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http://jjhale.com/blog/2010/10/pupil-geometry/
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# Pupil Geometry
I’ve been trying to reduce the number of free variables in the mapping between the position of the pupil in a eye-camera image and the point of gaze on a screen, given the head location of a person. Previously I was just fitting an ellipse to the pupil and using the location of its centre. This method did not use all of the information I’d just obtained about the ellipse. In this post I explore one theoretical method of using the ellipse parameters.
We can approximate the eyeball as a sphere with a circle on which corresponds to the pupil. We’ll describe the size of the pupil by the radius of the sphere $r$ and the angle $rho$ between the lines from the centre of the sphere to the centre of the pupil and to any point on the edge of the pupil, see figure 1.
Figure 1 – The major axis
If the eye is looking directly into the camera the pupil will appear to be circular (assuming minimal or corrected camera lens distortion). We will define the center of the pupil in the image plane in this case to be $(x_o, y_o)$. Since the pupil will appear to be a circle all its axes are equal. The red and the blue lines in figure 1 have the same length. This length $a$ can be calculated by:
$a = 2 \sin(\rho)$.
Figure 2 – The rotated eye
As the eye turns away for the camera the pupil image will become elliptical. Consider the eye following some line on the image plane which passes through $(x_o, y_o)$ by rotating the eye through an angle $\tau$ in the plane defined by our line and the center of rotation of the eye. The pupil will now appear to be an ellipse with its major axis the same length as before (the red line) but with a shorter minor axis.
The length of the minor axis $b$ can be calculated from the internal angles of the quadrilateral to calculate the angle $\lambda$. Note that $\lambda$ is the angle BED and is equal to the angle BFC
$2 \pi = \frac{\pi}{2} + \frac{\pi}{2} + \tau +\frac{\pi}{2} + \lambda$
$\lambda = \frac{\pi}{2} - \tau$
so we have
$b = 2sin(\rho) sin(\lambda)$
We consider it in 3D by just rotating orthogonally to the plane of rotation of $\tau$.
Figure 3 – Blow up of the angles from Figure 2 and  rotation by the second angle
These two rotations can move to pupil to any place on our model eyeball. There are two nice properties of this circle on a sphere. Firstly the line passing through the minor axis of the ellipse in the image passes through the centre of the eye in the image $(x_o, y_o)$. This fact means that the center of the eye can be determined from two images of the pupil in different positions by finding the intercept of the two lines defined by the minor axis.
This property allows us to calculate the angle $\gamma$ from the angle of the minor axis and the position of the pupil relative to $(x_o, y_o)$
The second useful property is that can calculate gamma from the ratio of the minor axis.
We can will consider the ratio of the major and minor axes. This measure will be invariant as the pupil size varies.
$\frac{a}{b} = \frac{2 sin(\rho)}{2sin(\rho) sin(\lambda)} = \frac{1}{sin(\lambda)}$
From this we can directly calculate $\tau$ as follows:
$\tau = \frac{\pi}{2} - sin^{-1}(\frac{b}{a})$
If we have at least two images of the eye we can describe the rotation of the eye by $(\tau, \gamma)$. These angles can be used to generate the unit vector $g'$ where
$g' = \left( \begin{array}{c} cos(\gamma) sin(\tau) \\ sin(\gamma) sin(\tau) \\ cos(\tau) \end{array} \right)$
These unit vectors then have to be rotated to align with the real world. The rotation matrix can be described by three scalars.
### Further work
In this analysis we assume that the eye is a sphere, rotating about it center: a fixed point relative to the camera and that we are able to accurately extract the ellipse describing the pupil. We do not consider the effects of noise, error or perspective. I’ve also assumed that the edge of the pupil is always on the edge of the sphere – when in fact it is more like a circle on a plane cutting through the eye (ouch). I have a feeling that this is not a problem as it is equivalent to the eyeball changing size as the pupil changes size.
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2016-07-28 14:24:00
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http://soft-matter.seas.harvard.edu/index.php?title=Profile_of_a_large_drop&oldid=5307
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Profile of a large drop
Zach Wissner-Gross
Discussion
Figure 1: Drop profiles for various contact angles and using the physical constants given in class.
In class we derived the profile $z(x)$ of a large drop as being:
$\sigma_{lv}(\frac{1}{\sqrt{1+\dot{z}^2}}-\cos\theta_e)=\frac{1}{2}\rho g(2ez-z^2)$
where we had defined $e$ as being the maximum height of the drop:
$e=2\kappa^{-1}\sin{\frac{\theta_e}{2}}$
and $\kappa=\sqrt{\frac{\rho g}{\sigma_{lv}}}$ was the inverse capillary length.
