dispersion relation does not agree with the result obtained by Li et al.
7. Conditional and unconditional Gaussian quantum dynamics
Genoni, Marco G.; Lami, Ludovico; Serafini, Alessio
2016-07-01
This article focuses on the general theory of open quantum systems in the Gaussian regime and explores a number of diverse ramifications and consequences of the theory. We shall first introduce the Gaussian framework in its full generality, including a classification of Gaussian (also known as 'general-dyne') quantum measurements. In doing so, we will give a compact proof for the parametrisation of the most general Gaussian completely positive map, which we believe to be missing in the existing literature. We will then move on to consider the linear coupling with a white noise bath, and derive the diffusion equations that describe the evolution of Gaussian states under such circumstances. Starting from these equations, we outline a constructive method to derive general master equations that apply outside the Gaussian regime. Next, we include the general-dyne monitoring of the environmental degrees of freedom and recover the Riccati equation for the conditional evolution of Gaussian states. Our derivation relies exclusively on the standard quantum mechanical update of the system state, through the evaluation of Gaussian overlaps. The parametrisation of the conditional dynamics we obtain is novel and, at variance with existing alternatives, directly ties in to physical detection schemes. We conclude our study with two examples of conditional dynamics that can be dealt with conveniently through our formalism, demonstrating how monitoring can suppress the noise in optical parametric processes as well as stabilise systems subject to diffusive scattering.
8. Quantum Brownian motion with inhomogeneous damping and diffusion
Massignan, Pietro; Lampo, Aniello; Wehr, Jan; Lewenstein, Maciej
2015-03-01
We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of additional terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (superlocalized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time-dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas.
9. Efficient self-consistent Schrödinger-Poisson-rate equation iteration method for the modeling of strained quantum cascade lasers
Li, Jian; Ma, Xunpeng; Wei, Xin; Jiang, Yu; Fu, Dong; Wu, Haoyue; Song, Guofeng; Chen, Lianghui
2016-05-01
We present an efficient method for the calculation of the transmission characteristic of quantum cascade lasers (QCLs). A fully Schrödinger-Poisson-rate equation iteration with strained term is presented in our calculation. The two-band strained term of the Schrödinger equation is derived from the eight-band Hamiltonian. The equivalent strain energy that affects the effective mass and raises the energy level is introduced to include the biaxial strain into the conduction band profile. We simplified the model of the electron-electron scattering process and improved the calculation efficiency by about two orders of magnitude. The thermobackfilling effect is optimized by replacing the lattice temperature with the electron temperature. The quasi-subband-Fermi level is used to calculate the electron density of laser subbands. Compared with the experiment results, our method gives reasonable threshold current (depends on the assumption of waveguide loss and scattering processes) and more accurate wavelength, making the method efficient and practical for QCL simulations.
10. Nonlinear thermoelectric response due to energy-dependent transport properties of a quantum dot
Svilans, Artis; Burke, Adam M.; Svensson, Sofia Fahlvik; Leijnse, Martin; Linke, Heiner
2016-08-01
Quantum dots are useful model systems for studying quantum thermoelectric behavior because of their highly energy-dependent electron transport properties, which are tunable by electrostatic gating. As a result of this strong energy dependence, the thermoelectric response of quantum dots is expected to be nonlinear with respect to an applied thermal bias. However, until now this effect has been challenging to observe because, first, it is experimentally difficult to apply a sufficiently large thermal bias at the nanoscale and, second, it is difficult to distinguish thermal bias effects from purely temperature-dependent effects due to overall heating of a device. Here we take advantage of a novel thermal biasing technique and demonstrate a nonlinear thermoelectric response in a quantum dot which is defined in a heterostructured semiconductor nanowire. We also show that a theoretical model based on the Master equations fully explains the observed nonlinear thermoelectric response given the energy-dependent transport properties of the quantum dot.
11. Master Study: Ceramics
ERIC Educational Resources Information Center
Clark, Kelly
2004-01-01
In painting and drawing classes, it is typical to ask students to work directly from a master. It is one way to study composition techniques, and to become familiar with classical style firsthand. In museums, easels are set up as artists work, not in an attempt to copy or plagiarize, but in an attempt to be part of history by participating in it.…
12. Cavity QED Deutsch quantum computer
Hollenberg, Lloyd C. L.; Salgueiro, A. N.; Nemes, M. C.
2001-10-01
The two-atom correlation scheme originally proposed by Davidovich, Brune, Raimond, and Haroche for measuring the decoherence of a mesoscopic superposition of coherent states of a QED cavity field is shown to be equivalent to a quantum computer solving Deutsch's problem. Using the existing analysis of decoherence in the Master equation formalism, and other important losses in this system, the final probability for obtaining the correct result for the computation is found in terms of the time period between atom traversals, the number of photons in the cavity, and the precision of the atomic velocity. The error due to decoherence in this system amounts to a phase error, and in the Master equation approach is a linear effect at small time scales. By explicitly considering the dynamics of the decoherence process when the system is coupled to a bath of oscillators with finite mode cutoff the error due to decoherence is found to decrease significantly and becomes a quadratic effect at short-time scales.
13. Integral Equation Theory of Molecular Solvation Coupled with Quantum Mechanical/Molecular Mechanics Method in NWChem Package
SciTech Connect
Chuev, Gennady N.; Valiev, Marat; Fedotova, Marina V.
2012-04-10
We have developed a hybrid approach based on a combination of integral equation theory of molecular liquids and QM/MM methodology in NorthWest computational Chemistry (NWChem) software package. We have split the evaluations into conse- quent QM/MM and statistical mechanics calculations based on the one-dimensional reference interaction site model, which allows us to reduce signicantly the time of computation. The method complements QM/MM capabilities existing in the NWChem package. The accuracy of the presented method was tested through com- putation of water structure around several organic solutes and their hydration free energies. We have also evaluated the solvent effect on the conformational equilibria. The applicability and limitations of the developed approach are discussed.
14. Geometric phases and quantum correlations dynamics in spin-boson model
SciTech Connect
Wu, Wei; Xu, Jing-Bo
2014-01-28
We explore the dynamics of spin-boson model for the Ohmic bath by employing the master equation approach and obtain an explicit expression of reduced density matrix. We also calculate the geometric phases of the spin-boson model by making use of the analytical results and discuss how the dissipative bosonic environment affects geometric phases. Furthermore, we investigate the dynamics of quantum discord and entanglement of two qubits each locally interacting with its own independent bosonic environments. It is found that the decay properties of quantum discord and entanglement are sensitive to the choice of initial state's parameter and coupling strength between system and bath.
15. Heat transport through lattices of quantum harmonic oscillators in arbitrary dimensions.
PubMed
Asadian, A; Manzano, D; Tiersch, M; Briegel, H J
2013-01-01
In d-dimensional lattices of coupled quantum harmonic oscillators, we analyze the heat current caused by two thermal baths of different temperatures, which are coupled to opposite ends of the lattice, with a focus on the validity of Fourier's law of heat conduction. We provide analytical solutions of the heat current through the quantum system in the nonequilibrium steady state using the rotating-wave approximation and bath interactions described by a master equation of Lindblad form. The influence of local dephasing in the transition of ballistic to diffusive transport is investigated.
16. Fluctuation theorem for a double quantum dot coupled to a point-contact electrometer
SciTech Connect
Golubev, D.; Utsumi, Y.; Marthaler, M.; Schön, G.
2013-12-04
Motivated by recent experiments on the real-time single-electron counting through a semiconductor GaAs double quantum dot (DQD) by a nearby quantum point contact (QPC), we develop the full-counting statistics of coupled DQD and QPC system. By utilizing the time-scale separation between the dynamics of DQD and QPC, we derive the modified master equation with tunneling rates depending on the counting fields, which fulfill the detailed fluctuation theorem. Furthermore, we derive universal relations between the non-linear corrections to the current and noise, which can be verified in experiments.
17. Quantum dots as active material for quantum cascade lasers: comparison to quantum wells
Michael, Stephan; Chow, Weng W.; Schneider, Hans Christian
2016-03-01
We review a microscopic laser theory for quantum dots as active material for quantum cascade lasers, in which carrier collisions are treated at the level of quantum kinetic equations. The computed characteristics of such a quantum-dot active material are compared to a state-of-the-art quantum-well quantum cascade laser. We find that the current requirement to achieve a comparable gain-length product is reduced compared to that of the quantum-well quantum cascade laser.
18. Small quantum absorption refrigerator with reversed couplings
Silva, Ralph; Skrzypczyk, Paul; Brunner, Nicolas
2015-07-01
Small quantum absorption refrigerators have recently attracted renewed attention. Here we present a missing design of a two-qubit fridge, the main feature of which is that one of the two machine qubits is itself maintained at a temperature colder than the cold bath. This is achieved by "reversing" the couplings to the baths compared to previous designs, where only a transition is maintained cold. We characterize the working regime and the efficiency of the fridge. We demonstrate the soundness of the model by deriving and solving a master equation. Finally, we discuss the performance of the fridge, in particular the heat current extracted from the cold bath. We show that our model performs comparably to the standard three-level quantum fridge and thus appears appealing for possible implementations of nanoscale thermal machines.
19. Isotropy and control of dissipative quantum dynamics
Dive, Benjamin; Burgarth, Daniel; Mintert, Florian
2016-07-01
We investigate the problem of what evolutions an open quantum system described by a time-local master equation can undergo with universal coherent controls. A series of conditions is given which exclude channels from being reachable by any unitary controls, assuming that the coupling to the environment is not being modified. These conditions primarily arise by defining decay rates for the generator of the dynamics of the open system, and then showing that controlling the system can only make these rates more isotropic. This forms a series of constraints on the shape and nonunitality of allowed evolutions, as well as an expression for the time required to reach a given goal. We give numerical examples of the usefulness of these criteria and explore some similarities they have with quantum thermodynamics.
20. Quantum theory of a two-photon micromaser
SciTech Connect
Davidovich, L.; Raimond, J.M.; Brune, M.; Haroche, S.
1987-10-15
We present the quantum theory of a microscopic maser operating on a degenerate two-photon transition between levels of the same parity. We derive both a master equation and a Fokker-Planck equation for this system, and show that quantum effects may have a substantial influence on the behavior of the maser. They modify the oscillation threshold and make external triggering of this maser unnecessary, whereas, according to semiclassical theory, such a triggering is required to start up the maser oscillation. We derive the phase-diffusion properties of the field and show that the diffusion coefficient is complex in this case, its imaginary part being associated with a frequency shift of the field inside the cavity. We show that, in steady state, the photon-number statistics is sub-Poissonian for a wide range of pumping rates. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8459762334823608, "perplexity": 960.8984520605181}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607786.59/warc/CC-MAIN-20170524035700-20170524055700-00635.warc.gz"} |
https://socratic.org/questions/what-is-13-root-3-4-root-48-in-radical-form | Algebra
Topics
# What is 13 root 3 - 4 root 48 in radical form?
Jul 18, 2017
If the question is to simplify this expression: $13 \sqrt{3} - 4 \sqrt{48}$
Then see a solution process below:
#### Explanation:
First, rewrite the radical on the right as:
$13 \sqrt{3} - 4 \sqrt{16 \cdot 3}$
Now, use this rule of radicals to simplify the term on the right:
$\sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}}$
$13 \sqrt{3} - 4 \sqrt{\textcolor{red}{16} \cdot \textcolor{b l u e}{3}} \implies$
$13 \sqrt{3} - 4 \sqrt{\textcolor{red}{16}} \sqrt{\textcolor{b l u e}{3}} \implies$
$13 \sqrt{3} - \left(4 \cdot 4 \sqrt{\textcolor{b l u e}{3}}\right) \implies$
$13 \sqrt{3} - 16 \sqrt{\textcolor{b l u e}{3}}$
Next, factor our the common term to simplify the constants:
$\left(13 - 16\right) \sqrt{\textcolor{b l u e}{3}} \implies$
$- 3 \sqrt{3}$
##### Impact of this question
1061 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8738962411880493, "perplexity": 2319.6434575526378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337531.3/warc/CC-MAIN-20221005011205-20221005041205-00321.warc.gz"} |
https://socratic.org/questions/what-is-22-degrees-in-fahrenheit-in-celsius | Algebra
Topics
What is 22 degrees in Fahrenheit in Celsius?
Sep 21, 2015
$- 5. \overline{5}$ degrees Celsius
Explanation:
This is the formula for converting Fahrenheit to Celsius:
$C = \left(F - 32\right) \cdot \frac{5}{9}$
All you have to do is insert the value and solve.
$C = \left(F - 32\right) \cdot \frac{5}{9}$
$C = \left[\left(22\right) - 32\right] \cdot \frac{5}{9}$
$C = \left(- 10\right) \cdot \frac{5}{9}$
$C = - 5. \overline{5}$
$22$ degrees Fahrenheit is equal to $- 5. \overline{5}$ degrees Celsius.
Impact of this question
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https://www.physicsforums.com/threads/degrees-of-freedom.743295/ | # Degrees of Freedom
1. Mar 14, 2014
### emmasaunders12
Hi all
If my model consists of two steps,
e.g, multiple linear regression to get an estimate of an intermediate response variable
followed by a further regression to get the final estimate of response variable
To estimate the degrees of freedom for the total model can I simply sum the degrees of freedom for the individual models?
Thanks
Emma
2. Mar 14, 2014
### FactChecker
I always hate it when someone suggests a different approach instead of answering the original question, but I have to suggest this: Why not apply multiple linear regression directly to estimate the response variable? It seems like either way will end up with a linear estimator, but the direct approach will allow you to apply existing tools and obtain all the relevant statistical information directly.
3. Mar 15, 2014
### emmasaunders12
Hi Fact checker, the reason I phrased the question as such is because it's slightly more complicated than regression, I'm comparing two "models", one of which requires pre processing of the data so want to know if, during this pre processing step I can simply add the degrees of freedom for each individual step?
4. Mar 15, 2014
### FactChecker
Emma, I see. My opinion is that you can only add the independent degrees of freedom of the second process from variables that are not the result of the first process. Anything more than that is beyond my abilities.
5. Mar 15, 2014
### Stephen Tashi
Emma,
"Degrees of freedom" has no precise meaning until a particular context is specified. (This is the case with many mathematical terms like "dual", "conjugate", "closed", "homogeneous".) I assume you are using a particular statistic or procedure which requires a "degrees of freedom" number. Explain exactly what procedure or formula you intend to use.
6. Mar 15, 2014
### emmasaunders12
Hi I using the F test for the comparsion of two models
7. Mar 16, 2014
### Stephen Tashi
As I understand your first post, it talks about a single linear model that is created in stages. This model is not (in general) the same model that you would obtain by a least-squares linear regression because you did the fit in two stages. Your final model is something like z = A x1 + B y + C where x1 is from the data, and y is not. The "intermediate" variable y is the result of a least squares fit to the data that gave y = D x2 + E x3 + F where x2 and x3 are values from the data.
In least squares regression, to obtain a model z = A x1 + By + C, we assume there are no "errors" in the x1 and y measurements. So you can't say that your procedure produces the same model as you would have obtained if you had done the regression in a single step using the data (x1, x2, x3) because there are "errors" in the y values. (The method of "total least squares" regression is often used when the model assumes errors exist in several of the variables.)
One technicality to investigate, is whether the F-test comparison of two linear models actually applies if one model is not the result of a least squares fit.
If you are comparing two models, then I assume the two models predict the same variable, which is z in my example. If so, my example describes only one of the models. Where does the other model come from?
8. Mar 17, 2014
### emmasaunders12
Thanks for the reply stephen, but without drifting off the topic too much, with respect to the degrees of freedom, is it legitimate to add the degrees of freedom for each individual step?
Thanks
Emma
9. Mar 17, 2014
### FactChecker
The math of "degrees of freedom" allows you to count up the number of variables in an equation that are independent of others and are free to vary. In that context, you can add them as long as you do not count the variables that are a result of your first step. Those variables are not free to vary since they are calculated in the first step.
However, using the degrees of freedom in statistics like F or chi-squared requires additional assumptions about the distribution of the free variables and about the equation of the statistic being calculated. Since your calculation is not one of the usual ones (sample mean, sample variance, linear regression, goodness of fit, etc.), it is not clear what statistics are valid to use, even if your degrees of freedom is correct. To use the standard distributions, you will have to use one of the processes that they apply to.
10. Mar 18, 2014
### Stephen Tashi
let's say you are dealing with a linear model and "degrees of freedom" in your context means the number of parameters in the model that were determined when you fit the model to data.
Using the previous example, z = A x1 + B y + C can be written as z = A x1 + B( D x2 + E x3 + F) + C = A x1 + BD x2 + BE x3 + BF + C. This amounts to a linear model with 4 parameters P1 = A, P2 = BD, P3 = BE and P4 = (BF + C). So there are 4 degrees of freedom.
There are 3 parameters in z = A x1 + B y + C and 3 parameters in y = D x2 + E x3 + F but there are only 4 parameters in the model that expresses z as a linear function of x1,x2,x3.
11. Mar 18, 2014
### emmasaunders12
Thanks stephen
Im a little confused as to why your not counting e.g, BD as two parameters. In MLR where e.g z=AX1, where A will now have many parameters, aren't all elements of A counted in this case?
In the example you give above the addition of number of parameters (=6 parameters above) would always result in more parameters when one simply adds them, resulting in even less degrees of freedom.
When comparing the sum of square residuals of two models using the F test, a simple model (S1) with degrees of freedom DF1 and a more complex model (S2) having less degrees (DF2):
F=[(S1-S2)/S2]/[(DF1-DF2)/DF2]
estimating less degrees of freedom in the complex model than may perhaps exist would give a smaller F ratio and thus favour the simpler model. Would this assumption be correct?
Emma
12. Mar 18, 2014
### Stephen Tashi
My undestanding of applying the F test to compare linear models is that we assume the models are nested and that they are each least squares fit to the data (...and a lot of other assumptions). So it's hard to answer your question because you are not fitting a model to data by a method that is guaranteed to produce a least squares fit. (and you have not mentioned a second model to which your first model is being compared.)
But suppose we have a linear model of the form z = P x + Q y + R where P,Q,R are constants. Suppose we fit this to data by some procedure that goes in stages and is guaranteed to produce a least squares fit in the end. We write the model as z = (A)(B)(C) + (D)(E)(F) y + (G)(H)(I) in stage 1, we find A,D,G. In stage 2 we find B,E,H. In stage 3 we find C,F,I. This does not change the fact that the final result of the process is a linear model that is a least squares fit to the data and has the form z = P x + Q y + R, which involves 3 constants.
13. Mar 18, 2014
### emmasaunders12
I was however under the impression that the F test can still be used if the models were not fitted using least squares. Quoting wiki
"It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact "F-tests" mainly arise when the models have been fitted to the data using least squares. "
I understand your proposition with respect to number of constants. Could you please however clarify my earlier point.
"Im a little confused as to why your not counting e.g, BD as two parameters. In MLR where e.g z=AX1, where A will now have many parameters, aren't all elements of A counted in this case?"
Thanks
Emma
14. Mar 18, 2014
### Stephen Tashi
We'd have to investigate whether "inexact" F-tests are good idea.. I don't know if the wiki is merely stating that they are a customary practice or whether it is stating they are a mathematically justifiable practice.
The counting of parameters in the linear model counts the number of constants in the model, with each numerical coefficient of a variable counted as single constant and the "constant term" of the model counted as a single constant. So for the term 16 x, the numerical value 16 is one constant even though it could be factored as (8)(2) or (2)(2)(2)(2).
15. Mar 19, 2014
### emmasaunders12
Hi Stephen thanks for your help:
The models I'm comparin are:
Model1: Y=A(B^-1)X
Model2: Y=A([ Z-CH]D+ C(B^-1)X)
How would you best proceed in this instance?
Thanks
Emma
16. Mar 19, 2014
### Stephen Tashi
Which of those letters represent independent variables? X and Z ? Z isn't a function of X?
17. Mar 19, 2014
### emmasaunders12
X is the independent variable, Z is a linear function of a projection of X, i.e S , in a lower dimensional domain, i.e an autoregressive describing the evolution in time of S.
S=(A^-1)X;
18. Mar 19, 2014
### Stephen Tashi
I suggest you give a precise definition of things involved.
I don't know what "an autoregressive" might be. The term "autoregressive" suggests your independent variables might be values indexed with time. Is X a vector of values indexed by a "time" ? Or is the kth component of X the value of something at time k? Are the values of the dependent variable Y also indexed by time?
19. Mar 20, 2014
### emmasaunders12
Hi Stephen,
Thanks for your patience with my problem:
Model1: Y=A(B^-1)X
A are the eigenvectors of Y,
B are the eigenvectors of X,
So the above is a total least squares type problem.
There are no time indices in the above method
Model2: Y=A([ Z-CH]D+ C(B^-1)X)
[ Z-CH]D+ C - is a Kalman filter
some preliminaries:
S=(A^-1)X;
D=(B^-1)Y
S is an estimate of X in a lower dimensional domain
D is an estimate of Y in a lower dimensional domain
C is the Kalman gain
H is a liner model between S and D
Z is an autoregressive fit of D
From the above A and B are fixed, all other parameters can vary, as the Kalman filter is adaptive, dependent upon Y using an EM algorithm.
20. Mar 20, 2014
### Stephen Tashi
It isn't clear what you are doing, because you haven't described the format of the observed data you are using.
One effort at mind reading says that your data consists of M ordered pairs of vectors (X,Y), so to exhibit one pair of vectors as scalars (X[k],Y[k]) = ( (X[k][1],X[k][2]...X[k][nx]), (Y[k][1],Y[k][2],...Y[k][ny])
Another effort at mind reading says your data consists of M ordered pairs of scalars (x[k],y[k]) and that there is a single vector Y = (y[1],y[2],...y[M]) and a single vector X = (x[1],x[2],...x[M]).
You haven't written an equation which shows any random errors, so it isn't clear why you say that the fit is a total least squares problem. I assume you mean that the model assumes a random additive error in both the X and Y terms.
What do X and Y represent in this model? (Are they the same variables in this model as they are in Model1?)
The term Kalman filter suggests that there are time indices involved in this model. Which index represents time?
21. Mar 21, 2014
### emmasaunders12
Hi Stephen,
your correct M ordered pairs of vectors (X,Y) represents the data.
The fit is total least squares as the eigen domain is used, i.e orthogonal regression
X and Y are the same in model 1 and model 2. Model 1 however is dynamic as mentioned previously
Yk=A([ Zk-CkHk]Dk+ Ck(B^-1)Xk) - - - with k as a time indicie
similarly model 1 can be written as
Yk=A(B^-1)Xk
if one wishes, just depends on how the model is being used, i,e with a batch of Xk's or just incremental data points?
Any idea how to proceed here?
22. Mar 22, 2014
### Stephen Tashi
Use simulation.
For simulation you need stochastic models, not mere curve fits. Each model, should specify a method for making a deterministic prediction, (such as Y = AX) but it also must specify a model for how the observed data arises in a stochastic manner (such as Y = AX + B* err(k}, where err(k) is an independent random draw at each time k from normal distribution with mean 0 and variance 1.)
I think your x-data is a time series of vectors. You need to generate representative examples of the x-data by simulation or have such examples from actual observations (i.e. one "example" is an entire time series of vectors). So you might need a stochastic model for the x-data.
I am assuming your predictive models give the predicted y-value as a function of the observed x-values , not as a function of the underlying "true" x-values. Of course a model may use the observed x-values to predict the "true" x-values and then make it's prediction based on those estimates.
Once you have the capability to do simulations, you can investigate various statistics by the Monte-Carlo method.
-------------
For example:
Let model_X be the stochastic model for generating the x-data.
Create a Mont-Carlo simulation involving two (possibly identical models) model_A and model_B as follows. One replication of the simulation is:
1) Generate the X-data using model_X
2) Generate the Y-data using the stochastic model associated with model_A
3) Generate the predicted Y-data using the deterministic model associated with model_A
4) Compute RSS_A = the sum of the squared residuals between the Y-data of step 2 and the predicted Y-values of step 3.
( I'm assuming that when using the F-test, your intent was to define the "residual" between two vectors as the euclidean distance between them. Whether this is wise depends on details of the real world problem.)
5) Generate the Y_data using the stochastic model associated with model_B
6) Generate the the predicted Y-data using the deterministic model associated with model_B
7) Compute RSS_B = the sum of the squares of the residuals between the Y_data from step 5 and the predictions of step 6.
G is an obvious imitation of the F-statistic. We don't know that G has the same distribution as any F-statistic, so we shouldn't call it one.
We can set model_A = model_B = your model2 and use the Monte-Carlo simulation to estimate the distribution of G. (When the stochastic model associated with model 2 is applied to the same X-data twice, it probably won't produce the same residuals due to the stochastic terms. Hence the value of G will vary on different replications.)
Take the "null hypothesis" to be that model1 is the same as model2 (as far as producing residuals goes). Compute the single numerical value G_obs = (RSS_1 - RSS_2)/RSS_2 by applying the two models to the actually observed X-data. Use the distribution of G to compute how likely it is to get a value of G equal or greater than G_obs. Then "accept" or "reject" the null hypothesis based on how this probability compares to whatever "significance level" you have chosen.
23. Mar 22, 2014
### emmasaunders12
wouldn't i still however be in the same predicament, as the testing of the null hypothesis requires the degrees of freedom for each model, which is what I am not sure about?
Thanks
Emma
24. Mar 22, 2014
### Stephen Tashi
No, you don't need to know any "degrees of freedom" information. You use the empirical distribution of G to do the test. There is never any need to deal with F statistics.
25. Mar 23, 2014
### emmasaunders12
I actually have 10 examples of the ground truth Y, would this be enough to not perform siulation as I am calculating the residuals (RSS_A - RSS_B) / RSS_B using them. Also the technique you mention would be dependent upon what one chooses as err{k}, in Y = AX + B* err(k}.
I'm also a little confused on how you would use the emprical distirbution of G to do the test, if you could clarify that would be great.
Thanks so much for your help
Emma | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.833814799785614, "perplexity": 706.16557648488}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867859.88/warc/CC-MAIN-20180526190648-20180526210648-00063.warc.gz"} |
http://mathhelpforum.com/pre-calculus/74180-solved-help-inequality.html | # Thread: [SOLVED] help with this inequality
1. ## [SOLVED] help with this inequality
Solve the following inequality. Write the answer in interval notation.
I got:
(-inifinity, 23/5)U(4,infinity)
&& got it wrong
does anyone know how to figure this out
2. How did you arrive at your answer? You added the 6 to the left-hand side, converted to the common denominator, combined the two fractions into one, found the zeroes of the numerator and denominator, tested the sign on each interval, and... then what?
3. huh?
i was left with
(-5x-23)/(x-4)
which was where i got my answer that is wrong
4. is it all real numbers?
5. no it is not all real numbers
because i kno by (-5x-23)/x-4
that the x in the denominator is 4.
but i am not sure if that fraction is right at all though
6. Originally Posted by lsnyder
no it is not all real numbers
because i kno by (-5x-23)/x-4
that the x in the denominator is 4.
but i am not sure if that fraction is right at all though
oops, isnt it
25/7<=x<4
x-1/x-4 <= -6
=x-1<= -6x+24
=7x-1<=24
=7x<=25
x<=25/7
and
x-1/x-4 <= -6
x-4<=-6x+24
7x<=28
x<=4
(i'm not sure why it's not equal to 4. When I worked it out on a calculator, it came up as x<4)
7. Hello, lsnyder!
Solve the following inequality. Write the answer in interval notation.
. . . $\frac{x-1}{x-4} \:\leq\:-6$
I got: . $\left(-\infty, \tfrac{23}{5}\right) \cup (4, \infty)$ . . . and got it wrong.
Does anyone know how to figure this out ?
After several false starts, I was forced to graph the function.
We have: . $y \:=\:\frac{x-1}{x-4}$
. . When is it below $y = -6$ ?
Code:
|
| :*
| :
| : *
| : *
| : *
| : *
- - - 1+ - - - - : - - - - - - - - -
* | :
* :4
--------+---*-----+--------------------
| * :
| o :
| :
| o:
| :
The function has a vertical asymptote $x = 4$ and horizontal asymptote $y = 1$.
We see that for values of $x$ slightly below 4, the graph is below $y = -6$
If the function equals -6, we have: . $\frac{x-1}{x-4} \:=\:-6 \quad\Rightarrow\quad x \:=\:\frac{25}{7}$
Therefore, the interval is: . $\left[\frac{25}{7},\,4\right)$
8. why is it not equal to 4? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9378771185874939, "perplexity": 1343.062425202207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698540909.75/warc/CC-MAIN-20161202170900-00079-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/did-i-rearrange-this-equation-correctly-circular-motion.727062/ | # Homework Help: Did i rearrange this equation correctly? (circular motion)
1. Dec 7, 2013
### SmallPub
d1. The problem statement, all variables and given/known data A bucket 2.00kg is whirled in a vertical circle of a radius 1.10m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N a) find the speed of the bucket b) how fast must the bucket move at the top of the circle so that the rope does not go slack?
2. Relevant equations v=√gr , FT = -mv2/r + mg ,
g=9.81m/s^2
m=2.00kg
r=1.10m
FT= 25.0 N
3. The attempt at a solution a) v=√rg = √1.10m x 9.81m/2^s = 3.28m/s
b) v=√FTr-mg/-m <----- Im not sure if i rearranged that correctly
2. Dec 7, 2013
### hjelmgart
Maybe you should show the steps in how you rearranged it. Go through every step, and you will see, that it is wrong.
3. Dec 7, 2013
### SmallPub
can you show the correct equation to find v?
4. Dec 7, 2013
### hjelmgart
FT = -mv^2/r + mg
FT - mg = -mv^2
(FT - mg)*r/m = -v^2
-(FT - mg)*r/m = v^2
v = sqrt(-(FT - mg)*r/m)
v = sqrt((mg*r - FT*r)/m)
5. Dec 7, 2013
### hjelmgart
Although I don't think that is the correct method for this problem, anyway. I didn't look too much into it, though, but I am guessing, you will get some complex number from this. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8765115737915039, "perplexity": 2029.819838475168}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676588972.37/warc/CC-MAIN-20180715203335-20180715223335-00375.warc.gz"} |
https://libris.nl/boek?authortitle=arnold-janssen-hartmut-milbrodt-helmut-strasser/infinitely-divisible-statistical-experiments--9780387960555 | # Infinitely Divisible Statistical Experiments
Taal: Engels
## € 110,00
This book is intended to give an account of the theory of infi nitely divisible statistical experiments which started from LeCam, 1974. It includes a presentation of LeCam's basic results as well as new developments in the field. The book consists of four chapters written by different authors. Chapters I, III and IV have been prepared in Bayreuth with the support of the Deutsche Forschungsgemeinschaft (DFG); Chapter II is part of its author's Habilitationsschrift, 1982 (Dortmund). For the reader's convenience, the chapters have been unified in presentation, without neglecting differences in the individual styles of writing. The authors are grateful to Dr. C. Becker for carefully reviewing the manuscript. Furthermore, acknowledgements are gratefully extended to the DFG for partly subsidizing Dr. Becker and the second author by a grant. Some special words of thanks are due to Mrs. Witzigmann, who typed the final manuscript and its predecessors with patience and skill. Universitat Bayreuth und A. Janssen Universitat Dortmund, H. Milbrodt Dezember 1984 H. Strasser CONTENTS Preface Introduction L~its of Triangular Arrays of 14 I. EXEeriments (H. Milbrodt and H. Strasser) 1. Basic Concepts 14 19 2. Gaussian Exper~ents 3. Introduction to Poisson Experiments 25 4. Convergence of Poisson Experiments 32 5. Convergence of Triangular Arrays 38 6. Identification of Limit Experiments 47 The Levy-Khintchine Formula for Infinitely 55 II.
ISBN
9780387960555
Vorm
Paperback
Uitgever
Springer-Verlag New York Inc.
Druk
1e
Verschenen
01-04-1985
Taal
Engels
Pagina's
164 pp.
Genre
Geografie & Landbouw
Geen recensies beschikbaar. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8030558228492737, "perplexity": 4230.41215241847}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323586043.75/warc/CC-MAIN-20211024142824-20211024172824-00544.warc.gz"} |
https://math.stackexchange.com/questions/2759967/what-does-subset-of-a-topological-vector-space-is-compact-w-r-t-to-the-weak-top | # What does “subset of a topological vector space is compact w.r.t to the weak topology” mean?
I've been reading about functional analysis and topology and I came across this sentence:
subset of a topological vector space is compact with respect to the weak topology
I have been trying to search the web for an explanation to understand what this means, and I also tried the chat rooms without finding a satisfactory answer. English is not my native language so that might be one of the reasons I'm not getting the sentence. I do understand the concepts of topological space, compactness and weak topology.
My question is: What does this sentence mean, explicitly?
• Shouldn't be your "topological" be "linear"? – Przemysław Scherwentke Apr 30 '18 at 9:03
• I'm sorry @PrzemysławScherwentke but I'm not 100% sure what you're asking. I added the "vector" term to my question. – jjepsuomi Apr 30 '18 at 9:08
• And this "vector" makes the difference. :-) – Przemysław Scherwentke Apr 30 '18 at 9:13
• Excellent, thank you :-) – jjepsuomi Apr 30 '18 at 9:15
• If you know weak topology the statement simply means any open cover of the subset by open sets in the weak topology has a finite subcover. For example, the closed unit ball of an infinite dimensional Hilbert space is not compact in the norm topology but it is compact with respect to the weak topology. – Kavi Rama Murthy Apr 30 '18 at 9:29
• Hi @KaviRamaMurthy okay, thank you very much! So every subset $t\in\tau_w$ is compact (where $\tau_w$ is the weak topology). Got it! :-) – jjepsuomi May 1 '18 at 17:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8136850595474243, "perplexity": 277.5592664943445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178385378.96/warc/CC-MAIN-20210308143535-20210308173535-00129.warc.gz"} |
https://www.physicsforums.com/threads/radiative-equilibrium-temperature-of-a-satellite.536058/ | # Radiative equilibrium temperature of a satellite
1. Oct 2, 2011
### guitarstorm
1. The problem statement, all variables and given/known data
A small, perfectly black, spherical satellite is in orbit around the Earth. If the Earth radiates as a black body at an equivalent blackbody temperature $T_{E}$ = 255 K, calculate the radiative equilibrium temperature of the satellite when it is in the Earth’s shadow. Start by setting dQ as function of solid angle dω and let integration over the arc of solid angle be 2.21.
2. Relevant equations
$F=\sigma T_{E}^{4}=\pi I$
$\int dE=\int \int I\, d\omega \, dA$
3. The attempt at a solution
First, I set up the integral of dE as:
$\int dE=\int_{A=\pi R_{E}^{2}}\int_{2.21}I\, d\omega \, dA$
My only question here is my limit of integration for dA... Should it be over the area of Earth's disk or the entire surface area (which would be $4\pi R_{E}^{2}$)?
Assuming the way I have it is right, I get:
$E=2.21\pi IR_{E}^{2}$
Substituting in for I and then $F_{E}$, it becomes:
$E=2.21R_{E}^{2}\sigma T_{E}^{4}$
And plugging in the values $R_{E} = 6.37 * 10^{6}m$, $\sigma = 5.67 * 10^{-8} Wm^{-2}K^{-4}$, and $T_{E} = 255 K$,
$E = 2.15 * 10^{16}W$, which is the energy transfer to the satellite per unit time.
Using the Steffan-Boltzmann Law again, I set up the equation for the temperature of the satellite as:
$F_{s}=\sigma T_{s}^{4}$, which I believe is the same as $\frac{E}{4\pi R_{E}^{2}}=\sigma T_{s}^{4}$.
Rearranging, $T_{s}=\frac{E}{4\pi \sigma R_{E}^{2}}^{1/4}$.
Plugging in the values and calculating, I get $T_{s} = 165 K$.
I was a bit uncertain about whether I did this last step correctly, and my answer seems a bit low...
Can you offer guidance or do you also need help?
Draft saved Draft deleted
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http://mathhelpforum.com/calculus/33219-integration-parts-adult-way-print.html | # Integration by Parts the Adult Way
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last
• April 4th 2008, 12:05 PM
ThePerfectHacker
Integration by Parts the Adult Way
A lot of people do integration by parts by defining the variables $u=...$ and $v'=...$ and flip them around.
There is a more adult way of doing this, which looks nicer and is a lot faster.
Say we have the integral, $\int xe^x dx$
The idea is to turn one of the factors into a derivative. For example, we know that $(e^x)' = e^x$.
Thus, we can think of the integral as,
$\int x \left( e^x \right) ' dx$
The next step is to take the function inside the differenciation operator and multiply it with the function unaffected with the differenciation and multiply them together. That is the $uv$ part that you get.
Thus, we get $xe^x - \int ...$
The next step is to take the derivative of the function which was unaffected by differenciation and multiply it by the function inside the differenciation sign. This is our $u'v$ part.
In this case we get, $xe^x - \int (x) ' e^x dx = xe^x - e^x + C$.
-----
Here is another example,
$\int \ln x dx = \int \ln x \left( x \right) ' dx = x\ln x - \int 1 dx = x\ln x - x + C$.
Look how fast that is.
Here is another example,
$\int x^2 \sin 2x dx = \int x^2 \left( -\frac{1}{2} \cos 2x \right) ' dx = -\frac{1}{2}x^2 \cos 2x + \int x\cos 2x dx$
$=-\frac{1}{2}x^2\cos 2x + \int x \left( \frac{1}{2} \sin 2x \right)' dx = - \frac{1}{2}x^2\cos 2x + \frac{1}{2}x\sin 2x - \int \frac{1}{2}\sin 2x dx =$ $-\frac{1}{2}x^2 \cos 2x + \frac{1}{2}x\sin 2x + \frac{1}{4}\cos 2x + C$
My point is that it is a lot easier to keep track of everything doing integration this way. Because you do not need to go out of your way to write $u$ and $v'$.
• April 4th 2008, 04:48 PM
topsquark
Quote:
Originally Posted by ThePerfectHacker
My point is that it is a lot easier to keep track of everything doing integration this way. Because you do not need to go out of your way to write $u$ and $v'$.
I agree and I think this is quite a clever observation. However, from an educational standpoint, I think teaching this to a student who has never seen integration by parts, ie. Freshman level college, would be a mistake. The students would not likely be able to follow the "reverse product rule" formulation that you worked out.
-Dan
• April 4th 2008, 05:00 PM
Krizalid
Quote:
Originally Posted by topsquark
I think teaching this to a student who has never seen integration by parts, ie. Freshman level college, would be a mistake.
That's pretty obvious. This method is mechanical, it's for people who have covered integration by parts. (I'm not sayin' the method doesn't work, I'm answering to Dan's post.)
The method is cool and I recently translated to post it in my spanish forum. (Not mine, of course.)
• April 5th 2008, 01:12 AM
Gusbob
Quote:
Originally Posted by topsquark
I agree and I think this is quite a clever observation. However, from an educational standpoint, I think teaching this to a student who has never seen integration by parts, ie. Freshman level college, would be a mistake. The students would not likely be able to follow the "reverse product rule" formulation that you worked out.
-Dan
I was taught integration by parts this way... I'm in the 11th grade. What is the other way of doing this?
• April 5th 2008, 01:17 AM
janvdl
Quote:
Originally Posted by ThePerfectHacker
Here is another example,
$\int \ln x dx = \int \ln x \left( x \right) ' dx = x\ln x - \int 1 dx = x\ln x - x + C$.
Look how fast that is.
We already do it this way with problems like this. (Nod)
Basically it seems your just skipping the writing of "Let u = ...", and doing that part in your head. I do it often too. (I'm lazy! :D )
EDIT: No not quite skipping it... But I see what you're doing. It is much faster.
• April 5th 2008, 02:59 AM
Moo
Well, i agree with topsquark's message, it jumps steps for people who learn it (generally).
Another way to do it is to put f(x) g'(x) dx = f(x) d(g(x)) but it's quite too far for me to remember how the teacher made it...
Some kind of g(x)d(f(x))
• April 5th 2008, 05:24 PM
ThePerfectHacker
I also want to add if you have limits of integration you just carry them through. For example,
$\int_0^{\pi} x\cos x dx = \int_0^{\pi} x \left( \sin x \right)' dx = x\sin x \bigg|_0^{\pi} - \int_0^{\pi} \sin x dx = \mbox{ whatever}$.
• April 19th 2008, 08:50 AM
Boris B
Dear Perfect H:
I like your method of integration by parts a lot. I haven't tested it yet, so I am still left with one question: is this a complete replacement for u substitution? If so I am very pleased!
Another thing I'm wondering is, are there ever cases in which we can't do the first step, i.e. in which we can't turn one of the factors into a (solvable) derivative? If so, what then?
• April 19th 2008, 08:54 AM
colby2152
TPH,
I like the explanation, and like any alternate method or shortcut, it is meant to be learned by the student after they learned the rigorous theorem in the text. Cheers!
-Colby
• April 19th 2008, 10:41 AM
Sean12345
Nice, however I'll stick to the original method for my exam (Yes)
• April 19th 2008, 01:18 PM
wingless
Quote:
Originally Posted by Boris B
Dear Perfect H:
I like your method of integration by parts a lot. I haven't tested it yet, so I am still left with one question: is this a complete replacement for u substitution? If so I am very pleased!