To visualize what the edge of a large drop then looks like, you can use the following two MATLAB programs (see below), which will solve and plot the above differential equation. Figure 1 also shows what some of these plots look like.
In class we had a brief discussion as to the nature of the drop's periphery on superhydrophobic surfaces. In the superhydrophobic limit (i.e., as $\theta_e$ approaches 180 degrees), will the edge of the drop have a semicircular shape? The shape is apparently not semicircular at 179 degrees, since the horizontal range of the drop is only 1 mm, while it vertically extends to a height of
$2\kappa^{-1}\approx 3.5 \text{mm}$
Nevertheless, the bottom left corner of the 179 degree profile does resemble a quarter-circle.
MATLAB code
To visualize your own large drop profiles, copy and paste the following two MATLAB scripts into two corresponding files. Then run dropshape.m -- it will automatically plot the profile. You can adjust the contact angle by variying the parameter "angle" in dropshape.m.
dropshapeODE.m
function df = dropshapeODE(dt,y)
global angle;
theta = pi/180 * angle;
sigma = 30*10^(-3);
rho = 10^3;
g = 9.8;
kappa = sqrt(rho*g/sigma);
E = 2/kappa*sin(theta/2);
X = cos(theta) + kappa^2/2*(2*E*y-y^2);
df = sqrt(1/X^2-1);
dropshape.m
global angle;
angle = 100;
theta = pi/180 * angle;
sigma = 30*10^(-3);
rho = 10^3;
g = 9.8;
kappa = sqrt(rho*g/sigma);
E = 2/kappa*sin(theta/2);
y0 = [0];
[Xout,Yout] = ode45('dropshapeODE',[0 5*kappa^(-1)],y0);
dydx = zeros(size(Xout)-1);
if theta > pi/180 * 90
dydx = (Yout(2:end)-Yout(1:end-1))./(Xout(2:end)-Xout(1:end-1));
[p q] = max(dydx);
for i = 1:q
Xout(i) = 2*Xout(q)-Xout(i);
end
Xout = Xout-Xout(1);
end
figure(1);
clf;
hold on;
plot(Xout,Yout);
axis equal;
A = axis;
axis([A(1) A(2) 0 A(4)]);
box on;
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2020-01-19 09:17:10
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http://proteinsandwavefunctions.blogspot.ru/2015/03/
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## Monday, March 30, 2015
### Definition of Gly HA2 and HA3 nomenclature in PDB and BMRB files
Note to self:
HA2 is pro-R and HA3 is Pro-S. HA2 corresponds to HA in the other amino acids
Thanks to +Lars Bratholm for figuring this out.
## Saturday, March 28, 2015
### Selecting full residues within a certain distance of another residue or atom in PyMOL
Note to self:
To select all atoms in a residue (plus any HETATMs) that is within 3 Å of any atom in residue 63 type:
select br. all within 3 of resi 63
To select all atoms in a residue (plus any HETATMs) that is within 3 Å of the CA atom in residue 63 type:
select br. all within 3 of 63/CA
If you want to exclude the HETAMS type "pol." (short for polymer) instead of "all".
if you want to name the selection type
select Ala63, br. all within 3 of resi 63
## Thursday, March 26, 2015
### Second reviews of the PCCP paper - almost there
PCCP writes: "I am pleased to inform you that your revised manuscript has been recommended for publication in Physical Chemistry Chemical Physics subject to further revision in line with the attached reports."
Referee: 1
However, I am still concerned about the recommendations for treating low frequency vibrations. The harmonic oscillator approximation is not appropriate for low frequencies because the entropy term goes to infinity. As recommended by Grimme (2012), a hindered / free rotor may be a better treatment (for related methods and discussion see TRUHLAR, DG, J COMP CHEM 1991, 12, 266-270 DOI: 10.1002/jcc.540120217 and McClurg, RB, Flagan, RC, Goddard, WA JCP 1997, 106, 6675-6680 DOI: 10.1063/1.473664)
My comment about explicit water molecules also concerned entropy. While including explicit water molecules may increase the CPU time, this can be overcome. What extra CPU time cannot overcome is the fact that the explicit waters in reality can exchange with the bulk water increasing their entropy compared to a RRHO calculation of their entropy. It is best to include explicitly only tightly bound
Immediate reactions
Point 1:
* The paper lists options for how to treat low frequency vibrations but there is not a specific recommendation because the issue as it applies to binding free energies has not been thoroughly compared.
* Grimme's approach is not derived from first principles and is therefore not a priori better than other approaches. It will in some cases treat modes that are clearly stretch vibration as a free rotor
* The free rotor entropy also goes to infinity as the frequency approaches zero.
Point 2:
* Any such effect is included implicitly in the parameterization of the solvation free energy of H2O.
* Any error in this parameterization is largely cancelled by using the water cluster approach recommended in the paper.
* Bryantsev et al. have shown that using the water cluster approach leads to smooth convergence of solvation free energies of H+ and Cu2+ that are in good agreement with experiment.
## Tuesday, March 24, 2015
### Should I become an editorial board member at Scientific Reports?
This is one of those "trying to get my head around something" posts.
Pros
* It's open access
* Uses CC-BY
* no perceived importance criteria
Cons
* It's too expensive. The APC is of $1495 and no option for waivers +PeerJ has shown it can be done for less. If you need to charge more you are either doing it wrong or you are too greedy. * Because of this, I wouldn't publish there. Anything bio-related would go to +PeerJ and the very few papers that aren't would to to PLoS ONE, usually with a full or partial APC waiver (unless it is directly related to an active grant). * Why should I work for Macmillan? On the face of it, I should decline the invitation. Comments, as always, welcome. This work is licensed under a Creative Commons Attribution 4.0 ## Sunday, March 15, 2015 ### Predicting amide nitrogen (15N) chemical shifts in proteins: where do we go from here? source CC As I wrote over on CCH: "The H-N HSQC spectrum of protein amide groups is one of the most frequently recorded experiments in protein NMR. 15N labeling is comparatively inexpensive and the spectrum can be acquired in a relatively short period of time." In fact I have sometimes seen HSQC spectra used "simply" as a proof that the protein is folded and then "thrown away". But can we extract more information? For example can we deduce the structural changes due to mutations or ligand binding? Can we discriminate between two similar predicted structures as part of protein structure determination? Unfortunately predicting amide N chemical shifts in proteins appears to be very difficult Zhu, He and Zhang predicted amide N chemical shifts at the B3LYP/6-31G(d,p) level of theory (using an implicit solvation model) for Protein G and ubiquitin with mean unsigned errors (MUEs) of 4.8 and 8.0 ppm, respectively. And this after a linear fit to experiment, with$R^2$values of 0.71 and 0.69. For comparison the MUEs for CA are 2.1 and 1.4 ppm, with$R^2$values of 0.79 and 0.88. This was done using a single AMBER optimized structure. Exner et al. used 500 structures from a 5 ns explicit solvent MD of the HA2 domain to predict amide N chemical shifts at the B3LYP/6-31G(d). The MUE was 14.6 ppm and the$R^2$value was 0.81. The corresponding value for C atoms (note: not just CA) is 5.2 and 0.99 and the$R^2$value for aliphatic carbons is 0.99 (couldn't find separate MUE). Has an amide N chemical shift ever been predicted accurately from first principles? Short answer: No - as far as I can tell. A few N chemical shifts have been measured in the gas phase and these can be reproduced to within 1.5 - 2.0 ppm using CCSD(T)/CBS//CCSD(T)/cc-pVTZ. Here vibrational contributions and basis set extrapolation were key with contributions as large as 5 ppm. Unfortunately, none of the molecules contained an amide group. While amide N chemical shifts have not been measured in the gas phase, there is plenty of data in solution - including data for small prototypical amides such as formamide, acetamide, and N-methyl acetamide that are within reach of CCSD(T)/CBS. However, one conclusion from this data (and similar data for ammonia) is that solvent effects can be very large (5-15 ppm) and that a low dielectric solvent is not representative of the gas phase when it comes to N chemical shifts. So (1) one cannot simply use a continuum representation of the solvent and (2) one might as well go for aqueous solution since that is most relevant for proteins and not necessarily harder than other solvents. Exner, Möller, and co-workers attempted a N chemical shift-prediction for N-methyl acetamide in aqueous solution based on 5000 explicit solvent CPMD snapshots, but the results for N is ambiguous since predicting the chemical shift requires an equal treatment of the reference, which they didn't do (se more on this below). However, one very encouraging result was that, for example, the H(N) chemical was predicted to be 5.2 ppm higher than the neighboring H(C), which is in excellent agreement with the experimental value of 5.1 ppm. For comparison the corresponding predicted value in the gas phase is 2.7 ppm. Where do we go from here? It's important to figure out what it takes to predict accurate amide N chemical shifts in aqueous solution. One option is small model systems and here we either have to deal with the reference problem or look at molecules with more than one N atom. Another is to look more closely at select protein residues. Either way, we need to get some basics straight so we are not fumbling in the dark. 1. The Basics a. CCSD(T)/CBS + vibrational correction (a la Teale et al.) benchmark N chemical shift values for formamide, acetamide, and N-methyl acetamide. Moon and Case have done this for the latter, but using a MP2/6-31G(d) structure, which I am not sure is good enough. b. Corresponding benchmark values for the effect of hydrogen bonding (e.g. to water molecules) and dihedral angle changes. Can we assume that the vibrational effects are unchanged? 