Another thing I'm wondering is, are there ever cases in which we can't do the first step, i.e. in which we can't turn one of the factors into a (solvable) derivative? If so, what then?
This is another way to do integration by parts, not u substitution (Well, integration by parts includes u substitution). This method is the same as the normal integration by parts, but here we don't have to write u, v, du, dv. If there's an insoluble case, then it means integration by parts can't be used there.
There's also tabular integration (tic-tac-toe method) which makes it easier to work with the cases where you need to apply integration by parts more than once.
• April 19th 2008, 08:51 PM
Boris B
I think I've been using this formula wrong. Do we ever actually take the derivative of the thing we originally turned into the derivative? This would be clearer if the example hadn't used a base e exponent; I think that is what may be screwing me up.
I tried to take the antiderivative of:
$f(x) = \frac{2.5(200)^{2.5}}{x^{3.5}}
$
First I decided that I could take the derivative of $x^{3.5}$. Since it was in the denominator first, I had to turned into a $x^{-3.5}$ and make it part of the numerator. Then I multiplied that by $2.5(200)^{2.5}$
$2.5(200)^{2.5} \cdot x^{-3.5} - \int ...$
For the other side of the integral, I took the derivative of $2.5(200)^{2.5}$, which, lacking a variable, is 1. That left $x^{3.5}$, which integrates to $\frac{x^{4.5}}{4.5}$
That leaves me with
$2.5(200)^{2.5} \cdot x^{-3.5} - \frac{x^{4.5}}{4.5} =$
$1,414,213 x^{-3.5} - \frac{x^{4.5}}{4.5} =$
The definite integrals I take are all quite preposterous (e.g. 5.02 billion minus negative infinity), implying my antiderivative is wrong. (My end goal is to find the difference of the 70th and 30th percentiles of X; I assume I'll need the antiderivative for this but I haven't quite worked out the endgame.)
• April 19th 2008, 09:26 PM
Jhevon
I hate using the formula as well. But my way isn't nearly as formal as TPH's. I just always thought of it as: "the integral of one function times the other, minus the $\int$ of the same function times the derivative of the other" sounds confusing in words, but it helps my weird mind to remember. i just remember to integrate one function, and that appears in both factors. then i put the other function in the first factor, and its derivative in the second factor.
• April 20th 2008, 04:01 PM
Boris B
To no one in particular
I found the short form of the formula for integration by parts at Wikipedia.
I think I'm finally starting to get it. The short form is:
$\int u dv = uv - \int v du$
Okay ... processing. I think this means that where there is a "u dv" you are actually multiplying variable u by variable dv. Also, dv is the derivative of v. (Why they didn't include a multiplier dot between u and dv is unknown. Edit: it is also unknown why the math script is taking the space out from between u and dv in the above.)
I'm still not sure if I have the correct intepretation here, because in every other case of d_ following an integration symbol the formula did not call for multiplication.
If I'm not mistaken, dx never means multiply, it's just this little thing that follows up integrations or antiderivations (presumably to make the multivariable calculus mavens happy). An example of dx not meaning "multiply by the derivative of x" is two lines up on the Wikipedia page (again "if I'm not mistaken").
I'm really jonesing to get this figured out. Got my fingers crossed.
• April 20th 2008, 04:56 PM
Mathstud28
Quote:
Originally Posted by Boris B
I found the short form of the formula for integration by parts at Wikipedia.
I think I'm finally starting to get it. The short form is:
$\int u dv = uv - \int v du$
Okay ... processing. I think this means that where there is a "u dv" you are actually multiplying variable u by variable dv. Also, dv is the derivative of v. (Why they didn't include a multiplier dot between u and dv is unknown. Edit: it is also unknown why the math script is taking the space out from between u and dv in the above.)
I'm still not sure if I have the correct intepretation here, because in every other case of d_ following an integration symbol the formula did not call for multiplication.
If I'm not mistaken, dx never means multiply, it's just this little thing that follows up integrations or antiderivations (presumably to make the multivariable calculus mavens happy). An example of dx not meaning "multiply by the derivative of x" is two lines up on the Wikipedia page (again "if I'm not mistaken").
I'm really jonesing to get this figured out. Got my fingers crossed.
You are almost right except u and v are functions of x...if they were different variables you would have to assume one is a constant...making the integration exceedingly simple (Wink)
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http://mathoverflow.net/questions/96777/non-isomorphic-stably-isomorphic-fields | # non-isomorphic stably isomorphic fields
Q1: What is the simplest example of two non-isomorphic fields $L$ and $K$ of characteristic $0$ such that $L(x)\simeq K(x)$ (here $x$ is an indeterminate)?
Q2: Do we have a sufficient criterion for a general field $K$ of characteristic $0$ which guarantees that if $K(x_1,\ldots,x_n)\simeq L(x_1,\ldots, x_n)$ (here $L$ is a field and the $x_i$'s are indeterminates) then $K\simeq L$?
-
Just a comment: The question $R[x] \cong S[x] \Rightarrow R \cong S$ has been studied in the literature since the 70s (google for "isomorphic polynomial rings" or see math.stackexchange.com/questions/13504); Hochster has constructed counterexamples. On the other hand, this is true for fields (consider units). So in Q1, we cannot expect $L[x] \cong K[x]$ to hold. By the way: 1+, since I don't know an example for Q1 at all. – Martin Brandenburg May 12 '12 at 15:15
Concerning Q2: A sufficient condition is that $K,L$ are algebraic extensions of the prime field. – Ralph May 12 '12 at 16:26
Here is one possible way of constructing such examples: Let $\iota_1:G\rightarrow S_{n}$ and $\iota_2:G\rightarrow S_{m}$ be two embeddings of a finite group $G$ where $S_k$ denotes the symmetric group of degree $k$. Let $K_n$ be the field of rational functions over $\mathbf{Q}$ in $n$ variables then it is easy to see that $K_n^{\iota_1(G)}$ and $K_{m}^{\iota_2(G)}$ are stable isomorphic, but in general I don't see any reason why they should be isomorphic. Of course one needs to choose the group $G$ carefully since for "many" $G$'s $K_n^{\iota_1(G)}$ will always be purely transcendental. – Hugo Chapdelaine May 12 '12 at 17:50
Thanks Ralph for the answer, yes indeed, an isomorphism takes algebraic elements over the prime field to algebraic elements over the prime field. – Hugo Chapdelaine May 12 '12 at 17:57
I don't know if it's of practical help, but $K(x) \cong L(x)$ implies $K \cong L$ iff there is an isomorphism $K(x) \xrightarrow[]{\sim} L(x)$ that maps a transcendence base of $K|F$ onto a transcendence base of $L|F$ (where $F$ denotes the prime field). – Ralph May 12 '12 at 20:43
I don't think that there are any really easy examples. In the famous paper of Beauville, Colliot-Thélène, Sansuc and Swinnerton-Dyer "Variétés stablement rationnelles non rationnelles" they construct surfaces $S$ over $\mathbb Q$ that are not rational, but such that the products $S \times \mathbb P^3$ are rational. You get an example by taking $K$ to be a purely transcendental extension of the function field of $S$ of transcendence degree $d$, and a purely transcendental extension of $\mathbb Q$ of transcendence degree $d+2$, for some $d$ between $0$ and $3$ (I don't know the correct value of $d$).
Silly question: what is wrong with the example $K=\mathbb{Q}$ and $L=\mathbb{Q}(x)$? – Mahdi Majidi-Zolbanin May 14 '12 at 4:20
@Mahdi: Well, $K(t) \not\cong L(t)$. – Martin Brandenburg May 14 '12 at 14:53
@Martin: I see. What I had in mind was to say $K(x)\cong L(x)$ (same $x$ as in $L=\mathbb{Q}(x)$), but I see now, $x$ is not an indeterminate over $\mathbb{Q}(x)$. Thanks! – Mahdi Majidi-Zolbanin May 14 '12 at 15:38
An answer to Q2, generalizing Ralph's comment: "$K$ is algebraically closed" is a sufficient condition. Indeed, you can characterize $K$ inside $K(x_1,\dots,x_n)$ as the set of elements having $m$-th roots for infinitely many integers $m$. More generally, it is enough to assume that for some $m>1$, the $m$-th power map on $K$ is onto. Examples: $K$ perfect of positive characteristic, or $K=\mathbb{R}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9461268782615662, "perplexity": 192.24859518954955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500823333.10/warc/CC-MAIN-20140820021343-00043-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.computer.org/csdl/proceedings/pads/2002/1608/00/16080079-abs.html | Proceedings 16th Workshop on Parallel and Distributed Simulation (2002)
Washington, D.C.
May 12, 2002 to May 15, 2002
ISSN: 1087-4097
ISBN: 0-7695-1608-4
pp: 79
Jason Liu , Dartmouth College
David M. Nicol , Dartmouth College
ABSTRACT
Rapid growth in wireless communication systems motivates the development of technology supporting the simulation of large-scale wireless systems. However, it is widely recognized that wireless communications do not have substantial lookahead'' needed by conservative synchronization protocols. This paper focuses on identifying and exploiting lookahead for such models. We find lookahead in three ways, exploiting characteristics of low power networks, the transceiver logic, and the way in which protocol stacks are typically constructed. We show how these observations allow a variety of conservative synchronization protocols to take advantage of lookahead, describe a synchronization method we use, and empirically examine the performance this method offers on a large-scale simulation of a sensor network intended for homeland defense scenarios.
INDEX TERMS
parallel simulation, parallel discrete-event simulation, synchronization, performance, lookahead, conservative parallel simulation, wireless networks, ad hoc networks, sensor networks.
CITATION
Jason Liu, David M. Nicol, "Lookahead Revisited in Wireless Network Simulations", Proceedings 16th Workshop on Parallel and Distributed Simulation, vol. 00, no. , pp. 79, 2002, doi:10.1109/PADS.2002.1004203 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8422297835350037, "perplexity": 4147.347585295369}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948542031.37/warc/CC-MAIN-20171214074533-20171214094533-00200.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=4160732 | # Abstract Algebra HW
by nateHI
Tags: abstract, algebra
P: 62 1. The problem statement, all variables and given/known data Suppose $N \lhd G$ and $K \vartriangleleft G$ and $N \cap K = \{e\}$. Show that if $n \in N$and $k \in K$, then $nk = kn$. Hint: $nk = kn$ if and only if $nkn^{-1}k^{-1} = e$. 2. Relevant equations These "relevant equations" were not provided with the problem I'm just putting them here to make my solution more clear. $e=k_1^{-1}k_1$ $e=n_1^{-1}n_1$ 3. The attempt at a solution Let $n_1,n_2\in N$ and let $k_1,k_2\in K$ Then $(n_1)(k_1)(n_2)(k_2)=(n_1)(k_1)(n_2)e(k_2)=n_1(k_1n_2k_1^{-1})(k1k2)=n1NK=NK$ But $(n_1)(k_1)(n_2)(k_2)=(n_1)(k_1)e(n_2)(k_2)=(n_1k_1n_1^{-1})(n_1n_2)k_2=KNk_2=Kk_2N=KN$ where we used the fact that in this case, $k_2 \notin N$. Therefore $NK=KN$ for all $n\in N$ and $k\in K$ This seems correct to me but I didn't use the hint and my usage of $N \cap K = \{e\}$ seems a little hand wavey. Please help.
Sci Advisor HW Helper PF Gold P: 3,288 I didn't follow your proof. I think you are confusing things by using two elements from each subgroup. You only need the elements given in the problem statement: $n \in N$ and $k \in K$. Now consider the element $nkn^{-1}k^{-1}$. The goal is to show that this equals $e$. One promising way to do this would be to show that it is an element of $N \cap K$.
P: 62 OK, I'll try your method next since I would like to know how to use the hint since the teacher is obviously trying to convey something he deems important. However, I modified my original method slightly and I think it works now. If someone has the time please tell if it does or how it doesn't work. Let $n_1,n_2\in N$ and let $k_1,k_2\in K$ Then ${\bf (n_1)(k_1)(n_2)(k_2)}=(n_1)(k_1)(n_2)e(k_2)=n_1(k_1n_2k_1^{-1})(k1k2)=n1NK=NK$ But ${\bf (n_1)(k_1)(n_2)(k_2)}=(n_1)(k_1)e(n_2)(k_2)=(n_1k_1n_1^{-1})(n_1n_2)k_2=KNk_2=Kk_2N=KN$ Where we used the facts that $Nk_2=k_2N$, $n_1N=N$ and $Kk_2=K$. Therefore $NK=KN$ for all $n\in N$ and $k\in K$
HW Helper
PF Gold
P: 3,288
Abstract Algebra HW
Quote by nateHI OK, I'll try your method next since I would like to know how to use the hint since the teacher is obviously trying to convey something he deems important. However, I modified my original method slightly and I think it works now. If someone has the time please tell if it does or how it doesn't work. Let $n_1,n_2\in N$ and let $k_1,k_2\in K$ Then ${\bf (n_1)(k_1)(n_2)(k_2)}=(n_1)(k_1)(n_2)e(k_2)=n_1(k_1n_2k_1^{-1})(k1k2)=n1NK=NK$
What does this string of equalities even mean? If we look at just the leftmost and rightmost expressions, you are asserting that
$$n_1 k_1 n_2 k_2 = NK$$
The left hand side is an element, but the right hand side is a group. How can an element equal a group?
P: 62
Quote by jbunniii What does this string of equalities even mean? If we look at just the leftmost and rightmost expressions, you are asserting that $$n_1 k_1 n_2 k_2 = NK$$ The left hand side is an element, but the right hand side is a group. How can an element equal a group?
Hmm, good point. I'll give this one last try using my method. Even if I don't get it this is good I feel like I'm learning a lot.
Here is my modification that I hope fixes it.
Let $n_1,n_2\in N$ and let $k_1,k_2\in K$
Then
${\bf (n_1)(k_1)(n_2)(k_2)}=(n_1)(k_1)(n_2)e(k_2)=n_1(k_1n_2k_1^{-1})(k_1k_2)=n_1nk=nk$
But
${\bf (n_1)(k_1)(n_2)(k_2)}=(n_1)(k_1)e(n_2)(k_2)=(n_1k_1n_1^{-1})(n_1n_2)k_2=knk_2=kk_2n=kn$
I guess the flaw now is that we can't assume that $nk_2=k_2n$. Well I'll just use your method cause this probably won't work.
HW Helper
PF Gold
P: 3,288
OK, this is better. But the problem is that the n and k in this equation:
${\bf (n_1)(k_1)(n_2)(k_2)}=(n_1)(k_1)(n_2)e(k_2)=n_1(k_1n_2k_1^{-1})(k_1k_2)=n_1nk=nk$
need not be the same as the n and k in this equation:
${\bf (n_1)(k_1)(n_2)(k_2)}=(n_1)(k_1)e(n_2)(k_2)=(n_1k_1n_1^{-1})(n_1n_2)k_2=knk_2=kk_2n=kn$
You have the right idea, but you are making it more complicated than it needs to be. Try similar manipulations with the idea I suggested earlier.
P: 62 I guess since the intersection of n and k is {e} for both of my n's of N and k's of K then intuitively nk=e and nk=e which implies nkn^-1k^-1=e and I'm done. I just need to formalize that idea.
HW Helper
PF Gold
P: 3,288
Quote by nateHI I guess since the intersection of n and k is {e} for both of my n's of N and k's of K then intuitively nk=e and nk=e which implies nkn^-1k^-1=e and I'm done. I just need to formalize that idea.
I didn't follow your argument. You have $n \in N$ and $k \in K$. Now what can you say about the element $nkn^{-1}k^{-1}$? Hint: what if you add some parentheses: $(nkn^{-1})k^{-1}$?
Math Emeritus Sci Advisor Thanks PF Gold P: 39,497 Where have you used the fact that N and K are normal subgroups of G?
P: 62 OK, let me try to make it a little more precise. Since $N$ is normal in $G$, any element $k\in K$ which also happens to be in $G$ since $K$ is a subgroup (the fact that K is normal isn't important until the next step) of $G$ we get $knk^{-1}=n \implies nk=kn$ for all $n \in N$. Since $K$ is normal in $G$, any element $n\in N$ which also happens to be in $G$ since $N$ is a subgroup (the fact that N is normal is only important in the last step) of $G$ we get $nkn^{-1}=k \implies kn=nk$ for all $k \in K$. However, the intersection of N and K can only be {e} therefore $nk=e=kn \implies nkn^{-1}k^{-1}=en^{-1}k^{-1}=ee=e$. Furthermore $(nkn^{-1})k^{-1} \in K$ and $n(kn^{-1}k^{-1}) \in N$ Am I done? It seems like there should be a more concise way to say it regardless.
P: 62 Wait wait I get it now.... If $(nkn^{-1})k^{-1} \in K$ and $n(kn^{-1}k^{-1}) \in N$ where we have used the fact that N ad K are normal in G then $(nkn^{-1})k^{-1}=e$ and $(nkn^{-1})k^{-1}=e \implies nk=kn$ Took me a while but I got it.
Quote by nateHI Wait wait I get it now.... If $(nkn^{-1})k^{-1} \in K$ and $n(kn^{-1}k^{-1}) \in N$ where we have used the fact that N ad K are normal in G then $(nkn^{-1})k^{-1}=e$ and $(nkn^{-1})k^{-1}=e \implies nk=kn$ Took me a while but I got it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9554589986801147, "perplexity": 201.42325085269414}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500823333.10/warc/CC-MAIN-20140820021343-00143-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://nrich.maths.org/5605/clue?nomenu=1 | So $i^2 = i \times i= 1 cis 90 \times 1 cis 90 = 1 cis 180 = -1.$ Now you should have enough information to multiply out (z-i)(z+i) . | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9922863841056824, "perplexity": 515.1221773645244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427132518903.47/warc/CC-MAIN-20150323174158-00152-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://hal.inria.fr/hal-00705045 | Parameter-field estimation for atmospheric dispersion: application to the Chernobyl accident using 4D-Var
Abstract : Atmospheric chemistry and air-quality numerical models are driven by uncertain forcing fields: emissions, boundary conditions, wind fields, vertical turbulent diffusivity, kinetic chemical rates, etc. Data assimilation can help to assess these parameters or fields of parameters. Because such parameters are often much more uncertain than the fields diagnosed in meteorology and oceanography, data assimilation is much more of an inverse modelling challenge in this context. In this article these ideas are experimented with by revisiting the Chernobyl accident dispersion event over Europe. A fast four-dimensional variational scheme (4D-Var) is developed, which seems appropriate for the retrieval of large parameter fields from large observation sets and the retrieval of parameters that are nonlinearly related to concentrations. The 4D-Var, and especially an approximate adjoint of the transport model, is tested and validated using several advection schemes that are influential on the forward simulation as well as on the data-assimilation results. Firstly, the inverse modelling system is applied to the assessment of the dry and wet deposition parameters. It is then applied to the retrieval of the emission field alone, the joint optimization of removal-process parameters and source parameters and the optimization of larger parameter fields such as horizontal and vertical diffusivities or the dry-deposition velocity field. The physical parameters used so far in the literature for the Chernobyl dispersion simulation are partly supported by this study. The crucial question of deciding whether such an inversion is merely a tuning of parameters or a retrieval of physically meaningful quantities is discussed. Even though inversion of parameter fields may fail to determine physical values for the parameters, it achieves statistical adaptation that partially corrects for model errors and, using the inverted parameter fields, leads to considerable improvement in the simulation scores. Copyright c 2011 Royal Meteorological Society
Type de document :
Article dans une revue
Quarterly Journal of the Royal Meteorological Society, Wiley, 2012, 138 (664), pp.664-681. 〈10.1002/qj.961〉
Domaine :
https://hal.inria.fr/hal-00705045
Contributeur : Nathalie Gaudechoux <>
Soumis le : mercredi 6 juin 2012 - 16:59:09
Dernière modification le : vendredi 25 mai 2018 - 12:02:03
Citation
Marc Bocquet. Parameter-field estimation for atmospheric dispersion: application to the Chernobyl accident using 4D-Var. Quarterly Journal of the Royal Meteorological Society, Wiley, 2012, 138 (664), pp.664-681. 〈10.1002/qj.961〉. 〈hal-00705045〉
Métriques
Consultations de la notice | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9043522477149963, "perplexity": 3636.3393041621975}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590074.12/warc/CC-MAIN-20180718080513-20180718100513-00510.warc.gz"} |
https://www.physicsforums.com/threads/conservation-of-mechanical-energy.227862/ | # Conservation Of Mechanical Energy
1. Apr 10, 2008
### tizzful
1. The problem statement, all variables and given/known data
Two blocks with different mass are attached to either end of a light rope that passes over a light, frictionless pulley that is suspended from the ceiling. The masses are released from rest, and the more massive one starts to descend. After this block has descended a distance 1.30 , its speed is 1.00 .
If the total mass of the two blocks is 18.0 , what is the mass of the more massive block?
Take free fall acceleration to be 9.80 .
I set the heavier block as m1.
2. Relevant equations
m1gh1+0.5m1v1^2=m2gh2+0.5m2v2^2
3. The attempt at a solution
They both start at height 0 and velocity 0 and so the initial PE and KE is going to be 0, and so the initial Mechanical energy is also 0 (I'm pretty sure this is wrong but don't know how to fix it). Then m1 drops 1.30m so thats h1 and m2 goes up -1.30m=h2. v should be equal between both, m1=1, m2=-1.
m1(gh1+0.5v^2)=m2(gh2+0.5v^2)
m1/m2=(gh2+0.5v^2)/(gh1+0.5v^2)
=(9.8*-1.30+1/2*-1^2)/(9.8*1.30+1/2*1^2)
Therefore
m1=-m2
So its wrong ahah I was wondering if someone could help me?
2. Apr 10, 2008
### tiny-tim
Hi tizzful!
erm … it's not "-1^2" …
No wonder they came out minus each other!
3. Apr 10, 2008
### tizzful
actually the -1 gets squared and so it becomes one.. Its negative because the height is negative because down is positive and up is negative.. But from what you're saying why isn't it -1? It's also in the opposite direction...
4. Apr 10, 2008
### tiny-tim
Noooooo … the kinetic energy mv^2/2 is always positive!
It depends only on speed, not direction!
You have too much imagination!
5. Apr 10, 2008
### tizzful
ahahah thank you! But I know KE is always positive because if velocity is negative it gets squared making it positive.. And thats what happened in this case.. But i still can't figure out the answer.. I think there's something wrong with me saying initial ME = 0...
6. Apr 10, 2008
### tiny-tim
ah … I see now … your basic equation is wrong …
KE gained is (m1 + m2)v^2/2
PE gained is (m1 - m2)gh.
7. Apr 10, 2008
### Kurret
When you have gotten the answer using that method (which is probably the easiest), you can also try doing it by finding the acceleration on the big mass, and then use s=(at^2)/2. You will end up with the exactly same equation. :)
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http://nrich.maths.org/4768/index?nomenu=1 | Full Screen Version
This text is usually replaced by the Flash movie.
Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. The bar strangely has no weight of its own.
Move the bar from side to side until you find a balance point. Is it possible to predict that position
(you might find it helps to look at the problem called "Inside Outside" first)
Which arrangements produce balance points outside the central interval (between the two inner attachment points for weights)?
If the bar now does have weight, what is the least weight it could have with no arrangements that produce balance points outside the central interval.
You only need the 1, 2, 4 and 8 unit weights for the problem set but the others are there to let you explore, conjecture and test - have fun. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.805916965007782, "perplexity": 667.778274019215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131298660.78/warc/CC-MAIN-20150323172138-00000-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://tel.archives-ouvertes.fr/tel-00008730/en/ | # deux contributions à l'étude semi-paramétrique d'un modèle de régression
Abstract : In this thesis we are interested in two semiparametric regression models which allow to get rid with the so-called curse of dimensionality problem linked with the nonparametrical approach. The first part of this work deals with the study of the regression model called partialy-linear ; the goal is the identification of the regressor that appear nonparametricaly in the regression function and we also estimate the parameters defining the model. Then we define a linearity measure and we derive a test of the number of non-linear components based on this measure. The second part is devoted to the study of the so-called single index model and its aim is to estimate the axis using geometrical properties and to construct a test of the whole model consistent under the null hypothesis and powerfull under a sequence of local alternatives.
Mots-clés :
Document type :
Theses
Mathematics [math]. Université Rennes 1, 2003. French
Domain :
https://tel.archives-ouvertes.fr/tel-00008730
Contributor : Céline Vial <>
Submitted on : Tuesday, March 8, 2005 - 11:02:24 PM
Last modification on : Friday, March 27, 2015 - 9:59:53 AM
### Identifiers
• HAL Id : tel-00008730, version 1
### Citation
Céline Roget-Vial. deux contributions à l'étude semi-paramétrique d'un modèle de régression. Mathematics [math]. Université Rennes 1, 2003. French. <tel-00008730>
Consultation de
la notice
## 136
Téléchargement du document | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8168489933013916, "perplexity": 2619.142049081919}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988924.75/warc/CC-MAIN-20150728002308-00275-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://library.kiwix.org/wikipedia_en_computer_novid_2018-10/A/Well-formed_formula.html | # Well-formed formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.[1] A formal language can be identified with the set of formulas in the language.
A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic.
## Introduction
A key use of formulas is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven.
Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. Thus the same formula may be written more than once, and a formula might in principle be so long that it cannot be written at all within the physical universe.
Formulas themselves are syntactic objects. They are given meanings by interpretations. For example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as a formula.
## Propositional calculus
The formulas of propositional calculus, also called propositional formulas,[2] are expressions such as . Their definition begins with the arbitrary choice of a set V of propositional variables. The alphabet consists of the letters in V along with the symbols for the propositional connectives and parentheses "(" and ")", all of which are assumed to not be in V. The formulas will be certain expressions (that is, strings of symbols) over this alphabet.
The formulas are inductively defined as follows:
• Each propositional variable is, on its own, a formula.
• If φ is a formula, then ¬φ is a formula.
• If φ and ψ are formulas, and • is any binary connective, then ( φ • ψ) is a formula. Here • could be (but is not limited to) the usual operators ∨, ∧, →, or ↔.
This definition can also be written as a formal grammar in Backus–Naur form, provided the set of variables is finite:
<alpha set> ::= p | q | r | s | t | u | ... (the arbitrary finite set of propositional variables)
<form> ::= <alpha set> | ¬<form> | (<form>∧<form>) | (<form>∨<form>) | (<form>→<form>) | (<form>↔<form>)
Using this grammar, the sequence of symbols
(((p q) (r s)) (¬q ¬s))
is a formula, because it is grammatically correct. The sequence of symbols
((p q)(qq))p))
is not a formula, because it does not conform to the grammar.
A complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules (akin to the standard mathematical order of operations) are assumed among the operators, making some operators more binding than others. For example, assuming the precedence (from most binding to least binding) 1. ¬ 2. 3. 4. . Then the formula
(((p q) (r s)) (¬q ¬s))
may be abbreviated as
p q r s ¬q ¬s
This is, however, only a convention used to simplify the written representation of a formula. If the precedence was assumed, for example, to be left-right associative, in following order: 1. ¬ 2. 3. 4. , then the same formula above (without parentheses) would be rewritten as
(p (q r)) (s ((¬q) (¬s)))
## Predicate logic
The definition of a formula in first-order logic is relative to the signature of the theory at hand. This signature specifies the constant symbols, relation symbols, and function symbols of the theory at hand, along with the arities of the function and relation symbols.
The definition of a formula comes in several parts. First, the set of terms is defined recursively. Terms, informally, are expressions that represent objects from the domain of discourse.
1. Any variable is a term.
2. Any constant symbol from the signature is a term
3. an expression of the form f(t1,...,tn), where f is an n-ary function symbol, and t1,...,tn are terms, is again a term.
The next step is to define the atomic formulas.
1. If t1 and t2 are terms then t1=t2 is an atomic formula
2. If R is an n-ary relation symbol, and t1,...,tn are terms, then R(t1,...,tn) is an atomic formula
Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds:
1. is a formula when is a formula
2. and are formulas when and are formulas;
3. is a formula when is a variable and is a formula;
4. is a formula when is a variable and is a formula (alternatively, could be defined as an abbreviation for ).
If a formula has no occurrences of or , for any variable , then it is called quantifier-free. An existential formula is a formula starting with a sequence of existential quantification followed by a quantifier-free formula.
## Atomic and open formulas
An atomic formula is a formula that contains no logical connectives nor quantifiers, or equivalently a formula that has no strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; for propositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term.
According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers.[3] This has not to be confused with a formula which is not closed.
## Closed formulas
A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first-order language in which the variables v1, ..., vn have free occurrences, then A preceded by v1 ... vn is a closure of A.
## Usage of the terminology
In earlier works on mathematical logic (e.g. by Church[4]), formulas referred to any strings of symbols and among these strings, well-formed formulas were the strings that followed the formation rules of (correct) formulas.
Several authors simply say formula.[5][6][7][8] Modern usages (especially in the context of computer science with mathematical software such as model checkers, automated theorem provers, interactive theorem provers) tend to retain of the notion of formula only the algebraic concept and to leave the question of well-formedness, i.e. of the concrete string representation of formulas (using this or that symbol for connectives and quantifiers, using this or that parenthesizing convention, using Polish or infix notation, etc.) as a mere notational problem.
While the expression well-formed formula is still in use,[9][10][11] these authors do not necessarily use it in contradistinction to the old sense of formula, which is no longer common in mathematical logic.
The expression "well-formed formulas" (WFF) also crept into popular culture. WFF is part of an esoteric pun used in the name of the academic game "WFF 'N PROOF: The Game of Modern Logic," by Layman Allen,[12] developed while he was at Yale Law School (he was later a professor at the University of Michigan). The suite of games is designed to teach the principles of symbolic logic to children (in Polish notation).[13] Its name is an echo of whiffenpoof, a nonsense word used as a cheer at Yale University made popular in The Whiffenpoof Song and The Whiffenpoofs.[14]
## Notes
1. Formulas are a standard topic in introductory logic, and are covered by all introductory textbooks, including Enderton (2001), Gamut (1990), and Kleene (1967)
2. First-order logic and automated theorem proving, Melvin Fitting, Springer, 1996
3. Handbook of the history of logic, (Vol 5, Logic from Russell to Church), Tarski's logic by Keith Simmons, D. Gabbay and J. Woods Eds, p568 .
4. Alonzo Church, [1996] (1944), Introduction to mathematical logic, page 49
5. Hilbert, David; Ackermann, Wilhelm (1950) [1937], Principles of Mathematical Logic, New York: Chelsea
6. Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6
7. Barwise, Jon, ed. (1982), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN 978-0-444-86388-1
8. Cori, Rene; Lascar, Daniel (2000), Mathematical Logic: A Course with Exercises, Oxford University Press, ISBN 978-0-19-850048-3
9. Enderton, Herbert [2001] (1972), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-238452-3
10. R. L. Simpson (1999), Essentials of Symbolic Logic, page 12
11. Mendelson, Elliott [2010] (1964), An Introduction to Mathematical Logic (5th ed.), London: Chapman & Hall
12. Ehrenburg 2002
13. More technically, propositional logic using the Fitch-style calculus.
14. Allen (1965) acknowledges the pun.
## References
• Allen, Layman E. (1965), "Toward Autotelic Learning of Mathematical Logic by the WFF 'N PROOF Games", Mathematical Learning: Report of a Conference Sponsored by the Committee on Intellective Processes Research of the Social Science Research Council, Monographs of the Society for Research in Child Development, 30 (1): 29–41
• Boolos, George; Burgess, John; Jeffrey, Richard (2002), Computability and Logic (4th ed.), Cambridge University Press, ISBN 978-0-521-00758-0
• Ehrenberg, Rachel (Spring 2002). "He's Positively Logical". Michigan Today. University of Michigan. Retrieved 2007-08-19.
• Enderton, Herbert (2001), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-238452-3
• Gamut, L.T.F. (1990), Logic, Language, and Meaning, Volume 1: Introduction to Logic, University Of Chicago Press, ISBN 0-226-28085-3
• Hodges, Wilfrid (2001), "Classical Logic I: First-Order Logic", in Goble, Lou, The Blackwell Guide to Philosophical Logic, Blackwell, ISBN 978-0-631-20692-7
• Hofstadter, Douglas (1980), Gödel, Escher, Bach: An Eternal Golden Braid, Penguin Books, ISBN 978-0-14-005579-5
• Kleene, Stephen Cole (2002) [1967], Mathematical logic, New York: Dover Publications, ISBN 978-0-486-42533-7, MR 1950307
• Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: Springer Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9538587331771851, "perplexity": 1312.6966028478153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578529472.24/warc/CC-MAIN-20190420080927-20190420102927-00224.warc.gz"} |
http://www.maplesoft.com/support/help/Maple/view.aspx?path=DifferentialGeometry/Library/MetricSearch | a Maplet for searching the DifferentialGeometry libraries of metrics - Maple Help
Home : Support : Online Help : Mathematics : DifferentialGeometry : Library : DifferentialGeometry/Library/MetricSearch
Library[MetricSearch] - a Maplet for searching the DifferentialGeometry libraries of metrics
Calling Sequences
MetricSearch()
MetricSearch(PropList)
Parameters
PropList - a list of spacetime metric properties to search for.
Description
• The DifferentialGeometry Library package contains an extensive database of spacetime metrics and matter fields which give solutions to the Einstein equations of general relativity. The command MetricSearch allows the Maple user to search the database for metrics with specified properties. The first calling sequence launches an easy to use Maplet. The second calling sequence provides the same search capabilities in a command line format.
• The command MetricSearch() initializes a maplet which searches the DifferentialGeometry Library for metrics with user-specified properties. The current search criteria are summarized in the following table.
Physical Properties Primary Description. Search for solutions to the Einstein equations with a prescribed class of energy-momentum tensors. Secondary Description. Search for metrics with a prescribed class of mathematical or physical properties. Keyword. Keyword descriptions include metric names or authors. Algebraic Properties Petrov Type. Algebraic classification of the Weyl tensor. Plebanksi-Petrov Type. Algebraic classification of the Ricci tensor. This is the Petrov type of the Plebanksi tensor. Segre Type. Algebraic classification of the Ricci tensor. The Segre type specifies the normal form of a linear transformation which is self-adjoint with respect to a 4-dimensional Lorentz signature metric. Isometry Properties Isometry Dimension. The number of Killing vectors. KillingVectors Orbit Dimension. The number of pointwise independent Killing vectors; or the dimension of the orbit of the group of isometries. Orbit Type. The signature of the induced metric on the orbits. SubspaceType Isotropy Type. For any Lorentz signature 4-dimensional spacetime, the infinitesimal isotropy representation defines a subalgebra of $\mathrm{so}\left(3,1\right)$. These subalgebras have been classified and are labeled .
• Once properties are selected by checking appropriate boxes, pressing the Search button will return all metrics in the DifferentialGeometry library which possess all of the indicated properties. The format of the result is a string representing the source reference and a sequence of lists indicating the equation numbers in the reference where the metrics appear. All the information in the DifferentialGeometry library for each metric can be obtained with the Retrieve command.
• Information regarding the metric found by a search of the DifferentialGeometry library can be retrieved in one of two ways. One method is to enter the reference name (string), equation number, and the name of an initialized manifold (created in the calling worksheet using DGsetup) into the text boxes in the Retrieve section of the MetricSearch Maplet. Then pressing the Retrieve button will return a list of the spacetime fields defining the solution (e.g., metric, electromagnetic field, etc.) to the calling worksheet. All the information in the DifferentialGeometry library for each metric can be obtained with the Retrieve command.
• The Clear button resets all check boxes and clears all text boxes. The Close button will close the Maplet.
• The second calling sequence accepts a list of desired properties, each specified by an equation . The possibilities are:
metricproperty propertyvalue Related Commands "PrimaryDescription" "Dust", "Einstein", "EinsteinMaxwell", "PerfectFluid", "PureRadiation", "Vacuum" "SecondaryDescription" "Homogeneous", "PlaneWave", "PPWave", "PureRadiation", "RobertsonWalker", "SimplyTransitive", "Static" "KeywordDescription" a list of strings: author names, metric names, etc. "PetrovType" "I", "II", "III", "D", "N", "O" "PlebanskiPetrovType" "I", "II", "III", "D", "N", "O" "SegreType" For example, "[1,(111)]" "IsometryDimension" 0, 1, 2, 3, 4, 5, 6, 7, 10 "OrbitDimension" 0, 1, 2, 3, 4 "OrbitType" "Null", "Riemannian", "PseudoRiemannian" "IsotropyType "F1", "F2", "F3", "F4", "F5", "F6", "F7", "F8", "F9", "F10", "F11", "F12", "F13", "F14", "F15"
• Currently the DifferentialGeometry library contains selected metrics from:
1 Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; and Herlt, E. Exact Solutions to Einstein's Field Equations. 2nd ed. Cambridge Monographs on Mathematical Physics, 2003.
2. Hawking, Stephen; and Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press.
Examples
> $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$$\mathrm{with}\left(\mathrm{Library}\right):$
Example 1
We find examples of metrics which are homogeneous Einstein metrics of Petrov type III. First initialize a manifold with coordinates, e.g.,
> $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],M\right)$
${\mathrm{frame name: M}}$ (2.1)
Start the MetricSearch Maplet.
M > $F:=\mathrm{MetricSearch}\left(\right):$
Check the Einstein box in the PhysicalProperties-Primary Description section, check type III in the Algebraic Properties-Petrov Type section, check 4 in the Orbit Dimension section.
The result is "Stephani": [12, 35, 1].
Enter "Stephani" into the Reference textbox, enter [12, 35, 1] into the equation number textbox, and enter M into the manifold textbox. Press the Retrieve button. The metric is assigned to $F.$ (If matter fields were present, they would also be assigned to $F$.)
M > $F$
Example 2
We calculate some properties of a given metric and identify the metrics with the same properties in the library database.
M > $\mathrm{DGsetup}\left(\left[t,x,y,\mathrm{φ}\right],M\right)$
${\mathrm{frame name: M}}$ (2.2)
M > $g:=\mathrm{evalDG}\left(\frac{1\left(\mathrm{dx}&t\mathrm{dx}+\mathrm{dy}&t\mathrm{dy}\right)}{{x}^{2}}+{x}^{2}\mathrm{dphi}&t\mathrm{dphi}-\left(\mathrm{dt}-2y\mathrm{dphi}\right)&t\left(\mathrm{dt}-2y\mathrm{dphi}\right)\right)$
${g}{:=}{-}{\mathrm{dt}}{}{\mathrm{dt}}{+}{2}{}{y}{}{\mathrm{dt}}{}{\mathrm{dphi}}{+}\frac{{\mathrm{dx}}{}{\mathrm{dx}}}{{{x}}^{{2}}}{+}\frac{{\mathrm{dy}}{}{\mathrm{dy}}}{{{x}}^{{2}}}{+}{2}{}{y}{}{\mathrm{dphi}}{}{\mathrm{dt}}{+}\left({-}{4}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dphi}}{}{\mathrm{dphi}}$ (2.3)
We use the command RainichConditions to see if the space-time is a solution of the Einstein-Maxwell equations (electrovac spacetime).
M > $\mathrm{RainichConditions}\left(g\right)$
${\mathrm{true}}$ (2.4)
We use the command KillingVectors to determine the dimension of the group of isometries.
M > $\mathrm{KV}:=\mathrm{KillingVectors}\left(g\right)$
${\mathrm{KV}}{:=}\left[\frac{{1}}{{4}}{}{x}{}{\mathrm{D_x}}{+}\frac{{1}}{{4}}{}{y}{}{\mathrm{D_y}}{-}\frac{{1}}{{4}}{}{\mathrm{φ}}{}{\mathrm{D_phi}}{,}\frac{{1}}{{2}}{}{\mathrm{φ}}{}{\mathrm{D_t}}{+}\frac{{1}}{{4}}{}{\mathrm{D_y}}{,}{-}\frac{{1}}{{4}}{}{\mathrm{D_phi}}{,}\frac{{1}}{{2}}{}{\mathrm{D_t}}\right]$ (2.5)
M > $\mathrm{nops}\left(\mathrm{KV}\right)$
${4}$ (2.6)
Next we find the Petrov type of the metric.
M > $\mathrm{PetrovType}\left(g\right)$
${"I"}$ (2.7)
Search for this metric in the data-base using the command line version of MetricSearch. We find that this is the metric in Stephani, Kramer et al. equation [12, 21, 1].