2. Small models a. Internal reference. Look for experimental data for small amide containing compounds with only one conformation and no significant pH or tautomer effects. I had a very quick look at some tables and found one (not ideal) candidate: hydantoin. Kricheldorf has measured a difference in N chemical shift of 63.9 ppm in 20% w/w aqueous solution. It not ideal because neither N is strictly speaking in an amide group. Ideal model systems would be substituted diglycolyldiamides - like this study but in aqueous solution. b. External reference. Exner, Möller and co-worker predicted a N chemical shift for N-methyl acetamide (NMA) of 159.99 ppm, which they compared to an experimental value of 113.8 ppm. However, the computed shift was relative to computed gas phase ammonia, while the experimental value was referenced to liquid ammonia. One solution is to try to make a similar prediction for liquid ammonia. Another is to compare two aqueous phase prediction: e.g. repeat the study for acetamide and compare the two N chemical shifts to experimental measurements. Yet another approach is this one: the experimental chemical shielding difference between gas phase and liquid ammonia is 19.1, so the predicted chemical shift relative to liquid ammonia is 140.9, which is already a little better. The prediction was done with B3LYP/cc-pVTZ and that certainly contains some error. To find out how much we need CCSD(T)/CBS values which we don't have (Moon and Case don't appear to give the numerical values for NMA). The best I would find so far was MP2/cc-pVQZ which gives a NMA gas phase N chemical shift relative to gas phase ammonia of 114.97 ppm, which is 11.45 ppm lower than the corresponding B3LYP/cc-pVTZ values. So 140.0 - 11.4 = 129.6 ppm is a better estimate. Finally, because of the CPMD the predicted amide chemical shielding included some vibrational averaging, but ammonia does not. Teale et al. predict that vibrational effects lowers the chemical shield of ammonia by 8.7 ppm, so 129.6 - 8.7 = 120.8 ppm, which is starting to approach the experimental value of 113.8 ppm. If we had the CCSD(T)/CBS value and vibration corrections for NMA it is not inconceivable that we could get closer to experiment. The corresponding result for NMA based on the approach applied typically applied to proteins (classical MD using fixed bond lengths and B3LYP/6-31G(d) chemical shifts) is 77.7 ppm, so the CPMD study already tells us something about why this usual approach taken for proteins might fail. 3. Select protein residues I also think that the approach we took for amide protons could yield some insights for amide N atom: i.e. build some detailed structural models (including high level geometry optimizations) of protein regions with well defined secondary structure and (at least initially) as far away from the solvent and charged groups as possible. De Dios, Pearson and Oldfield took a similar approach and got promising results for relative amide N chemical shifts of select Val residues in SNase. If we consistently can get accurate (e.g. within 2 ppm) predictions for amide N chemical shifts, we can use the approach to try to understand the outliers. This work is licensed under a Creative Commons Attribution 4.0 ## Sunday, March 8, 2015 ### Experimental chemical shifts of nitrogen (15N) atoms in formamide, acetamide, and N-methyl acetamide I recently lamented the lack of experimental nitrogen chemical shifts for small prototypical amides such as formamide, acetamide, and N-methylacetamide. I did some digging and here is what I have came up with so far. Before we get to the amides this very interesting paper show the effects of various solvents on the N chemical shift of ammonia relative to the gas phase value. Lower dielectric solvents do not necessarily result in a more gas-phase like chemical shifts. So I only discuss chemical shifts measured in water for the amides since that is most relevant for protein NMR. I use anhydrous liquid ammonia at 25$^o$C as the reference for the same reason. Formamide. Martin et al. measured a N chemical (downfield) shift of 260.0 ppm relative to an external sodium nitrate reference solution. This translates to a chemical shift of (376.5 - 260.0 =) 116.5 ppm. This value is for 0.2 molar fraction formamide in water. Martin et al. show a plot of the chemical shift wrt to mole fraction and the "nitrate" chemical shift appears to drop to about 259 ppm due further dilution, so 117.5 ppm is perhaps a better value. As I've mentioned in this post other values have been measured. Also, the proton chemical shifts have been measured in the gas phase (thanks to +Anders Steen Christensen for the tip. Acetamide. Kricheldorf and Haupt measured a (upfield) N chemical shift of -263.3 ppm relative to nitrate in a 25 wt% sodium nitrate reference solution. This translates to a chemical shift of (376.2 - 263.3 =) 112.9 ppm. The value is for 0.001 M and is within 0.01 ppm of the value measured for 0.01 M and within 0.4 ppm of the value measured for 4.0 M. N-methyl acetamide (NMA). Kricheldorf measured a (upfield) N chemical shift of -263.4 ppm relative to nitrate in a 30 wt% external sodium nitrate reference solution. This translates to a chemical shift of (376.2 - 263.4 =) 112.8 ppm. The value is for 1.5 M at pH 