M > $\mathrm{MetricSearch}\left(\left["PrimaryDescription"="EinsteinMaxwell","PetrovType"="I","IsometryDimension"=4\right]\right)$
$\left[\left[{"Stephani"}{,}{1}{,}\left[{12}{,}{21}{,}{1}\right]\right]\right]$ (2.8)
M > $\mathrm{Retrieve}\left("Stephani",1,\left[12,21,1\right],\mathrm{manifoldname}=M,\mathrm{output}=\left["Metric"\right]\right)$
$\left[{-}{\mathrm{dt}}{}{\mathrm{dt}}{+}{2}{}{y}{}{\mathrm{dt}}{}{\mathrm{dphi}}{+}\frac{{{\mathrm{_a}}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}}{{{x}}^{{2}}}{+}\frac{{{\mathrm{_a}}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{{x}}^{{2}}}{+}{2}{}{y}{}{\mathrm{dphi}}{}{\mathrm{dt}}{+}\left({-}{4}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dphi}}{}{\mathrm{dphi}}\right]$ (2.9)
Example 3
Find the Goedel metric in the data-base and then retrieve it:
M > $S:=\mathrm{MetricSearch}\left(\left["Keywords"=\left["Goedel"\right]\right]\right)$
${S}{:=}\left[\left[{"Stephani"}{,}{1}{,}\left[{12}{,}{26}{,}{1}\right]\right]\right]$ (2.10)
M > $g:={\mathrm{Retrieve}\left(\mathrm{op}\left({S}_{1}\right),\mathrm{manifoldname}=M,\mathrm{output}=\left["Metric"\right]\right)}_{1}$
${g}{:=}{-}{{\mathrm{_a}}}^{{2}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{{\mathrm{_a}}}^{{2}}{}{{ⅇ}}^{{x}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{+}{{\mathrm{_a}}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{\mathrm{_a}}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}{{\mathrm{_a}}}^{{2}}{}{{ⅇ}}^{{x}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{-}\frac{{1}}{{2}}{}{{\mathrm{_a}}}^{{2}}{}{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}$ (2.11) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 31, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8771886825561523, "perplexity": 4597.644811626911}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701152130.53/warc/CC-MAIN-20160205193912-00142-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://zbmath.org/?q=an%3A1219.05181 | ×
# zbMATH — the first resource for mathematics
Note on the structure of Kruskal’s algorithm. (English) Zbl 1219.05181
Let $$G=(V,E)$$ be a connected edge-weighted graph and let $$(V,F)$$ be its minimal spanning tree constructed by Kruskal’s algorithm [J.B. Kruskal jun., “On the shortest spanning subtree of a graph and the traveling salesman problem,” Proc. Am. Math. Soc. 7, 48–50 (1956; Zbl 0070.18404)]. We capture the evolution of the spanning forest from $$(V,\emptyset)$$ to $$(V,F)$$ by a rooted binary tree $$R$$ with leaves in $$V$$ and internal nodes in $$F$$. Let $$h(G)$$ denote the height of $$R$$. In case of Prim’s algorithm we would have $$h(G_n) = n-1$$ for every connected graph $$G_n$$ on $$n$$ vertices. In case of Kruskal’s algorithm there is a constant $$c>0$$ such that the probability of $$h(G_n) \geq cn$$ tends to $$1$$ for $$n\to\infty$$, and therefore the expected value of $$h(G_n)$$ is in $$\Theta(n)$$, for three choices of random edge-weights:
(1)
$$G_n$$ is a complete graph on $$n$$ independently uniformly distributed random points in $$[0,1]^d$$ and the edges are weighted by the Euclidean distance,
(2)
$$G_n$$ is a complete graph on $$n$$ vertices and the edge-weights are independently uniformly distributed in $$[0,1]$$,
(3)
$$G_n$$ is the Cartesian product of $$d$$ paths $$P_k$$, $$n=k^d$$, and the edge-weights are independently uniformly distributed in $$[0,1]$$.
##### MSC:
05C85 Graph algorithms (graph-theoretic aspects) 05C80 Random graphs (graph-theoretic aspects) 68R10 Graph theory (including graph drawing) in computer science
Full Text:
##### References:
[1] Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Boston (1974) · Zbl 0326.68005 [2] Aldous, D.: A random tree model associated with random graphs. Random Struct. Algorithms 4, 383–402 (1990) · Zbl 0747.05077 · doi:10.1002/rsa.3240010402 [3] Aldous, D., Steele, J.M.: Asymptotics for euclidean minimum spanning trees on random points. Probab. Theory Relat. Fields 92, 247–258 (1992) · Zbl 0767.60005 · doi:10.1007/BF01194923 [4] Bollobás, B.: Random Graphs, 2nd edn. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2001) [5] Bollobás, B., Simon, I.: On the expected behavior of disjoint set union algorithms. In: STOC ’85: Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, pp. 224–231. ACM, New York (1985) [6] Bollobás, B., Simon, I.: Probabilistic analysis of disjoint set union algorithms. SIAM J. Comput. 22, 153–1074 (1993) · Zbl 0789.68069 [7] Borgs, C., Chayes, J.T., Kesten, H., Spencer, J.: The birth of the infinite cluster: Finite-size scaling in percolation. Commun. Math. Phys. 224, 153–204 (2001) · Zbl 1038.82035 · doi:10.1007/s002200100521 [8] Boruvka, O.: O jistém problému minimálním. Práce Mor. Přírodověd. Spol. v Brně (Acta Soc. Sci. Natur. Moravicae) 3, 37–58 (1926) [9] Chernoff, H.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493–507 (1952) · Zbl 0048.11804 · doi:10.1214/aoms/1177729330 [10] Dijkstra, E.: A note on two problems in connection with graphs. Numer. Math. 1, 269–271 (1959) · Zbl 0092.16002 · doi:10.1007/BF01386390 [11] Erdos, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960) · Zbl 0103.16301 [12] Frieze, A.M.: On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10, 47–56 (1985) · Zbl 0578.05015 · doi:10.1016/0166-218X(85)90058-7 [13] Grimmett, G.R.: Percolation, 2nd edn. A Series of Comprehensive Studies in Mathematics, vol. 321. Springer, New York (1999) [14] Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley, New York (2000) [15] Jarńik, V.: O jistém problému minimálnim. Práce Mor. Přírodověd. Spol. v Brně (Acta Soc. Sci. Natur. Moravicae) 6, 57–63 (1930) [16] Kesten, H.: Percolation Theory for Mathematicians. Birkhäuser, Boston (1980) · Zbl 0522.60097 [17] Knuth, D.E., Schönhage, A.: The expected linearity of a simple equivalence algorithm. Theor. Comput. Sci. 6, 281–315 (1978) · Zbl 0377.68024 · doi:10.1016/0304-3975(78)90009-9 [18] Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 2, 48–50 (1956) · Zbl 0070.18404 · doi:10.1090/S0002-9939-1956-0078686-7 [19] McDiarmid, C., Johnson, T., Stone, H.S.: On finding a minimum spanning tree in a network with random weights. Random Struct. Algorithms 10(1–2), 187–204 (1997) · Zbl 0872.60008 · doi:10.1002/(SICI)1098-2418(199701/03)10:1/2<187::AID-RSA10>3.3.CO;2-Y [20] Meester, R., Roy, R.: Continuum Percolation. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1996) · Zbl 0858.60092 [21] Penrose, M.: The longest edge of a random minimal spanning tree. Ann. Appl. Probab. 7(2), 340–361 (1997) · Zbl 0884.60042 · doi:10.1214/aoap/1034625335 [22] Penrose, M.: Random minimal spanning tree and percolation on the n-cube. Random Struct. Algorithms 12, 63–82 (1998) · Zbl 0899.60083 · doi:10.1002/(SICI)1098-2418(199801)12:1<63::AID-RSA4>3.0.CO;2-R [23] Penrose, M.: A strong law for the longest edge of the minimal spanning tree. Ann. Probab. 27(1), 246–260 (1999) · Zbl 0944.60015 · doi:10.1214/aop/1022677261 [24] Penrose, M.: Random Geometric Graphs. Oxford Studies in Probability. Oxford University Press, Oxford (2003) · Zbl 1029.60007 [25] Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36, 1389–1401 (1957) [26] Steele, J.M.: Probability Theory and Combinatorial Optimization. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1997) · Zbl 0916.90233 [27] Yao, A.C.: On the average behavior of set merging algorithms. In: STOC ’76: Proceedings of the 8th Annual ACM Symposium on Theory of Computing, pp. 192–195 (1976) · Zbl 0365.68034 [28] Yukich, J.E.: Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Mathematics, vol. 1675. Springer, New York (1998) · Zbl 0902.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8359979391098022, "perplexity": 2452.333222816666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178360293.33/warc/CC-MAIN-20210228054509-20210228084509-00148.warc.gz"} |
https://www.math.bgu.ac.il/teaching/fall2016/courses/partial-differential-equations | ## Prof Yitzchak Rubinstein
##### יום ב 11:00 - 09:00 בגוטמן [32] חדר 210 יום ה 14:00 - 12:00 בבנין 90 (מקיף ז’) [90] חדר 227
1. Linear first order partial differential equations; characteristics - particles trajectories in a continuum; the Cauchy problem, propagation of singularities; complete integral and general solution.
2. System of two linear first order partial differential equations; classification; normal and canonical form; solution of the Cauchy problem for a hyperbolic system.
3. Classification of second order partial differential equations with a linear main part; canonical form; characteristics; propagation of singularities; Cauchy-Kovalevskaya theorem; physical phenomena leading to equations of various types.
4. One dimensional wave equation - example of a hyperbolic equation; initial and boundary conditions; Cauchy and boundary value problems; propagating waves method; D’Alambert’s formula; boundary value problems on a semi-axis and a segment; propagation of singularities; separation of variables; non-homogeneous problems; Duhamel’s principle.
5. One dimensional heat equation - example of a parabolic equation; typical problems - the Cauchy and boundary value problem; moments; solutions of heat equation on the axis, similarity variable and solution, fundamental solution and it’s properties, solution of the Cauchy problem; boundary value problems on a semi-axis, on a segment, separation of variables; non-homogeneous problems, Duhamel’s principle; Green’s functions; maximum principle and comparison theorems.
6. The Laplace’s equation - example of an elliptic equation; harmonic, sub- and super- harmonic functions and their properties, mean value theorem, maximum principle, Hopf’s lemma; Hadamard’s example and typical boundary value problems for elliptic equations; comparison theorems for linear and quasi-linear elliptic equations; fundamental solutions an their physical meaning; Green’s functions, method of images, inversion; separation of variables. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8402742147445679, "perplexity": 2906.910427877197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718296.19/warc/CC-MAIN-20161020183838-00468-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://www.purplemath.com/learning/viewtopic.php?p=4629 | ## Radicals and Rational exponents: 5*cbrt[16] + cbrt[54]
Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
EricA
Posts: 2
Joined: Mon Sep 06, 2010 3:40 pm
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### Radicals and Rational exponents: 5*cbrt[16] + cbrt[54]
This is from my Algebra/preCalc class and I do not understand how to breakdown and solve this problem. I have thus far been able to solve most problems in this section by using the book examples to work it out and understand them. This problem does not have an example, and I can't figure out how to break it down to the book given answer of: $13\sqrt[3]{2}$
If I could see this worked out step-by-step I feel I would be able to understand the rest like this.
Book Instructions: Add or subtract terms.
$5\sqrt[3]{16}+\sqrt[3]{54}$
I have another problem I am stuck on as well, but will wait until seeing this one to see if I can figure it out from there. Much appreciated.
Stranger_1973
Posts: 21
Joined: Sun Feb 22, 2009 9:56 pm
### Re: Radicals and Rational exponents
EricA wrote:Book Instructions: Add or subtract terms.
$5\sqrt[3]{16}+\sqrt[3]{54}$
Hint:
$\sqrt[3]{16}\, =\, \sqrt[3]{8\times 2}\, =\, \sqrt[3]{2^3\, \times\, 2}\, =\, \sqrt[3]{2^3}\,\times\, \sqrt[3]{2}$
$\sqrt[3]{54}\, =\, \sqrt[3]{27\times 3}\, =\, \sqrt[3]{3^3\, \times\, 2}\, =\,\sqrt[3]{3^3}\,\times\, \sqrt[3]{2}$
EricA
Posts: 2
Joined: Mon Sep 06, 2010 3:40 pm
Contact:
### Re: Radicals and Rational exponents: 5*cbrt[16] + cbrt[54]
Got it! Thanks for the hint! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 5, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8747833371162415, "perplexity": 956.8332519780507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645258858.62/warc/CC-MAIN-20150827031418-00269-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/37462-descartes-rule-signs.html | # Math Help - Descartes Rule of Signs
1. ## Descartes Rule of Signs
Use Descartes's Rule of Signs to determine the possible number of negative real zeros for the given function.
$g(x)=-3x^7+x^5-x^2+4$
So far I have this:
$g(-x)=-3(-x)^7+1(-x)^5-1(x)^2+4$
$g(-x)=-3x^7-x^5-x^2+4$
1 negative real root.
2. g(x) has -+-+ That's 3, so 3 or 1 positive.
g(-x) has +--+ That's 2, so 2 or 0 negative.
ALL the negative powers change sign for g(-x). You missed the $x^{7}$ term. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8380571603775024, "perplexity": 1824.2350591508932}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701153323.32/warc/CC-MAIN-20160205193913-00116-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.hepdata.net/search/?q=&collaboration=ATLAS&page=1&observables=ACC | Showing 21 of 21 results
#### Version 2 Search for direct stau production in events with two hadronic $\tau$-leptons in $\sqrt{s} = 13$ TeV $pp$ collisions with the ATLAS detector
The collaboration Aad, Georges ; Abbott, Brad ; Abbott, Dale Charles ; et al.
Phys.Rev.D 101 (2020) 032009, 2020.
Inspire Record 1765529
A search for the direct production of the supersymmetric partners of $\tau$-leptons (staus) in final states with two hadronically decaying $\tau$-leptons is presented. The analysis uses a dataset of $pp$ collisions corresponding to an integrated luminosity of $139$ fb$^{-1}$, recorded with the ATLAS detector at the Large Hadron Collider at a center-of-mass energy of 13 TeV. No significant deviation from the expected Standard Model background is observed. Limits are derived in scenarios of direct production of stau pairs with each stau decaying into the stable lightest neutralino and one $\tau$-lepton in simplified models where the two stau mass eigenstates are degenerate. Stau masses from 120 GeV to 390 GeV are excluded at 95% confidence level for a massless lightest neutralino.
26 data tables
The observed upper limits on the model cross-section in units of pb for simplified models with combined ${\tilde{\tau}}^{+}_{R,L} {\tilde{\tau}}^{-}_{R,L}$ production. Three points at ${M({\tilde{\chi}}^{0}_{1})}=200GeV$ were removed from the plot but kept in the table because they overlapped with the plot's legend and are far from the exclusion contour.
The observed upper limits on the model cross-section in units of pb for simplified models with ${\tilde{\tau}}_L {\tilde{\tau}}_L$ only production. Three points at $M({\tilde{\chi}}^{0}_{1})=200GeV$ were removed from the plot but kept in the table because they overlapped with the plot's legend and are far from the exclusion contour.
The observed 95\% CL exclusion contours for the combined fit of SR-lowMass and SR-highMass for simplified models with combined ${\tilde{\tau}}^{+}_{R,L} {\tilde{\tau}}^{-}_{R,L}$ production.
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#### Version 2 Search for heavy charged long-lived particles in the ATLAS detector in 36.1 fb$^{-1}$ of proton-proton collision data at $\sqrt{s} = 13$ TeV
Phys.Rev.D 99 (2019) 092007, 2019.
Inspire Record 1718558
A search for heavy charged long-lived particles is performed using a data sample of 36.1 fb$^{-1}$ of proton-proton collisions at $\sqrt{s} = 13$ TeV collected by the ATLAS experiment at the Large Hadron Collider. The search is based on observables related to ionization energy loss and time of flight, which are sensitive to the velocity of heavy charged particles traveling significantly slower than the speed of light. Multiple search strategies for a wide range of lifetimes, corresponding to path lengths of a few meters, are defined as model-independently as possible, by referencing several representative physics cases that yield long-lived particles within supersymmetric models, such as gluinos/squarks ($R$-hadrons), charginos and staus. No significant deviations from the expected Standard Model background are observed. Upper limits at 95% confidence level are provided on the production cross sections of long-lived $R$-hadrons as well as directly pair-produced staus and charginos. These results translate into lower limits on the masses of long-lived gluino, sbottom and stop $R$-hadrons, as well as staus and charginos of 2000 GeV, 1250 GeV, 1340 GeV, 430 GeV and 1090 GeV, respectively.
30 data tables
Lower mass requirement for signal regions.
Lower mass requirement for signal regions.
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#### Version 2 Search for squarks and gluinos in final states with hadronically decaying $\tau$-leptons, jets, and missing transverse momentum using $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector
Phys.Rev.D 99 (2019) 012009, 2019.
Inspire Record 1688943
A search for supersymmetry in events with large missing transverse momentum, jets, and at least one hadronically decaying $\tau$-lepton is presented. Two exclusive final states with either exactly one or at least two $\tau$-leptons are considered. The analysis is based on proton-proton collisions at $\sqrt{s}$ = 13 TeV corresponding to an integrated luminosity of 36.1 fb$^{-1}$ delivered by the Large Hadron Collider and recorded by the ATLAS detector in 2015 and 2016. No significant excess is observed over the Standard Model expectation. At 95% confidence level, model-independent upper limits on the cross section are set and exclusion limits are provided for two signal scenarios: a simplified model of gluino pair production with $\tau$-rich cascade decays, and a model with gauge-mediated supersymmetry breaking (GMSB). In the simplified model, gluino masses up to 2000 GeV are excluded for low values of the mass of the lightest supersymmetric particle (LSP), while LSP masses up to 1000 GeV are excluded for gluino masses around 1400 GeV. In the GMSB model, values of the supersymmetry-breaking scale are excluded below 110 TeV for all values of $\tan\beta$ in the range $2 \leq \tan\beta \leq 60$, and below 120 TeV for $\tan\beta>30$.
52 data tables
1$\tau$ Compressed SR eff.
1$\tau$ MediumMass SR eff.
2$\tau$ Compressed SR eff.
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#### Search for additional heavy neutral Higgs and gauge bosons in the ditau final state produced in 36 fb$^{−1}$ of pp collisions at $\sqrt{s}=13$ TeV with the ATLAS detector
JHEP 01 (2018) 055, 2018.
Inspire Record 1624690
A search for heavy neutral Higgs bosons and $Z^{\prime}$ bosons is performed using a data sample corresponding to an integrated luminosity of 36.1 fb$^{-1}$ from proton-proton collisions at $\sqrt{s}$ = 13 TeV recorded by the ATLAS detector at the LHC during 2015 and 2016. The heavy resonance is assumed to decay to $\tau^+\tau^-$ with at least one tau lepton decaying to final states with hadrons and a neutrino. The search is performed in the mass range of 0.2-2.25 TeV for Higgs bosons and 0.2-4.0 TeV for $Z^{\prime}$ bosons. The data are in good agreement with the background predicted by the Standard Model. The results are interpreted in benchmark scenarios. In the context of the hMSSM scenario, the data exclude $\tan\beta > 1.0$ for $m_A$ = 0.25 TeV and $\tan\beta > 42$ for $m_A$ = 1.5 TeV at the 95% confidence level. For the Sequential Standard Model, $Z^{\prime}_\mathrm{SSM}$ with $m_{Z^{\prime}} < 2.42$ TeV is excluded at 95% confidence level, while $Z^{\prime}_\mathrm{NU}$ with $m_{Z^{\prime}} < 2.25$ TeV is excluded for the non-universal $G(221)$ model that exhibits enhanced couplings to third-generation fermions.
29 data tables
Observed and predicted mTtot distribution in the b-veto category of the 1l1tau_h channel. Despite listing this as an exclusive final state (as there must be no b-jets), there is no explicit selection on the presence of additional light-flavour jets. Please note that the bin content is divided by the bin width in the paper figure, but not in the HepData table. In the paper, the first bin is cut off at 60 GeV for aesthetics but contains underflows down to 50 GeV as in the HepData table. The last bin includes overflows. The combined prediction for A and H bosons with masses of 300, 500 and 800 GeV and $\tan\beta$ = 10 in the hMSSM scenario are also provided.
Observed and predicted mTtot distribution in the b-tag category of the 1l1tau_h channel. Despite listing this as an exclusive final state (as there must be at least one b-jets), there is no explicit selection on the presence of additional light-flavour jets. Please note that the bin content is divided by the bin width in the paper figure, but not in the HepData table. In the paper, the first bin is cut off at 60 GeV for aesthetics but contains underflows down to 50 GeV as in the HepData table. The last bin includes overflows. The combined prediction for A and H bosons with masses of 300, 500 and 800 GeV and $\tan\beta$ = 10 in the hMSSM scenario are also provided.
Observed and predicted mTtot distribution in the b-veto category of the 2tau_h channel. Despite listing this as an exclusive final state (as there must be no b-jets), there is no explicit selection on the presence of additional light-flavour jets. Please note that the bin content is divided by the bin width in the paper figure, but not in the HepData table. The last bin includes overflows. The combined prediction for A and H bosons with masses of 300, 500 and 800 GeV and $\tan\beta$ = 10 in the hMSSM scenario are also provided.
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#### Version 7 Search for a scalar partner of the top quark in the jets plus missing transverse momentum final state at $\sqrt{s}$=13 TeV with the ATLAS detector
JHEP 12 (2017) 085, 2017.
Inspire Record 1623207
A search for pair production of a scalar partner of the top quark in events with four or more jets plus missing transverse momentum is presented. An analysis of 36.1 fb$^{-1}$ of $\sqrt{s}$=13 TeV proton-proton collisions collected using the ATLAS detector at the LHC yields no significant excess over the expected Standard Model background. To interpret the results a simplified supersymmetric model is used where the top squark is assumed to decay via $\tilde{t}_1 \rightarrow t^{(*)} \tilde\chi^0_1$ and $\tilde{t}_1\rightarrow b\tilde\chi^\pm_1 \rightarrow b W^{(*)} \tilde\chi^0_1$, where $\tilde\chi^0_1$ ($\chi^\pm_1$) denotes the lightest neutralino (chargino). Exclusion limits are placed in terms of the top-squark and neutralino masses. Assuming a branching ratio of 100% to $t \tilde\chi^0_1$, top-squark masses in the range 450-950 GeV are excluded for $\tilde\chi^0_1$ masses below 160 GeV. In the case where $m_{\tilde{t}_1}\sim m_t+m_{\tilde\chi^0_1}$, top-squark masses in the range 235-590 GeV are excluded.
83 data tables
Distribution of $E_\text{T}^\text{miss}$ for SRA-TT after the likelihood fit. The stacked histograms show the SM expectation and the hatched uncertainty band around the SM expectation shows the MC statistical and detector-related systematic uncertainties. A representative signal point is shown for each distribution.
Distribution of $m_\text{T2}^{\chi^2}$ for SRA-T0 after the likelihood fit. The stacked histograms show the SM expectation and the hatched uncertainty band around the SM expectation shows the MC statistical and detector-related systematic uncertainties. A representative signal point is shown for each distribution.
Distribution of $m_\text{T}^{b,\text{max}}$ for SRB-TW after the likelihood fit. The stacked histograms show the SM expectation and the hatched uncertainty band around the SM expectation shows the MC statistical and detector-related systematic uncertainties. A representative signal point is shown for each distribution.
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#### Version 2 Search for a new heavy gauge boson resonance decaying into a lepton and missing transverse momentum in 36 fb$^{-1}$ of $pp$ collisions at $\sqrt{s} =$ 13 TeV with the ATLAS experiment
Eur.Phys.J.C 78 (2018) 401, 2018.
Inspire Record 1605396
The results of a search for new heavy $W^\prime$ bosons decaying to an electron or muon and a neutrino using proton-proton collision data at a centre-of-mass energy of $\sqrt{s} = 13$ TeV are presented. The dataset was collected in 2015 and 2016 by the ATLAS experiment at the Large Hadron Collider and corresponds to an integrated luminosity of 36.1 fb$^{-1}$. As no excess of events above the Standard Model prediction is observed, the results are used to set upper limits on the $W^\prime$ boson cross-section times branching ratio to an electron or muon and a neutrino as a function of the $W^\prime$ mass. Assuming a $W^\prime$ boson with the same couplings as the Standard Model $W$ boson, $W^\prime$ masses below 5.1 TeV are excluded at the 95% confidence level.
6 data tables
Transverse mass distribution for events satisfying all selection criteria in the electron channel.
Transverse mass distribution for events satisfying all selection criteria in the muon channel.
Upper limits at the 95% CL on the cross section for SSM W' production and decay to the electron+neutrino channel as a function of the W' pole mass.
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#### Search for new phenomena in events containing a same-flavour opposite-sign dilepton pair, jets, and large missing transverse momentum in $\sqrt{s}=$ 13 $pp$ collisions with the ATLAS detector
Eur.Phys.J.C 77 (2017) 144, 2017.
Inspire Record 1498566
Two searches for new phenomena in final states containing a same-flavour opposite-lepton (electron or muon) pair, jets, and large missing transverse momentum are presented. These searches make use of proton--proton collision data, collected during 2015 and 2016 at a centre-of-mass energy $\sqrt{s}=13$ TeV by the ATLAS detector at the Large Hadron Collider, which correspond to an integrated luminosity of 14.7 fb$^{-1}$. Both searches target the pair production of supersymmetric particles, squarks or gluinos, which decay to final states containing a same-flavour opposite-sign lepton pair via one of two mechanisms: a leptonically decaying Z boson in the final state, leading to a peak in the dilepton invariant-mass distribution around the Z boson mass; and decays of neutralinos (e.g. $\tilde{\chi}_2^0 \rightarrow \ell^+\ell^- \tilde{\chi}_1^0$), yielding a kinematic endpoint in the dilepton invariant-mass spectrum. The data are found to be consistent with the Standard Model expectation. Results are interpreted in simplified models of gluino-pair (squark-pair) production, and provide sensitivity to gluinos (squarks) with masses as large as 1.70 TeV (980 GeV).
36 data tables
Dilepton invariant mass distribution in SRZ.
Dilepton transverse momentum distribution in SRZ.
Missing transverse momentum distribution in SRZ.
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#### Search for squarks and gluinos in events with hadronically decaying tau leptons, jets and missing transverse momentum in proton–proton collisions at $\sqrt{s}=13$ TeV recorded with the ATLAS detector
Eur.Phys.J.C 76 (2016) 683, 2016.
Inspire Record 1477209
A search for supersymmetry in events with large missing transverse momentum, jets, and at least one hadronically decaying tau lepton has been performed using 3.2 fb$^{-1}$ of proton-proton collision data at $\sqrt{s}=13$ TeV recorded by the ATLAS detector at the Large Hadron Collider in 2015. Two exclusive final states are considered, with either exactly one or at least two tau leptons. No excess over the Standard Model prediction is observed in the data. Results are interpreted in the context of gauge-mediated supersymmetry breaking and a simplified model of gluino pair production with tau-rich cascade decays, substantially improving on previous limits. In the GMSB model considered, supersymmetry-breaking scale ($\Lambda$) values below 92 TeV are excluded at the 95% confidence level, corresponding to gluino masses below 2000 GeV. For large values of $\tan\beta$, values of $\Lambda$ up to 107 TeV and gluino masses up to 2300 GeV are excluded. In the simplified model, gluino masses are excluded up to 1570 GeV for neutralino masses around 100 GeV. Neutralino masses up to 700 GeV are excluded for all gluino masses between 800 GeV and 1500 GeV, while the strongest exclusion of 750 GeV is achieved for gluino masses around 1400 GeV.
32 data tables
mTtau distributions for "extended SR selections" of the 1 tau channel, for the Compressed SR selection without the mTtau > 80 GeV requirement. The last bin includes overflow events. Uncertainties are statistical only. Signal predictions are overlaid for several benchmark models, normalised to their predicted cross sections. For the simplified model, "LM" refers to a low mass splitting, or compressed scenario, with m(gluino)=665 GeV and m(neutralino)=585 GeV; "MM" stands for a medium mass splitting, with m(gluino)=1145 GeV and m(neutralino)=265 GeV; "HM" denotes a high mass splitting scenario, with m(gluino)=1305 GeV and m(neutralino)=105 GeV.
mTtau distributions for "extended SR selections" of the 1 tau channel, for the Medium Mass SR selection without the mTtau > 200 GeV requirement. The last bin includes overflow events. Uncertainties are statistical only. Signal predictions are overlaid for several benchmark models, normalised to their predicted cross sections. For the simplified model, "LM" refers to a low mass splitting, or compressed scenario, with m(gluino)=665 GeV and m(neutralino)=585 GeV; "MM" stands for a medium mass splitting, with m(gluino)=1145 GeV and m(neutralino)=265 GeV; "HM" denotes a high mass splitting scenario, with m(gluino)=1305 GeV and m(neutralino)=105 GeV.
mTtau distributions for "extended SR selections" of the 1 tau channel, for the High Mass SR selection without the mTtau > 200 GeV requirement. The last bin includes overflow events. Uncertainties are statistical only. Signal predictions are overlaid for several benchmark models, normalised to their predicted cross sections. For the simplified model, "LM" refers to a low mass splitting, or compressed scenario, with m(gluino)=665 GeV and m(neutralino)=585 GeV; "MM" stands for a medium mass splitting, with m(gluino)=1145 GeV and m(neutralino)=265 GeV; "HM" denotes a high mass splitting scenario, with m(gluino)=1305 GeV and m(neutralino)=105 GeV.
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#### Search for bottom squark pair production in proton–proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector
Eur.Phys.J.C 76 (2016) 547, 2016.
Inspire Record 1472822
The result of a search for pair production of the supersymmetric partner of the Standard Model bottom quark ($\tilde{b}_1$) is reported. The search uses 3.2 fb$^{-1}$ of $pp$ collisions at $\sqrt{s}=$13 TeV collected by the ATLAS experiment at the Large Hadron Collider in 2015. Bottom squarks are searched for in events containing large missing transverse momentum and exactly two jets identified as originating from $b$-quarks. No excess above the expected Standard Model background yield is observed. Exclusion limits at 95% confidence level on the mass of the bottom squark are derived in phenomenological supersymmetric $R$-parity-conserving models in which the $\tilde{b}_1$ is the lightest squark and is assumed to decay exclusively via $\tilde{b}_1 \rightarrow b \tilde{\chi}_1^0$, where $\tilde{\chi}_1^0$ is the lightest neutralino. The limits significantly extend previous results; bottom squark masses up to 800 (840) GeV are excluded for the $\tilde{\chi}_1^0$ mass below 360 (100) GeV whilst differences in mass above 100 GeV between the $\tilde{b}_1$ and the $\tilde{\chi}_1^0$ are excluded up to a $\tilde{b}_1$ mass of 500 GeV.
37 data tables
Expected exclusion limit at 95% CL in the $m(\tilde b_1)$-$m(\tilde\chi^0_1)$ plane for the sbottom pair production scenario.
Observed exclusion limit at 95% CL in the $m(\tilde b_1)$-$m(\tilde\chi^0_1)$ plane for the sbottom pair production scenario.
Signal region (SR) providing the best expected sensitivity in the $m(\tilde b_1)$-$m(\tilde\chi^0_1)$ plane.
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#### Search for new resonances in events with one lepton and missing transverse momentum in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector
Phys.Lett.B 762 (2016) 334-352, 2016.
Inspire Record 1469070
A search for $W^\prime$ bosons in events with one lepton (electron or muon) and missing transverse momentum is presented. The search uses 3.2 fb$^{-1}$ of $pp$ collision data collected at $\sqrt{s} = 13$ TeV by the ATLAS experiment at the LHC in 2015. The transverse mass distribution is examined and no significant excess of events above the level expected from Standard Model processes is observed. Upper limits on the $W^\prime$ boson cross-section times branching ratio to leptons are set as a function of the $W^\prime$ mass. Assuming a $W^\prime$ boson as predicted by the Sequential Standard Model, $W^\prime$ masses below 4.07 TeV are excluded at the 95% confidence level. This extends the limit set using LHC data at $\sqrt{s}=8$ TeV by around 800 GeV.
4 data tables
Observed and predicted electron channel transverse mass (MT) distribution in the search region. The bin width is constant in log(MT).
Observed and predicted muon channel transverse mass (MT) distribution in the search region. The bin width is constant in log(MT).
W' Product of acceptance and efficiency for the electron and muon selections as a function of the SSM W' pole mass.
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#### Summary of the searches for squarks and gluinos using $\sqrt{s}=8$ TeV pp collisions with the ATLAS experiment at the LHC
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
JHEP 10 (2015) 054, 2015.
Inspire Record 1383884
A summary is presented of ATLAS searches for gluinos and first- and second-generation squarks in final states containing jets and missing transverse momentum, with or without leptons or b-jets, in the $\sqrt{s}$ = 8 TeV data set collected at the Large Hadron Collider in 2012. This paper reports the results of new interpretations and statistical combinations of previously published analyses, as well as a new analysis. Since no significant excess of events over the Standard Model expectation is observed, the data are used to set limits in a variety of models. In all the considered simplified models that assume R-parity conservation, the limit on the gluino mass exceeds 1150 GeV at 95% confidence level, for an LSP mass smaller than 100 GeV. Furthermore, exclusion limits are set for left-handed squarks in a phenomenological MSSM model, a minimal Supergravity/Constrained MSSM model, R-parity-violation scenarios, a minimal gauge-mediated supersymmetry breaking model, a natural gauge mediation model, a non-universal Higgs mass model with gaugino mediation and a minimal model of universal extra dimensions.
30 data tables
Acceptance for the loose channel of the Razor analysis for the direct squark-squark model.
Acceptance times efficiency for the loose channel of the Razor analysis for the direct squark-squark model.
Acceptance for the tight channel of the Razor analysis for the direct squark-squark model.
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#### A search for high-mass resonances decaying to $\tau^{+}\tau^{-}$ in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS detector
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
JHEP 07 (2015) 157, 2015.
Inspire Record 1346398
A search for high-mass resonances decaying into $\tau^{+}\tau^{-}$ final states using proton-proton collisions at $\sqrt{s}= 8$ TeV produced by the Large Hadron Collider is presented. The data were recorded with the ATLAS detector and correspond to an integrated luminosity of 19.5-20.3 fb$^{-1}$. No statistically significant excess above the Standard Model expectation is observed; 95% credibility upper limits are set on the cross section times branching fraction of $Z^{\prime}$ resonances decaying into $\tau^+\tau^-$ pairs as a function of the resonance mass. As a result, $Z^{\prime}$ bosons of the Sequential Standard Model with masses less than 2.02 TeV are excluded at 95% credibility. The impact of the fermionic couplings on the $Z^{\prime}$ acceptance is investigated and limits are also placed on a $Z^{\prime}$ model that exhibits enhanced couplings to third-generation fermions.
9 data tables
Signal acceptance times efficiency (ACC*EFF) for Z'L, Z'R, Z'narrow and Z'wide divided by ACC*EFF for Z'SSM as a function of the Z' mass, separately for the had-had and lep-had channels.
Ratio of the Z'NU to Z'SSM cross section times tau+tau- branching fraction (SIG*BR) as a function of sin^2phi and the Z' mass.
Ratio of the Z'NU to Z'SSM acceptance times efficiency (ACC*EFF) in the had-had channel as a function of sin^2phi and the Z' mass.
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#### Search for direct pair production of a chargino and a neutralino decaying to the 125 GeV Higgs boson in $\sqrt{s} = 8$ TeV ${pp}$ collisions with the ATLAS detector
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
Eur.Phys.J.C 75 (2015) 208, 2015.
Inspire Record 1341609
A search is presented for the direct pair production of a chargino and a neutralino $pp\to\tilde{\chi}^\pm_1\tilde{\chi}^0_2$, where the chargino decays to the lightest neutralino and the $W$ boson, $\tilde{\chi}^\pm_1 \to \tilde{\chi}^0_1 (W^{\pm}\to\ell^{\pm}\nu)$, while the neutralino decays to the lightest neutralino and the 125 GeV Higgs boson, $\tilde{\chi}^0_2 \to \tilde{\chi}^0_1 (h\to bb/\gamma\gamma/\ell^{\pm}\nu qq)$. The final states considered for the search have large missing transverse momentum, an isolated electron or muon, and one of the following: either two jets identified as originating from bottom quarks, or two photons, or a second electron or muon with the same electric charge. The analysis is based on 20.3 fb$^{-1}$ of $\sqrt{s}=8$ TeV proton-proton collision data delivered by the Large Hadron Collider and recorded with the ATLAS detector. Observations are consistent with the Standard Model expectations, and limits are set in the context of a simplified supersymmetric model.
62 data tables
Distribution of contransverse mass $m_{\rm CT}$ in CRlbb-T, central $m_{bb}$ bin. The background histograms are obtained from the background-only fit, and their uncertainty represents the total background uncertainty after the fit. The last bin includes overflow.
Distribution of contransverse mass $m_{\rm CT}$ in SRlbb-1 and SRlbb-2, $m_{bb}$ sideband. The background histograms are obtained from the background-only fit, and their uncertainty represents the total background uncertainty after the fit. The last bin includes overflow.
Distribution of the transverse mass of the $W$-candidate $m_{\rm T}^{W}$ for the one lepton and two $b$-jets channel in VRlbb-2, central $m_{bb}$ bin. The background histograms are obtained from the background-only fit, and their uncertainty represents the total background uncertainty after the fit. The last bin includes overflow.
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#### Search for squarks and gluinos in events with isolated leptons, jets and missing transverse momentum at $\sqrt{s}=8$ TeV with the ATLAS detector
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
JHEP 04 (2015) 116, 2015.
Inspire Record 1339376
The results of a search for supersymmetry in final states containing at least one isolated lepton (electron or muon), jets and large missing transverse momentum with the ATLAS detector at the Large Hadron Collider (LHC) are reported. The search is based on proton-proton collision data at a centre-of-mass energy $\sqrt{s} = 8$ TeV collected in 2012, corresponding to an integrated luminosity of 20 fb$^{-1}$. No significant excess above the Standard Model expectation is observed. Limits are set on the parameters of a minimal universal extra dimensions model, excluding a compactification radius of $1/R_c=950$ GeV for a cut-off scale times radius ($\Lambda R_c$) of approximately 30, as well as on sparticle masses for various supersymmetric models. Depending on the model, the search excludes gluino masses up to 1.32 TeV and squark masses up to 840 GeV.
112 data tables
Observed and expected $E_T^{miss}/m_{eff}$ distribution in soft single-lepton 3-jet signal region. The last bin includes the overflow.
Observed and expected $E_T^{miss}/m_{eff}$ distribution in soft single-lepton 5-jet signal region. The last bin includes the overflow.
Observed and expected $E_T^{miss}/m_{eff}$ distribution in soft single-lepton 3-jet inclusive signal region. The last bin includes the overflow.
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#### Search for $W' \rightarrow tb \rightarrow qqbb$ decays in $pp$ collisions at $\sqrt{s}$ = 8 TeV with the ATLAS detector
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
Eur.Phys.J.C 75 (2015) 165, 2015.
Inspire Record 1309877
A search for a massive $W'$ gauge boson decaying to a top quark and a bottom quark is performed with the ATLAS detector in $pp$ collisions at the LHC. The dataset was taken at a centre-of-mass energy of $\sqrt{s} = 8$ TeV and corresponds to 20.3 fb$^{-1}$ of integrated luminosity. This analysis is done in the hadronic decay mode of the top quark, where novel jet substructure techniques are used to identify jets from high-momentum top quarks. This allows for a search for high-mass $W'$ bosons in the range $1.5 - 3.0$ TeV. $b$-tagging is used to identify jets originating from $b$-quarks. The data are consistent with Standard Model background-only expectations, and upper limits at 95% confidence level are set on the $W' \rightarrow tb$ cross section times branching ratio ranging from $0.16$ pb to $0.33$ pb for left-handed $W'$ bosons, and ranging from $0.10$ pb to $0.21$ pb for $W'$ bosons with purely right-handed couplings. Upper limits at 95% confidence level are set on the $W'$-boson coupling to $tb$ as a function of the $W'$ mass using an effective field theory approach, which is independent of details of particular models predicting a $W'$ boson.
6 data tables
m_tb distributions in data in the one b-tag and the two b-tag category, together with background-only fits excluding the region 4-5 TeV which is beyond the range considered for this analysis. Potential WPRIME_L signal shapes in the hadronic top-quark decay channel with gPRIME = gSM are also given for resonance masses of 1.5, 2.0, 2.5 and 3.0 TeV.
Limits at 95% CL on the cross section times branching ratio to TOP BOTTOM for the left-handed and for the right-handed WPRIME model. The expected cross section for WPRIME production with gprime = gSM is also shown.
Observed and expected 95% CL limits on the ratio of coupling gWPRIME_L/gSM (gWPRIME_R/gSM) of the WPRIME_L (WPRIME_R) model as a function of the WPRIME mass.
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#### Search for supersymmetry at $\sqrt{s}$=8 TeV in final states with jets and two same-sign leptons or three leptons with the ATLAS detector
The collaboration Aad, Georges ; Abbott, Brad ; Abdallah, Jalal ; et al.
JHEP 06 (2014) 035, 2014.
Inspire Record 1289225
A search for strongly produced supersymmetric particles is conducted using signatures involving multiple energetic jets and either two isolated leptons ($e$ or $\mu$) with the same electric charge, or at least three isolated leptons. The search also utilises jets originating from b-quarks, missing transverse momentum and other observables to extend its sensitivity. The analysis uses a data sample corresponding to a total integrated luminosity of 20.3 fb$^{-1}$ of $\sqrt{s} =$ 8 TeV proton-proton collisions recorded with the ATLAS detector at the Large Hadron Collider in 2012. No deviation from the Standard Model expectation is observed. New or significantly improved exclusion limits are set on a wide variety of supersymmetric models in which the lightest squark can be of the first, second or third generations, and in which R-parity can be conserved or violated.
106 data tables
Numbers of observed and background events for SR0b for each bin of the distribution in Meff. The table corresponds to Fig. 4(b). The statistical and systematic uncertainties are combined for the expected backgrounds.
Numbers of observed and background events for SR1b for each bin of the distribution in Meff. The table corresponds to Fig. 4(c). The statistical and systematic uncertainties are combined for the predicted numbers.
Numbers of observed and background events for SR3b for each bin of the distribution in Meff. The table corresponds to Fig. 4(a). The statistical and systematic uncertainties are combined for the predicted numbers. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9879243969917297, "perplexity": 2065.549816174751}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400206329.28/warc/CC-MAIN-20200922161302-20200922191302-00028.warc.gz"} |
https://math.stackexchange.com/questions/2801330/mathematical-induction-and-peano-arithmetic | # Mathematical Induction and Peano Arithmetic
Peano Arithmetic cannot employ Induction for any ε0 ordering. My question is too easy to be interesting and there is a reason obviously for why it has a negative answer. Can you please provide it for me? Is there no way to build Induction upon the simple ω order that the towers of ε0 represent as written down? E.g. Prove something for the elements of the first ω at the bottom of the tower. Then assume that all elements up to some ω exponent have the same property, and manage (?) to prove that on this assumption all elements up the next exponent have it as well. Is there anything wrong with this approach in principle? This is a follow up after Andrés' illuminating reply. So, if I've got that right, on the assumption that the Church-Turing thesis is true, there must be no algorithm that represents how the Inductive proofs of some level in the tower behave relative to how they behave at the previous level. Otherwise, it would have being possible to assume that all elements up the nth omega level have the property, and, then, by using the algorithm establish the presence of an Inductive proof for all the elements of level n+1. (Proofs for all finite towers of omegas are assumed to be recursive here.) Thank you Andrés, I think I've got it now. One cannot built/code the entire Σ(n) inductive schema within any of the particular instances of it, and therefore nor within an ω exponent.
• – Cameron Buie May 30 '18 at 2:16
• Gentzen showed that PA + induction up to $\epsilon_0$ proves the consistency of PA. Does that constitute an answer to your question or were you aware of that and looking for something different? – spaceisdarkgreen May 30 '18 at 2:57
• What you suggest is quite reasonable. When trying to formalize it to prove concrete statements (such as "every Goodstein sequence terminates") you usually find that the proof at the $n$th level of the tower requires something like a $\Sigma_n$ instance of the induction schema, so that the whole process is not quite formalizable in PA. – Andrés E. Caicedo May 30 '18 at 10:43
• I think it may be benefitial to work in detail through a concrete example to see how the problem manifests. If you haven't, I suggest you read the Kirby-Paris paper to see this in action (model-theoretically). For a proof-theoretic approach, take a look at some of the papers in the AMS Contemporary Mathematics volume on "Logic and combinatorics". – Andrés E. Caicedo May 31 '18 at 2:30
• No, it is not necessary. But my paper relies on some proof-theoretic results, so it is not self-contained. – Andrés E. Caicedo Jun 6 '18 at 12:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8918651342391968, "perplexity": 483.4443810756774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998325.55/warc/CC-MAIN-20190616222856-20190617004856-00514.warc.gz"} |
https://www.ms.u-tokyo.ac.jp/journal/number_e/jms1004_e.html | ## Vol. 10 (2003) No. 04
Journal of Mathematical Sciences The University of Tokyo
1. Hazama, Fumio
Hodge Cycles on Abelian Varieties with Complex Multiplication by Cyclic CM-Fields
Vol. 10 (2003), No. 4, Page 581--598.
2. Masuoka, Akira
More Homological Approach to Composition of Subfactors
Vol. 10 (2003), No. 4, Page 599--630. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8222952485084534, "perplexity": 3184.487516740026}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362879.45/warc/CC-MAIN-20211203121459-20211203151459-00330.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-algebra/chapter-5-exponents-and-polynomials-5-3-multiplying-polynomials-problem-set-5-3-page-208/90 | Elementary Algebra
$4x^2+36x+81$
Using $(a+b)^2=a^2+2ab+b^2$, or the square of a binomial, the given expression, $(2x+9)^2 ,$ is equivalent to \begin{array}{l}\require{cancel} (2x)^2+2(2x)(9)+(9)^2 \\\\= 4x^2+36x+81 .\end{array} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9993239641189575, "perplexity": 4456.479393064504}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583514437.69/warc/CC-MAIN-20181021224001-20181022005501-00120.warc.gz"} |
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Tutorials_(Rioux)/Quantum_Optics/295%3A_Another_Example_of_a_Two-photon_Quantum_Eraser | # 295: Another Example of a Two-photon Quantum Eraser
Greenberger, Horne and Zeilinger (GHZ) surveyed the then relatively new field of multiparticle interferometry in an August 1993 Physics Today article, ʺMultiparticle Interferometry and the Superposition Principle.ʺ This tutorial will use Mathcad and tensor algebra to analyze the results associated with Figure 5, which dealt with a two‐photon quantum eraser. A parametric down converter (PDC) produces two horizontally polarized, entangled photons, one taking the upper path and the other the lower path. The beams are combined at a beam splitter as shown below.
As the figure shows both photons arrive at either the U detector or the D detector. This result will now be confirmed using tensor algebra.
The photons emerging from the PDC are entangled and can be moving up or down with horizontal polarization. Later we will consider rotating the polarization in the lower arm to the vertical orientation in order to explore the consequences of providing path information. We use the following vectors to represent the motional and polarization states of the photons.
$u = \begin{pmatrix} 1 \\ 0 \end{pmatrix} ~~~ d = \begin{pmatrix} 0 \\ 1 \end{pmatrix} ~~~ h = \begin{pmatrix} 1 \\ 0 \end{pmatrix} ~~~ v = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$
The four possible photon states are expressed using vector tensor multiplication.
$uh = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix}$
$uh = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}$
$uh = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix}$
$uh = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$
$uh = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix}$
$uh = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}$
$uh = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix}$
$uh = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$
There are sixteen photon measurement (output) states at the detectors. These are also represented using tensor algebra. The first two letters refer to photon 1, the second two refer to photon 2. The uhdv (|uh>1|dv>2) state is constructed as an example.
$uhdv = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$uhuh = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$uhuv = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$uhdh = \begin{pmatrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$uhdv = \begin{pmatrix} = 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$uvuh = \begin{pmatrix} 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$uvuv = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$uvdh = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$uvdv = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$dhuh = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$dhuv = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$dhdh = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$dhdv = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{pmatrix} ^T$
$dvuh = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{pmatrix} ^T$
$dvuv = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{pmatrix} ^T$
$dvdh = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{pmatrix} ^T$
$dvdv = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix} ^T$
Matrix operators:
Identity:
$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
Mirror:
$M = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
Beam splitter:
$BS = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}$
The mirrors and the beam splitter operate on the motional degree of freedom. Their operators as configured in the apparatus are constructed using matrix tensor multiplication, implemented with Mathcadʹs kronecker command.
$MI = \text{kronecker} (M,~ \text{kronecker} (I,~ \text{kronecker} (M,~I)))$
$BSI = \text{kronecker} (BS,~ \text{kronecker} (I,~ \text{kronecker} (BS,~I)))$
The entangled state produced by the PDC is:
$\Psi_{boson} = \frac{1}{ \sqrt{2}} (uhdh + dhuh)$
The output state after the photons interact with the mirrors and the beam splitter is:
$\Psi _{out} = BSI (MI) \Psi_{boson}$
An equivalent algebraic analysis clearly shows the constructive and destructive interference between the probability amplitudes for the measurement states.
$\Psi _{out} = \frac{1}{ 2 \sqrt{2}} (i (uh)uh - uh (dh) + dh(uh) + i(dh)dh + i (uh)uh + uh (dh) - dh(uh) + i (dh) dh = \frac{i}{ \sqrt{2}} (uh (uh) - dh (dh))$
We now calculate a matrix of all possible experimental outcomes, recognizing at this point that because the photons are horizontally polarized we could have just calculated a 2x2 matrix, eliminating columns 2 and 4, and rows 2 and 4.
$P_{boson} = \begin{bmatrix} \left[ \left| ( \overline{uhuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{uvuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{dhuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{dvuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvdv} )^T \Psi _{out} \right| \right]^2 \\ \end{bmatrix} = \begin{pmatrix} \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$
$P_{boson} = P_{uhuh} = P_{dhdh} = \frac{1}{2}$
We see that this calculation is in agreement with the experimental results represented in the figure above: both photons always arrive at the same detector. 50 % of the time it is U and 50% of the time it is D.
Now assume that a 90 degree polarization rotator is placed in the lower arm which rotates the horizontal state to the vertical polarization orientation. This provides path information and even though polarization is not measured in this experiment it has a significant affect on the measurement results.
The entangled photon state after the PDC now is:
$\Psi _{boson} = \frac{1}{ \sqrt{2}} (uhdv + dvuh)$
This leads to the following output state after interaction with the mirrors and the beam splitter:
$\Psi _{out} = BSI(MI) \Psi _{boson}$
The measurement outcome matrix now shows that the photons arrive at different detectors 50% of the time and the same detector 50% of the time. Remember that polarization is not being measured in this experiment. But the fact that polarization information exists changes the experimental results. The following algebraic expression for the output states shows that the interence effects that were seen previous do not occur because of the h/v polarization markers on the motional states.
$\Psi _{out} = \frac{1}{ \sqrt{2}} (i (uh) uv - uh (dv) + dh (uv) + i (dh) dv + i (uv) uh - uv (dh) - dv (uh) + i (dv) dh)$
$P_{boson} = P_{uhuv} = P_{uhdv} = P_{dhdv} = P_{uvuh} = P_{uvdh} = P_{dvuh} = P_{dvdh} = \frac{1}{8}$
$P_{boson} = \begin{bmatrix} \left[ \left| ( \overline{uhuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{uvuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{dhuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{dvuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvdv} )^T \Psi _{out} \right| \right]^2 \\ \end{bmatrix} = \begin{pmatrix} 0 & \frac{1}{8} & 0 & \frac{1}{8} \\ \frac{1}{8} & 0 & \frac{1}{8} & 0 \\ 0 & \frac{1}{8} & 0 & \frac{1}{8} \\ \frac{1}{8} & 0 & \frac{1}{8} & 0 \end{pmatrix}$
The path information provided by the h/v polarization states of the photons can be ʺerasedʺ by placing diagonally (45 degrees to the veritcal and labelled s for slant) oriented polarizers after the beam splitter and before the detectors. Polarizers are projection operators, consequently only diagonally polarized photons reach the detectors. There are only four possible final measurement states,
$usus = \frac{1}{2} \begin{pmatrix} 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}$
$usds = \frac{1}{2} \begin{pmatrix} 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}$
$dsus = \frac{1}{2} \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \end{pmatrix}$
$dsds = \frac{1}{2} \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \end{pmatrix}$
where for example dsus is calculated as follows. Tensor multiplication is implied between the vector states.
$dsus = \left[ \begin{pmatrix} 0 \\ 1 \end{pmatrix} \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \right]^T$
where
$s = \frac{1}{ \sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}$
The recalculated experimental outcome matrix shows that the photons again always arrive at the same detector. The accompanying algebraic analysis reveals the revived interference effects that lead to the final result.
$\begin{bmatrix} \left( \left| usus \Psi_{out} \right| \right)^2 & \left( \left| usds \Psi_{out} \right| \right)^2 \\ \left( \left| dsus \Psi_{out} \right| \right)^2 & \left( \left| dsds \Psi_{out} \right| \right)^2 \end{bmatrix} = \begin{pmatrix} \frac{1}{8} & 0 \\ 0 & \frac{1}{8} \end{pmatrix}$
$\Psi_s = \frac{1}{ 4 \sqrt{2}} = (i (us) us - us (ds) + ds (us) + i (ds) ds + i (us) us +us (ds) - ds (us) + i (ds) ds) = \frac{1}{2 \sqrt{2}} (us (us) + ds (ds))$
Photons are bosons and therefore have symmetric wave functions. This is why in the initial experiment they always arrive at the same detector. We will now assume that they are fermions, which have anti‐symmetric wave functions, and repeat the calculations and observe the consequences.
The fermionic entangled photon wave function:
$\Psi_{fermion} = \frac{1}{ \sqrt{2}} (uhuh - dhdh)$
The output state after the photons interact with the mirrors and the beam splitter is:
$\Psi _{out} = BMI (MI) \Psi_{fermion}$
The measurement outcome matrix shows that the fermions, as expected, always arrive at different detectors:
$P_{fermion} = \begin{bmatrix} \left[ \left| ( \overline{uhuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{uvuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{dhuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{dvuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvdv} )^T \Psi _{out} \right| \right]^2 \\ \end{bmatrix} = \begin{pmatrix} 0 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$
Naturally an algebraic analysis yields the same result.
$\Psi _{out} = \frac{1}{2 \sqrt{2}} (i (uh) uh - uh (dh) + dh (uh) + i (dh) dh - i (uh) uh - uh (dh) + dh (uh) - i (dh) dh) = \frac{1}{ sqrt{2}} (dh (uh) - uh(dh))$
$P_{fermion} = P_{uhdh} = P_{dhuh} = \frac{1}{2}$
As algebraic and matrix calculations show, introduction of path information for fermions yields the same result as for bosons, the photons sometimes arrive at the same detector and sometimes at different detectors.
$\Psi _{fermion} = \frac{1}{ \sqrt{2}} = \frac{1}{ sqrt{2}} (uhdv = dvuh)$
$\Psi _{out} = BSI(MI) \Psi_{fermion}$
$\Psi_{out} = \frac{1}{2 \sqrt{2}} (i (uh) uv - uh (dv) + dh (uv) + i (dh) dv- i (uv) uh - uv (dh) + dv (uh) - i (dv)dh)$
$P_{fermion} = P_{uhuv} = P_{uhdv} = P_{dhuv} = P_{dhdv} = P_{uvuh} = P_{uvdh} = P_{dvuh} = P_{dvdh} = \frac{1}{8}$
$P_{fermion} = \begin{bmatrix} \left[ \left| ( \overline{uhuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uhdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{uvuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{uvdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{dhuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dhdv} )^T \Psi _{out} \right| \right]^2 \\ \left[ \left| ( \overline{dvuh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvuv} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvdh} )^T \Psi _{out} \right| \right]^2 & \left[ \left| ( \overline{dvdv} )^T \Psi _{out} \right| \right]^2 \\ \end{bmatrix} = \begin{pmatrix} 0 & \frac{1}{8} & 0 & \frac{1}{8} \\ \frac{1}{8} & 0 & \frac{1}{8} & 0 \\ 0 & \frac{1}{8} & 0 & \frac{1}{8} \\ \frac{1}{8} & 0 & \frac{1}{8} & 0 \end{pmatrix}$
With erasure of path information the fermionic photons again always arrive at different detectors.
$\begin{bmatrix} \left( \left| usus \Psi_{out} \right| \right)^2 & \left( \left| usds \Psi_{out} \right| \right)^2 \\ \left( \left| dsus \Psi_{out} \right| \right)^2 & \left( \left| dsds \Psi_{out} \right| \right)^2 \end{bmatrix} = \begin{pmatrix} 0 & \frac{1}{8} \\ \frac{1}{8} & 0 \end{pmatrix}$
$\Psi _s = \frac{1}{4 \sqrt{2}} (i (us) us - us (ds) + ds (us) + i (ds) ds - i (us) us - us (ds) + ds(us) - i (ds) ds = \frac{1}{2 \sqrt{2}} (ds (us) - us (ds))$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9732818007469177, "perplexity": 891.4962296362241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107891228.40/warc/CC-MAIN-20201026115814-20201026145814-00440.warc.gz"} |
https://hal.archives-ouvertes.fr/hal-00730353 | # Mercury exosphere. III: Energetic characterization of its sodium component
1 HEPPI - LATMOS
LATMOS - Laboratoire Atmosphères, Milieux, Observations Spatiales
Abstract : Mercury's sodium exosphere has been observed only few times with high spectral resolution from ground based observatories enabling the analysis of the emission spectra. These observations highlighted the energetic state of the sodium exospheric atoms relative to the surface temperature. More recently, the Doppler shift of the exospheric Na atoms was measured and interpreted as consistent with an exosphere moving outwards from the subsolar point (Potter et al. 2009). Using THEMIS Solar telescope, we observed Mercury's sodium exosphere with very high spectral resolution at two opposite positions of its orbit. Using this very high spectral resolution and the scanning capabilities of THEMIS, we were able to reconstruct the 2D spatial distributions of the Doppler shifts and widths of the sodium atomic Na D2 and D1 lines. These observations revealed surprisingly large Doppler shift as well as spectral width consistent with previous observations. Starting from our 3D model of Mercury Na exosphere (Mercury Exosphere Global Circulation Model, Leblanc and Johnson 2010), we coupled this model with a 3D radiative transfer model described in a companion paper (Chaufray and Leblanc 2012) which allows us to properly treat the non-maxwellian state of the simulated sodium exospheric population. Comparisons between THEMIS observations and simulations suggest that the previously observed energetic state of the Na exosphere might be essentially explained by a state of the Na exospheric atoms far from thermal equilibrium along with the Doppler shift dispersion of the Na atoms induced by the solar radiation pressure. However, the Doppler shift of the spectral lines cannot be explained by our modelling, suggesting either an exosphere spatially structured very differently than in our model or the inaccuracy of the spectral calibration when deriving the Doppler shift.
Keywords :
Type de document :
Article dans une revue
Icarus, Elsevier, 2013, 223 (2), pp.963-974. 〈10.1016/j.icarus.2012.08.025〉
Domaine :
https://hal.archives-ouvertes.fr/hal-00730353
Contributeur : Catherine Cardon <>
Soumis le : dimanche 9 septembre 2012 - 19:22:48
Dernière modification le : vendredi 16 novembre 2018 - 02:11:11
### Citation
François Leblanc, Jean-Yves Chaufray, Alain Doressoundiram, Jean-Jacques Berthelier, Valeria Mangano, et al.. Mercury exosphere. III: Energetic characterization of its sodium component. Icarus, Elsevier, 2013, 223 (2), pp.963-974. 〈10.1016/j.icarus.2012.08.025〉. 〈hal-00730353〉
### Métriques
Consultations de la notice | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8975290656089783, "perplexity": 4280.95686149228}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039749562.99/warc/CC-MAIN-20181121173523-20181121195523-00546.warc.gz"} |
https://test.routledgehandbooks.com/doi/10.4324/9781315152318-3 | 151
Handbook of Optoelectronic Device Modeling and Simulation
Print publication date: October 2017
Online publication date: October 2017
Print ISBN: 9781498749565
eBook ISBN: 9781315152318
10.4324/9781315152318-3
Abstract
The brightness of an optical source is commonly defined as the emitted power per unit of emitting area and per unit of the solid angle into which the power is emitted (Walpole 1996). Therefore, a high-brightness source requires not only a high value of the emitted power but also a high “beam quality” in terms of a low product of the beam size and the beam divergence. The product of the beam radius at waist and the beam divergence half angle is called “beam parameter product” and based on it, the most widely used figure of merit for beam quality, the beam propagation ratio M 2 , is defined as the ratio of the beam parameter product of the beam of interest to the beam parameter product of a diffraction-limited, perfect Gaussian beam ( TEM 0 0 ) of the same wavelength ? (ISO 2005; Siegman et al. 1998). Therefore, a value M 2 = 1 represents an ideal diffraction-limited source, while values higher than unity indicate a degradation of the beam quality. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9347541332244873, "perplexity": 1489.3539413676226}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499758.83/warc/CC-MAIN-20230129180008-20230129210008-00448.warc.gz"} |
https://brilliant.org/problems/tic-tac-toe-to-the-nth-dimension/ | # Tic-Tac-Toe to the Nth Dimension
Probability Level 3
In a normal $3 \times 3$ tic-tac-toe board, there are 8 winning lines.
How many winning lines are there in a $4\times 4\times 4$ tic-tac-toe board? (In this case, a winning line consists of four boxes in a row, either on the surface or inside the cube.)
Bonus: How many winning lines are there in a $n^d$ tic-tac-toe hypercube, where $n$ is the number of cells per side and $d$ is the number of dimensions?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8067920207977295, "perplexity": 691.5648449791694}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178361510.12/warc/CC-MAIN-20210228145113-20210228175113-00352.warc.gz"} |
https://arxiv.org/list/math.FA/new | # Functional Analysis
## New submissions
[ total of 16 entries: 1-16 ]
[ showing up to 2000 entries per page: fewer | more ]
### New submissions for Fri, 3 Jul 20
[1]
Title: Products of positive operators
Comments: 32 pages. Dedicated to Henk de Snoo, on his 75th birthday
Subjects: Functional Analysis (math.FA)
On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class $\LL$ of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in $\LL$ are developed, and membership in $\LL$ among special classes, including algebraic and compact operators, is examined.
[2]
Title: Property($K^*$) Implies the Weak Fixed Point Property
Authors: Tim Dalby
Subjects: Functional Analysis (math.FA)
It is shown that if the dual of a separable Banach space has Property($K^*$) then the original space has the weak fixed point property. This is an improvement of previously results.
[3]
Title: Canonical graph contractions of linear relations on Hilbert spaces
Subjects: Functional Analysis (math.FA)
Given a closed linear relation $T$ between two Hilbert spaces $\mathcal H$ and $\mathcal K$, the corresponding first and second coordinate projections $P_T$ and $Q_T$ are both linear contractions from $T$ to $\mathcal H$, and to $\mathcal K$, respectively. In this paper we investigate the features of these graph contractions. We show among others that $P_T^{}P_T^*=(I+T^*T)^{-1}$, and that $Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$. The ranges $\operatorname{ran} P_T^{*}$ and $\operatorname{ran} Q_T^{*}$ are proved to be closely related to the so called `regular part' of $T$. The connection of the graph projections to Stone's decomposition of a closed linear relation is also discussed.
[4]
Title: Fractional Fourier transforms on $L^p$ and applications
Subjects: Functional Analysis (math.FA)
This paper is devoted to the $L^p(\mathbb R)$ theory of the fractional Fourier transform (FRFT) for $1\le p < 2$. In view of the special structure of the FRFT, we study FRFT properties of $L^1$ functions, via the introduction of a suitable chirp operator. However, in the $L^1(\mathbb{R})$ setting, problems of convergence arise even when basic manipulations of functions are performed. We overcome such issues and study the FRFT inversion problem via approximation by suitable means, such as the fractional Gauss and Abel means. We also obtain the regularity of fractional convolution and results on pointwise convergence of FRFT means. Finally we discuss $L^p$ multiplier results and a Littlewood-Paley theorem associated with FRFT.
[5]
Title: Besov spaces in multifractal environment and the Frisch-Parisi conjecture
Subjects: Functional Analysis (math.FA); Mathematical Physics (math-ph); Metric Geometry (math.MG)
We give a solution to the so-called Frisch-Parisi conjecture by constructing a Baire functional space in which typical functions satisfy a multifractal formalism, with a prescribed singularity spectrum. This achievement combines three ingredients developed in this paper. First we prove the existence of almost-doubling fully supported Radon measure on $\R^d$ with a prescribed multifractal spectrum. Second we define new \textit{heterogeneous} Besov like spaces possessing a wavelet characterization; this uses the previous doubling measures. Finally, we fully describe the multifractal nature of typical functions in these functional spaces.
### Cross-lists for Fri, 3 Jul 20
[6] arXiv:2007.00976 (cross-list from math.OC) [pdf, ps, other]
Title: Optimal Transport losses and Sinkhorn algorithm with general convex regularization
Subjects: Optimization and Control (math.OC); Functional Analysis (math.FA)
We introduce a new class of convex-regularized Optimal Transport losses, which generalizes the classical Entropy-regularization of Optimal Transport and Sinkhorn divergences, and propose a generalized Sinkhorn algorithm. Our framework unifies many regularizations and numerical methods previously appeared in the literature. We show the existence of the maximizer for the dual problem, complementary slackness conditions, providing a complete characterization of solutions for such class of variational problems. As a consequence, we study structural properties of these losses, including continuity, differentiability and provide explicit formulas for the its gradient. Finally, we provide theoretical guarantees of convergences and stability of the generalized Sinkhorn algorithm, even in the continuous setting. The techniques developed here are directly applicable also to study Wasserstein barycenters or, more generally, multi-marginal problems.
[7] arXiv:2007.01024 (cross-list from math.DG) [pdf, ps, other]
Title: Curve shortening flow on singular Riemann surfaces
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Functional Analysis (math.FA)
In this paper, we study curve shortening flow on Riemann surfaces with singular metrics. It turns out that this flow is governed by a degenerate quasilinear parabolic equation. Under natural geometric assumptions, we prove short-time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and prove some collapsing and convergence results.
[8] arXiv:2007.01101 (cross-list from math.MG) [pdf, ps, other]
Title: A Prékopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality
Authors: Yuchi Wu
Subjects: Metric Geometry (math.MG); Functional Analysis (math.FA)
In this paper, we prove a Pr\'ekopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Pr\'ekopa-Leindler inequality. In addition, we prove a functional $L_p$ Minkowski inequality.
[9] arXiv:2007.01175 (cross-list from math.CO) [pdf, other]
Title: Stirling operators in spatial combinatorics
Subjects: Combinatorics (math.CO); Mathematical Physics (math-ph); Functional Analysis (math.FA); Probability (math.PR)
We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol $(m)_n$ can be extended from a natural number $m\in\mathbb N$ to the falling factorials $(z)_n=z(z-1)\dotsm (z-n+1)$ of an argument $z$ from $\mathbb F=\mathbb R\text{ or }\mathbb C$, and Stirling numbers of the first and second kinds are the coefficients of the expansions of $(z)_n$ through $z^k$, $k\leq n$ and vice versa. When taking into account spatial positions of elements in a locally compact Polish space $X$, we replace $\mathbb N$ by the space of configurations---discrete Radon measures $\gamma=\sum_i\delta_{x_i}$ on $X$, where $\delta_{x_i}$ is the Dirac measure with mass at $x_i$.The spatial falling factorials $(\gamma)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}\delta_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ can be naturally extended to mappings $M^{(1)}(X)\ni\omega\mapsto (\omega)_n\in M^{(n)}(X)$, where $M^{(n)}(X)$ denotes the space of $\mathbb F$-valued, symmetric (for $n\ge2$) Radon measures on $X^n$. There is a natural duality between $M^{(n)}(X)$ and the space $\mathcal {CF}^{(n)}(X)$ of $\mathbb F$-valued, symmetric continuous functions on $X^n$ with compact support. The Stirling operators of the first and second kind, $\mathbf{s}(n,k)$ and $\mathbf{S}(n,k)$, are linear operators, acting between spaces $\mathcal {CF}^{(n)}(X)$ and $\mathcal {CF}^{(k)}(X)$ such that their dual operators, acting from $M^{(k)}(X)$ into $M^{(n)}(X)$, satisfy $(\omega)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*\omega^{\otimes k}$ and $\omega^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(\omega)_k$, respectively. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.
[10] arXiv:2007.01250 (cross-list from math-ph) [pdf, ps, other]
Title: Ground States for translationally invariant Pauli-Fierz Models at zero Momentum
Subjects: Mathematical Physics (math-ph); Functional Analysis (math.FA)
We consider the translationally invariant Pauli-Fierz model describing a charged particle interacting with the electromagnetic field. We show under natural assumptions that the fiber Hamiltonian at zero momentum has a ground state.
### Replacements for Fri, 3 Jul 20
[11] arXiv:1904.05239 (replaced) [pdf, ps, other]
Title: On Matrix Rearrangement Inequalities
Subjects: Functional Analysis (math.FA); Optimization and Control (math.OC)
[12] arXiv:1909.08316 (replaced) [pdf, ps, other]
Title: Approximation of the average of some random matrices
Journal-ref: Journal of Functional Analysis. (2020) 279:7, 108684
Subjects: Functional Analysis (math.FA); Metric Geometry (math.MG)
[13] arXiv:2003.11364 (replaced) [pdf, ps, other]
Title: Relative compactness of orbits and geometry of Banach spaces
Subjects: Functional Analysis (math.FA)
[14] arXiv:2005.08299 (replaced) [pdf, ps, other]
Title: Operator inequalities and characterizations
Authors: Ameur Seddik
Subjects: Functional Analysis (math.FA)
[15] arXiv:2007.00118 (replaced) [pdf, other]
Title: Approximation with Tensor Networks. Part I: Approximation Spaces
Comments: For part II see arXiv:2007.00128
Subjects: Functional Analysis (math.FA); Machine Learning (cs.LG); Numerical Analysis (math.NA)
[16] arXiv:2007.00128 (replaced) [pdf, other]
Title: Approximation with Tensor Networks. Part II: Approximation Rates for Smoothness Classes
Comments: For part I see arXiv:2007.00118
Subjects: Functional Analysis (math.FA); Machine Learning (cs.LG); Numerical Analysis (math.NA)
[ total of 16 entries: 1-16 ]
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Links to: arXiv, form interface, find, math, recent, 2007, contact, help (Access key information) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9328790903091431, "perplexity": 1159.5748804038928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886706.29/warc/CC-MAIN-20200704201650-20200704231650-00558.warc.gz"} |
http://mathhelpforum.com/math-topics/56833-physics.html | # Math Help - Physics
1. ## Physics
I have a test soon and I don't understand these types of problems very well...can anyone please explain these?
1) A carton of eggs rests on the seat of a car moving at 22.5 m/s. What is the least distance in which the cart can be uniformly slowed to a stop if the eggs are not to slide? The coefficient of friction between the seat and the carton is 0.24
2) Two blocks are connected by string of negligible mass that passes over a frictionless pulley. The mass of the block on the table is 43.1 kg and the mass of the block in the air is 18.9 kg. Assume the table is frictionless. Find the acceleration of the two blocks and the tension in the string
Answers are 2.98 m/s^2 & 129N
3) 2 masses 5 kg and 10 kg are tied to the opposite ends of a massless rope, and the rope is hung over a massless and friictionless pulley. Find the acceleration of the masses
4) Block 1, 8 kg, is moving on a 33 degree incline ( coefficient of kinetic friction is 0.25. This block is connected to block 2, 22 kg, by a cord that passes over a massless and frictionless pulley. Find the acceleration of each block and the tension in the cord
Answers are 5.2 m/s^2 and 101N
2. Originally Posted by realintegerz
I have a test soon and I don't understand these types of problems very well...can anyone please explain these?
1) A carton of eggs rests on the seat of a car moving at 22.5 m/s. What is the least distance in which the cart can be uniformly slowed to a stop if the eggs are not to slide? The coefficient of friction between the seat and the carton is 0.24
[snip]
$F_{\text{net}} = ma$.
If carton of eggs are on the point of sliding then $F_{\text{net}} = - \mu N = - (0.24) (m) (9.8)$.
Therefore $m a = - \mu N = - (0.24) (m) (9.8) \Rightarrow a = - (0.24)(9.8) = - 2.352$.
a = - 2.352 m/s^2
u = 22.5 m/s
v = 0 m/s
x = ?
Solve for x using an appropriate motion-under-constant-acceleration formula.
3. Originally Posted by realintegerz
[snip]
2) Two blocks are connected by string of negligible mass that passes over a frictionless pulley. The mass of the block on the table is 43.1 kg and the mass of the block in the air is 18.9 kg. Assume the table is frictionless. Find the acceleration of the two blocks and the tension in the string
Answers are 2.98 m/s^2 & 129N
[snip]
Did you draw a force diagram? Equations of motion:
43.1 kg mass:
(43.1)(a) = T .... (1)
18.9 kg mass:
(18.9) (a) = (18.9)(9.8) - T .... (2)
Solve equations (1) and (2) simultaneously for a and T.
4. Originally Posted by realintegerz
[snip]
3) 2 masses 5 kg and 10 kg are tied to the opposite ends of a massless rope, and the rope is hung over a massless and friictionless pulley. Find the acceleration of the masses
[snip]
Did you draw a force diagram? Equations of motion:
10 kg mass:
10 a = (10)(9.8) - T .... (1)
5 kg mass:
5a = T - (5)(9.8) .... (2)
Solve equations (1) and (2) simultaneously for a.
5. Originally Posted by realintegerz
[snip]
4) Block 1, 8 kg, is moving on a 33 degree incline ( coefficient of kinetic friction is 0.25. This block is connected to block 2, 22 kg, by a cord that passes over a massless and frictionless pulley. Find the acceleration of each block and the tension in the cord
Answers are 5.2 m/s^2 and 101N
Did you draw a force diagram? Resolve the forces on the 8 kg mass? Equations of motion:
22 kg mass:
22 a = (22)(9.8) - T .... (1)
8 kg mass:
Perpendicular to slope: 0 = R - (8)(9.8) cos 33 => R = (8)(9.8) cos 33 .... (2)
Parallel to slope: 8a = T - (8)(9.8) sin 33 - (0.25) R .... (3)
Substitute equation (2) into equation (3) and solve equations (1) and (3) simultaneously for a and T.
6. I don't know what formulas to use to solve for a & t....
7. Originally Posted by realintegerz
I don't know what formulas to use to solve for a & t....
I'm not solving the equations for you as well.
If you're studying dynamics you must surely have been taught basic things like how to solve simultaneous equations. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8159211874008179, "perplexity": 792.2803526262796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500834258.45/warc/CC-MAIN-20140820021354-00139-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://jmlr.csail.mit.edu/papers/v20/18-395.html | ## Best Arm Identification for Contaminated Bandits
Jason Altschuler, Victor-Emmanuel Brunel, Alan Malek; 20(91):1−39, 2019.
### Abstract
This paper studies active learning in the context of robust statistics. Specifically, we propose a variant of the Best Arm Identification problem for contaminated bandits, where each arm pull has probability epsilon of generating a sample from an arbitrary contamination distribution instead of the true underlying distribution. The goal is to identify the best (or approximately best) true distribution with high probability, with a secondary goal of providing guarantees on the quality of this distribution. The primary challenge of the contaminated bandit setting is that the true distributions are only partially identifiable, even with infinite samples. To address this, we develop tight, non-asymptotic sample complexity bounds for high-probability estimation of the first two robust moments (median and median absolute deviation) from contaminated samples. These concentration inequalities are the main technical contributions of the paper and may be of independent interest. Using these results, we adapt several classical Best Arm Identification algorithms to the contaminated bandit setting and derive sample complexity upper bounds for our problem. Finally, we provide matching information-theoretic lower bounds on the sample complexity (up to a small logarithmic factor).
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Chemistry is the study of ______ and the changes that it undergoes matter ______ is anything that has mass and taks up space. Matter Each _______ is made up of the same kind of atom. element A ________ is made up of two or more different kinds of elements. compound The three states of matter are _____, ______, and ___. solid, liquid, gas A ___________ mixture is not uniform throughout. heterogeneous An example of a ________ property is its boiling point. physical ________ properties can only be observed when a substance is changed into another substance. Chemical The SI units of temperature are expressed in degrees ______. Kelvin The term __________ ______ refers to digits that were actually measured. Significant digits The smallest particle of matter is the ____. atom Atoms are neither _______ nor _________ in chemical reactions. created destroyed _________ are formed when atoms of more than one element combine. compounds The elemental ___________ of a pure substance never varies. composition The law of concervation of mass states that the mass of substance present at the end of a chemical process is _____ to the mass of subtances present before the reaction took place. equal The charge on an electron is a ________ ___. negative one The charge on a proton is a ________ ___ positive one A _______ has a mass of one amu and is neutral neutron _____________ is the spontaneous emission of radiation by an atom radioactivity Protons and neutrons are located in the _______ of an atom. nucleus The mass of an atom is calculated by adding thenumber of _______ to the number of ________. protons, neutrons 12/6 C, the 6 in the symbol represents the ____ ______ atomic number Atoms of the same element with different masses are called ________ and only differ by the number of ________ isotopes, neutrons The ________ ______ is a systematic catalog of the elements which are arranged in the order of their ______ ________ periodic table, atomic numbers _________ formulas give the lowest whole-number ratio of atoms of each element in a compound while _________ formulas give the exact number of atoms of each element in a compound. Empirical, molecular Name two different types of reactions. combustion reaction, decomposition reaction The sum of the atomic weights of the atoms in a molecule is called the ________ ______. molecular weight C6H12o6 = . CO2 =6 H20. The underlined 6 is called the ___________. coefficient The units of molar mass are _____ ____. grams per mole What is the definition of energy? The capacity to do work or produce heat. When energy is used to cause an object that has mass to move, it is called ____. work What is the definition of heat? The energy used to cause the temperature of an object to increase. Distinguish between PE and KE. PE is inergy of position, KE is energy of movement The formula for KE is 1/2MV2 The SI unit of energy is the _____. Joule One J is equal to 1kgm2/sec2 An older non-SI unit still used is the _______, which is equal to _____J. calorie, 4.184J The formula for work is ________explain each symbol. W=F*D w=work f=force d=distance Heat flows from ____ objects to ____ objects. warm cold What is the first law of thermodynamics? Energy cannot be created nor destroyed. The ________ ______ of a system is the sum of all kinetic and potentiol energies of all components of the system. Its symbol is _____. internal energy, E When heat is absorbed by the system from the surroundings the process is __________. endothermic When heat is released by the system to the surroundings the process is __________. exothermic The internal energy of a system is independent of the path by which the system achieved that state, it is therefore known as a _____ ________. state function The thing that students hate to do is called ____. It can be expressed in terms of pressure and volume by the equation ________ W=-P(delta)V If a reaction is exothermic, its Delta H is always ________. positive If a reaction is endothermic, the Delta H is always ________. negative The experimental science for measuring the flow of heat is known as ___________. calorimetry The amount of energy required to raise the temperature of a substance by 1K is its ____ ________. heat capacity _______ law states that as a reaction is carried out in a series of steps, delta H for the overall reaction will be equal to the sum of the enthalpy changs for the individual steps. Hess's Standard enthalpies of fomation delta Hf are measured under standard conditions which are _____ degrees and ______ pressure. 298K 25C, atmospheric 1STM Name three different types of reactions. combustion, neutralization, decomposition The sum of the atomic weights of the atoms in a molecule is called the _________ ______. molecular weight In a chemical reaction, if there is more of one reactant than is necessary, that reactant is said to be __ ______. The other reactant is known as the _________ _______ and is used in all molar calculations. in excess, limiting reagent Solutions are ____________ mixtures of two or more pure substances. homogeneous the dissolving medium is called the ________ while the cvhamical dissolved is known as the ______. solvent, solute ____________ occurs when an ionic substance dissolves in water. Dissociation Give five examples of strong acids. HCL, HBR,HI,HNO3,H2SO4,HCLO3,HCLO4 Give four examples of strong bases. LiOH, NaOH, KOH, Ca(OH)2 If ions are mixed and react to form an insoluble species, that compound is called a ___________. precipitate Metathesis is simply an ________ reaction. exchange In an ionic reaction equation the things that don't change are called the _________ ____. spectator ions Both definitions of acis involve the ______ _____ or _______. proton donor H+ ion Substances that increase the concentration of OH ions are called _____. bases The products of the reaction of an acid and a base are always a ____ and _____. salt, water. Oxidation is when an atom or ion _____ electrons while recuction is when an atom or ion _____ electrons. loses, gains The oxidation state of an element from the periodic table is always ____. zero The sum of the oxidation numbers in a neutral compound is equal to ____. zero Write the equation for molarity. M= moles of solvent --------------- volume of solution in liters Two important properties of waves are their __________ and ________ (words and symbols). wavelength , frequency λν The equation for the speed of light is _____________--which is equal to _________ c=wavelength*frequency, c=hv, 3.00*10/8 m/s Waves do not explain all the properties of light and Max Planck explained this by assuming that energy comes in ______ called ______. packets quanta Einstein concluded that energy is proportional to ________, the equation being E=hv. frequency h is equal to _______ constant which numerically is _________. Planck s, 6.63*10-34J-s Neils Bohr is credited with the theory of ______ ________. atomic orbitals Energy is only absorbed or emitted in such a way as to move an electron from one allowed energy state to another, the energy is difined by _________ (equation) E=hv de Broglie demonstrated that the relationship between mass and wavelength was ____________(equation). wavelength=h/ mv Erwin Schrodinger developed a mathematical treatment that took into account both the wave and particle nature of matter which became known as_______ _________. quantum mechanics An orbital is decribed by a set of quantum numbers. Theste quantum numbers are the ______ _______ _____ abbv. _, the _________ _______ ______ abb, L, the ________ _______ ______ abbv __, and finally the ____ _______ ______ abbv. __ principal quantum number, n azimutal quantum number, l magnetic quantum number, ml spin quantum number, ms For a one electron hydrogen atom, orbitals on the same energy level have the same energy, that is , they are, _________. degenerate. No two electrons in the same atom can have exactly the same energy, this is known as the_____ _________ _________. Pauli exclusion principle 4p5: 4 stands for the ___________, p stands for the _______ and 5 stands for the _________. energy level, orbital, number of electrons in orbital For degenerate orbitals, the lowest energy is attained when the number of electrons with the same spin is _________. This is known as ______ ____. maximized, Hundt's rule I have told you that I will never give you any exceptions to the rule which your book calls_________. anomalies Dmitri Mendeleev and Lathar Meyer were both given credit for creating the ________ _____. periodic table Three trends that can be observed using the table are _____ __ _____ ___ ____, _________ _____, _______ ________. sizes of atoms and ions, ionization energy, electron affinity The nuclear charge that an electron experiances depends on the following two factors: Electrons are both attracted by the nucleus and repelled by other electrons. The effective nuclear charge Zeff is equal to Z-S, where Z is the _____ ______ and S is a ________ _______. atomic number, screening constant. The bonding atomic radius is defined as ____ of the distance between __________ bonded nuclei. half, covalently Cations are _______ than their parent atoms because the outermost electron is ________ and __________ are reduced. smaller, removed, repulsions Anions are ______ than their parent atom because electrons are _____ and __________ are increased. larger, added, repulsions What is the definition of ionization energy? The energy required to remove an electron from the ground state of a gaseous atom. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8705107569694519, "perplexity": 2025.8261396111784}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720941.32/warc/CC-MAIN-20161020183840-00035-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.cut-the-knot.org/arithmetic/algebra/CosSinInequality.shtml | # cos/sin inequality
Here's a problem from the 2004 Russian Mathematical Olympiad [Lecture Notes, p 61]:
Let $a,b,c$ be positive numbers, satisfying $\displaystyle a+b+c=\frac{\pi}{2}$, prove that
$\mbox{cos}(a)+\mbox{cos}(b)+\mbox{cos}(c)\gt \mbox{sin}(a)+\mbox{sin}(b)+\mbox{sin}(c)$
Solution
### References
1. Xu Jiagu, Lecture Notes on Mathematical Olympiad Courses, v 8, (For senior section, v 1), World Scientific, 2012
Let $a,b,c$ be positive numbers, satisfying $\displaystyle a+b+c=\frac{\pi}{2}$, prove that
$\mbox{cos}(a)+\mbox{cos}(b)+\mbox{cos}(c)\gt \mbox{sin}(a)+\mbox{sin}(b)+\mbox{sin}(c)$
It appears that the problem admits a brute-force, a rather straightforward, solution (Proof 1), which caused me to wonder why it was offered as an olympiad problem. However, the book solution (Proof 2) makes a very elegant shortcut that makes the problem certainly worth looking into.