7. Marchal and Carnet have measured a (upfield) N chemical shift of -266.4 ppm relative to neat nitromethane. This translates to a chemical shift of (380.2 - 266.4 =) 113.8 ppm. The value is for 2.0 M. Finally, Exner et al. have recently measured a value of 114.2 ppm for NMA dissolved in phosphate buffer (H2O/D2O = 95:5, 50 mM phosphate, 150 mM NaCl, pH 8) to a final concentration of 50 mM. The same paper also describes a prediction of the N chemical shift, which was followed by this study a year later. NMA has the additional complication of having two conformations E and Z where the methyl group is opposite and next to the O atom, respectively. Fritz et al. have shown that 98.5% of NMA is in the biologically relevant Z conformation in DMSO. Referencing for QM calculations Can we reproduce these values computationally? One of the many problems to address is the referencing. The experimentally measured chemical shielding of liquid ammonia is known (244.4 ppm) so one option is$\delta_x = \sigma_{\mathrm{NH_3(liq)}} - \sigma_x^{\mathrm{comp}} $However, to increase error cancellation one could use another molecule for which gas phase N chemical shielding has been measured, e.g. ammonia (263.5 ppm):$\delta_x = \sigma_{\mathrm{NH_3(gas)}}^{\mathrm{comp}} - \sigma_x^{\mathrm{comp}} + \left( \sigma_{\mathrm{NH_3(liq)}} - \sigma_{\mathrm{NH_3(gas)}} \right)\$
One could also use other molecules and take an average.
Finally, I can recommend this interesting paper entitled Nitrogen NMR and Molecular Interactions
## Wednesday, March 4, 2015
### Reviews of PCCP paper
The reviews of my PCCP paper arrived on Feb 25th and I forgot to post them. I have started working on the rebuttal. Comments welcome.
Referee: 1
This perspective article discusses the many factors that can make small but important contributions to absolute host-guest binding energies computed with electronic structure methods. This is a valuable contribution to the literature in that it gathers in one place various aspects of solvation and thermodynamics for binding energies. There are a number of items that I would like to see addressed before this paper is accepted.
(a) State whether ZPE is included in E_gas or in G_gas,RRHO. I presume in the former, the latter accounting for only the thermal corrections using RRHO. (however, when I see RRHO, I automatically think it includes ZPE)
(b) Recently, some have advocated calculating the gas-phase thermochemistry at an elevated pressure to simulate the decreased translational freedom encountered in solution. Does this affect the thermal and entropy corrections beyond a simple change in volume? Is this something that should be encouraged?
(c) For delta G_solv(H+), I find it very risky to compute this directly using explicit solvent molecules. Better to put the H+ on another molecule of known pKa and use continuum solvation to compute the energy difference.
(d) Other ions: If the ion concentrations are high (experimentally), is it necessary to consider the effect of ionic strength on the activity in calculating the binding energies? Is this a consideration in the computational simulations as well?
(e) Including more than a few explicit waters in a binding energy calculation can also mess up the entropy term, since the positions of these extra water molecules are not sampled adequately. However, this should not be a problem if only a few tightly bound waters are included. It would be good to add a comment.
(f) Minor matters:
Pg 3: The sentence before “Molecular Thermodynamics” seems out of place. Should it be part of the previous paragraph?
Pg 4: “volume of and ideal gas”
Pg 6: “double-differencing” central difference of double numerical difference?
Pg 6: “better to pretend that the imaginary frequency is real” – a very bad idea if the frequency is small, since the entropy blows up. Maybe better to “pretend” that it is a free rotor, which has a well defined entropy.
Pg 9: “van der Waals interactions with the solvent” -> “van der Waals and dispersion interactions with the solvent” (just so there is no misunderstanding)
Pg 10 - Eq 21 and the sentence after it: V_solv or Delta V_solv? (as in Eq 22 and 23)
Pg 11: “numerical instability”? (is this more a matter of numerical noise due to the discretization of the surface elements of the cavity leading to discontinuities in the PES that are problematic for the optimizer – a number of codes have overcome this problem)
Pg 12: For an interesting paper on thermodynamic cycles and solution phase optimization, see DOI: 10.1039/c4cp04538f)
Pg 15: “if protonations states”
Pg 15 – 19, abstract: The first person is normally not used in scientific writing
Referee: 2
This is a perspective about using QM methods to estimate ligand-binding free energies, using approaches originating from QM-cluster studies of enzyme reactions. The perspective is concentrated on the treatment of multiple conformations and pKa effects, although other effects are also mentioned. It is somewhat surprising that the author has not published a single paper on the subject of the perspective; consequently numerical results are very few and discussion is much concentrated on a few publications of the Grimme group. Still, the subject is of general interest. However, the scope needs to be better defined and all methods and formulae need to much better defined before the paper can be accepted.