Both proofs use the fact that $y=\mbox{cos}(x)$ is monotone decreasing on interval $\displaystyle (0,\frac{\pi}{2})$:
### Proof 1
\displaystyle \begin{align} \mbox{cos}(a) &= \mbox{cos}(\frac{\pi}{2}-b-c) \\ &= \mbox{sin}(b+c) = \mbox{sin}(b)\mbox{cos}(c)+\mbox{sin}(c)\mbox{cos}(b), \end{align}
and similarly for $\mbox{cos}(b)$ and $\mbox{cos}(c)$. Summing up the three give
\begin{align} \mbox{cos}(a)+\mbox{cos}(b)+\mbox{cos}(c) =& \mbox{sin}(a)[\mbox{cos}(b)+\mbox{cos}(c)] \\ &+ \mbox{sin}(b)[\mbox{cos}(c)+\mbox{cos}(a)] \\ &+ \mbox{sin}(c)[\mbox{cos}(a)+\mbox{cos}(b)]. \end{align}
Let's focus on one of the terms, say, $\mbox{sin}(a)[\mbox{cos}(b)+\mbox{cos}(c)]$:
$\mbox{sin}(a)[\mbox{cos}(b)+\mbox{cos}(c)] = \mbox{sin}(a)\cdot 2\cdot \mbox{cos}(\frac{b+c}{2})\cdot \mbox{cos}(\frac{b-c}{2}).$
Now, since $a\gt 0$, $b+c\lt\frac{\pi}{2}$ and, therefore, $\frac{b+c}{2}\lt\frac{\pi}{4}$. As we observed at the outset, function $y=\mbox{cos}(x)$ is monotone decreasing on interval $(0,\frac{\pi}{2})$ such that $\mbox{cos}(\frac{b+c}{2})\gt \mbox{cos}(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$. For the other factor, we also have $\mbox{cos}(\frac{b-c}{2})\gt \mbox{cos}(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$, because of the triangle inequality $|b-c|\le |b|+|c|=b+c$. It follows that
$\mbox{sin}(a)[\mbox{cos}(b)+\mbox{cos}(c)] \gt \mbox{sin}(a)\cdot 2\cdot \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{2}}{2} = \mbox{sin}(a).$
For the other two terms we similarly have $\mbox{sin}(b)[\mbox{cos}(c)+\mbox{cos}(a)]\gt\mbox{sin}(b)$ and $\mbox{sin}(c)[\mbox{cos}(a)+\mbox{cos}(b)]\gt\mbox{sin}(c)$. Adding the three up gives the desired inequality.
### Proof 2
Observe that, say, $a+b\lt\frac{\pi}{2}$ which implies $a\lt \frac{\pi}{2} - b$, and since $y=\mbox{cos}(x)$ is monotone decreasing on interval $(0,\frac{\pi}{2})$,
$\mbox{cos}(a) \gt \mbox{cos}(\frac{\pi}{2} - b) = \mbox{sin}(b).$
Similarly $\mbox{cos}(b) \gt \mbox{sin}(c)$ and $\mbox{cos}(c) \gt \mbox{sin}(a)$. The sum of the three inequalities gives the desired one.
Looking back at the two proofs, it may occur to you that the inequality that has been proved is actually rather weak. Furthermore, as we've seen on the last step of the first proof, $\mbox{cos}(b)+\mbox{cos}(c)\gt 1$, implying by analogy that $\mbox{cos}(c)+\mbox{cos}(a)\gt 1$ and $\mbox{cos}(a)+\mbox{cos}(b)\gt 1$ and so $\mbox{cos}(a)+\mbox{cos}(b)+\mbox{cos}(c)\gt \frac{3}{2}$. The olympiad inequality would also have been proved had we shown that $\mbox{sin}(a)+\mbox{sin}(b)+\mbox{sin}(c)\le \frac{3}{2}$, making more precise the notion of the weakness of that inequality.
### Proof 3
The proof is based on the
### Lemma
Let $a,b,c$ be positive numbers, satisfying $\displaystyle a+b+c=\frac{\pi}{2}$, prove that
$\mbox{sin}(a)+\mbox{sin}(b)+\mbox{sin}(c)\le \frac{3}{2}$
### Proof of Lemma
Instead of an algebraic derivation, I'll base the proof on a geometric insight. First of all note that the graph of $y = \mbox{sin}(x)$ on interval $(0,\frac{\pi}{2})$ is concave.
Then for three points on the graph that correspond to $a,b,c$ satisfying $\displaystyle a+b+c=\frac{\pi}{2}$, the center of gravity is below the graph at the point corresponding to $\displaystyle\frac{a+b+c}{3}=\frac{\pi}{6}$. It follows that
$\displaystyle\frac{\mbox{sin}(a)+\mbox{sin}(b)+\mbox{sin}(c)}{3}\le\mbox{sin}(\frac{a+b+c}{3})=\frac{1}{2}$,
with the equality only when $a=b=c=\frac{\pi}{6}$. Q.E.D.
### Corollary
Assume $\alpha,\beta,\gamma$ be the angles of an acute triangle, so that each is positive, less than $\frac{\pi}{2}$ and $\alpha+\beta+\gamma=\pi$. Then
$\mbox{cos}(\alpha)+\mbox{cos}(\beta)+\mbox{cos}(\gamma)\le \frac{3}{2}$
Indeed, consider $a=\frac{\pi}{2}-\alpha$, $b=\frac{\pi}{2}-\beta$, $c=\frac{\pi}{2}-\gamma$, so that $a+b+c=\frac{\pi}{2}$ and each is positive. From the discussion above,
$\mbox{sin}(\frac{\pi}{2}-\alpha)+\mbox{sin}(\frac{\pi}{2}-\beta)+\mbox{sin}(\frac{\pi}{2}-\gamma)\le \frac{3}{2}$,
which is equivalent to $\mbox{cos}(\alpha)+\mbox{cos}(\beta)+\mbox{cos}(\gamma)\le \frac{3}{2}$.
### Proof 4
This proof has been posted below in the comments area. I decided to have it on the page proper for completeness and fairness sake because at the same time two other proofs have been posted at the CutTheKnotMath facebook page which I habitually reproduce at this site.
Let $\alpha = \pi/2 - a$ and so on. So $\alpha + \beta + \gamma = \pi$ and all three angles are in $(0,\pi/2),$ that is, they are the angles of an acute triangle $ABC.$ The inequality is now equivalent to
$\sin\alpha + \sin\beta + \sin\gamma > \cos\alpha + \cos\beta + \cos\gamma.$
Multiplying by the circumdiameter $2R$ and using the law of sines and the fact that $AH = 2Rcos\alpha,$ in which $H$ is the orthocenter, we find that we need to prove that
$BC + AC + BC > AH + BH + CH.$
But $AB$ is longer than the altitude from $A$ of which $AH$ is only a part (here we are using that the triangle is acute), and similarly $BC > BH$ and $CA > CH,$ and we are done.
### Proof 5
The proof is by Leo Giugiuc.
First we prove
### Lemma
Let $\Delta ABC$ be acute angled. Then
$\sin\alpha + \sin\beta + \sin\gamma > \cos\alpha + \cos\beta + \cos\gamma,$
where $\alpha =\angle BAC,$ etc. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9958012104034424, "perplexity": 371.26338464869764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203865.15/warc/CC-MAIN-20190325092147-20190325114147-00041.warc.gz"} |
http://math.stackexchange.com/questions/246681/calculating-the-expected-value-of-free-tickets-in-a-paytable?answertab=votes | Calculating the expected value of free tickets in a paytable?
Lets says I have a lottery game where the ticket costs $1 and has the following probability/prize distribution: • 0.3 -> \$1
• 0.2 -> X
• 0.5 -> \$0 If X = \$1, then the expected value is: 0.3(\$1) + 0.2(\$1) = \$0.50 If X = FREE_TICKET, then I've calculated (and confirmed) via sampling that the EV is either: • 0.3(\$1) / (0.3+0.5) = 0.375
• 0.3(\$1) + 0.2 * \$1 * 0.3/(0.3+0.5) = 0.375
Obviously, a FREE_TICKET isn't the same as \$1, the price of a ticket. However, I'm stumped when calculating the EV of the table for the following two cases: • X = 2 FREE_TICKETS • X = \$5 + FREE_TICKET
How do I calculate them?
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You have the expected value as $Y = 0.3\cdot 1 + 0.2\cdot X + 0.5\cdot 0$. If, in addition, $X = 2Y$, or $X = 5 + Y$, you can solve for $X$ and $Y$ simultaneously. – mjqxxxx Nov 28 '12 at 18:24
Thanks, you're right, it's just basic algebra. – AlexD Nov 29 '12 at 14:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8780689239501953, "perplexity": 1254.4131307120895}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860122902.86/warc/CC-MAIN-20160428161522-00178-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://www.ams.org/samplings/feature-column/fcarc-primes2 | ## Primes
2. Basic ideas
The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. Although Euclid's proof that the primes are infinite is quite well known, his argument is so neat that I will repeat it here in modern language. Suppose there were only a finite number of primes. Form the number M obtained by multiplying the numbers in this finite collection and adding 1. M is either prime or has a prime factor, yet M or a prime factor of M cannot be in our original list (because none of the primes in the original list divides M without remainder). This contradiction means that the collection of primes cannot be finite.
When a result is very important, it is not surprising that there are various approaches to proving the same thing. This is true with the infinitude of the primes for which a variety of proofs are known.
Many interesting questions about primes arise. We mentioned before that 60 is 5 x 12 or 6 x 10. These factorizations, breaking down the number into smaller numbers, are different, but in each case one or more of the factors is not a prime. Thus, 12 is 2 x 2 x 3 and 6 is 2 x 3 and 10 is 2 x 5. The first factorization leads to the product 2 x 2 x 3 x 5, as does the second. This suggests that even though one may break down a number into primes in different ways, when one gets down to primes and lists these in increasing order, perhaps the result one gets is always the same. This is the content of the Fundamental Theorem of Arithmetic which states:
Every positive integer other than 1 can be written in a unique way as a product of primes.
Again, the reference to a unique factoring into primes refers to the fact that we are not interested in the order in which the primes are listed, only in which primes appear and how often each prime occurs.
Just as artists must learn to use the tools of their "trade," such as brushes, charcoal, or watercolors, mathematicians have tools they put to use regularly. These tools include proof techniques such as mathematical induction or parity arguments, as well the knowledge of "workhorse" theorems. "Workhorse" theorems are results which always seem to be useful in proving new results no matter how often they have been put to use in the past. A good example of a workhorse theorem in the area of number theory is known as Fermat's Little Theorem. This result enables one to show that certain numbers must be composite, that is, they are not prime.
To explain Fermat's Little Theorem first let me briefly review the idea of the congruence of two numbers. This notion, which goes back at least to Carl Friedrich Gauss (1777-1855), calls two numbers equivalent or congruent modulo m (where m is a positive integer which is at least 2) if the numbers leave the same remainder when divided by m. Thus, 24 and 57 are congruent modulo 11 because when divided by 11 they both leave the remainder of 2. (Note that 24 = 2(11) + 2 while 57 = 5(11) + 2). When two numbers are congruent modulo m, their difference is divisible by m with a zero remainder. Thus, since 57 and 24 are congruent modulo 11, we notice that 57 - 24 = 33 is exactly divisible by 11. Because of its many similarities to the properties of the equal sign (=), the symbol " " was used by Gauss for congruence, and we can write 57 24 mod 11. (Mod is short for "modulo.")
We can now state the result that Pierre Fermat (1601-1665) showed: If p is a prime and a is not a multiple of p, then ap-1 1 mod p. (From this it immediately follows that when p is a prime, ap a mod p for any integer a.)
Here is an example to illustrate Fermat's wonderful result. Suppose p is 11, and a is 2. (Since 2 is not a multiple of 11, the theorem applies.) We need to verify that 211-1 which is 210, leaves the remainder of 1 when divided by 11. Now 210 = 1024, and 1024 = 11(93) + 1, so we indeed have 1 as a remainder when we divide 210 by 11. Notice that to do a calculation which involves Fermat's Little Theorem it is helpful to be able to compute the remainder for numbers raised to large powers easily.
Unfortunately, the converse of Fermat's Little Theorem is not true. For example, you can check for yourself that for m = 341 (which is composite because 341 = 31(11)), 2m-1 1 mod 341. Because of this we say that 341 is a pseudoprime for the base 2. More generally, we say that a number m is a pseudoprime for base b (b a positive integer) if m is composite and bm-1 1 mod m. It turns out that for each choice of a base b, there are pseudoprimes for b. However, things are much worse. When am-1 1 mod m for every choice of positive a (where a and m are such that the only number which divides both is 1), it does not follow that m must be prime. Such composite numbers m are called Carmichael Numbers, after Robert Daniel Carmichael (1879-1967). An example of a Carmichael Number is 561. Not only do there exist Carmichael Numbers, but like the primes themselves, there are infinitely many of the "varmints."
What the discovery of Fermat's Little Theorem set in motion is typical of many parts of mathematics. An interesting result is proven and it encourages a detailed investigation of ideas in the intellectual vicinity of that result. For example, Euler (1707-1783) studied an important relative of the Little Theorem. To state Euler's result we need a very useful concept, that of two positive integers being relatively prime. We say that j and k are relatively prime if the only integer that divides both j and k is 1. Thus, any two primes are relatively prime and, for example, 14 and 15 are relatively prime, while 10 and 14 are not, since 2 divides both 10 and 14. Euler's result states that for any positive integers n and a with the property that the largest integer which divides them both is 1, then aphi(n) 1 mod n. Here, phi(n) (n at least 1) denotes the number of positive integers no more than n which are relatively prime to n. Hence, for a prime p, phi(p) = p-1, while for, say, the composite number 20, phi(20) = 8. (The numbers relatively prime to 20 are 1, 3, 7, 9, 11, 13, 17, and 19.)
If one looks at a moderately long list of primes, many questions about primes come to mind. For example, after noting that there are infinitely many primes, it is natural to ask if there are infinitely many primes of special types. There are many interesting things here to explore once one has this new idea. For example, 3, 7, 11, 19, 23, 31, are all primes as are 5, 13, 17, 29, 37. The first set of primes has the form 4k +3 while the second has the form 4k + 1. Numbers which have the form as + b where s can take on the values 1, 2, 3, etc., are called arithmetic progressions. Here we will be interested only in the special class of arithmetic progressions where the numbers a and b are relatively prime, which I will call a relatively prime arithmetic progression. Are there infinitely many primes in the arithmetic progressions 4k +1 and 4k + 3? It turns out that with some work one can show the answer is "yes." However, what is truly amazing is that the following more general observation is true: There are infinitely many primes in any relatively prime arithmetic progression! This remarkable theorem (1837) is due to the mathematician Lejeune Dirichlet (1805-1859). For those interested in connections between music and mathematics, Dirichlet was married to composer Felix Mendelssohn's sister Rebecca. Unquestionably, Dirichlet's Theorem is one of the major landmarks in the history of number theory.
Other special sequences that have been searched for primes, include, for example, the Mersenne primes, named for the cleric Marin Mersenne, pictured below.
Mersenne primes are primes of the form 2n - 1. It remains unknown whether or not there are infinitely many Mersenne primes, though very large examples are known. It is also unknown if there are infinitely many primes which are one more than a perfect square (e.g. have the form n2 + 1). Another tantalizing problem is whether or not for every integer n there is always a prime in the interval from n2 to (n+1)2. (For example, when n is 10 this question asks if between 100 and 121 there is a prime, and in fact, there are several.) There are many other simple-to-state open problems about primes.
Another pattern that quickly becomes apparent when one carefully looks at the sequence of primes is the occurrence of "twins." Twin primes are primes which differ by 2. Thus, 3 and 5; 11 and 13; 17 and 19; 29 and 31; 41 and 43 are examples of twin primes. The Norwegian mathematician Viggo Brun (1885-1978), shown below, raised the specific question as to whether or not there are infinitely many twin primes.
(This photograph was made available with the kind permission of Harald Hanche-Olsen)
His work makes more specific the earlier family of problems raised by Alphonse de Polignac, who raised the question in 1849 if, for any even integer s, there are infinitely many pairs of primes that differ by s? The remarkable answer to Brun's question about twin primes is that no one knows. For the primes, if one sums the series whose terms are 1/p where p is a prime, the sum turns out not to be a finite number. Perhaps surprisingly, if one sums the series whose terms are 1/s and 1/(s+2) where s and s+2 are both primes, Brun showed the result is a finite number. To use slightly more technical terminology, the first series diverges and the second converges to the constant 1.9021605... , regardless of whether it has a finite number of terms (which as I noted, is still not known). The sum of the series of twin prime pairs is now known as Brun's constant. Finding its exact value has served as a challenge for mathematicians interested in computation. Thomas Nicely, during his effort in 1995 to find a more precise value for Brun's constant, discovered a flaw in the design of one of Intel's Pentium series of chips! Who would have thought that trying to solve an arcane mathematics problem would save many individuals from improperly relying on calculations done by a microprocessor? Nicely is still involved with trying to find more exact values of Brun's constant.
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These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9036021828651428, "perplexity": 222.04436964310446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424931.1/warc/CC-MAIN-20170724222306-20170725002306-00369.warc.gz"} |
http://projecteuclid.org/euclid.adjm/1388953704 | ## African Diaspora Journal of Mathematics
### Existence and Attractivity Results for Some Fractional Order Partial Integro-differential Equations with Delay
#### Abstract
In this paper we study some existence, uniqueness, estimates and global asymptotic stability results for some functional integro-differential equations of fractional order with finite delay. To achieve our goals we make extensive use of some fixed point theorems as well as the so-called Pachpatte techniques.
#### Article information
Source
Afr. Diaspora J. Math. (N.S.) Volume 15, Number 2 (2013), 87-100.
Dates
First available: 5 January 2014
Mathematical Reviews number (MathSciNet)
MR3161669
#### Citation
Abbas, S.; Benchohra, M.; Diagana, T. Existence and Attractivity Results for Some Fractional Order Partial Integro-differential Equations with Delay. African Diaspora Journal of Mathematics. New Series 15 (2013), no. 2, 87--100. http://projecteuclid.org/euclid.adjm/1388953704.
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• S. Abbas, and M. Benchohra, Fractional order Riemann-Liouville integral equations with multiple time delay, Appl. Math. E-Notes 12 (2012), pp. 79–87.
• S. Abbas, M. Benchohra and J. Henderson, On global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order, Comm. Appl. Nonlinear Anal. 19 (2012), pp. 79–89.
• S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in fractional differential equations, Developments in Mathematics, 27, Springer, New York, 2012.
• J. M. Appell, A. S. Kalitvin, and P. P. Zabrejko, Partial integral operators and integrodifferential equations, 230, Marcel and Dekker, Inc., New York, 2000.
• D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional calculus models and numerical methods, World Scientific Publishing, New York, 2012.
• J. Caballero, A.B. Mingarelli and K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electron. J. Differential Equations 2006 (57) (2006), pp. 1–11.
• K.M. Case and P.F. Zweifel, Linear transport theory, Addison-Wesley, Reading, MA 1967.
• S. Chandrasekher, Radiative transfer, Dover Publications, New York, 1960.
• K. Diethelm, The Analysis of fractional differential equations. Springer, Berlin, 2010.
• S. Hu, M. Khavani and W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal. 34 (1989), pp. 261–266.
• C.T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Eq. 4 (1982), pp. 221–237.
• A. A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations. Elsevier Science B.V., Amsterdam, 2006.
• K. S. Miller and B. Ross, An Introduction to the fractional calculus and differential equations, John Wiley, New York, 1993.
• B. G. Pachpatte, Volterra integral and integrodifferential equations in two variables, J. Inequ. Pure Appl. Math. 10 (4) (2009), pp. 1–21.
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
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• A. N. Vityuk and A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (2004), pp. 318–325. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8165358304977417, "perplexity": 3304.9861972867434}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00426-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/199626/what-is-this-operation-on-random-variables-called | # What is this operation on random variables called?
Let $X$ be a random variable and let $N$ be a discrete random variable which takes values in the non-negative integers. Let $X_1, X_2, ...$ be a sequence of i.i.d. random variables with the same distribution as $X$, all of which are also independent of $N$. Is there a name for the random variable
$$Y = X_1 + X_2 + ... + X_N?$$
The only hint I've found is that this appears to be what actuaries call the aggregate risk model. One reason I ask is that there is a very nice expression for the cumulant generating function $C_Y$ of $Y$ in terms of the cumulant generating functions $C_X, C_N$ of $X, N$, namely
$$C_Y = C_N \circ C_X.$$
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In case where $N$ has a Poisson distribution and is independent of $X_i$'s, it is called a compound Poisson random variable. – Sangchul Lee Sep 20 '12 at 9:36
It is interesting and reminds me sequential detection. Does it have only mathematical importance or used for some sort of modeling of some physical phenomena? – Seyhmus Güngören Sep 20 '12 at 10:19
@Seyhmus: it models something fairly natural for actuaries. Imagine you have some known probability distribution for bad things happening per unit time (e.g. car crashes) and also some known probability distribution for how much each bad thing costs you; then this construction describes the distribution of your total cost due to bad things happening. – Qiaochu Yuan Sep 20 '12 at 17:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8766686320304871, "perplexity": 156.04412877252204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042981576.7/warc/CC-MAIN-20150728002301-00173-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://astarmathsandphysics.com/index.php?option=com_content&view=article&id=5476:the-negative-binomial-distribution&catid=206&tmpl=component&print=1&layout=default&Itemid=1736 | ## The Negative Binomial Distribution
The negative binomial models the number of trials
$n$
up to and including a given number of successes
$x$
, where the probability of success is a fixed
$p$
. The first
$x-1$
successes may be any of the first
$n-1$
trials, but the
$n$
th trial must be the
$x$
th success.
We may model the first
$x-1$
successes from
$n-1$
by the binomial
$B(n-1,p)$
distribution:
\begin{aligned}P(X=x-1) &={}^{n-1}C_{x-1}p^{x-1}(1-p)^{(n-1)-(x-1)} \\ &={}^{n-1}C_{x-1}p^{x-1}(1-p)^{n-x}{}\end{aligned}
The
$n$
th trial is the
$x$
th success with probability
$p$
, so the probability of the needing
$n$
trials to obtain
$x$
successes is
${}^{n-1}C_{x-1}p^x(1-p)^{n-x}$
.
It is important to realise that the variable being modelled here is
$n$
.
$x$
is the fixed number of successes.
It is often convenient to reparametrize in terms of number of successes and failures, so let
$k$
be number of failures, then
$x+k=n$
and the expression becomes
${}^{x+k-1}C_{x-1}p^x(1-p)^{k}$
. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9684234261512756, "perplexity": 1977.7787307504032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347409337.38/warc/CC-MAIN-20200530133926-20200530163926-00114.warc.gz"} |
https://www.physicsforums.com/threads/sound-waves.150362/ | # Sound Waves
1. Jan 7, 2007
### ubiquinone
Hi I have a question here from a chapter on sound. I'm not sure on how to solve this, so I was wondering if someone here could please give me a hand. Thank You!!
Question: A point source at $$A$$ emits sound uniformly in all directions. At point $$B$$, the listener measures the sound intensity to be $$50.0dB$$. At point $$C$$, the sound intensity level is $$45.3dB$$. The distance $$\overline{BC}$$ is $$10.0m$$. Calculate the distance $$\overline{AB}$$.
Diagram:
Code (Text):
A----------B
\ |
\ |
\ |
\ |
\ |
C
2. Jan 7, 2007
### Hootenanny
Staff Emeritus
Have you any thoughts yourself?
3. Jan 7, 2007
### ubiquinone
This question is from an old physics exercise book which does not have any examples. I have tried solving it by reading up the concepts on waves and sound from a conceptual physics book. I'm hoping if someone here can show me how to solve this problem, so I can see and understand how a physicist or a good problem solver applies physics concepts and theory into problem solving. Again, any help with this would be greatly appreciated. Thanks!
4. Jan 7, 2007
### Hootenanny
Staff Emeritus
5. Jan 8, 2007
### ubiquinone
Hi Hootenanny, thank you for the reference! I think I got it now.
Let $$x$$ be the distance of $$\overline{AB}$$
Calculate the intensities:
$$\displaystyle 50.0dB=10\log\left (\frac{I_1}{10^{-12}}\right )$$
$$\displaystyle\Leftrightarrow 10^5=\frac{I_1}{10^-12}$$
$$\displaystyle\Leftrightarrow 10^{-7}=I_1$$
$$\displaystyle 45.3dB=10\log\left (\frac{I_2}{10^{-12}}\right )$$
$$\displaystyle\Leftrightarrow 10^{4.53}=\frac{I_2}{10^-12}$$
$$\displaystyle\Leftrightarrow 10^{-7.47}=I_2$$
Since $$\displaystyle\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}\Rightarrow \frac{10^{-7}}{10^{-7.47}}=\frac{\sqrt{x^2+10^2}^2}{x^2}$$
$$\displaystyle\Leftrightarrow 10^{0.47}x^2=x^2+100$$
$$\displaysytle\Leftrightarrow x^2(-1+10^{0.47})=100$$
$$\displaystyle\Rightarrow x=\sqrt{\frac{100}{10^{0.47}-1}}\approx 7.16m$$
Similar Discussions: Sound Waves | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8960002064704895, "perplexity": 833.0289262645674}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189032.76/warc/CC-MAIN-20170322212949-00518-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://www.ck12.org/book/CK-12-Algebra-I-Honors/section/9.9/ | <meta http-equiv="refresh" content="1; url=/nojavascript/"> Chapter Test | CK-12 Foundation
9.9: Chapter Test
Created by: CK-12
0 0 0
Multiple Choice – Please circle the letter of the correct answer and write that letter in the space provided to the left of each question.
1. _____ What is the general form of the following quadratic equation? $\frac{1}{2}(y-1)=(x+3)^2$
1. $2x^2+6x+10$
2. $2x^2+12x+17$
3. $2x^2+12x+19$
4. $2x^2+6x+17$
2. _____ What are the zeros of the following quadratic equation? $(m+2)^2=25$
1. $m=7$; $m=-3$
2. $m=-7$; $m=3$
3. $m=7$; $m=3$
4. $m=-7$; $m=-3$
3. _____ $\boxed{h(t)=-2(t-3)^2+21}$ This quadratic function represents height $(h)$ in yards, reached by a ball thrown into the air after $t$ seconds. From what height was the ball thrown?
1. 2yd
2. 5yd
3. 21yd
4. 3yd
4. _____ What is the value of the discriminant for the following quadratic equation? $3y^2-5y=1$
1. $\sqrt{13}$
2. $\sqrt{37}$
3. 37
4. 13
5. _____ Simplify the following and express the result as a complex number. $(4-5i)(3+2i)$
1. $22-7i$
2. $2-7i$
3. $-22+7i$
4. $22-7 \sqrt{i}$
6. _____ For what values of ‘$m$’ does the following equation have imaginary roots? $4x^2+mx+9=0$
1. $m=12$; $m=-12$
2. $m>12$; $m<-12$
3. $m<12$; $m<-12$
4. $m<12$; $m>-12$
7. _____ The length of a driveway is 5yd longer than the width. If the area of the driveway is $300 \ yd^2$, what are its length and width?
1. $l=10 \ yd$; $w=15 \ yd$
2. $l=20 \ yd$; $w=15 \ yd$
3. $l=10 \ yd$; $w=3 \ yd$
4. $l=25 \ yd$; $w=20 \ yd$
8. _____ What are the exact roots of the following quadratic equation? $2x^2=2x+3$
1. $x=\frac{1 \pm \sqrt{7}}{2}$
2. $x=\frac{2 \pm \sqrt{28}}{4}$
3. $x=\frac{1 \pm \sqrt{7}}{4}$
4. $x=\frac{2 \pm \sqrt{-20}}{4}$
9. _____ What is the solution for the following radical equation? $\sqrt{x+2}=x$
1. the roots are 2 and 1
2. the roots are –2 and 1
3. the root is 2
4. the root is 1
10. _____ The following quadratic equation was solved using the quadratic formula. $x^2-4x=3$ Which step contains the first mathematical error?
11. Step 1:
12. $x=\frac{- \left(-4\right) \pm \sqrt{\left(-4\right)^2-4 \left(1\right) \left(-3\right)}}{2 \left(1\right)}$
13. Step 2:
14. $x=\frac{4 \pm \sqrt{16-12}}{4}$
15. Step 3:
16. $x=\frac{4 \pm \sqrt{4}}{2}$
17. Step 4:
18. $x=3;x=1$
1. Step 1
2. Step 2
3. Step 3
4. Step 4
Date Created:
Aug 20, 2013
Jun 04, 2014
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https://cbp.tnw.utwente.nl/PolymeerDictaat/node33.html | Next: Appendix B Up: Stochastic processes Previous: The Smoluchowski equation
# Appendix A
We shall derive the Fokker-Planck equation by looking at for any function . Because we are always interested in averages like this, equations that may be derived using this object are all we need. (In mathematical terms is a distribution or generalized function, not an ordinary function).
Our proof very much resembles the one in Appendix 3.C. Our starting point is again the Chapman-Kolmogorov equation
(4.40)
Multiplying by and integrating yields
(4.41)
Now we shall perform the integral with respect to z on the right hand side. Because differs from zero only when is in the neighbourhood of , we expand around
(4.42)
Introducing this into Eq. (4.41) we get
= (4.43)
Now we make use of
= 1 (4.44) = (4.45) = (4.46) = 0 (4.47) = 0 (4.48) = (4.49)
which hold true to the first order in . The first three of these equations are obvious. The last three easily follow from the Langevin equation in section 4.1 together with the fluctuation-dissipation theorem Eq. (4.21). 1 in Eq. (4.49) denotes the 3-dimensional unit matrix. Introducing everything into Eq. (4.43), dividing by and taking the limit , we get
= (4.50)
Next we change the integration variable into and perform some partial integrations, obtaining
= (4.51)
Because this has to hold true for all possible we conclude that the Fokker-Planck equation (4.14) must hold true.
Next: Appendix B Up: Stochastic processes Previous: The Smoluchowski equation
W.J. Briels | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9697288274765015, "perplexity": 897.3187165245391}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703497681.4/warc/CC-MAIN-20210115224908-20210116014908-00030.warc.gz"} |
https://cs.stackexchange.com/questions/133759/how-to-derive-the-worst-case-time-complexity-of-heapify-algorithm | # How to derive the worst case time complexity of Heapify algorithm?
I would like to know how to derive the time complexity for the Heapify Algorithm for Heap Data Structure.
I am asking this question in the light of the book "Fundamentals of Computer Algorithms" by Ellis Horowitz et al. I am adding some screenshots of the algorithm as well as the derivation given in the book.
procedure $$HEAPIFY(A,n)$$
//Readjust the elements in A(1:n) to form a heap//
integer $$n,i$$
for $$i\leftarrow\lfloor n/2 \rfloor$$ to $$1$$ by $$-1$$ do
call $$ADJUST(A, i, n)$$
repeat
end $$HEAPIFY$$
Derivation for worst case complexity:
I understood the first part and last part of this calculation but I cannot figure out how $$2^{i-1}(k-i)$$ changed into $$i 2^{k-i-1}$$.
All the derivations I can find in the internet takes a different approach by considering the height of the tree differently. I know that approach also leads to the same answer but I would like to know about this approach.
You might need the following information:
$$2^k-1 = n$$ or approximately $$2^k = n$$, where $$k$$ is the number of levels, starting from the root node and the level of root is 1 (not 0) and $$n$$ is the number of nodes.
Also the worst case time complexity of the Adjust() function is proportional to the height of the sub-tree it is called, that is $$O(log n)$$, where $$n$$ is the total number of elements in the sub-tree.
• The alii being Sartaj Sahni and Sanguthevar Rajasekaran. – greybeard Dec 28 '20 at 15:45
## 1 Answer
It's a variable substitution.
First, realize that in the leftmost side of the equation, the last term of the sum is zero (because when $$i = k$$, $$k-i = 0$$). So, the range of the first summation can be written as $$1 \le i \le k-1$$. Now, substitute $$i$$ with $$k-i$$. $$i$$ iterates over the set $${1, 2, ... , k-1}$$ and $$k-i$$ iterates over the set $${k-1, ... 2, 1}$$, they are the same set, so, we can do this.
• Thank you for the answer. But, k is a constant right? But is it mathematically possible? – Midhunraj R Pillai Dec 28 '20 at 14:00
• my question is, is that substitution method purely mathematical or are we using some other facts like may be 'i' and 'k' are related? – Midhunraj R Pillai Dec 28 '20 at 14:10
• @MidhunrajRPillai I don't know about any mathematical rules for substituting variables in a summation expression (like the substitution rules for integrals). My reasoning was just that since both expressions run over the same set of values, their results would be equal. (because addition in real numbers are commutative). It is more clear if you remove the summation symbol (the sigma letter) and write down both expressions as a sum. (The left side is 2^1*(k-1) + 2^2*(k-2) ... 2^(k-1)*1, and the right side is the same thing but reversed. ) – yemre Dec 28 '20 at 17:25
• Yeah got it. On expanding, getting the answer. Thank you. – Midhunraj R Pillai Dec 28 '20 at 17:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9157637357711792, "perplexity": 330.2587182653921}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178378043.81/warc/CC-MAIN-20210307170119-20210307200119-00037.warc.gz"} |
http://math.stackexchange.com/questions/387062/what-are-examples-of-unexpected-algebraic-numbers-of-high-degree-occured-in-some | # What are examples of unexpected algebraic numbers of high degree occured in some math problems?
Recently I asked a question about a possible transcendence of the number $\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)/\left(\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)\right)$, which, to my big surprise, turned out to be an algebraic number, but not some decent algebraic number like $\left(\sqrt{5}-1\right)/2$, but an enormous one with the minimal polynomial of degree 120 and a coefficient exceeding $10^{15}$.
So, my question: are there other interesting examples of numbers occurred in some math problems that were expected likely to be transcendental, but later unexpectedly were proven to be algebraic with a huge minimal polynomial.
-
I don't know if Conway's constant is quite what you are looking for, as I'm not sure one would expect it initially to be transcendental or not. So, perhaps it's my bad intuition, but I was certainly surprised to learn that it is an algebraic number with minimal polynomial of degree 71.
-
Sometimes you can get unexpected algebraic values while working with hypergeometric functions. For example, the following absolute value of a complex-valued $_4F_3$ function: $$\left|\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\sqrt{\phi }\right)\right|,$$ where $\phi$ is the golden ratio, is actually an algebraic number with the minimal polynomial of degree 80 and a coefficient exceeding $10^{55}$: $$340282366920938463463374607431768211456 x^{80}+152118072027387528179604384645120000000000 x^{72}+202824096036516704239472512860160000000000 x^{70}-45334718235548594051481600000000000000000000 x^{62}+15111572745182864683827200000000000000000000 x^{60}-5629499534213120000000000000000000000000000000 x^{56}-24769797950537728000000000000000000000000000000 x^{54}-33776997205278720000000000000000000000000000000 x^{52}-9007199254740992000000000000000000000000000000 x^{50}+1006632960000000000000000000000000000000000000000 x^{46}+3523215360000000000000000000000000000000000000000 x^{44}+74161139200000000000000000000000000000000000000000 x^{40}+300000000000000000000000000000000000000000000000000 x^{38}+675000000000000000000000000000000000000000000000000 x^{36}+1050000000000000000000000000000000000000000000000000 x^{34}-2975290298461914062500000000000000000000000000000000 x^{32}-14701161193847656250000000000000000000000000000000000 x^{30}-37252902984619140625000000000000000000000000000000000 x^{28}-74505805969238281250000000000000000000000000000000000 x^{26}-59604644775390625000000000000000000000000000000000000 x^{24}-22351741790771484375000000000000000000000000000000000 x^{22}+7450580596923828125000000000000000000000000000000000 x^{20}-555111512312578270211815834045410156250000000000000000 x^{16}-1110223024625156540423631668090820312500000000000000000 x^{14}-1665334536937734810635447502136230468750000000000000000 x^{12}-2220446049250313080847263336181640625000000000000000000 x^{10}+82718061255302767487140869206996285356581211090087890625$$
I'm not sure if it is expressible in radicals.
-
Just an everyday polynomial, then. Looks simple enough.. – Thomas May 10 '13 at 3:55
Yes, it would be much more interesting if a number with a pretty simple definition turned out to be unexpectedly algebraic, but with a minimal polynomial so large that we could not explicitly write it (e.g. with degree and coefficients exceeding TREE(3)) – Vladimir Reshetnikov May 10 '13 at 20:55
Some numbers were found recently algebraic using the LLL and PSLQ algorithms : these algorithms return directly the integer coefficients of a polynomial starting with the numerical value provided with enough precision. They are implemented in many Computer algebra software (for example lindep and algdep of pari/gp).
Broadhurst found that the third and fourth bifurcation points (B3 and B4) of the logistic map were algebraic of degree $12$ and $240$ (page 3 from this paper and 5 from this one and this ps file for B4).
See too this MO thread 'What Are Some Naturally-Occurring High-Degree Polynomials?'.
-
Unless I'm missing something, the bifurcation points of a rational function being algebraic (and of high degree) doesn't seem unexpected. My (admittedly naive) take on Broadhurst's paper is that the challenge was in identifying the specific polynomial. – Erick Wong May 10 '13 at 0:36
@ErickWong: Yes you are right (even if I found these polynomials pleasant) and it would be more surprising if, for example, one of the Feigenbaum constants was proved algebraic... Anyway perhaps that the integer relation algorithms and the MO thread will be of interest. – Raymond Manzoni May 10 '13 at 8:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8365684151649475, "perplexity": 433.6563682518179}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510276250.57/warc/CC-MAIN-20140728011756-00278-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://liujing.neocities.org/newworld/Social-Economic-Networks/5.%20Diffusion%20on%20Networks/7%20.%20%20Solving%20the%20SIS%20Model%20-%20Ordering.html | # Solving for the Positive Steady State when it Exists¶
$\theta = \sum P(d) \frac{\lambda \theta d^2}{[\lambda \theta d + 1)E[d]]}$
$1 = \sum P(d) \frac{\lambda d^2}{[\lambda \theta d + 1)E[d]]}$
(1) Regular: so just plugging in that everybody has the same degree which is just the expected degree, then we can:
• get rid of $\sum P(d)$
• $d^2 \rightarrow E(d)^2$
• $\lambda \theta d \rightarrow \lambda \theta E(d)$
Regular: $1 = \frac{\lambda E[d]}{(\lambda \theta E[d] + 1)}$
then we can rearrange this in terms of $\theta$
$\theta = 1 - \frac{1}{\lambda E[d]}$
$\theta$ is increasing in $\lambda E[d]$ Need $\lambda E[d] > 1$
So if everybody had the same degree then we can solve explicitly for what the steady state expression is going to be for the infection rate.
$\theta = \sum P(d) \frac{\lambda \theta d^2}{[\lambda \theta d + 1)E[d]]}$
$1 = \sum P(d) \frac{\lambda d^2}{[\lambda \theta d + 1)E[d]]}$
(2) Power law degree distribution: $P(d) = 2d^{-3}$
If plug in the power law P(d) and integrate this out, then end up with an expression that can solve for $\theta$:
$\theta = \frac{1}{\lambda(e^{\frac{1}{\lambda}} - 1)}$
As $\lambda$ increases we get a very rapid increase and then eventually it asymptotes, it can't go above "1", but we're getting a very high neighbor infection rate as $\lambda$ increases, because then we've got these very high degree nodes, they become infected, they infect others and so forth, and as $\lambda$ is increasing, we get a very rapid infection increase.
# Ordering Networks - Jackson Rogers (07b)¶
$\theta = \sum P(d) \frac{\lambda \theta d^2}{[\lambda \theta d + 1)E[d]]}$
How does the right side shift with the degree distribution, P(d), if we want to do comparisons, if we go from regular to power law or a regular to Erdos Renyi, sort of graph?
Remember the way that we're solving this, we look at theta here, we have this right hand side, which is H(θ) and we're looking for the solution to this thing and if we can say that H(θ) goes up, right, so if we do something that changes H(θ) in a way that goes up then that's going to move the solution to this equation upwards. So, any kind of comparative static where we're making changes that change the distribution in a way that increases this overall expression on the right hand side for each theta gets a higher value then we can say something about what the resulting change is in theta. So, let's see what we can say about how this right hand side moves.
$\frac{\lambda \theta d^2}{[\lambda \theta d + 1)E[d]]}$ is increasing in d
So higher degree nodes are going to have higher relative expected infection rates and basically that's what we're getting
If $P'$ first order stochastic dominates $P$, then rhs increases at every $\theta$
If $P'$ is a mean-preserving spread of P, then rhs increases at every $\theta$
# Ideas¶
• Mean preserving spread - more high degree nodes and low degree nodes
• Higher degree nodes are more prone to infection
• Neighbors are more likely to be high degree
• So, either first order stochastic dominance, or mean-preserving spreads in $P$ increase $\theta$
• infection rate of neighbors is not the same as infection rate of the population
• $\theta$ is the chance when you're meeting somebody in the population that they're infected, and is increased as we increased in the senses of first or second order stochastic dominance.
• $\rho$ in the population is the acutally average, if we take expectations over all degrees, so the higher people are going to be infected at higher rates, so when you're meeting them at higher values that means that people you're going to meet are more infected, but if we're somebody who just cares about the average level in the population, so if I'm a government and I care about how infected my population is, ultimatly what I care about is what $\rho$ is, not what $\theta$ is, so $\theta$ is very important in determining what the steady state is going to be, but the thing I might be interested in terms of my policies is what fraction of my population ends up being infected
• Theorem JR(2007): If $P'$ is a mean preserving spread of P, then the highest steady state $\theta' > \theta$, but the corresponding $\rho' > \rho$ if $\lambda$ is low, while $\rho' < \rho$ if $\lambda$ is high.
• what's the intuition behind this? The intuition is that in situations where $\lambda$ is very high, the high interaction nodes are already going to be very infected. And so, actually increasing, in putting weight on higher degrees isn't going to matter that much because those nodes are going to be infected at such a high rate that they're going to already be infected. And so you're not changing that much but putting more weight on low degree nodes can actually decrease the, so now you've got some people who have very few interactions. Those people can actually end up being infected with lower rates. So, the actual overall infection rate in the society can be balanced by the fact that you are increasing some of the higher degree nodes but those people are already going to be infected even without this increase and the low degree nodes as you move them towards lower part they can actually end up with lower infection rates and so that counterbalances it. So, when you average across the population, not with the relative frequency of meetings, you actually end up with a decrease in the overall rate.
# Proof¶
$0 = \frac {d\rho(d)}{dt} = (1- \rho(d))v \theta d - \rho(d)\delta$
Expecting over d:
$0 = v\theta E[d] - \sum P(d)\rho(d) v \theta d - \rho \delta$
$= v\theta E[d] - v \theta^2 E[d] - \rho \delta$
$\rho = \lambda \theta E[d](1-\theta)$
rhs is increasing in $\theta$ iff $\theta < 1/2$, and is decreasing in $\theta$ iff $\theta > 1/2$
$\theta$ is increasing in $\lambda$
# SIS Diffusion Model¶
• Simple and tractable model
• Bring in relative meeting rates
• Can order infections by properties of "network"
The limitationis of this model:
• we lose the fact that a lot of applications are ones where you become infected but then if you recover you're actually immune to catching the disease again, which is actually true of some flus, so it's somewhat limited in term of the applications.
• Also the interactions that we talked about were completely random meeting processes. So, it was not as if we'd drawn out a network and actually had people located on the network. We just had people bumping into each other and meeting each other and so that's a special kind of process which gives rise to special kinds of conditions.
# Next¶
Now we, more generally what we'd have to do if we start to work with things where the network architecture is given, then it's going to be more important to use simulations and so forth. So, we did some calculations before where we talked about component science and so forth and that gives us some insights. But, more generally if we actually want to study these processes a lot of it's going to be done by simulation. So, if you give me a particular network and ask what's going to happen on it then I might have to write down a program and actually simulate what's going to happen there. So, the next thing we'll look at is a simple model of diffusion where we'll do some calculations and just simulate that model and see exactly how it works and that will allow us to actually fit something directly to data. And there's a large amount of that that goes on in epidemiology and marketing and other kinds of areas where you're trying to make predictions. If you know something about the network you're working with you can actually simulate things and that's going to go a long way towards improving your accuracy. So, the SIS model gives us nicer intuitions, simple ideas, but it's not one you're going to easily take to data. We're going to have to enrich the model to fit it to networks. And that's what we'll talk about next. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8798819780349731, "perplexity": 398.9242621202261}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337504.21/warc/CC-MAIN-20221004121345-20221004151345-00423.warc.gz"} |
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complex number to polar form calculator with steps
iR 2(: a+bi)p. Alternately, simply type in the angle in polar form by pressing 2qbZ330p. As you observed that polar and rectangular form conversions are very easy. A complex number can be represented in the form a + bi, where a and b are real numbers and i denotes the imaginary unit. Expressing a Complex Number in Trigonometric or Polar Form, Ex 2. Finding Products of Complex Numbers in Polar Form. This tool is used to perform the subtraction operation between two set of complex numbers. The complex number calculator is also called an imaginary number calculator. Convert the given complex number, into polar form. -/. Complex numbers, as any other numbers, can be added, subtracted, multiplied or divided, and then those expressions can be simplified. Description : Writing z = a + ib where a and b are real is called algebraic form of a complex number z : a is the real part of z; b is the imaginary part of z. Thus, The number in polar form is . COMPLEX FORM AND POLAR FORM. Consider the complex number in polar form, 5. Arithmetic operations with complex numbers. Figure 2. Comment(0) Chapter , Problem is solved. Just type your formula into the top box. Complex Numbers Calculator evaluates expressions with complex numbers and presents the result in rectangular and polar forms. Example: type in (2-3i)*(1+i), and see the answer of 5-i. It's interesting to trace the evolution of the mathematician opinions on complex number problems. Real numbers can be considered a subset of the complex numbers that have the form a + 0i. Obtain n distinct values. For example, you can convert complex number from algebraic to trigonometric representation form or from exponential back to algebraic, ect. Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. Polar Form of Complex Numbers; Convert polar to rectangular using hand-held calculator; Polar to Rectangular Online Calculator; 5. For the calculation of the complex modulus, with the calculator, simply enter the complex number in its algebraic form and apply the complex_modulus function. Step by step tutorial with examples, several practice problems plus a worksheet with an answer key Instructions. A polar form uses the magnitude of the number as the length of line and the angle at which a number extends. To convert a complex number from polar form to rectangular form on the Casio fx−250, you use the P→R and X↔Y keys. View a sample solution. Operations on Complex Numbers in Polar Form - Calculator. We can think of complex numbers as vectors, as in our earlier example. to 79 steps. Likewise, the conversions are. This calculator extracts the square root, calculate the modulus, finds inverse, finds conjugate and transform complex number to polar form.The calculator will generate a step by step … So, our points will lie on the unit circle and they will be equally spaced on the unit circle at every $$\frac{{2\pi }}{n}$$ radians. The standard form of a complex number is $a + bi$ where $$a$$ and $$b$$ are real numbers and they can be anything, positive, negative, zero, integers, fractions, decimals, it doesn’t matter. By … Some sample complex numbers are 3+2i, 4-i, or 18+5i. Finding the Absolute Value of a Complex Number. Press A r to toggle the display between the absolute value (r) and argument ( ). This online calculator will help you to convert rectangular form of complex number to polar and exponential form. Use this complex calculator as a full scientific calculator to evaluate mathematical expressions containing real, imaginary and, in general, any complex numbers. How to Understand Complex Numbers. Complex Numbers Subtraction Calculation . Thus, the polar form of 5 is. 4. A complex number, then, is made of a real number and some multiple of i. The argument of a complex number is the direction of the number from the origin or the angle to the real axis. This rectangular to exponential form conversion calculator converts a number in rectangular form to its equivalent value in exponential form. How to convert between Polar and Rectangular form . by M. Bourne. Complex numbers may be represented in standard from as In what follows, the imaginary unit $$i$$ is defined as: $$i^2 = -1$$ or $$i = \sqrt{-1}$$. Show Step-by-step Solutions. show help ↓↓ examples ↓↓). When in the standard form $$a$$ is called the real part of the complex number and $$b$$ is called the imaginary part of the complex number. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). An online real & imaginary numbers subtraction calculation. To convert a complex number into polar form, press 2+5bU. Get the free "Convert Complex Numbers to Polar Form" widget for your website, blog, Wordpress, Blogger, or iGoogle. Complex Numbers in Polar Form. This video lesson shows how to write a complex number in polar form. Recall from our discussion on the polar form (and hence the exponential form) that these points will lie on the circle of radius $$r$$. For example the number 7 ∠ 40°. Find more Mathematics widgets in Wolfram|Alpha. All Math Calculators :: Complex numbers:: Arithmetic operations with complex numbers; Complex numbers arithmetic. Complex Number Z(A+Bi) + i: Complex Number W(C+Di) + i . Exponential Form of Complex Numbers; Euler Formula and Euler Identity interactive graph; 6. Sometimes this function is designated as atan2(a,b). Note that CALC memory can be used in the COMP Mode and CMPLX Mode only. How do we understand the Polar representation of a Complex Number? Feature-rich, powerful, easy-to-use complex number calculator to calculate complex numbers in any form including rectangular and polar forms of complex numbers. There is another way by which we can represent the complex numbers. How to Add Complex numbers. After applying Moivre’s Theorem in step (4) we obtain which has n distinct values. When we first learned to count, we started with the natural numbers – 1, 2, 3, and so on. ... rectangular form complex number to its polar form, and a polar form complex number to its rectangular form. This solver can performs basic operations with complex numbers i.e., addition, subtraction, multiplication and division. Step 3. When b=0, z is real, when a=0, we say that z is pure imaginary. Answer to Use a calculator to convert each complex number to polar form. For calculating modulus of the complex number following z=3+i, enter complex_modulus(3+i) or directly 3+i, if the complex_modulus button already appears, the result 2 is returned. 4. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to convert rectangular form of complex number to polar and exponential form. To use the calculator, one need to choose representation form of complex number and input data to the calculator. That's why, to add two complex numbers in polar form, we can convert polar to canonical, add and then convert the result back to polar form. Subtracting Complex Numbers Calculation. Compare this with the standard notation of complex number in Cartesian form, Thus, Consider the following conversions from Cartesian to polar form. For instance, to convert the number 2∠60° to rectangular form, first make sure that your calculator is in degrees mode, and then type 2 P→R 60 = . [See more on Vectors in 2-Dimensions].. We have met a similar concept to "polar form" before, in Polar Coordinates, part of the analytical geometry section. The absolute value of a complex number is the same as its magnitude, or It measures the distance from the origin to a point in the plane. Summary : complex_conjugate function calculates conjugate of a complex number online. Complex Number Calculator. For example, the graph of in , shows. Absolute Value of a Complex Number. Raise index 1/n to the power of z to calculate the nth root of complex number. The complex symbol notes i. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Just another example of writing a complex number in polar form. Complex numbers in rectangular form are presented as a + b * %i, where a and b are real numbers.Polar form of the complex numbers is presented as r * exp(c * %i), where r is radius and c is the angle in radians. Step 2. Press C2qbZ330. Soon after, we added 0 to represent the idea of nothingness. Step 4. The complex number online calculator, allows to perform many operations on complex numbers. Starting from the 16th-century, mathematicians faced the special numbers' necessity, also known nowadays as complex numbers. An online calculator to add, subtract, multiply and divide complex numbers in polar form is presented. Try the free Mathway calculator and problem solver below to practice various math topics. Subtracting Complex Numbers. complex_conjugate online. For the rest of this section, we will work with formulas developed by French mathematician Abraham de Moivre (1667-1754). Apply De Moivre’s Theorem. Below we give some minimal theoretical background to be able to understand step by step solution given by our calculator. View this answer. The complex number calculator is able to calculate complex numbers when they are in their algebraic form. iR1(: r ∠q)p. To convert any polar form of a complex number, use the r theta command or type in the angle in polar form. In the complex number a + bi, a is called the real part and b is called the imaginary part. Addition of complex numbers is much more convenient in canonical form #z=a+i*b#. This online calculator finds -th root of the complex number with step by step solution.To find -th root, first of all, one need to choose representation form (algebraic, trigonometric or exponential) of the initial complex number. Instructions:: All Functions . Add 2kπ to the argument of the complex number converted into polar form. Step 5. Polar Form of a Complex Number. Step-by-step explanations are provided. Basic operations explained. Argument of a Complex Number Calculator. You need to apply special rules to simplify these expressions with complex numbers. The first step toward working with a complex number in polar form is to find the absolute value. A complex number is a number of the form a+bi, where a,b — real numbers, and i — imaginary unit is a solution of the equation: i 2 =-1.. and. This calculator allows one to convert complex number from one representation form to another with step by step solution. Show Step-by-step Solutions. Products and Quotients of Complex Numbers; Graphical explanation of multiplying and dividing complex numbers; 7. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Unlike the polar form, which is expressed in unit degrees, a complex exponential number is expressed in unit radians. Imaginary number calculator operations with complex numbers ; Graphical explanation of multiplying and dividing complex numbers arithmetic representation form from... The natural numbers – 1, 2, 3, and see the answer of 5-i and. Obtain which has n distinct values in Cartesian form, Thus, consider the complex number online calculator add... Evaluates expressions in the angle in polar form we will learn how to perform the subtraction operation between two of... Of 5-i free Mathway calculator and problem solver below to practice various Math topics some minimal theoretical background be... In ( 2-3i ) * ( 1+i ), and a polar form presented! A+Bi ) p. Alternately, simply type in ( 2-3i ) * 1+i... Z=A+I * b # COMP Mode and CMPLX Mode only the first step toward working with complex., consider the following conversions from Cartesian to polar form uses the magnitude of number. Some minimal theoretical background to be able to calculate complex numbers very easy number converted into form... Operations with complex numbers ; convert polar to rectangular online calculator, one need to choose representation form complex. One to convert a complex number in polar form in canonical form # z=a+i * b # minimal background... More convenient in canonical form # z=a+i * b # real axis Mode only z is real when. The set of complex number to polar and exponential form of complex numbers as vectors, in! The argument of a real number and some multiple of i i.e. addition! ( a+bi ) + i 1667-1754 ) numbers to polar form by pressing 2qbZ330p subtract, multiply and complex. For the rest of this section, we added 0 to represent the idea of nothingness or exponential! Sample complex numbers ; 7. to 79 steps obtain which has n distinct.... Products and Quotients of complex numbers ; Graphical explanation of multiplying and dividing complex and... ' necessity, also known nowadays as complex numbers calculator to convert a complex number W C+Di... Set of complex number to polar and rectangular form to rectangular online calculator add... Operations with complex numbers and evaluates expressions with complex numbers ; complex numbers to polar form the evolution the!, the graph of in, shows the P→R and X↔Y keys, consider the complex number in polar by! Perform many operations on complex numbers ; Graphical explanation of multiplying and dividing complex number to polar form calculator with steps... Converts a number in polar form uses the magnitude of the mathematician opinions complex number to polar form calculator with steps complex numbers are 3+2i 4-i... The best experience to understand step by step solution ( a+bi ) Alternately... B ) standard notation of complex numbers ; complex numbers in polar form number to its rectangular form conversions very...: complex_conjugate function calculates conjugate of a complex number, into polar form is presented given by calculator. Atan2 ( a, b ) divide complex numbers ; complex numbers that have the a! This tool is used to perform many operations on complex numbers:: complex numbers as the length line!, we started with the standard notation of complex numbers two set of numbers... # z=a+i * b # you complex number to polar form calculator with steps the calculator are in their algebraic form number in form... More convenient in canonical form # z=a+i * b # i: complex number to and... Forms of complex numbers in any form including rectangular and polar forms by which we can convert complex numbers -! In any form including rectangular and polar forms is called the real part and b is called the axis. Which is expressed in unit degrees, a is called the imaginary part ( 1+i ), and the! To rectangular form conversions are very easy by our calculator mathematician Abraham de Moivre ( 1667-1754 ) P→R! Another with step by step solution a=0, we added 0 to represent the idea of nothingness think of number. Some sample complex numbers understand the polar form complex number converted into polar form we will work with developed. And see the answer of 5-i exponential form conversion calculator converts a number in form... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9487655758857727, "perplexity": 745.9826119196931}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039596883.98/warc/CC-MAIN-20210423161713-20210423191713-00621.warc.gz"} |
http://math.stackexchange.com/questions/37869/fisher-information-of-a-function-of-a-parameter?answertab=votes | # Fisher Information of a function of a parameter
Suppose that $X$ is a random variable for which the p.d.f. or the p.f. is $f(x|\theta)$, where the value of the parameter $\theta$ is unknown but must lie in an open interval $\Omega$. Let $I_0(\theta)$ denote the Fisher information in $X.$ Suppose now that the parameter $\theta$ is replaced by a new parameter $\mu$, where $\theta = \psi(\mu)$ and $\psi$ is a differentiable function. Let $I_1(\mu)$ denote the Fisher information in $X$ when the parameter is regarded as $\mu.$ Show that $$I_1(\mu) = [\psi'(\mu)]^2 I_0[\psi(\mu)].$$
How would I do this? Do I need to use a Taylor expansion? Regardless, I would appreciate a written proof. This isn't for class but the above statement has been mentioned in texts without any detail whatsoever.
Thanks!
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Hint: What's the definition of $I_0(\theta)$? Under standard regularity conditions, what is $\mathbb{E}\big(\frac{\partial \log f(x;\theta)}{\partial \theta}\big)$. Put this together with the chain rule of differentiation. – cardinal May 8 '11 at 20:15
By definition $I_{0}(\theta)=-\mathbb{E}\left[\frac{d^{2}\log f\left(X\vert\theta\right)}{d\theta^{2}}\right].$ So $I_{1}(\mu)=-\mathbb{E}\left[\frac{d^{2}\log f\left(X\vert\mu\right)}{d\mu^{2}}\right]$.By the chain rule we have $$I_{1}\left(\mu\right) = -\mathbb{E}\left[\frac{d^{2}\log f\left(X\vert\theta\right)}{d\theta^{2}}\left(\frac{d\theta}{d\mu}\right)^{2}+\frac{d\log f\left(X\vert\theta\right)}{d\theta}\frac{d^{2}\theta}{d\mu^{2}}\right]$$ $$= -\mathbb{E}\left[\frac{d^{2}\log f\left(X\vert\theta\right)}{d\theta^{2}}\right]\left(\frac{d\theta}{d\mu}\right)^{2}+\mathbb{E}\left[\frac{d\log f\left(X\vert\theta\right)}{d\theta}\right]\frac{d^{2}\theta}{d\mu^{2}}.$$
But $\mathbb{E}\left[\frac{d\log f\left(X\vert\theta\right)}{d\theta}\right]=0.$ So we get
$$I_{1}\left(\mu\right)=I_{0}\left(\theta\right)\left(\frac{d\theta}{d\mu}\right)^{2}.$$
The functions involved are $\frac{d}{d\theta}\log f\left(X\vert\theta\right)$ and $\frac{d\theta}{d\mu}$ It is a product and you differentiating wrt $\mu$ – Nana May 9 '11 at 3:10
NB: $\frac{d^{2}\log f\left(X\vert\mu\right)}{d\mu^{2}}=\frac{d}{d\mu}\left[\frac{d}{d\mu}\log f\left(X\vert\mu\right)\right]=\frac{d}{d\mu}\left[\frac{d}{d\theta}\log f\left(X\vert\theta\right)\cdot\frac{d\theta}{d\mu}\right]$ – Nana May 9 '11 at 3:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9833754897117615, "perplexity": 130.66441791450862}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398459875.44/warc/CC-MAIN-20151124205419-00140-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://dorigo.wordpress.com/2009/03/13/cdf-discovers-a-new-hadron/ | ## CDF discovers a new hadron! March 13, 2009
Posted by dorigo in news, physics, science.
Tags: , , , ,
This morning CDF released the results of a search for narrow resonances produced in B meson decays, and in turn decaying into a pair of vector mesons: namely, $Y \to J/\psi \phi$. This Y state is a new particle whose exact composition is as of yet unknown, except that CDF has measured its mass (4144 MeV) and established that its decay appears to be mediated by strong interactions, given that the natural width of the state is in the range of a few MeV. I describe succintly the analysis below, but first let me make a few points on the relevance of area of investigation.
Heavy meson spectroscopy appears to be a really entertaining research field these days. While all eyes are pointed at the searches for the Higgs boson and supersymmetric particles, if not at even more exotic high-mass objects, and while careers are made and unmade on those uneventful searches, it is elsewhere that action develops. Just think about it: the $\Xi_b$ baryon, the $\Omega_b$, those mysterious X and Y states which are still unknown in their quark composition. Such discoveries tell the tale of a very prolific research field: one where there is really a lot to understand.
Low-energy QCD is still poorly known and not easily calculable. In frontier High-Energy Physics we bypassed the problem for the sake of studying high-energy phenomena by tuning our simulations such that their output well resembles the result of low-energy QCD processes in all cases where we need them -such as the details of parton fragmentation, or jet production, or transverse momentum effects in the production of massive bodies. However, we have not learnt much with our parametrizations: those describe well what we already know, but they do not even come close to guessing whatever we do not know. Our understanding of low-energy QCD is starting to be a limiting factor in cosmological studies, such as in baryogenesis predictions. So by all means, let us pursue low-energy QCD in all the dirty corners of our produced datasets at hadron colliders!
CDF is actively pursuing this task. The outstanding spectroscopic capabilities of the detector, combined with the huge size of the dataset collected since 2002, allow searches for decays in the one-in-a-million range of branching ratios. The new discovery I am discussing today has indeed been made possible by pushing to the limit our search range.
The full decay chain which has been observed is the following: $B^+ \to Y K^+ \to J/\psi \phi K^+ \to \mu^+ \mu^- K^+ K^- K^+$. That $J/\psi$ mesons decay to muon pairs is not a surprise, as is the decay to two charged kaons of the $\phi$ vector meson. Also the original decay of the B hadron into the $J/\psi \phi K$ final state is not new: it had been in fact observed previously. What had not been realized yet, because of the insufficient statistics and mass resolution, is that the $J/\psi$ and $\phi$ mesons produced in that reaction often “resonate” at a very definite mass value, indicating that in those instances the $B \to J/\psi \phi K$ decay actually takes place in two steps as the chain of two two-body decays: $B \to Y K$ and $Y \to J/\psi \phi$.
The new analysis by CDF is a pleasure to examine, because the already excellent momentum resolution of the charged particle tracking system gets boosted when constraints are placed on the combined mass of multi-body systems. Take the B meson, reconstructed with two muons and three charged tracks, each assumed to be a kaon: if you did not know that the muon pair comes from a $J/\psi$ nor that two of the kaons come from a $\phi$, the mass resolution of the system would be in the few tens of MeV range. Instead, by forcing the momenta of the two muons to be consistent with the World average mass of the $J/\psi$, $M_{J/\psi}=3096.916 \pm 0.011 MeV$ , and by imposing that the two kaons make exactly the extremely well-known $\phi$ mass ($M_\phi=1019.455 \pm 0.020 MeV$), much of the uncertainty on the daughter particle momenta disappears, and the B meson becomes an extremely narrow signal: its mass resolution is just 5.9 MeV, a per-mille measurement event-by-event!
The selection of signal events requires several cleanup cuts, including mass window cuts around the J/Psi and phi masses, a decay length of the reconstructed B+ meson longer than 500 microns, and a cut on the log-likelihood ratio fed with dE/dx and time-of-flight information capable of discriminating kaon tracks from other hadrons. After those cuts, the B+ signal really stands above the flat background. There is a total of 78+-10 events in the sample after these cuts, and this is the largest sample of such decays ever isolated. It is shown above (left), together with the corresponding distribution in the $\phi \to KK$ candidate mass (right).
A Dalitz plot of the reconstructed decay candidates is shown in the figure on the right. A Dalitz plot is a scatterplot of the squared invariant mass of a subset of the particles emitted in the decay, versus the squared invariant mass of another subset. If the decay proceeds via the creation of an intermediate state, one may observe a horizontal or vertical cluster of events. Judge by yourself: do the points appear to spread evenly in the allowed phase space of the B+ decays ?
The answer is no: a significant structure is seen corresponding to a definite mass of the $J/\psi \phi$ system. A histogram of the difference between the reconstructed mass of the $J/\psi \phi$ system and the $J/\psi$ mass is shown in the plot below: a near-threshold structure appears at just above 1 GeV energy. An unbinned fit to a relativistic Breit-Wigner signal shape on top of the expected background shape shows a signal at a mass difference of $\Delta M=1046.3 \pm 2.9 MeV$, with a width of 11.7+-5.7 MeV.
The significance of the signal is, after taking account of trial factors, equal to 3.8 standard deviations. For the non-zero width hypothesis, the significance is of 3.4 standard deviations, implying that the newfound structure has strong decay. The mass of the new state is thus of 4143+-2.9 MeV.
The new state is above the threshold for decay to pair of charmed hadrons. The decay of the state appears to occur to a pair of vector mesons, $J/\psi \phi$, in close similarity to a previous state found at 3930 MeV, the Y(3930), which also decays to two vector mesons in $Y \to J/\psi \omega$. Therefore, the new state can be also called a Y(4140).
Although the significance of this new signal has not reached the coveted threshold of 5 standard deviations, there are few doubts about its nature. Being a die-hard sceptic, I did doubt about the reality of the signal shown above for a while when I first saw it, but I must admit that the analysis was really done with a lot of care. Besides, CDF now has tens of thousands of fully reconstructed B meson decays available, with which it is possible to study and understand even the most insignificant nuances to every effect, including reconstruction problems, fit method, track characteristics, kinematical biases, you name it. So I am bound to congratulate with the authors of this nice new analysis, which shows once more how the CDF experiment is producing star new results not just in the high-energy frontier, but as well as in low-energy spectroscopy. Well done, CDF!
1. Eric Weinstein - March 13, 2009
Wow. That seems very exciting and of interest to readers beyond the community of practicing professionals. As you write very clearly, this blog is a favorite of several people such as myself who do not speak this language of experimental accelerator physics as natives. What can be surmised from this about the nature of the hadron at this point beyond the inference that there is a non-trivial SU(3) representation involved? Is there any ability at this early stage to translate this for visitors to this field who are more comfortable with descriptions of group representations and internal quantum numbers?
Should this question be mal-formed, please feel free to answer the nearest good question available!
Thanks for the write-up,
Eric
2. mfrasca - March 13, 2009
Tommaso,
some formulae do not appear to parse correctly.
Cheers,
Marco
dorigo - March 13, 2009
Hi Marco, do you have a Mac ? I see them fine with my windoze Pc.
Thanks anyways… Will try to figure out what’s going on, another person with a Mac does not see the formulas.
T.
3. mfrasca - March 13, 2009
Tommaso,
I am Windows addicted since the pioneering times of PC-AT. It seems that the problem is still there.
Marco
4. Andrea Giammanco - March 13, 2009
I’m using firefox with windows, and I have the same problem.
5. Sumar Ongi - March 13, 2009
So, another meson! This one with strangeness -1, no charm or beauty, JP=0+,1-, 2+, … mass 4 MeV, an decays to J/psi phi…
Well, beats me… I have no idea what this is…
6. Sumar Ongi - March 13, 2009
(I mean, 4GeV)
7. dorigo - March 13, 2009
Dear Eric,
the question is well-put, but there is not a clear answer. What I think is that it is most likely some charm-anticharm excitation, above the DD threshold, and possibly some form of bound state of those particles.
Sumar, no, the newfound object has zero strangeness, since it decays strongly to a S=0 state.
I do not know what is wrong with the formulas, I will ask the experts of wordpress.
Cheers,
T.
8. Markk - March 13, 2009
Could these types of studies be called Strong Chemistry? That is, looking at collections of particles and their excitations with the Strong Force as the dominant thing? Just like what chemistry does with new molecules and their excited states? Looking at excited bound states of particles and their decays. its like, molecules, atomic nuclei, and hadrons themselves all have slightly similar happenings with different forces acting. (Well the last two I guess are both the strong force, but in a different way). You get collections of particles in bound states with excitations internal excitations that have coherent properties from the “outside”.
It looks like we are getting to the point of doing this at the hadron level. Was this the kind of thing that ILS (? name) accelerator would be trying to do? Really tune things to look for resonances or other fine details through a range of energies?
9. Dr BDO Adams - March 13, 2009
There so many mystery mesons (especially) in the JP=0+ states, that it might be time to invent some new quarks. Lets invent an exicted S* quark with opposite parity to normal S quarks. Then Y could be d (bar S*), which letter decays via bar S* -> bar c + c + s. S* would be around 4000 GeV, and there be a full spectrum of S* meson around there. Could that fit?
10. Carl Brannen - March 13, 2009
My Mozilla browser loses most of the LaTeX (I get them back by running the mouse over them), but the windows browser gets them all okay. Interesting.
Meanwhile, Phys Math Central has rejected my paper, for its lack of use of the dyanmics of QCD. I think this is unfair in that QCD dynamics is perturbative field theory and I’m looking at a QM approximation. A similar calculation is the Lamb shift in hydrogen-like atoms, where one uses QM and Schroedinger’s equation for the first order approximation to the bound states, and only then uses QED to make corrections. So I’ll send a note back to that effect.
11. dorigo - March 13, 2009
Hi Carl,
I am sorry to hear your paper has been rejected. I cannot judge on that decision by the journal, but I hope you can find some publisher who is less constrained by mainstream thinking.
Cheers,
T.
12. dorigo - March 13, 2009
Markk, yes, actually a colleague at the conference today was calling it the same way, hadronic chemistry. But no, the ILC (if that is the accelerator you have in mind) would study higher-energy particles, like SUSY ones, not hadrons.
BDO, no, I do not think it would fit, for a number of reasons. An additional quark would contribute to R, change electroweak precision parameters, create havoc in the theory. In any case, the phi is a s-sbar
state, so we need something decaying to ccbar ssbar, strongly. Could be a molecular state of two D_s mesons, or an excited charmonium state of some kind. I have not studied in detail the problem of classifying these states yet…
Cheers,
T.
13. Sumar Ongi - March 14, 2009
Geeeezzzz!! The first decay is obviously weak, and I was silly enough to compute the strangeness from it. Anyway, it’s quite likely the Y is a D\bar{D} molecule of some type.
Regarding the problem with some of the equations, all I can say is that all of the problematic ones contain an URL with “&” instead of “&”. That usually leads to problems with links. Copy and paste one of the links to a problem formula into your browser, change all “&” for “&” and the formula is displayed correctly.
14. Sumar Ongi - March 14, 2009
Well, today is not my day… I meant “&” instead of “& amp;” (without the blank…)
15. Kea - March 14, 2009
Sheeeesh. Requiring QCD when it might not be the right theory to describe particle masses to first order seems a bit self defeating, Carl. Maybe another journal will be more sensible.
16. Tony Smith - March 15, 2009
Carl and Kea, since the rejected paper acknowledges substantial contributions by Kea, perhaps it could be resubmitted as a joint paper with Kea as co-author.
If so, maybe Kea’s Oxford affiliation might get it accepted.
Tony Smith
17. carlbrannen - March 15, 2009
Tony and Kea, while I appreciate your support, this is not the place to discuss this. I believe that my being an amateur did not have any effect on acceptance; professionals also complain when their revolutionary papers get rejected. The big difference between the pros and amateurs is that the pros have to publish and so work a lot harder at it than I have. I should have a dumbed down version that is easier to accept in a few days.
18. Streaming video for Y(4140) discovery « A Quantum Diaries Survivor - March 17, 2009
[…] Streaming video for Y(4140) discovery March 17, 2009 Posted by dorigo in news, physics, science. Tags: B physics, CDF, discoveries, QCD, standard model, Tevatron trackback The CDF collaboration will present at a public venue (Fermilab’s Wilson Hall) its discovery of the new Y(4140) hadron, a mysterious particle created in B meson decays, and observed to decay strongly into a state, a pair of vector mesons. I have described that exciting discovery in a recent post. […]
19. Neil B. - March 18, 2009
I used to figure, it’s obvious that every hadron is a straightforward combination of any 2 (well, for mesons typ. quark+antiquark) or 3 quarks from the set u d s c b t. But now I gather, it’s more complicated because of superpositions. Indeed, the proton was said to have a bit of “strangeness” despite canonical composition uud (well, does it?) So maybe this new particle can’t be adequately written up in the simple way. (Pardon any lack of attention to answers in the post or comments, or literature – I’m a nuclear dilettante despite savvy on neglected subjects like the dynamics of extended bodies subject to forces and accelerations, in relativity theory.)
20. Neil B. - March 18, 2009
Followup: I already can find some superpositions, from Wikipedia on mesons. I copied the (?tXt?) direct for two examples but you can figure what it shows:
Neutral rho meson[22] ρ0(770) Self \mathrm{\tfrac{u\bar{u}-d\bar{d}}{\sqrt 2}}\, 0,775.49 ± 0.34
Omega meson[23] ω(782) Self \mathrm{\tfrac{u\bar{u} + d\bar{d} – 2s\bar{s}}{\sqrt{6}}}\, 0,782.65 ± 0.12
If mesons were limited to all q+qbar combos, there wouldn’t be as many – so that leads to, how many superposition variations are possible? “Where does it all end”?
21. La partícula Y(4140) descubierta en el Fermilab podría ser un error de cálculo « Francis (th)E mule Science’s News - March 19, 2009
[…] El 13 de marzo de 2009, la colaboración CDF del Fermilab anunció el descubrimiento de una nueva partícula algo “rara” (artículo técnico aparecido en el ArXiv). El 17 de marzo se ha realizado el anuncio oficial, fichero de transparencias Powerpoint y vídeo de la presentación, momento en el que muchos se han hecho eco de este “gran” descubrimiento, por ejemplo, Ciencia Kanija “Extraña bola de partículas sorprende a los físicos del Fermilab,” poco Meneada quizás porque un titular sobre una “extraña bola” no es tan espectacular como debería, una “nueva partícula fuera del Modelo Estándar” hubiera sido más meneado. […]
22. anonymous - March 19, 2009
The previous comment links to an article on a spanish blog. If I understand it correctly it says the following (very briefly):
“CDF found that new hadron. DZero does not see it, although it should be able to see it, i.e. DZero contradicts the CDF findings.”
In the article there are links to presentations of the CDF analysis, however I can’t find any link to a reference for the DZero point of view.
Does anyone know what is going on?
23. dorigo - March 19, 2009
Anon, I think the person who wrote that piece misunderstood totally my blog post about the CDF multi-muon excess not seen by DZERO. In other words, he thought my post was about the Y(4140), and he got confused and carried away. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 25, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8215467929840088, "perplexity": 1833.1827625596377}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645298065.68/warc/CC-MAIN-20150827031458-00262-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/electric-field-between-two-infinite-sheets-of-opposite-charge.254450/ | # Electric Field between two infinite sheets of Opposite charge.
1. Sep 7, 2008
### Hells_Kitchen
1. The problem statement, all variables and given/known data
I am having trouble with an E&M conceptual problem. Suppose we have two sheets of infinite charge, one positive and one negative (the positive is left of the negative and they have charge densities given respectively alpha and - alpha.). The problem asks to find the direction and magnitude of the electric field at 4 points in the same line, 2 points outside the sheets of charge, 1 near the positive sheet of charge (from the inside) and 1 in the center of the two infinite sheets of charge.
3. The attempt at a solution
Now, i know that the direction of the electric field on the outside will be towards the positive sheet of charge and away from the negative sheet of charge. What about the magnitude?
In the center the E-field should be in the -x direction since the y components cancel out but still i cant find the magnitude because there is no distance between them given.
Lastly near the positive sheet of charge(from the inside) i think the e-field should be in the -x direction with a lesser magnitude than in the center.
I am not sure how to get the magnitudes without the distance separation between them.
2. Sep 7, 2008
### alphysicist
Hi Hells_Kitchen,
What is the formula for the electric field of an infinite sheet of charge?
3. Sep 7, 2008
### Hells_Kitchen
E = charge density/ epsilon zero. Well that might be the magnitude for the outside points what about for the center and the point inside (in between the sheets)
4. Sep 7, 2008
### alphysicist
That's not be the total magnitude for outside points. That formula is the electric field from a single sheet of charge. To find the total field at a single point, you'll need to add the two separate electric field vectors.
So look at some point between the sheets first. What is the electric field at that point from the positive sheet? (magnitude and direction) What about from the negative sheet? Once you have the magnitude and direction of the two individual vectors, what is their vector sum?
Similar Discussions: Electric Field between two infinite sheets of Opposite charge. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.926354706287384, "perplexity": 367.4471304514134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806708.81/warc/CC-MAIN-20171122233044-20171123013044-00192.warc.gz"} |
http://www.zora.uzh.ch/53860/ | # Comparison of FBAR and QCM-D sensitivity dependence on adlayer thickness and viscosity
Nirschl, M; Schreiter, M; Vörös, J (2011). Comparison of FBAR and QCM-D sensitivity dependence on adlayer thickness and viscosity. Sensors and Actuators A: Physical, 165(2):415-421.
## Abstract
Unlike the quartz crystal microbalance, which has been used extensively for the analysis of biochemical interactions, only few measurements with biochemical adsorbent have been done with film bulk acoustic resonators (FBAR). In this paper, the FBAR behaviour on exposure to a lipid vesicle solution and the formation of a polyelectrolyte multilayer structure is investigated and compared with the results obtained with the quartz crystal microbalance. Differences in the resonator response were found between the two techniques and depending on the resonators resonance frequency ranging from the MHz to the GHz regime. As an explanation, we suggest that the penetration depth and the influence on viscoelastic properties, which are both known to be frequency dependent, cause the variations in the results. As a consequence, the higher operating resonance frequencies of the FBAR increase the sensitivity to changes in the viscoelasticity of the adsorbent and also decrease the sensing length of the device.
Unlike the quartz crystal microbalance, which has been used extensively for the analysis of biochemical interactions, only few measurements with biochemical adsorbent have been done with film bulk acoustic resonators (FBAR). In this paper, the FBAR behaviour on exposure to a lipid vesicle solution and the formation of a polyelectrolyte multilayer structure is investigated and compared with the results obtained with the quartz crystal microbalance. Differences in the resonator response were found between the two techniques and depending on the resonators resonance frequency ranging from the MHz to the GHz regime. As an explanation, we suggest that the penetration depth and the influence on viscoelastic properties, which are both known to be frequency dependent, cause the variations in the results. As a consequence, the higher operating resonance frequencies of the FBAR increase the sensitivity to changes in the viscoelasticity of the adsorbent and also decrease the sensing length of the device.
## Citations
11 citations in Web of Science®
13 citations in Scopus®
## Altmetrics
Detailed statistics
Item Type: Journal Article, refereed, original work 04 Faculty of Medicine > Institute of Biomedical Engineering 170 Ethics 610 Medicine & health English 2011 12 Jan 2012 17:22 05 Apr 2016 15:18 Elsevier 0924-4247 https://doi.org/10.1016/j.sna.2010.11.003
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https://soi.ch/wiki/dp-optimization/ | DP optimization
Written by Daniel Rutschmann. Translated by Johannes Kapfhammer.
Convex Hull Trick
The convex hull trick applies to any DP that looks like this:
\begin{aligned} \mathtt{DP}[0]&=0\\ \mathtt{DP}[i]&=a[i] + \max_{0 \le k < i}\{m[k]\cdot x[i]+q[k]\} \quad \text{for } 1 \leq i \leq n \end{aligned}
$m[k]$ and $q[k]$ are constants only depending on $k$ (not $i$) and have to be known as soon as $i=k$.
$a[i]$ and $x[i]$ are constants only depending on $i$ (not $k$) and have to be known right before $\mathtt{DP}[i]$ is computed.
Example: Commando
Before we show how the trick works, we solve the task Commando.