1. In general, the author should go through all equations and ensure that all terms are defined.
2. The scope of the perspective must be better defined. QM methods have been used for over 10 years for ligand binding to proteins, typically using MM/PBSA-like approaches (cf. publications and reviews by Merz, Hobza and Ryde, for example). Likewise, the author ignores attempts to using QM post-processing FEP calculations.
3. The introduction should start with a more general discussion of available methods to calculate ligand-binding energies and why QM is needed.
4. Different types of QM methods should be described and it should be explained why the author concentrate on DFT and SQM methods.
5. What is TPSS27 (p.3)
6. HF-3c should be explained
7. “by fitting against ∆∆H_f,gas to ∆E_gas values” does not make sense to me.
8. Regarding the low-frequency vibrations, Grimme uses a scaling function so that there are smooth transition between vibrations and free rotation (making the actual value of the frequency unimportant below ~100 cm-1). Truhlar et al. have used a similar approach (but not for ligand binding).
9. The prime problem with conformations is not to use Eqn. 7 but to find all low-energy conformations, including the global minimum.
10. What is the accuracy of computationally estimated pKa values (i.e. what does “fairly accurately” mean quantitatively)? Is it enough for ligand binding?
11. The meaning of dG_solv(H+) should be explained and in general the difference between upper- and lower-case delta should be clarified.
12. What do the over-bar X and L in Eqns 14 and 16, etc. signify?
13. It should be “van der Waals”.
14. I think the selection of the reference state is primarily determined by what experimental results you want to reproduce.
15. A short description of available CM approaches would be appropriate, referring to Table 1.
I suppose you need to specify the variant of PCM also (IEF or C or what?).
16. COSMO-RS is parametrized for many more levels of theory than BP/TZVP.
17. References to the accuracy of solvation energies should be given. In the SAMPL competitions, appreciably worse results are typically seen.
18. I think problem with converging solution-phase optimizations is a problem special to the implementation in Gaussian. With COSMO in Turbomole, no such problems are ever seen.
19. A recent update to Ho et al. 2010 is PCCP 2015, 17, 2859.
20. Since the author only considers water solvation, he should consider changing “solvation” to “hydration”.
21. What is meant by “(dispersion and free energy contributions to the binding free energy” on p.19.
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2017-08-17 08:02:19
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https://www.nature.com/articles/s41467-019-08445-1?error=cookies_not_supported&code=84622a04-080f-47f1-8f8a-81c25bfded52
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# Kink far below the Fermi level reveals new electron-magnon scattering channel in Fe
## Abstract
Many properties of real materials can be modeled using ab initio methods within a single-particle picture. However, for an accurate theoretical treatment of excited states, it is necessary to describe electron-electron correlations including interactions with bosons: phonons, plasmons, or magnons. In this work, by comparing spin- and momentum-resolved photoemission spectroscopy measurements to many-body calculations carried out with a newly developed first-principles method, we show that a kink in the electronic band dispersion of a ferromagnetic material can occur at much deeper binding energies than expected (Eb = 1.5 eV). We demonstrate that the observed spectral signature reflects the formation of a many-body state that includes a photohole bound to a coherent superposition of renormalized spin-flip excitations. The existence of such a many-body state sheds new light on the physics of the electron-magnon interaction which is essential in fields such as spintronics and Fe-based superconductivity.
## Introduction
Spin-flip excitations, including single-particle Stoner and collective spin-wave excitations (magnons), schematically shown in Fig. 1a, are fundamental for the description of ferromagnetic materials1,2,3,4. The interaction between conduction electrons and magnons is critical for fundamental physical properties, such as the temperature dependence of the resistivity5 and magnetotransport6. It also plays an essential role in models that describe the laser-induced ultrafast demagnetization7. On the more applied side, electron-magnon interactions are the basis of the field of magnonics, which offers prospects of faster and more energy-efficient computation8. While magnons have a well defined dispersion relation with excitation energies up to a few hundred meV, the Stoner excitations form a quasi-continuum in the magnetic excitation spectrum and their excitation energies are typically in the order of a few eV (Fig. 1b). Although the spin of majority electrons is more likely flipped than that of minority electrons (Fig. 1b), a minority spin flip can have a strong effect on the electronic dispersions, as this article reveals.