The DP looks like this:
$\mathtt{DP}[i] = \max_{0 \le k < i} \left\{a\cdot\Big(\sum_{j=k+1}^i x[j]\Big)^2 + b\cdot\Big(\sum_{j=k+1}^i x[j]\Big) + c + \mathtt{DP}[k]\right\}$
This looks quadratic which and not at all in the form above, but we can simplify it by computing the prefix sums $s[i]=x[i]+s[i-1]$ (with $s[0]=0$), multiplying out and collecting the terms:
\begin{align*} \mathtt{DP}[i] &= \max_{0 \le k < i} \{a\cdot (s[i] - s[k])^2 + b\cdot(s[i] - s[k]) + c + \mathtt{DP}[k]\}\\ &= \max_{0 \leq k < i} \{a\cdot s[i]^2 + a\cdot s[k]^2 - 2\cdot a\cdot s[i]\cdot s[k] + b\cdot s[i] - b\cdot s[k] + c + \mathtt{DP}[k]\}\\ &= \underbrace{a\cdot s[i]^2 + b\cdot s[i] + c}_{a[i]} + \max_{0 \le k < i} \{\underbrace{-2\cdot a\cdot s[k]}_{m[k]}\cdot \underbrace{s[i]}_{x[i]} + \underbrace{a\cdot s[k]^2 - b\cdot s[k] + \mathtt{DP}[k]}_{q[k]}\}\\ \end{align*}
Convex Hull Data Structure
To speed up the DP equation
$\mathtt{DP}[i]=a[i] + \max_{0 \le k< i}\{m[k]\cdot x[i]+q[k]\}$
we implement a hull data structure that supports the following queries:
• $\mathtt{insert}(m, q)$: inserting a new line $y=m\cdot x+q$ in $\mathcal O(\log n)$
• $\mathtt{query}(x)$: computing $\max_\ell\{m_\ell\cdot x+q_\ell\}$ in $\mathcal O(\log n)$
This allows us to solve the recurrence by repeatedly adding new lines and querying for the maximum:
hull h;
vector<long long> dp(n);
for (int i = 1; i <= n; ++i) {
h.insert({m[i-1], q[i-1]});
dp[i] = a[i] + h.query(x[i]);
}
This data structure works by dynamically storing the convex hull of linear functions and finding the value at a given $x$ by binary searching to find the line that is currently dominating.
Deque Implementation
Quite often, the DP equation of the convex hull trick
$\mathtt{DP}[i]=a[i] + \max_{0 \le k < i}\{m[k]\cdot x[i]+q[k]\}$
• $m[j+1]>m[j]$ (slopes are increasing) and
• $x[j+1]\ge x[j]$ (queries are increasing).
This is the case in our example problem above: The queries at $x[i]=s[i]$ are increasing because $s$ is the prefix sum of positive values. Because $a<0$, the slope $m[i]=-2\cdot a\cdot s[i]$ is also increasing.
The increasing slopes simplify the insertion: We know that the line must be added to the right side. So we can remove lines as long as they are hidden (similar to the convex hull idea), and then add the new line. An update is amortized $\mathcal O(1)$.
The same trick can be applied for the query: We know that whereever the current maximal line is, we can never “go back” afterwards, so we can just remove the lines at the beginning. Such a query is also amortized $\mathcal O(1)$.
Below is code that case using a std::queue.
struct line {
long long m, q;
long long eval(long long x) const { return m*x + q; }
};
// check if l2 is completely below max(l1, l3)
// requires that l1.m < l2.m < l2.m
bool bad(line const &l1, line const &l2, line const &l3) {
// or long double if __int128 is not available
return ((__int128)l2.q - l3.q) * (l2.m - l1.m) <=
((__int128)l1.q - l2.q) * (l3.m - l2.m);
}
struct hull {
deque<line> slopes;
// insert line to hull
void insert(line l) {
assert(slopes.empty() || l.m > slopes.back().m);
while (slopes.size() > 1 &&
slopes.pop_back();
slopes.push_back(l);
}
// maximum at x: max_l l.m*x + l.q
long long query(long long x) {
assert(!slopes.empty());
while (slopes.size() > 1 && slopes[0].eval(x) < slopes[1].eval(x))
slopes.pop_front();
return slopes.front().eval(x);
}
};
To support arbitrary queries, the query function can perform a binary search with upper_bound, similar to the std::multiset implementation.
In case $m[j+1]=m[j]$ can happen, there needs to be a collinearity check at the beginning of insert.
Arbitrary Queries and Inserts
To support inserts in the middle and arbitrary queries, we need something more dynamic. We can avoid implementing our own balanced binary search tree by exploiting that multiset supports different comparison operators for insert and calls to upper_bound.
As we have to binary search over the lines, we additionally need to remember xleft, the left endpoint of each segment in the hull. As we need to change that during an insert, we make it mutable.
struct line {
long long m, q; // y = m*x + q
// x from which this line becomes relevant in the hull
// mutable means we can change xleft on a const-reference to line
mutable long double xleft = numeric_limits<long double>::quiet_NaN();
};
bool operator<(line const& a, line const& b) { // sort lines after m
return make_pair(a.m, a.q) < make_pair(b.m, b.q);
}
bool operator<(long long x, line const& l) { // binary search for x
return x < l.xleft;
}
// x coordinate of the intersection between l1 and l2
long double intersect(line const &l1, line const &l2) {
return (l2.q - l1.q) / (long double)(l1.m - l2.m);
}
// check if l2 is completely below max(l1, l3)
// requires that l1.m < l2.m < l2.m
bool bad(line const &l1, line const &l2, line const &l3) {
// or long double if __int128 is not available
return (__int128)(l2.q - l3.q) * (l2.m - l1.m) <=
(__int128)(l1.q - l2.q) * (l3.m - l2.m);
}
struct hull {
multiset<line, less<>> slopes; // less<> to support upper_bound on long long's
// insert line to hull
void insert(line const &l) {
// insert l and then fix the hull until it is convex again
auto e = slopes.insert(l);
// delete collinear lines
if (e != slopes.begin() && prev(e)->m == e->m) {
slopes.erase(prev(e));
} else if (next(e) != slopes.end() && e->m == next(e)->m) {
slopes.erase(e);
return;
}
// delete l again if it is hidden by the lines to the left and the right
if (e != slopes.begin() && next(e) != slopes.end() &&
slopes.erase(e);
return;
}
// delete lines to the right of l and adjust their xleft
if (next(e) != slopes.end()) {
while (next(e, 2) != slopes.end() && bad(*e, *next(e), *next(e, 2)))
slopes.erase(next(e));
next(e)->xleft = intersect(*e, *next(e));
}
// delete lines to the left of l and adjust xleft of l
if (e != slopes.begin()) {
while (prev(e) != slopes.begin() && bad(*prev(e, 2), *prev(e), *e))
slopes.erase(prev(e));
e->xleft = intersect(*e, *prev(e));
} else {
e->xleft = -numeric_limits<long double>::infinity();
}
}
// maximum at x: max_l l.m*x + l.q
long long query(long long x) {
assert(!slopes.empty());
line const& l = *prev(slopes.upper_bound(x)); // upper_bound can never return begin() because it is -inf
return l.m * x + l.q;
}
};
Generalized Deque Trick
The idea of storing lines in a deque can be generalized to any DP formula of the form
\begin{aligned} \mathtt{DP}[0] &= 0\\ \mathtt{DP}[i] &= a[i] + \max_{0 \leq k < i} f[k][i] \qquad \text{for } 1 \leq i \leq n \end{aligned}
where the cost function $f[k][i]$ defined on $0 \leq k < i \leq n$ satisfies the convex quadrangle inequality
$f[a][d] + f[b][c] \leq f[a][c] + f[b][d] \qquad \forall a\leq b\leq c\leq d$
The intuition behind this inequality is the following: If for the current $i_{1}$ a larger value $k_{2}$ is better than a smaller value $k_{1}$, then $k_{2}$ will be a better option than $k_{1}$ for all $i_{2}\geq i_{1}$, i.e. for all future values of $i$. This is some sense generalizes the fact that two distinct lines intersect at at most one point, which was crucial for the convex hull trick to work.
In general, checking this inequality can be quite annoying. Fortunately, it often is intuitively clear that if a newer option (larger $k$) is better at some $i$, then it will also be better for larger $i$. As a last resort option, one could always write a brute-force solution and check the inequality on some examples.
We are interested in the point where $k_{2}$ becomes a better option than $k_{1}$, so let $D(k_{1}, k_{2})$ be the smallest $i$ for which $k_{2}$ is better than $k_{1}$, i.e.
$D(k_1, k_2) = \min \left\{k_2 < i \leq n \Big| f[k_1][i]\leq f[k_2][i]\right\}$
$D$ can be computed in $\mathcal{O}(\log n)$ time with binary search. Sometimes, special properties of $f$ allow you to compute it in $\mathcal{O}(1)$.
If there are three options $k_{1}< k_{2}< k_{3}$ with $D(k_{1}, k_{2}) \geq D(k_{2}, k_{3})$, then by the time $k_{2}$ is better than $k_{1}$, $k_{3}$ will already be better than $k_{2}$. In this case, $k_{2}$ is never optimal, so we way ignore it. (Similar to lines which are hidden below two other lines in the convex hull case.)
We maintain our candidates $k_{1}< k_{2}< \dots < k_m$ in a deque where we have $D(k_{1}, k_{2}) < D(k_{2}, k_{3}) < \dots < D(k_{m-1} k_m)$. $k_{1}$ is the value of $k$ this is currently optimal and $k_{2}, \dots, k_m$ are values that will be optimal for some larger $i$.
• When inserting $i$ as a new candidate for $k$, if $D(k_{m-1}, k_m) \geq D(k_m, i)$, we pop $k_m$ from the deque. Repeat this until $D(k_{m-1}, k_m) < D(k_m, i)$ and then push $i$ to the back of the deque.
• When computing $\mathtt{DP}[i]$, we pop $k_{1}$ from the deque until $k_{1}$ is better than $k_{2}$ (in other words, until $D(k_{1}, k_{2}) > i$). We then know that $k_{1}$ is the optimal value of $k$.
This ensure that after any operation, we still have $k_{1}< k_{2}< \dots < k_m$ and $D(k_{1}, k_{2}) < D(k_{2}, k_{3}) < \dots < D(k_{m-1}, k_m)$.
The total runtime is $\Theta(n \log n)$, as we need to evaluate $D$ $\Theta(n)$ times.
Implementation
struct Deque_Optimization {
const int n;
deque<int> cands;
function<int64_t(int, int)> > f;
Deque_Optimization (int n_, function<int64_t(int, int)> f_) : n(n_), f(f_) {}
// minimum i s.t. f(k_1, i) <= f(k_2, i)
// returns n+1 if D(k_1, k_2) would be infinite.
int D(int k_1, int k_2){
// k_1 is better at l, k_2 is better at r
int l = k_2, r = n+1;
while(l+1 < r){
const int m = l + (r-l)/2;
if(f(k_1, i) <= f(k_2, i)){
r = m;
} else {
l = m;
}
}
return r;
}
void insert(int i){
assert(cands.empty() || i > cands.back()); // candidates are added in increasing order.
while(cands.size() > 1 && D(cands.rbegin()[1], cands.rbegin()[0]) >= D(cands.rbegin()[0], i)){
cands.pop_back();
}
cands.push_back(i);
}
int64_t query(int i){
assert(!cands.empty());
while(cands.size() > 1 && f(cands[0], i) <= f(cands[1], i)){
cands.pop_front();
}
return f(cands[0], i);
}
};
Some calculations with the quadrangle inequality
In this part, we show how the quadrangle inequality implies the intuition that if a larger value of $k$ is better now, it will also be better in the future.
Recall that our DP has the form
$\mathtt{DP}[i] = a[i] + \max_{0 \leq k < i} (f[k][i] + q[k])$
where $f$ satisfies the quadrangle inequality
$f[a][d] + f[b][c] \leq f[a][c] + f[b][d] \qquad \forall a\leq b\leq c\leq d$
To see this, rewrite the inequality as
$f[b][c] - f[a][c] \leq f[b][d] - f[a][d]$
and set $a = k_{1}$, $b = k_{2}$, $c = i_{1}$, $d=i_{2}$ for some $k_{1}\leq k_{2}$ and $i_{1}\leq i_{2}$. Then we get
$f[k_2][i_1] - f[k_1][i_1] \leq f[k_2][i_2] - f[k_1][i_2]$
If at $i = i_{1}$, $k_{2}$ is a better option than $k_{1}$, then we have
$f[k_1][i_1] \leq f[k_2][i_1]$
i.e.
$0 \leq f[k_2][i_1] - f[k_1][i_1]$
so the quadrangle inequality tells us that
$0 \leq f[k_2][i_1] - f[k_1][i_1] \leq f[k_2][i_2] - f[k_1][i_2]$
hence
$f[k_1][i_2] \leq f[k_2][i_2]$
so $k_{2}$ is a better option than $k_{1}$ at $i = i_{2}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 111, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8945950269699097, "perplexity": 3015.0146122636847}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337322.29/warc/CC-MAIN-20221002115028-20221002145028-00519.warc.gz"} |
https://sinews.siam.org/About-the-Author/gromovs-non-squeezing-theorem-and-optics-1 | # Gromov’s Non-squeezing Theorem and Optics
Figure 1.
A remarkable theorem discovered by Gromov ([1], [2]) states that it is impossible to map the unit ball $$q_1 ^2 + p_1 ^2 + q_2 ^2 + p_2 ^2 \leq 1$$ in $${\mathbb R}^4$$ into a cylinder $$q_1 ^2 + p_1 ^2 \leq r ^2$$ of smaller radius $$r < 1$$ by a symplectic mapping.1 This general statement has a surprising implication in optics.
Any optical device (a system of lenses and mirrors) gives rise to a symplectic map in $${\mathbb R}^4$$ as follows. Placing the device between two parallel planes (Figure 1), we specify an incoming ray by the coordinates $$(q_1,q_2)$$ of its crossing the first plane, and by the direction sines $$(\sin \theta_1, \sin \theta_2) =(p_1,p_2)$$. The ray is thus characterized by the point $$x= (q_1,q_2, p_1,p_2) \in {\mathbb R} ^4$$. Let $$X= (Q_1,Q_2, P_1,P_2)$$ be the similarly-defined “exit data” of the same ray. Now it turns out that the mapping $$\varphi$$ from $$x$$ to $$X$$ is symplectic, preserving the symplectic form $$\omega$$ mentioned in the footnote. (An intuitive proof, along with a physical interpretation of $$\omega$$, can be found in [3]).
Figure 2.
With such parametrization of rays, the unit ball in $${\mathbb R} ^4$$ can be seen on the left side of Figure 1; at each point $$q =(q_1,q_2)$$ we have a cone of rays whose direction sines satisfy $$p ^2 \leq 1-q ^2$$ (in particular, the cones become narrower near the edge of the unit disk). The cylinder $$Q_1 ^2 + P_1^2 \leq r^2$$(Figure 2) corresponds to the set of rays exiting through the slit $$| Q_1|\leq r$$ and confined to the dihedral angle with$$|P_1|= |\sin \Theta_1|\leq \sqrt{ 1-r ^2}$$. Again, the aperture of this angle decreases with the distance to the edge of the slit.
And thus Gromov’s theorem implies the surprising fact that no optical device can shepherd the unit “ball” of incoming rays (Figure 1) through a narrow slit and with a narrow dihedral angle, as described more precisely in the preceding sentence.
Speaking of applications, the first application of Gromov’s theorem to PDEs can be found in the remarkable paper [4].
1 More precisely, by the map preserving the form $$\omega = dq_1\wedge dp_1+dq_2\wedge dp_2$$. Geometrically, this amounts to the requirement that for any infinitesimal parallelogram, the sum of signed areas of its projections onto the planes $$(q_1,p_1)$$ and $$(q_2,p_2)$$ is preserved under the map.
Acknowledgements: The work from which these columns are drawn is funded by NSF grant DMS-1412542.
References
[1] Gromov, M. (1985). Pseudo holomorphic curves in symplectic manifolds. Inventiones Mathematicae, 82, 307-347.
[2] Hofer, H., & Zehnder, E. (1994).Symplectic Invariants and Halmiltonian Dynamics. Birkhauser.
[3] Levi, M. (2014). Classical Mechanics with Calculus of Variations and Optimal Control: an Intuitive Introduction. AMS.
[4] Kuksin, S. (1995). Infinite Dimensional Symplectic Capacities and a Squeezing Theorem for Hamiltonian PDEs. Commun. Math. Phys. 167, 531-552.
Mark Levi ([email protected]) is a professor of mathematics at the Pennsylvania State University. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9641167521476746, "perplexity": 524.4592805829799}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743854.48/warc/CC-MAIN-20181117205946-20181117231946-00399.warc.gz"} |
https://lmfa.ec-lyon.fr/~lmfa/spip.php?article1510&lang=fr | # Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509
LMFA - UMR 5509
Laboratoire de Mécanique des Fluides et d’Acoustique
Lyon
France
## Nos partenaires
Article dans J. Fluid Mech. (2020)
## The turbulent Faraday instability in miscible fluids
Antoine Briard, Louis Gostiaux & Benoît-Joseph Gréa
Experiments of a turbulent mixing zone created by the Faraday instability at the statically stable interface between salt and fresh water are presented. The two-layer system, contained in a cuboidal tank of large dimensions, is accelerated vertically and periodically at various frequencies and amplitudes for three different density contrasts. We have developed a linear approach accounting for the full inhomogeneous and viscous problem, that is applied to a linear piecewise background density profile, and recovers the limiting cases of interface and homogeneous turbulence with a fully developed mixing layer. At onset, the wavelength of the most amplified modes and the corresponding Floquet exponent of the interface both verify our predictions. The dynamics is rather different when the instability is triggered from a sharp or diffuse interface : in the latter case, a change of characteristic wavelengths can be observed experimentally and explained by the theory. In the turbulent regime, the time evolution of the mixing zone size $L(t)$ for various experimental configurations compares well with confined direct numerical simulations. For some initial conditions, a short harmonic response of the instability is observed before the usual subharmonic one. Finally, the ultimate size of the mixing layer $L_\mathit{end}$, measured with a probe after the saturation of the instability and end of the forcing, is in excellent agreement with the recent theoretical prediction $L_\mathit{sat}=2{\mathcal A}g_0(2F+4)/\omega^2$, where $g_0$ is the gravitational acceleration, ${\mathcal A}$ the Atwood number, $\omega/2\pi$ the frequency and $F$ the acceleration ratio. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8230512142181396, "perplexity": 1183.145686431676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488560777.97/warc/CC-MAIN-20210624233218-20210625023218-00224.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=971258&postcount=15 | View Single Post
Sci Advisor HW Helper P: 9,398 Now, Nimz, this is where you've got it wrong. Some people do include 0 as a natural number, some people do not. Some people define 0 to be positive and negative and use the phrases strictly positive and strictly negative to exclude it. Others take it to be neither and use the phrase non-negative to mean 0,1,2... It is all a matter of what your conventions are. You have assumed that yours are absolutely correct, and that is not the case. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9030248522758484, "perplexity": 515.2310823751507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510256757.9/warc/CC-MAIN-20140728011736-00082-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://xianblog.wordpress.com/tag/path-sampling/ | ## computational methods for statistical mechanics [day #3]
Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on June 6, 2014 by xi'an
The third day [morn] at our ICMS workshop was dedicated to path sampling. And rare events. Much more into [my taste] Monte Carlo territory. The first talk by Rosalind Allen looked at reweighting trajectories that are not in an equilibrium or are missing the Boltzmann [normalizing] constant. Although the derivation against a calibration parameter looked like the primary goal rather than the tool for constant estimation. Again papers in J. Chem. Phys.! And a potential link with ABC raised by Antonietta Mira… Then Jonathan Weare discussed stratification. With a nice trick of expressing the normalising constants of the different terms in the partition as solution(s) of a Markov system
$v\mathbf{M}=v$
Because the stochastic matrix M is easier (?) to approximate. Valleau’s and Torrie’s umbrella sampling was a constant reference in this morning of talks. Arnaud Guyader’s talk was in the continuation of Toni Lelièvre’s introduction, which helped a lot in my better understanding of the concepts. Rephrasing things in more statistical terms. Like the distinction between equilibrium and paths. Or bias being importance sampling. Frédéric Cérou actually gave a sort of second part to Arnaud’s talk, using importance splitting algorithms. Presenting an algorithm for simulating rare events that sounded like an opposite nested sampling, where the goal is to get down the target, rather than up. Pushing particles away from a current level of the target function with probability ½. Michela Ottobre completed the series with an entry into diffusion limits in the Roberts-Gelman-Gilks spirit when the Markov chain is not yet stationary. In the transient phase thus.
## controlled thermodynamic integral for Bayesian model comparison [reply]
Posted in Books, pictures, Running, Statistics, University life with tags , , , , , , , , , , , , on April 30, 2014 by xi'an
Chris Oates wrotes the following reply to my Icelandic comments on his paper with Theodore Papamarkou, and Mark Girolami, reply that is detailed enough to deserve a post on its own:
Thank you Christian for your discussion of our work on the Og, and also for your helpful thoughts in the early days of this project! It might be interesting to speculate on some aspects of this procedure:
(i) Quadrature error is present in all estimates of evidence that are based on thermodynamic integration. It remains unknown how to exactly compute the optimal (variance minimising) temperature ladder “on-the-fly”; indeed this may be impossible, since the optimum is defined via a boundary value problem rather than an initial value problem. Other proposals for approximating this optimum are compatible with control variates (e.g. Grosse et al, NIPS 2013, Friel and Wyse, 2014). In empirical experiments we have found that the second order quadrature rule proposed by Friel and Wyse 2014 leads to substantially reduced bias, regardless of the specific choice of ladder.
(ii) Our experiments considered first and second degree polynomials as ZV control variates. In fact, intuition specifically motivates the use of second degree polynomials: Let us presume a linear expansion of the log-likelihood in θ. Then the implied score function is constant, not depending on θ. The quadratic ZV control variates are, in effect, obtained by multiplying the score function by θ. Thus control variates can be chosen to perfectly correlate with the log-likelihood, leading to zero-variance estimators. Of course, there is an empirical question of whether higher-order polynomials are useful when this Taylor approximation is inappropriate, but they would require the estimation of many more coefficients and in practice may be less stable.
(iii) We require that the control variates are stored along the chain and that their sample covariance is computed after the MCMC has terminated. For the specific examples in the paper such additional computation is a negligible fraction of the total computational, so that we did not provide specific timings. When non-diffegeometric MCMC is used to obtain samples, or when the score is unavailable in closed-form and must be estimated, the computational cost of the procedure would necessarily increase.
For the wide class of statistical models with tractable likelihoods, employed in almost all areas of statistical application, the CTI we propose should provide state-of-the-art estimation performance with negligible increase in computational costs.
## controlled thermodynamic integral for Bayesian model comparison
Posted in Books, pictures, Running, Statistics, University life with tags , , , , , , , , , , on April 24, 2014 by xi'an
Chris Oates, Theodore Papamarkou, and Mark Girolami (all from the University of Warwick) just arXived a paper on a new form of thermodynamic integration for computing marginal likelihoods. (I had actually discussed this paper with the authors on a few occasions when visiting Warwick.) The other name of thermodynamic integration is path sampling (Gelman and Meng, 1998). In the current paper, the path goes from the prior to the posterior by a sequence of intermediary distributions using a power of the likelihood. While the path sampling technique is quite efficient a method, the authors propose to improve it through the recourse to control variates, in order to decrease the variance. The control variate is taken from Mira et al. (2013), namely a one-dimensional temperature-dependent transform of the score function. (Strictly speaking, this is an asymptotic control variate in that the mean is only asymptotically zero.) This control variate is then incorporated within the expectation inside the path sampling integral. Its arbitrary elements are then calibrated against the variance of the path sampling integral. Except for the temperature ladder where the authors use a standard geometric rate, as the approach does not account for Monte Carlo and quadrature errors. (The degree of the polynomials used in the control variates is also arbitrarily set.) Interestingly, the paper mixes a lot of recent advances, from the zero variance notion of Mira et al. (2013) to the manifold Metropolis-adjusted Langevin algorithm of Girolami and Calderhead (2011), uses as a base method pMCMC (Jasra et al., 2007). The examples processed in the paper are regression (where the controlled version truly has a zero variance!) and logistic regression (with the benchmarked Pima Indian dataset), with a counter-example of a PDE interestingly proposed in the discussion section. I quite agree with the authors that the method is difficult to envision in complex enough models. I also did not see mentions therein of the extra time involved in using this control variate idea.
## Pre-processing for approximate Bayesian computation in image analysis
Posted in R, Statistics, University life with tags , , , , , , , , , , , , , on March 21, 2014 by xi'an
With Matt Moores and Kerrie Mengersen, from QUT, we wrote this short paper just in time for the MCMSki IV Special Issue of Statistics & Computing. And arXived it, as well. The global idea is to cut down on the cost of running an ABC experiment by removing the simulation of a humongous state-space vector, as in Potts and hidden Potts model, and replacing it by an approximate simulation of the 1-d sufficient (summary) statistics. In that case, we used a division of the 1-d parameter interval to simulate the distribution of the sufficient statistic for each of those parameter values and to compute the expectation and variance of the sufficient statistic. Then the conditional distribution of the sufficient statistic is approximated by a Gaussian with these two parameters. And those Gaussian approximations substitute for the true distributions within an ABC-SMC algorithm à la Del Moral, Doucet and Jasra (2012).
Across 20 125 × 125 pixels simulated images, Matt’s algorithm took an average of 21 minutes per image for between 39 and 70 SMC iterations, while resorting to pseudo-data and deriving the genuine sufficient statistic took an average of 46.5 hours for 44 to 85 SMC iterations. On a realistic Landsat image, with a total of 978,380 pixels, the precomputation of the mapping function took 50 minutes, while the total CPU time on 16 parallel threads was 10 hours 38 minutes. By comparison, it took 97 hours for 10,000 MCMC iterations on this image, with a poor effective sample size of 390 values. Regular SMC-ABC algorithms cannot handle this scale: It takes 89 hours to perform a single SMC iteration! (Note that path sampling also operates in this framework, thanks to the same precomputation: in that case it took 2.5 hours for 10⁵ iterations, with an effective sample size of 10⁴…)
Since my student’s paper on Seaman et al (2012) got promptly rejected by TAS for quoting too extensively from my post, we decided to include me as an extra author and submitted the paper to this special issue as well.
## Importance sampling schemes for evidence approximation in mixture models
Posted in R, Statistics, University life with tags , , , , , , , , , on November 27, 2013 by xi'an
Jeong Eun (Kate) Lee and I completed this paper, “Importance sampling schemes for evidence approximation in mixture models“, now posted on arXiv. (With the customary one-day lag for posting, making me bemoan the days of yore when arXiv would give a definitive arXiv number at the time of submission.) Kate came twice to Paris in the past years to work with me on this evaluation of Chib’s original marginal likelihood estimate (also called the candidate formula by Julian Besag). And on the improvement proposed by Berkhof, van Mechelen, and Gelman (2003), based on averaging over all permutations, idea that we rediscovered in an earlier paper with Jean-Michel Marin. (And that Andrew seemed to have completely forgotten. Despite being the very first one to publish [in English] a paper on a Gibbs sampler for mixtures.) Given that this averaging can get quite costly, we propose a preliminary step to reduce the number of relevant permutations to be considered in the averaging, removing far-away modes that do not contribute to the Rao-Blackwell estimate and called dual importance sampling. We also considered modelling the posterior as a product of k-component mixtures on the components, following a vague idea I had in the back of my mind for many years, but it did not help. In the above boxplot comparison of estimators, the marginal likelihood estimators are
1. Chib’s method using T = 5000 samples with a permutation correction by multiplying by k!.
2. Chib’s method (1), using T = 5000 samples which are randomly permuted.
3. Importance sampling estimate (7), using the maximum likelihood estimate (MLE) of the latents as centre.
4. Dual importance sampling using q in (8).
5. Dual importance sampling using an approximate in (14).
6. Bridge sampling (3). Here, label switching is imposed in hyperparameters.
## Carnon [and Core, end]
Posted in Books, Kids, pictures, R, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , on June 16, 2012 by xi'an
Yet another full day working on Bayesian Core with Jean-Michel in Carnon… This morning, I ran along the canal for about an hour and at last saw some pink flamingos close enough to take pictures (if only to convince my daughter that there were flamingos in the area!). Then I worked full-time on the spatial statistics chapter, using a small dataset on sedges that we found in Gaetan and Guyon’s Spatial Statistics and Modelling. I am almost done tonight, with both path sampling and ABC R codes documented and working for this dataset. But I’d like to re-run both codes for longer to achieve smoother outcomes.
## Harmonic means, again again
Posted in Books, R, Statistics, University life with tags , , , , , , , , on January 10, 2012 by xi'an
Another arXiv posting I had had no time to comment is Nial Friel’s and Jason Wyse’s “Estimating the model evidence: a review“. This is a review in the spirit of two of our papers, “Importance sampling methods for Bayesian discrimination between embedded models” with Jean-Michel Marin (published in Jim Berger Feitschrift, Frontiers of Statistical Decision Making and Bayesian Analysis: In Honor of James O. Berger, but not mentioned in the review) and “Computational methods for Bayesian model choice” with Darren Wraith (referred to by the review). Indeed, it considers a series of competing computational methods for approximating evidence, aka marginal likelihood:
The paper correctly points out the difficulty with the naïve harmonic mean estimator. (But it does not cover the extension to the finite variance solutions found in”Importance sampling methods for Bayesian discrimination between embedded models” and in “Computational methods for Bayesian model choice“.) It also misses the whole collection of bridge and umbrella sampling techniques covered in, e.g., Chen, Shao and Ibrahim, 2000 . In their numerical evaluations of the methods, the authors use the Pima Indian diabetes dataset we also used in “Importance sampling methods for Bayesian discrimination between embedded models“. The outcome is that the Laplace approximation does extremely well in this case (due to the fact that the posterior is very close to normal), Chib’s method being a very near second. The harmonic mean estimator does extremely poorly (not a suprise!) and the nested sampling approximation is not as accurate as the other (non-harmonic) methods. If we compare with our 2009 study, importance sampling based on the normal approximation (almost the truth!) did best, followed by our harmonic mean solution based on the same normal approximation. (Chib’s solution was then third, with a standard deviation ten times larger.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8960392475128174, "perplexity": 1208.8545729613072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119647629.9/warc/CC-MAIN-20141024030047-00072-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://terrytao.wordpress.com/tag/harmonic-analysis/ | You are currently browsing the tag archive for the ‘harmonic analysis’ tag.
I’d like to begin today by welcoming Timothy Gowers to the mathematics blogging community; Tim’s blog will also double as the “official” blog for the Princeton Companion to Mathematics, as indicated by his first post
which also contains links to further material (such as sample articles) on the Companion. Tim is already thinking beyond the blog medium, though, as you can see in his second post
Anyway, this gives me an excuse to continue my own series of PCM articles. Some years back, Tim asked me to write a longer article on harmonic analysis – the quantitative study of oscillation, transforms, and other features of functions and sets on domains. At the time I did not fully understand the theme of the Companion, and wrote a rather detailed and technical survey of the subject, which turned out to be totally unsuitable for the Companion. I then went back and rewrote the article from scratch, leading to this article, which (modulo some further editing) is close to what will actually appear. (These two articles were already available on my web site, but not in a particularly prominent manner.) So, as you can see, the articles in the Companion are not exactly at the same level as the expository survey articles one sees published in journals.
I should also mention that some other authors for the Companion have put their articles on-line. For instance, Alain Connes‘ PCM article “Advice for the beginner“, aimed at graduate students just starting out in research mathematics, was in fact already linked to on one of the pages of this blog. I’ll try to point out links to other PCM articles in future posts in this series.
One of the oldest and most fundamental concepts in mathematics is the line. Depending on exactly what mathematical structures we want to study (algebraic, geometric, topological, order-theoretic, etc.), we model lines nowadays by a variety of standard mathematical objects, such as the real line ${\Bbb R}$, the complex line ${\Bbb C}$, the projective line $\Bbb{RP}^1$, the extended real line ${}[-\infty,+\infty]$, the affine line ${\Bbb A}^1$, the continuum $c$, the long line $L$, etc. We also have discrete versions of the line, such as the natural numbers ${\Bbb N}$, the integers ${\Bbb Z}$, and the ordinal $\omega$, as well as compact versions of the line, such as the unit interval ${}[0,1]$ or the unit circle ${\Bbb T} := {\Bbb R}/{\Bbb Z}$. Finally we have discrete and compact versions of the line, such as the cyclic groups ${\Bbb Z}/N{\Bbb Z}$ and the discrete intervals $\{1,\ldots,N\}$ and $\{0,\ldots,N-1\}$. By taking Cartesian products we then obtain higher-dimensional objects such as Euclidean space ${\Bbb R}^n$, the standard lattice ${\Bbb Z}^n$, the standard torus ${\Bbb T}^n = {\Bbb R}^n/{\Bbb Z}^n$, and so forth. These objects of course form the background on which a very large fraction of modern mathematics is set.
Broadly speaking, the line has three major families of structures on it:
1. Geometric structures, such as a metric or a measure, completeness, scales (coarse and fine), rigid motions (translations and reflection), similarities (dilation, affine maps), and differential structures (tangent bundle, etc.);
2. Algebraic structures, such group, ring, or field structures, and everything else that comes from those categories (e.g. subgroups, homomorphisms, involutions, etc.); and
3. One-dimensional structures, such as order, a length space structure (in particular, path-connectedness structure), a singleton generator, the Archimedean property, the ability to use mathematical induction (i.e. well-ordering), convexity, or the ability to disconnect the line by removing a single point.
Of course, these structures are inter-related, and it is an important phenomenon that a mathematical concept which appears to be native to one structure, can often be equivalently defined in terms of other structures. For instance, the absolute value $|n|$ of an integer $n$ can be defined geometrically as the distance from 0 to $n$, algebraically as the index of the subgroup $\langle n \rangle = n \cdot \Bbb Z$ of the integers ${\Bbb Z}$ generated by n, or one-dimensionally as the number of integers between 0 and $n$ (including 0, but excluding $n$). This equivalence of definitions becomes important when one wants to work in more general contexts in which one or more of the above structures is missing or otherwise weakened.
What I want to talk about today is an important toy model for the line (in any of its incarnations), in which the geometric and algebraic structures are enhanced (and become neatly nested and recursive), at the expense of the one-dimensional structure (which is largely destroyed). This model has many different names, depending on what field of mathematics one is working in and which structures one is interested in. In harmonic analysis it is called the dyadic model, the Walsh model, or the Cantor group model; in number theory and arithmetic geometry it is known as the function field model; in topology it is the Cantor space model; in probability it is the martingale model; in metric geometry it is the ultrametric, tree, or non-Archimedean model; in algebraic geometry it is the Puiseux series model; in additive combinatorics it is the bounded torsion or finite field model; in computer science and information theory it is the Hamming cube model; in representation theory it is the Kashiwara crystal model. Let me arbitrarily select one of these terms, and refer to all of these models as dyadic models for the line (or of objects derived from the line). While there is often no direct link between a dyadic model and a non-dyadic model, dyadic models serve as incredibly useful laboratories in which to gain insight and intuition for the “real-world” non-dyadic model, since one has much more powerful and elegant algebraic and geometric structure to play with in this setting (though the loss of one-dimensional structure can be a significant concern). Perhaps the most striking example of this is the three-line proof of the Riemann hypothesis in the function field model of the integers, which I will discuss a little later.
I am very saddened to find out (first via Wikipedia, then by several independent confirmations) that Paul Cohen died on Friday, aged 72.
Paul Cohen is of course best known in mathematics for his Fields Medal-winning proof of the undecidability of the continuum hypothesis within the standard Zermelo-Frankel-Choice (ZFC) axioms of set theory, by introducing the now standard method of forcing in model theory. (More precisely, assuming ZFC is consistent, Cohen proved that models of ZFC exist in which the continuum hypothesis fails; Gödel had previously shown under the same assumption that models exist in which the continuum hypothesis is true.) Cohen’s method also showed that the axiom of choice was independent of ZF. The friendliest introduction to forcing is perhaps still Timothy Chow‘s “Forcing for dummies“, though I should warn that Tim has a rather stringent definition of “dummy”.
But Cohen was also a noted analyst. For instance, the Cohen idempotent theorem in harmonic analysis classifies the idempotent measures $\mu$ in a locally compact abelian group G (i.e. the finite regular measures for which $\mu * \mu = \mu$); specifically, a finite regular measure $\mu$ is idempotent if and only if the Fourier transform $\hat \mu$ of the measure only takes values 0 and 1, and furthermore can be expressed as a finite linear combination of indicator functions of cosets of open subgroups of the Pontryagin dual $\hat G$ of G. (Earlier results in this direction were obtained by Helson and by Rudin; a non-commutative version was subsequently given by Host. These results play an important role in abstract harmonic analysis.) Recently, Ben Green and Tom Sanders connected this classical result to the very recent work on Freiman-type theorems in additive combinatorics, using the latter to create a quantitative version of the former, which in particular is suitable for use in finite abelian groups.
Paul Cohen’s legacy also includes the advisorship of outstanding mathematicians such as the number theorist and analyst Peter Sarnak (who, incidentally, taught me analytic number theory when I was a graduate student). Cohen was in fact my “uncle”; his advisor, Antoni Zygmund, was the advisor of my own advisor Elias Stein.
It is a great loss for the world of mathematics.
[Update, Mar 25: Added the hypothesis that ZFC is consistent to the description of Cohen’s result. Several other minor edits also.]
I’ve just uploaded the short story “Uchiyama’s constructive proof of the Fefferman-Stein decomposition“. In 1982, Uchiyama gave a new proof of the celebrated Fefferman-Stein theorem that expressed any BMO function as the sum of a bounded function, and Riesz transforms of bounded functions. Unlike the original proof (which relied, among other things, on the Hahn-Banach theorem), Uchiyama’s proof was very explicit, constructing the decomposition by building the bounded functions one Littlewood-Paley frequency band at a time while keeping the functions taking values on or near a sphere, and then iterating away the error. Here I have written some notes on how the proof goes. The notes are a little condensed, in that a number of standard computations involving estimations of Schwartz tails, Carleson measures, etc. have been omitted, but hopefully the gist of the argument is still clear. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 30, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8726741671562195, "perplexity": 452.50311490919194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936469016.33/warc/CC-MAIN-20150226074109-00309-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/combination-of-springs.138594/ | # Combination of springs
1. Oct 16, 2006
### emilykay
I dont understand how to calculate force constants of a combination of springs.
for example 2 spings parallel with the mass on one end lying on a table.
or 2 springs attached with mass at one end lying on a table.
i know that force constant = sum of forces/ extension
but just dont get how to combine springs!
Thanks!
EmilyKay
2. Oct 16, 2006
### rohit88
Hey Its Easy Man.just Assume The Springs As Resistance And Like We Calculate Net Resistances In Series And In Parallel Calculate The Net Spring Constant.ex-two Springs Of Spring Constant =k Conected In Series Are Equal Ti A Spring Of Constant 2k.the Masses Have Got No Role To Play.only Thing Is That U Have To Find How They Aye Joined In Seriesa Or Parallel.ok...
3. Oct 16, 2006
### tim_lou
when in Parallel configurations:
since they are parallel, the extension must be the same.
k=sum (F) / x
hence $$k=(F_1+F_2)/x = k_1+k_2$$
when in series configurations:
when in equilibrium, the tension in the two springs must be equal and the extension is the extension of the first spring + the extension of the second spring:
k=F/ (x1+x2)
take the inverse of both side:
$$\frac{1}{k}=\frac{x_1}{F}+\frac{x_2}{F}=\frac{1}{k_1}+\frac{1}{k_2}$$
4. Oct 16, 2006
### ZapperZ
Staff Emeritus
rohit88 response is incorrect. Please pay attention to the response given by tim_lou, which is the correct answer AND derivation.
Springs in series behave like resistors in parallel, which springs in parallel behave like resistors in series.
Zz.
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http://slideplayer.com/slide/4167557/ | # Operations on Complementary Edge Binary Decision Diagrams EE 552 Instructor: Dr. James Ellison Kenny Liu July 22 nd, 2003.
## Presentation on theme: "Operations on Complementary Edge Binary Decision Diagrams EE 552 Instructor: Dr. James Ellison Kenny Liu July 22 nd, 2003."— Presentation transcript:
Operations on Complementary Edge Binary Decision Diagrams EE 552 Instructor: Dr. James Ellison Kenny Liu [email protected] July 22 nd, 2003
Introduction From the text, we are familiar with performing various operations in the ROBDD (Reduced Order Binary Decision Diagram) form. There were various notations mentioned in the text and we will be focusing on the CCE BDD (Canonical Complementary Edge Binary Decision Diagram) form. The presentation will show and prove methods of performing the operations we are already familiar with in the ROBDD form, but now in the CCE BDD form.
Introduction The operations mentioned will include: complementation finding the product of sums (POS) and sum of products (SOP) representation reordering variables dual cofactor representation tautology and satisfiability disjunctive composition
Conversion from ROBDD to CCEBDD x y z 01 1)Change all dotted lines (0-edges) into a line with a clear circle. Here we have our basic ROBDD form. We will be converting to CCE BDD form through a series of steps. ROBDD x y z 01 After Step 1 f = xz’+x’yz’f
Conversion from ROBDD to CCEBDD 2)All lines going to the 1- leaf are represented with leaf-less notation 3)All lines going to the 0- leaf are represented as leaf-less and with a solid dot on that branch. x y z 0 After Step 2 x y z After Step 3 f f
Conversion from ROBDD to CCEBDD x y z After Step 3 4) The next step is to remove all solid dots that are on lines without clear dots, in our example we have just one such case. We are only allowed to remove this solid dot if we add solid dots to all other branches of the node. x y z After Step 4 ff
Conversion from ROBDD to CCEBDD x y z After Step 4 f Repeat Step 4 x y z After Step 4 f
Conversion from ROBDD to CCEBDD x y z After Step 4 f 5) In this step, any edge with 2 solid dots can cancel out. The text also takes another step in which to represent an edge with a solid dot and clear dot as just a solid dot since we know it can only appear on zero edges anyways, but we will skip this step to make it easier to understand as we perform operations later. x y z After Step 5 f
Conversion from ROBDD to CCEBDD Here we have our final CCEBDD. We can see a major difference from an ROBDD in that it has a root edge which is the branch above the root node. Through this method of notation, the solid dots complement the portion of the function from the child node down. We will go into more detail of how this works later. x y z After Step 5 f
Summary of CCEBDD Basic Properties of CCEBDD form: 1)Has a root edge which may have a solid dot. 2)0-edges are represented as lines with clear dot. 3)There are only 1 leaves, 0-leaves are replaces with 1-leaves with a solid dot. 4)No solid dots may appear on positive edge. Cancel out with method shown previously. 5)Two solid dots on the same branch cancels each other out. 6)There is one unique CCEBDD for a boolean function for a particular ordering of its variables.
Basic Operation on CCEBDD x y z f The main operation we must be familiar with before we proceed is how to obtain the boolean function from a given CCEBDD. For Sum of Products in ROBDD, we are used to taking each possible path down to the 1-leaf, but here the process will be a little different. Instead, the path we are looking for must contain an even number of solid dots (ex. 0,2,4,6 etc.). If there is a solid dot on the root edge, we must take that into account for each path. We can verify our results with our original ROBDD.
Basic Operation on CCEBDD x y z f In our simple example there is only two such paths which is highlighted in red and pink. Each path contains 2 dots and each path represents part of the function. Sum of Products (even number of dots): (red highlight, 2-dots, path = x, z’) = xz’ (pink highlight, 2-dots, path = x’, y, z’) = x’yz’ Therefore f = xz’ + x’yz’
Complementation Complementation in ROBDD form is simple because all that needs to be done is to switch the 1- leaf with the 0-leaf. Actually complementation in CCEBDD form is just as easy but we will show the process with an example. Here we have a function in ROBDD and its complement. x y z 01 f = x+x’y+x’y’z’ x y z 10 f’ = x’y’z
Complementation We will now convert the ROBDD into a CCEBDD both regular and complemented functions to compare. x y z 01 f = x+x’y+x’y’z’ x y z f x y z f
Complementation x y z 10 f’ = x’y’z x y z x y z
Complementation By visually comparing the two CCEBDD’s, we can see that the complemented version just has the added solid dot on the root edge. With the solid dot representing the complementation of everything below it, it makes sense that a solid dot above the root node will complement the entire function. x y z f = x+x’y+x’y’z’ x y z f’ = x’y’z
Product of Sums We are already familiar with finding the Sum of Products function of a given CCEBDD in which we take each path that contains an even number of solid dots as a group of products with each group summed together. To find the Product of Sums function, it’s basically the same process, just with the 0-leaf, and so in CCEBDD form we look for paths with odd number of solid dots. We also need to complement each node as we did when finding Product of Sums for ROBDD’s.
Product of Sums x y z f Going back to our original example, we have 3 paths with an odd number of solid dots. Product of Sums (odd number of dots): (red highlight, 1-dot, path = x, z) = x’+z’ (pink highlight, 1-dot, path = x’, y, z) = x+y’+z’ (yellow highlight, 1-dot, path = x’, y’) = x+y Therefore f = (x’+z’)(x+y’+z’)(x+y)
Reordering of Variables The process in which we reorder variables in CCEBDD is very much the same as when we reorder variables in ROBDD. The leaves and nodes which involve the 2 variables to be reordered must be fully expanded keeping in mind that the 0-edge is always on the left. Afterwards, the variables can be switched along with the 2 leaf values in the expanded tree. a bb N1N2N3N4 After swap b aa N1N3N2N4
Reordering of Variables In the following example, we will swap the y and z variable. x y z f We first expand the CCEBDD x y z f y zzz
Reordering of Variables x y z f y zzz There are 2 sets of mini y-z trees, and so the process must be applied to each tree individually. After swapping the variables, we keep in mind that the middle 2 branches must also be swapped. The solid dots on the z-node branches represents 0-leaves and so that solid dot must swap to the opposite branch.
Reordering of Variables x z y f z yyy We apply the rule that no solid dots can be on a positive edge, and so we remove them by adding solid dots to each adjacent branch.
Reordering of Variables x z y f z yyy After the reordering process and applying basic CCEBDD rules, we have the following graph. We can verify that functionality wise it is the same. The 2 highlighted paths are the only paths that contain 2 solid dots for SOP. They represent xz’y and xz’ for the red and pink respectively. Therefore the equation is: f = xz’y + xz’ (matches previous)
Dual Function Conversion To perform the dual function, we are accustomed to switching the 1-leaves with the 0-leaves, and then swapping 0-edges with 1- edges, and swapping 1-edges with 0-edges. There are not any special rules we must taken into account when performing the dual function conversion for CCEBDD’s but by just applying what we already know it is a simple process. The following steps summarize the procedure: Step 1)Complement the entire function, effectively swapping the 1 and 0-leaves. In terms of CCEBDD, we add the solid dot at the root edge. Step 2)Switch complemented branches and positive branches, in CCEBDD, that means moving the clear-circle to the opposite branch of that node.
Dual Function Conversion x y z f x y z f After Step 1 x y z f After Step 2Initial CCEBDD
Dual Function Conversion x y z f After Step 2 x y z f We must remember that at the end of the series of steps, the CCEBDD must still uphold the rule that no solid dots can appear on a positive edge, therefore we must perform an additional step for our example. Final Answer!
Cofactor Representation x y z f For cofactor representation, we eliminate the opposing edge of the variable worked with. For example if we are looking for f z we will eliminate the 0-edge at the z-node, and if it was f z’ we eliminate the 1-edge at the z-node. We eliminate the z-node itself and link the remaining edge with the branch above the z-node. In the following example we will try to find f z’.
Cofactor Representation x y z f z’ x y The f z’ cofactor replaces the z- decision with a path down the 0- branch. 2 solid dots on one line cancel each other out.
Cofactor Representation x y f z’ Here is our final cofactor representation of f z. Let’s verify our answer. We have: f = x + x’y + x’y’ Our original function was f = x+x’y+x’y’z’. f z’ = f(x,y,0) = x + x’y + x’y’(0)’ = x + x’y + x’y’ This verifies our answer and also brings up another subject that we will discuss next.
Tautology and Satisfiability In CCEBDD notation, a tautology is represented as a graph with either no solid dots at all, or a graph that has all paths containing an even number of dots. The simplest possible CCEBDD, the single edge, is a CCEBDD that satisfies this criterion. The CCEBDD is “canonical”, that is, there is only one CCEBDD for a given function. Therefore, the CCEBDD for a tautology must be a single edge. The only case in which a graph is not satisfiable is when every individual path on the graph contains an odd number of solid dots. Thus, the CCEBDD consisting of the dotted root, with a single edge, is not satisfiable. Therefore, it is the only unsatisfiable CCEBDD.
Disjunctive Composition The process of Disjunctive Composition is quite a bit more complicated than the previous topics. The steps will be shown that illustrate the procedure, and then it will be verified by working with just the equations. There are some main differences we recognize immediately because we are used to see the 1’s and 0 leaves, but if we approach the problem with the following rules in mind, it is straightforward: 1) Any path can be thought of as having a 0-leaf at the end of it if it has an odd number of total dots for that path. 2) Any path can be thought of as having a 1-leaf at the end of it if it has an even number of total dots for that path.
Disjunctive Composition f(x,y,z) = xz’ + x’yz’ g(a,b) = a’ + ab’ x y z f a b g Our goal is to find f(x,g,z) Substituting g in for y.
Disjunctive Composition When trying to perform disjunctive composition, the node at which we will be performing the substitution has of course a 1 and 0 edge. The destination of the 1-edge for the y node in our case, is the destination of all branches that has a 1-leaf, in the function to be substituted. The 0-edge for the y node is then the destination for any branch in the function that has a 0-leaf
Disjunctive Composition x y z f The area highlighted is our focus for this part. The y-node’s 0-edge is just dangling, and so that will be the destination for all branches in the function g that has a 0-leaf. This will be called Destination-0. The 1-edge for the y-node is the z-node, and so that will be the destination for all branches in the function g that has a 1-leaf. This will be called Destination-1.
Disjunctive Composition a b g Now let us determine which leaves in the function are 1-leaves or 0-leaves. Following the rule that any path that contains an odd number of dots has a 0-leaf, and any path that contains an even number of dots has a 1-leaf, we can label our function. (red highlight, 2-dots) = even, 1-leaf (pink highlight, 2-dots) = even, 1-leaf (green highlight, 1-dot) = odd, 0-leaf 0-leafs go to Destination-0. 1-leafs go to Destination-1. 1 1 0
Disjunctive Composition Now when substituting g back into f, we can remove all solid dots that had been on g. We have already used the solid dots to determine all paths down to the leaves, and so their destinations are set. By directing each branch to their correct destinations, we have the newly composed h function. Let us first find the formula of the new function in sum of products form. (red)(pink)(green) h = xz’ + x’a’z’ + x’ab’z’ x a z h(x,a,b,z) b
Disjunctive Composition Now let us verify our results by working with just function formulas. We have: f(x,y,z) = xz’ + x’yz’ g(a,b) = a’ + ab’ h(x,a,b,z) = xz’ + x’a’z’ + x’ab’z’ Let’s sub g in for y h(x,a,b,z) = xz’ + x’(a’ + ab’)z’ = xz’ + x’a’z’ + z’ab’z’ Which matches the equation we obtained from the composed CCEBDD, therefore verifying our answer.
Disjunctive Composition x y z f A case that we must consider that was not covered in the previous example, is if there is a solid dot on the 0-edge. This extra dot was added to our original f function to show what we must do in this situation. We automatically throw out the possibility of a solid dots on the 1-edge because we know it is not allowed. When substituting the g equation into f, we add that solid dot to any edge that ends with a 0-leaf.
Disjunctive Composition x a z h(x,a,b,z) b The final function has the extra highlighted solid dot because of the solid dot on the 0- edge of the original function. Because we only had that branch going to 0-leaf, we only have to add the solid dot to one place. Every other step remains the step in the procedure.
Conclusion We have covered how to perform many operations in CCEBDD form. Some of these operations were straightforward and basically followed steps as in ROBDD’s, while some operations were very different. It was interesting to see how some operations could become even simpler in the CCEBDD notation, while it also complicated others by forcing us to watch out for new rules and restrictions. The process in which these steps came about was by testing it in ROBDD first, and then converting to CCEBDD and observing changes that occurred.
References Techniques in Advanced Switching Theory by Professor Ellison BDD section
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https://hamronotes.com/lessons/solutions-chapter-6-linear-inequalities/ | MATHEMATICS XI
Solutions Chapter 6 – Linear Inequalities
Question 1:
Solve 24x < 100, when (i) x is a natural number (ii) x is an integer
The given inequality is 24x < 100.
(i) It is evident that 1, 2, 3, and 4 are the only natural numbers less than.
Thus, when x is a natural number, the solutions of the given inequality are 1, 2, 3, and 4.
Hence, in this case, the solution set is {1, 2, 3, 4}.
(ii) The integers less than are …–3, –2, –1, 0, 1, 2, 3, 4.
Thus, when x is an integer, the solutions of the given inequality are
…–3, –2, –1, 0, 1, 2, 3, 4.
Hence, in this case, the solution set is {…–3, –2, –1, 0, 1, 2, 3, 4}.
Question 2:
Solve –12x > 30, when
(i) x is a natural number (ii) x is an integer
The given inequality is –12x > 30.
(i) There is no natural number less than.
Thus, when x is a natural number, there is no solution of the given inequality.
(ii) The integers less than are …, –5, –4, –3.
Thus, when x is an integer, the solutions of the given inequality are
…, –5, –4, –3.
Hence, in this case, the solution set is {…, –5, –4, –3}.
Question 3:
Solve 5x– 3 < 7, when
(i) x is an integer (ii) x is a real number
The given inequality is 5x– 3 < 7.
(i) The integers less than 2 are …, –4, –3, –2, –1, 0, 1.
Thus, when x is an integer, the solutions of the given inequality are
…, –4, –3, –2, –1, 0, 1.
Hence, in this case, the solution set is {…, –4, –3, –2, –1, 0, 1}.
(ii) When x is a real number, the solutions of the given inequality are given by x < 2, that is, all real numbers x which are less than 2.
Thus, the solution set of the given inequality is x ∈ (–∞, 2).
Question 4:
Solve 3x + 8 > 2, when
(i) x is an integer (ii) x is a real number
The given inequality is 3x + 8 > 2.
(i) The integers greater than –2 are –1, 0, 1, 2, …
Thus, when x is an integer, the solutions of the given inequality are
–1, 0, 1, 2 …
Hence, in this case, the solution set is {–1, 0, 1, 2, …}.
(ii) When x is a real number, the solutions of the given inequality are all the real numbers, which are greater than –2.
Thus, in this case, the solution set is (– 2, ∞).
Question 5:
Solve the given inequality for real x: 4x + 3 < 5x + 7
4x + 3 < 5x + 7
⇒ 4x + 3 – 7 < 5x + 7 – 7
⇒ 4x – 4 < 5x
⇒ 4x – 4 – 4x < 5– 4x
⇒ –4 < x
Thus, all real numbers x,which are greater than –4, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–4, ∞).
Question 6:
Solve the given inequality for real x: 3x – 7 > 5x – 1
3x – 7 > 5x – 1
⇒ 3x – 7 + 7 > 5x – 1 + 7
⇒ 3x > 5x + 6
⇒ 3x – 5x > 5x + 6 – 5x
⇒ – 2> 6
Thus, all real numbers x,which are less than –3, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, –3).
Question 7:
Solve the given inequality for real x: 3(x – 1) ≤ 2 (– 3)
3(x – 1) ≤ 2(x – 3)
⇒ 3x – 3 ≤ 2x – 6
⇒ 3x – 3 + 3 ≤ 2x – 6 + 3
⇒ 3x ≤ 2x – 3
⇒ 3x – 2≤ 2x – 3 – 2x
⇒ x ≤ – 3
Thus, all real numbers x,which are less than or equal to –3, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, –3].
Question 8:
Solve the given inequality for real x: 3(2 – x) ≥ 2(1 – x)
3(2 – x) ≥ 2(1 – x)
⇒ 6 – 3x ≥ 2 – 2x
⇒ 6 – 3x + 2x ≥ 2 – 2+ 2x
⇒ 6 – x ≥ 2
⇒ 6 – x – 6 ≥ 2 – 6
⇒ –x ≥ –4
⇒ x ≤ 4
Thus, all real numbers x,which are less than or equal to 4, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 4].
Question 9:
Solve the given inequality for real x
Thus, all real numbers x,which are less than 6, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 6).
Question 10:
Solve the given inequality for real x
Thus, all real numbers x,which are less than –6, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, –6).
Question 11:
Solve the given inequality for real x
Thus, all real numbers x,which are less than or equal to 2, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 2].
Question 12:
Solve the given inequality for real x
Thus, all real numbers x,which are less than or equal to 120, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 120].
Question 13:
Solve the given inequality for real x: 2(2x + 3) – 10 < 6 (x – 2)
Thus, all real numbers x,which are greater than or equal to 4, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (4, ∞).
Question 14:
Solve the given inequality for real x: 37 – (3x + 5) ≥ 9x – 8(– 3)
Thus, all real numbers x,which are less than or equal to 2, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 2].
Question 15:
Solve the given inequality for real x
Thus, all real numbers x,which are greater than 4, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (4, ∞).
Question 16:
Solve the given inequality for real x
Thus, all real numbers x,which are less than or equal to 2, are the solutions of the given inequality.
Hence, the solution set of the given inequality is (–∞, 2].
Question 17:
Solve the given inequality and show the graph of the solution on number line: 3x – 2 < 2x +1
3x – 2 < 2x +1
⇒ 3– 2x < 1 + 2
⇒ x < 3
The graphical representation of the solutions of the given inequality is as follows.
Question 18:
Solve the given inequality and show the graph of the solution on number line: 5x – 3 ≥ 3x – 5
The graphical representation of the solutions of the given inequality is as follows.
Question 19:
Solve the given inequality and show the graph of the solution on number line: 3(1 – x) < 2 (x + 4)
The graphical representation of the solutions of the given inequality is as follows.
Question 20:
Solve the given inequality and show the graph of the solution on number line:
The graphical representation of the solutions of the given inequality is as follows.
Question 21:
Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.
Let x be the marks obtained by Ravi in the third unit test.
Since the student should have an average of at least 60 marks,
Thus, the student must obtain a minimum of 35 marks to have an average of at least 60 marks.
Question 22:
To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.
Let x be the marks obtained by Sunita in the fifth examination.
In order to receive grade ‘A’ in the course, she must obtain an average of 90 marks or more in five examinations.
Therefore,
Thus, Sunita must obtain greater than or equal to 82 marks in the fifth examination.
Question 23:
Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
Let x be the smaller of the two consecutive odd positive integers. Then, the other integer is x + 2.
Since both the integers are smaller than 10,
x + 2 < 10
⇒ x < 10 – 2
⇒ x < 8 … (i)
Also, the sum of the two integers is more than 11.
x + (x + 2) > 11
⇒ 2x + 2 > 11
⇒ 2x > 11 – 2
⇒ 2x > 9
From (i) and (ii), we obtain .
Since is an odd number, x can take the values, 5 and 7.
Thus, the required possible pairs are (5, 7) and (7, 9).
Question 23:
Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
Let x be the smaller of the two consecutive odd positive integers. Then, the other integer is x + 2.
Since both the integers are smaller than 10,
x + 2 < 10
⇒ x < 10 – 2
⇒ x < 8 … (i)
Also, the sum of the two integers is more than 11.
x + (x + 2) > 11
⇒ 2x + 2 > 11
⇒ 2x > 11 – 2
⇒ 2x > 9
From (i) and (ii), we obtain
4.5<x<8.
Since is an odd number, x can take the values, 5 and 7.
Thus, the required possible pairs are (5, 7) and (7, 9).
Question 24:
Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.
Let x be the smaller of the two consecutive even positive integers. Then, the other integer is x + 2.
Since both the integers are larger than 5,
x > 5 … (1)
Also, the sum of the two integers is less than 23.
x + (x + 2) < 23
⇒ 2x + 2 < 23
⇒ 2x < 23 – 2
⇒ 2x < 21
From (1) and (2), we obtain 5 < x < 10.5.
Since is an even number, x can take the values, 6, 8, and 10.
Thus, the required possible pairs are (6, 8), (8, 10), and (10, 12).
Question 25:
The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
Let the length of the shortest side of the triangle be cm.
Then, length of the longest side = 3x cm
Length of the third side = (3x – 2) cm
Since the perimeter of the triangle is at least 61 cm,
Thus, the minimum length of the shortest side is 9 cm.
Question 26:
A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second?
[Hint: If x is the length of the shortest board, then x, (x + 3) and 2x are the lengths of the second and third piece, respectively. Thus, x = (+ 3) + 2≤ 91 and 2x ≥ (+ 3) + 5]
Let the length of the shortest piece be x cm. Then, length of the second piece and the third piece are (x + 3) cm and 2x cm respectively.
Since the three lengths are to be cut from a single piece of board of length 91 cm,
x cm + (x + 3) cm + 2x cm ≤ 91 cm
⇒ 4x + 3 ≤ 91
⇒ 4x ≤ 91 – 3
⇒ 4x ≤ 88
Also, the third piece is at least 5 cm longer than the second piece.
∴2x ≥ (+ 3) + 5
⇒ 2x ≥ + 8
⇒ x ≥ 8 … (2)
From (1) and (2), we obtain
8 ≤ x ≤ 22
Thus, the possible length of the shortest board is greater than or equal to 8 cm but less than or equal to 22 cm.
Question 1:
Solve the given inequality graphically in two-dimensional plane: x + y < 5
The graphical representation of x + y = 5 is given as dotted line in the figure below.
This line divides the xy-plane in two half planes, I and II.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
0 + 0 < 5 or, 0 < 5, which is true
Therefore, half plane II is not the solution region of the given inequality. Also, it is evident that any point on the line does not satisfy the given strict inequality.
Thus, the solution region of the given inequality is the shaded half plane I excluding the points on the line.
This can be represented as follows.
Question 2:
Solve the given inequality graphically in two-dimensional plane: 2x + y ≥ 6
The graphical representation of 2x + y = 6 is given in the figure below.
This line divides the xy-plane in two half planes, I and II.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
2(0) + 0 ≥ 6 or 0 ≥ 6, which is false
Therefore, half plane I is not the solution region of the given inequality. Also, it is evident that any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the shaded half plane II including the points on the line.
This can be represented as follows.
Question 3:
Solve the given inequality graphically in two-dimensional plane: 3x + 4y ≤ 12
3x + 4y ≤ 12
The graphical representation of 3x + 4y = 12 is given in the figure below.
This line divides the xy-plane in two half planes, I and II.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
3(0) + 4(0) ≤ 12 or 0 ≤ 12, which is true
Therefore, half plane II is not the solution region of the given inequality. Also, it is evident that any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the shaded half plane I including the points on the line.
This can be represented as follows.
Question 4:
Solve the given inequality graphically in two-dimensional plane: y + 8 ≥ 2x
The graphical representation of y + 8 = 2x is given in the figure below.
This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
0 + 8 ≥ 2(0) or 8 ≥ 0, which is true
Therefore, lower half plane is not the solution region of the given inequality. Also, it is evident that any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the half plane containing the point (0, 0) including the line.
The solution region is represented by the shaded region as follows.
Question 5:
Solve the given inequality graphically in two-dimensional plane: x – y ≤ 2
The graphical representation of x – y = 2 is given in the figure below.
This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
0 – 0 ≤ 2 or 0 ≤ 2, which is true
Therefore, the lower half plane is not the solution region of the given inequality. Also, it is clear that any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the half plane containing the point (0, 0) including the line.
The solution region is represented by the shaded region as follows.
Question 6:
Solve the given inequality graphically in two-dimensional plane: 2x – 3y > 6
The graphical representation of 2x – 3y = 6 is given as dotted line in the figure below. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
2(0) – 3(0) > 6 or 0 > 6, which is false
Therefore, the upper half plane is not the solution region of the given inequality. Also, it is clear that any point on the line does not satisfy the given inequality.
Thus, the solution region of the given inequality is the half plane that does not contain the point (0, 0) excluding the line.
The solution region is represented by the shaded region as follows.
Question 7:
Solve the given inequality graphically in two-dimensional plane: –3x + 2y ≥ –6
The graphical representation of – 3x + 2y = – 6 is given in the figure below.
This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
– 3(0) + 2(0) ≥ – 6 or 0 ≥ –6, which is true
Therefore, the lower half plane is not the solution region of the given inequality. Also, it is evident that any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the half plane containing the point (0, 0) including the line.
The solution region is represented by the shaded region as follows.
Question 8:
Solve the given inequality graphically in two-dimensional plane: 3y – 5x < 30
The graphical representation of 3y – 5x = 30 is given as dotted line in the figure below.
This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
3(0) – 5(0) < 30 or 0 < 30, which is true
Therefore, the upper half plane is not the solution region of the given inequality. Also, it is evident that any point on the line does not satisfy the given inequality.
Thus, the solution region of the given inequality is the half plane containing the point (0, 0) excluding the line.
The solution region is represented by the shaded region as follows.
Question 9:
Solve the given inequality graphically in two-dimensional plane: y < –2
The graphical representation of y = –2 is given as dotted line in the figure below. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
0 < –2, which is false
Also, it is evident that any point on the line does not satisfy the given inequality.
Hence, every point below the line, y = –2 (excluding all the points on the line), determines the solution of the given inequality.
The solution region is represented by the shaded region as follows.
Question 10:
Solve the given inequality graphically in two-dimensional plane: x > –3
The graphical representation of x = –3 is given as dotted line in the figure below. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
0 > –3, which is true
Also, it is evident that any point on the line does not satisfy the given inequality.
Hence, every point on the right side of the line, x = –3 (excluding all the points on the line), determines the solution of the given inequality.
The solution region is represented by the shaded region as follows.
Question 1:
Solve the following system of inequalities graphically: x ≥ 3, y ≥ 2
x ≥ 3 … (1)
y ≥ 2 … (2)
The graph of the lines, x = 3 and y = 2, are drawn in the figure below.
Inequality (1) represents the region on the right hand side of the line, x = 3 (including the line x = 3), and inequality (2) represents the region above the line, = 2 (including the line = 2).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question 2:
Solve the following system of inequalities graphically: 3x + 2y ≤ 12, x ≥ 1, y ≥ 2
3x + 2y ≤ 12 … (1)
x ≥ 1 … (2)
y ≥ 2 … (3)
The graphs of the lines, 3x + 2y = 12, x = 1, and y = 2, are drawn in the figure below.
Inequality (1) represents the region below the line, 3x + 2y = 12 (including the line 3x + 2y = 12). Inequality (2) represents the region on the right side of the line, x = 1 (including the line x = 1). Inequality (3) represents the region above the line, y = 2 (including the line y = 2).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question 3:
Solve the following system of inequalities graphically: 2x + y≥ 6, 3x + 4y ≤ 12
2x + y≥ 6 … (1)
3x + 4y ≤ 12 … (2)
The graph of the lines, 2x + y= 6 and 3x + 4= 12, are drawn in the figure below.
Inequality (1) represents the region above the line, 2x + y= 6 (including the line 2x + y= 6), and inequality (2) represents the region below the line, 3x + 4y =12 (including the line 3x + 4y =12).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question 4:
Solve the following system of inequalities graphically: x + y≥ 4, 2x – y > 0
x + y≥ 4 … (1)
2x – y > 0 … (2)
The graph of the lines, x + y = 4 and 2x – y = 0, are drawn in the figure below.
Inequality (1) represents the region above the line, x + y = 4 (including the line x + y = 4).
It is observed that (1, 0) satisfies the inequality, 2x – y > 0. [2(1) – 0 = 2 > 0]
Therefore, inequality (2) represents the half plane corresponding to the line, 2x – y = 0, containing the point (1, 0) [excluding the line 2x – y > 0].
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on line x + y = 4 and excluding the points on line 2x – y = 0 as follows.
Question 5:
Solve the following system of inequalities graphically: 2x – y > 1, x – 2y < –1
2x – y > 1 … (1)
x – 2y < –1 … (2)
The graph of the lines, 2x – y = 1 and x – 2= –1, are drawn in the figure below.
Inequality (1) represents the region below the line, 2x – y = 1 (excluding the line 2x – y = 1), and inequality (2) represents the region above the line, x – 2= –1 (excluding the line x – 2= –1).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region excluding the points on the respective lines as follows.
Question 6:
Solve the following system of inequalities graphically: x + y ≤ 6, x + y ≥ 4
x + y ≤ 6 … (1)
x + y ≥ 4 … (2)
The graph of the lines, x + y = 6 and x + y = 4, are drawn in the figure below.
Inequality (1) represents the region below the line, x + y = 6 (including the line x + y = 6), and inequality (2) represents the region above the line, x + y = 4 (including the line x + y = 4).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question 7:
Solve the following system of inequalities graphically: 2x + y≥ 8, x + 2y ≥ 10
2x + y= 8 … (1)
x + 2= 10 … (2)
The graph of the lines, 2x + y= 8 and x + 2= 10, are drawn in the figure below.
Inequality (1) represents the region above the line, 2x + = 8, and inequality (2) represents the region above the line, x + 2= 10.
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question 8:
Solve the following system of inequalities graphically: x + y ≤ 9, y > xx ≥ 0
x + y ≤ 9 … (1)
y > x … (2)
x ≥ 0 … (3)
The graph of the lines, x + y= 9 and y = x, are drawn in the figure below.
Inequality (1) represents the region below the line, x + = 9 (including the line x + y = 9).
It is observed that (0, 1) satisfies the inequality, y > x. [1 > 0]
Therefore, inequality (2) represents the half plane corresponding to the line, y = x, containing the point (0, 1) [excluding the line y = x].
Inequality (3) represents the region on the right hand side of the line, x = 0 or y-axis (including y-axis).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the lines, x + y = 9 and x = 0, and excluding the points on line y = x as follows.
Question 9:
Solve the following system of inequalities graphically: 5x + 4y ≤ 20, x ≥ 1, y ≥ 2
5x + 4y ≤ 20 … (1)
x ≥ 1 … (2)
y ≥ 2 … (3)
The graph of the lines, 5x + 4y = 20, x = 1, and y = 2, are drawn in the figure below.
Inequality (1) represents the region below the line, 5x + 4y = 20 (including the line 5x + 4y = 20). Inequality (2) represents the region on the right hand side of the line, x = 1 (including the line x = 1). Inequality (3) represents the region above the line, y = 2 (including the line y = 2).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question 10:
Solve the following system of inequalities graphically: 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0
3x + 4y ≤ 60 … (1)
x + 3y ≤ 30 … (2)
The graph of the lines, 3x + 4y = 60 and x + 3y = 30, are drawn in the figure below.
Inequality (1) represents the region below the line, 3x + 4y = 60 (including the line 3x + 4y = 60), and inequality (2) represents the region below the line, x + 3y = 30 (including the line x + 3y = 30).
Since x ≥ 0 and y ≥ 0, every point in the common shaded region in the first quadrant including the points on the respective line and the axes represents the solution of the given system of linear inequalities.
Question 11:
Solve the following system of inequalities graphically: 2x + y≥ 4, x + y ≤ 3, 2x – 3y ≤ 6
2x + y≥ 4 … (1)
x + y ≤ 3 … (2)
2x – 3y ≤ 6 … (3)
The graph of the lines, 2x + y= 4, x + y = 3, and 2x – 3y = 6, are drawn in the figure below.
Inequality (1) represents the region above the line, 2x + y= 4 (including the line 2x + y= 4). Inequality (2) represents the region below the line,
x + y = 3 (including the line x + y = 3). Inequality (3) represents the region above the line, 2x – 3y = 6 (including the line 2x – 3y = 6).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question 12:
Solve the following system of inequalities graphically:
x – 2y ≤ 3, 3x + 4y ≥ 12, x ≥ 0, y ≥ 1
x – 2y ≤ 3 … (1)
3x + 4y ≥ 12 … (2)
y ≥ 1 … (3)
The graph of the lines, x – 2y = 3, 3x + 4y = 12, and y = 1, are drawn in the figure below.
Inequality (1) represents the region above the line, x – 2y = 3 (including the line x – 2y = 3). Inequality (2) represents the region above the line, 3x + 4y = 12 (including the line 3x + 4y = 12). Inequality (3) represents the region above the line, y = 1 (including the line y = 1).
The inequality, x ≥ 0, represents the region on the right hand side of y-axis (including y-axis).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines and y– axis as follows.
Question 13:
[[Q]]
Solve the following system of inequalities graphically:
4x + 3y ≤ 60, y ≥ 2xx ≥ 3, xy ≥ 0
4x + 3y ≤ 60 … (1)
y ≥ 2x … (2)
x ≥ 3 … (3)
The graph of the lines, 4x + 3y = 60, y = 2x, and x = 3, are drawn in the figure below.
Inequality (1) represents the region below the line, 4x + 3y = 60 (including the line 4x + 3y = 60). Inequality (2) represents the region above the line, y = 2(including the line y = 2x). Inequality (3) represents the region on the right hand side of the line, x = 3 (including the line x = 3).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question 14:
Solve the following system of inequalities graphically: 3x + 2y ≤ 150, x + 4y ≤ 80, x ≤ 15, y ≥ 0, x ≥ 0
3x + 2y ≤ 150 … (1)
x + 4y ≤ 80 … (2)
x ≤ 15 … (3)
The graph of the lines, 3x + 2y = 150, x + 4y = 80, and x = 15, are drawn in the figure below.
Inequality (1) represents the region below the line, 3x + 2y = 150 (including the line 3x + 2y = 150). Inequality (2) represents the region below the line, x + 4y = 80 (including the line x + 4y = 80). Inequality (3) represents the region on the left hand side of the line, x = 15 (including the line x = 15).
Since x ≥ 0 and y ≥ 0, every point in the common shaded region in the first quadrant including the points on the respective lines and the axes represents the solution of the given system of linear inequalities.
Question 15:
Solve the following system of inequalities graphically: x + 2y ≤ 10, x + y ≥ 1, x – y ≤ 0, x ≥ 0, y ≥ 0
x + 2y ≤ 10 … (1)
x + y ≥ 1 … (2)
x – y ≤ 0 … (3)
The graph of the lines, x + 2y = 10, x + y = 1, and x – y = 0, are drawn in the figure below.
Inequality (1) represents the region below the line, x + 2y = 10 (including the line x + 2y = 10). Inequality (2) represents the region above the line, x + y = 1 (including the line x + y = 1). Inequality (3) represents the region above the line, x – y = 0 (including the line x – y = 0).
Since x ≥ 0 and y ≥ 0, every point in the common shaded region in the first quadrant including the points on the respective lines and the axes represents the solution of the given system of linear inequalities.
Question 1:
Solve the inequality 2 ≤ 3x – 4 ≤ 5
2 ≤ 3x – 4 ≤ 5
⇒ 2 + 4 ≤ 3x – 4 + 4 ≤ 5 + 4
⇒ 6 ≤ 3x ≤ 9
⇒ 2 ≤ ≤ 3
Thus, all the real numbers, x, which are greater than or equal to 2 but less than or equal to 3, are the solutions of the given inequality. The solution set for the given inequalityis [2, 3].
Question 2:
Solve the inequality 6 ≤ –3(2x – 4) < 12
6 ≤ – 3(2x – 4) < 12
⇒ 2 ≤ –(2x – 4) < 4
⇒ –2 ≥ 2x – 4 > –4
⇒ 4 – 2 ≥ 2> 4 – 4
⇒ 2 ≥ 2x > 0
⇒1 ≥ > 0
Thus, the solution set for the given inequalityis (0, 1].
Question 3:
Solve the inequality
Thus, the solution set for the given inequalityis [–4, 2].
Question 4:
Solve the inequality
⇒ –75 < 3(x – 2) ≤ 0
⇒ –25 < x – 2 ≤ 0
⇒ – 25 + 2 < ≤ 2
⇒ –23 < x ≤ 2
Thus, the solution set for the given inequalityis (–23, 2].
Question 5:
Solve the inequality
Thus, the solution set for the given inequalityis.
Question 6:
Solve the inequality
Thus, the solution set for the given inequalityis.
Question 7:
Solve the inequalities and represent the solution graphically on number line: 5x + 1 > –24, 5x – 1 < 24
5x + 1 > –24
⇒ 5> –25
⇒ > –5 … (1)
5x – 1 < 24
⇒ 5< 25
⇒ x < 5 … (2)
From (1) and (2), it can be concluded that the solution set for the given system of inequalities is (–5, 5). The solution of the given system of inequalities can be represented on number line as
Question 8:
Solve the inequalities and represent the solution graphically on number line: 2(x – 1) < x + 5, 3(x + 2) > 2 – x
2(x – 1) < x + 5
⇒ 2x – 2 < x + 5
⇒ 2x – x < 5 + 2
⇒ x < 7 … (1)
3(x + 2) > 2 – x
⇒ 3x + 6 > 2 – x
⇒ 3x + > 2 – 6
⇒ 4x > – 4
⇒ x > – 1 … (2)
From (1) and (2), it can be concluded that the solution set for the given system of inequalities is (–1, 7). The solution of the given system of inequalities can be represented on number line as
Question 9:
Solve the following inequalities and represent the solution graphically on number line:
3x – 7 > 2(x – 6), 6 – x > 11 – 2x
3x – 7 > 2(x – 6)
⇒ 3x – 7 > 2x – 12
⇒ 3x – 2> – 12 + 7
⇒ x > –5 … (1)
6 – x > 11 – 2x
⇒ –x + 2x > 11 – 6
⇒ x > 5 … (2)
From (1) and (2), it can be concluded that the solution set for the given system of inequalities is. The solution of the given system of inequalities can be represented on number line as
Question 10:
Solve the inequalities and represent the solution graphically on number line: 5(2x – 7) – 3(2x + 3) ≤ 0, 2x + 19 ≤ 6x + 47
5(2x – 7) – 3(2x + 3) ≤ 0
⇒ 10x – 35 – 6x – 9 ≤ 0
⇒ 4x – 44 ≤ 0
⇒ 4x ≤ 44
⇒ x ≤ 11 … (1)
2x + 19 ≤ 6x + 47
⇒ 19 – 47 ≤ 6x – 2x
⇒ –28 ≤ 4x
⇒ –7 ≤ x … (2)
From (1) and (2), it can be concluded that the solution set for the given system of inequalities is [–7, 11]. The solution of the given system of inequalities can be represented on number line as
Question 11:
A solution is to be kept between 68°F and 77°F. What is the range in temperature in degree Celsius (C) if the Celsius/Fahrenheit (F) conversion formula is given by
Since the solution is to be kept between 68°F and 77°F,
68 < F < 77
Putting we obtain
Thus, the required range of temperature in degree Celsius is between 20°C and 25°C.
Question 12:
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?
Let x litres of 2% boric acid solution is required to be added.
Then, total mixture = (x + 640) litres
This resulting mixture is to be more than 4% but less than 6% boric acid.
∴2%x + 8% of 640 > 4% of (x + 640)
And, 2% + 8% of 640 < 6% of (x + 640)
2%x + 8% of 640 > 4% of (x + 640)
⇒ 2x + 5120 > 4x + 2560
⇒ 5120 – 2560 > 4x – 2x
⇒ 5120 – 2560 > 2x
⇒ 2560 > 2x
⇒ 1280 > x
2% + 8% of 640 < 6% of (x + 640)
⇒ 2x + 5120 < 6x + 3840
⇒ 5120 – 3840 < 6x – 2x
⇒ 1280 < 4x
⇒ 320 < x
∴320 < x < 1280
Thus, the number of litres of 2% of boric acid solution that is to be added will have to be more than 320 litres but less than 1280 litres.
Question 13:
How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?
Let x litres of water is required to be added.
Then, total mixture = (x + 1125) litres
It is evident that the amount of acid contained in the resulting mixture is 45% of 1125 litres.
This resulting mixture will contain more than 25% but less than 30% acid content.
∴30% of (1125 + x) > 45% of 1125
And, 25% of (1125 + x) < 45% of 1125
30% of (1125 + x) > 45% of 1125
25% of (1125 + x) < 45% of 1125
⇒251001125 + x<45100×1125 ⇒251125 + x<45×1125⇒25×1125 +25x<45×1125 ⇒25x<45×1125 -25×1125 ⇒25x<22500⇒x<900
∴ 562.5 < x < 900
Thus, the required number of litres of water that is to be added will have to be more than 562.5 but less than 900.
Question 14:
IQ of a person is given by the formula
Where MA is mental age and CA is chronological age. If 80 ≤ IQ ≤ 140 for a group of 12 years old children, find the range of their mental age. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8414466381072998, "perplexity": 602.035978636829}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400212959.12/warc/CC-MAIN-20200923211300-20200924001300-00036.warc.gz"} |
https://math.stackexchange.com/questions/2151368/how-can-you-show-that-if-0%E2%89%A4a-n%E2%89%A4b-n-then-if-sum-a-n-diverge-so-does-sum-b | # How can you show that if $0≤a_n≤b_n$ then if $\sum a_n$ diverge so does $\sum b_n$
if $0≤a_n≤b_n$ then if $\sum a_n$ diverge so does $\sum b_n$
My approach :
Let $A_n = \sum_{1 \to k} a_k$ and $B_n = \sum_{1 \to k} b_k$
$A_n$ is an increasing sequence, so is $B_n$ We also have $A_n≤B_n$ for all n
Since $A_n$ diverges,$B_n$
I'm stuck here. Any help?
• This is Comparison Test, of which the proof is well known – Juniven Feb 19 '17 at 13:11
## 1 Answer
Hint: As $A_n$ diverges to $\infty$ as increasing sequence, by definition for every $k \in \mathbb R$ exists $n \in \mathbb N$, such that $\forall m \geq n$ we have $A_m > k$. What does this imply for $B_m$? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.989890992641449, "perplexity": 133.98933020027602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250606975.49/warc/CC-MAIN-20200122101729-20200122130729-00523.warc.gz"} |
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