Electron dispersion anomalies, such as kinks, are regarded as signatures of an electron-boson interaction, expected to occur at the scale of the boson energy (typically up to few hundred meV)9,10,11,12,13. In the case of superconducting materials, the appearance of kinks is a priceless clue pointing to the origin of the electron-electron coupling14,15,16,17. In ferromagnetic materials, kinks observed by photoemission at binding energies of 100–300 meV were interpreted as originating from the electron-magnon interaction because the involved energy scale was regarded as too large to reflect electron-phonon interaction12,13,18. Up to now, this interpretation was merely a suggestion, as no ab initio method has been able so far to reproduce magnon-induced kinks.
In this work, we have experimentally mapped the electronic band structure of an Fe(001) thin film and identified a characteristic kink located 1.5 eV below the Fermi level, which can be reproduced by ab initio calculations based on a diagrammatic expansion of the self-energy, a quantity that describes the deviation of the quasiparticle spectrum from the ‘undressed’ electron picture. This GT self-energy (Fig. 1c) accounts for the coupling of electrons or holes (Green function G) to the correlated many-body system through the creation and absorption of spin excitations (T matrix), taking into account the full nonlocal excitation spectrum with magnons and Stoner excitations treated on an equal footing. The T matrix, which describes the correlated motion of an electron-hole pair with opposite spins, is a mathematically complex quantity because it depends on four points in space (two incoming and two outgoing particles) and time (or frequency). The method is a first-principles approach, therefore, apart from the atomic composition, no additional parameters are used. It naturally takes into account nonlocal electron correlations (momentum dependence of self-energy), which were recently experimentally shown to be important for 3d ferromagnets19. Details of the theory are presented in refs. 20,21.
## Results
### Momentum-resolved photoemission
To get experimental access to the bulk electronic structure of Fe, we have used a thin Fe film (38 ML) deposited on a Au(001) single crystal. The photoemission measurements have been performed at the NanoESCA beamline of Elettra, the Italian synchrotron radiation facility, using a modified FOCUS NanoESCA photoemission electron microscope (PEEM) in the k-space mapping mode22. The experiment is shown schematically in Fig. 1d. We will discuss here the results obtained using s-polarized light of = 70 eV, which according to the free-electron final state model induces transitions from the initial states located close to the Γ point of the bulk Brillouin zone. Complementary results of the measurements with p-polarized light, as well as spin-resolved measurements are shown in the Supplementary Figs. 1 and 2 and discussed in the Supplementary Notes 1 and 2.
Figure 2a presents a comparison between experiment and a mean-field band structure of bulk Fe obtained with the local-spin-density approximation (LSDA)23 of density functional theory (DFT). The blue (red) labels: Δ1, Δ2, etc. identify the symmetry of the orbital part of the wavefunctions for minority (majority) states along Fe(001) direction24. Here, we neglect the spin-orbit coupling (SOC), as we are interested in the general shape of the bands within a wide binding energy range. The effect of the SOC on the Fe(001) electronic states close to the Fermi level was discussed earlier25. In the Supplementary Fig. 1, we also compare the experimental spectrum to GW calculations26,27,28, which include quasiparticle renormalization effects beyond DFT. The identification of the experimentally observed electronic states is possible thanks to the results of the spin-polarized measurements (Supplementary Fig. 2) and the consideration of the dipole selection rules that depend on the photon polarization (Supplementary Note 1). Specifically, we identify a minority band of Δ2 symmetry, which is particularly sharp, especially in contrast to the majority bands (e.g., Δ1), which become broad and diffuse directly below the Fermi level (Fig. 2a). Importantly, in contrast to the prediction of LSDA and GW calculations, the experimentally observed minority band Δ2 exhibits a peculiar anomaly near a binding energy of Eb = 1.5 eV marked by arrows in Fig. 2a.
### Ab initio calculations
Figure 2b shows the theoretical spectral function as obtained from the GT calculation summed over the spins on the left and only for the minority spin on the right. We observe a strong renormalization and lifetime broadening of the band structure, in particular, for the majority bands. For example, the majority Δ2 band loses its quasiparticle character completely below Eb = 1 eV, which explains why this band is not visible in the experiment (Fig. 2a) despite favorable dipole selection rules. Such spin dependence of the electron-electron correlation effects is in line with earlier theoretical reports29,30,31 and experimental findings19,32. In the minority channel, the calculated band dispersions remain relatively sharp. However, the minority Δ2 band exhibits an anomaly that seems to coincide with the kink observed in the photoemission experiment.
### Experiment vs. theory
In order to compare the theoretical prediction with the experimental measurement, we have fitted experimental momentum distribution curves (MDC) with a Lorentzian function on a linear background (Fig. 2e) for binding energies between 0.8 and 2.0 eV and superimposed the fitted peak positions on the experimental and theoretical spectral functions in Fig. 2c, d. Both curves, the experimental and the theoretical one, strikingly show the band anomaly at roughly the same energy and momentum. For further analysis, we compare in Fig. 2f the experimental (circles) and theoretical spectral functions (lines) taken at k = 1.3 Å−1. We observe a characteristic double-peak structure in both, which indicates a transfer of spectral weight from one branch of the quasiparticle band to another resulting in the appearance of a kink. The Lorentzians fitted to the experimental dispersion can be used to derive the experimental self-energy, which compares remarkably well with the calculated self-energy (see Supplementary Note 3 for the discussion of the self-energy and Supplementary Fig. 3 for the comparison between theoretical and experimental self-energy).
## Discussion
Interestingly, the binding energy at which the anomaly appears is much higher than what one would normally expect for electron-magnon scattering. Specifically, it is larger than typical magnon energies. The reason for this is twofold. First, self-energy resonances appear not at the boson energy, but rather at the sum of two energies: the boson (e.g., magnon) energy (from T) and a single-particle energy (from G). Second, we consider a coupling of a propagating minority spin hole to excitations that, due to spin conservation, would have to carry a spin of +1, which is just opposite to the respective spin transfer of magnon excitations. The coupling is thus predominantly with renormalized Stoner excitations, whose energies are typically larger than magnon energies. In fact, Fig. 1b (left panel) shows a particularly strong resonance around 0.7 eV close to the H point, which, together with a peak in the majority density of states of bulk iron (Supplementary Fig. 4 and Supplementary Note 4) at 0.8 eV, produces a self-energy resonance at around 1.5 eV. This resonance is a manifestation of a broadened many-body state that consists of a majority hole and a superposition of correlated (electron-hole) Stoner excitations and that, by interaction with the minority band, is ultimately responsible for the appearance of the band anomaly. In other words, a minority photohole is created in the photoemission process, which becomes dressed with electron-hole pairs of opposite spins (Stoner excitations), and flips its spin in the process. This broadened many-body scattering state has a resonance at around Eb = 1.5 eV in bcc iron. It bears similarities to the spin polaron33 in halfmetallic ferromagnets and to the Fermi polaron34 in ultracold fermion gases.
By examining other k-space directions, we find that the appearance of the high-energy kink is very sensitive to the value of the self-energy, and therefore strongly dependent on the k direction (see Supplementary Note 5 and Supplementary Fig. 5).
The high-energy kink identified in our work seems similar to the results of F. Mazzola et al.35,36, who found a kink in the σ band of graphene close to Eb = 3 eV. In this case, however, the authors attribute the kink to the strong electron-phonon coupling near the top of the σ band, which effectively places the kink exactly at the boson energy. It is also important to note that some high-energy anomalies, observed especially for cuprates37,38,39, were later interpreted to be the result of the photoemission matrix elements40,41,42. We can rule out such an explanation in our experiment, as it is not related to the suppressed photoelectron intensity near a high-symmetry direction40,41,42. It is interesting to note, however, that we observe a similar suppressed intensity for the majority band Δ1 near the Γ point (Fig. 2a).
While the experimental position and shape of the kink match the prediction by the GT renormalization very well, it should be mentioned that the calculation has been carried out without SOC. The SOC gives rise to an avoided crossing (a spin-orbit gap) between the minority Δ2 band and both the majority Δ5 and $${\mathrm{\Delta }}_2^\prime$$ bands of the size equal to 60 meV and 100 meV, respectively25. However, experimentally, we observe only one kink along the Δ2 minority band, with the separation in the double-peak structure as large as 600 meV (shown in Fig. 2f), which is why we can rule out that SOC is responsible for the observed band anomaly. Furthermore, the surface states that could potentially interfere with the bulk electronic dispersion are not visible in our experiment (also when measured with other photon energies)25. However, to unambiguously prove that the observed kink is not a result of an anticrossing with the surface state, we have analyzed the orbital character of the Fe(001) majority surface state based on relativistic DFT slab calculations (Supplementary Fig. 6). The details of this analysis can be found in the Supplementary Note 6.
The many-body scattering state observed in our experiment can be compared to the ‘plasmaron’, a bound state of an electron (or hole) and a plasmon43, which can appear as satellite resonances in photoemission spectra. However, there are important differences, too. First, from a formal point of view, the plasmon propagator is a two-point function (in space and time), while the T matrix is a four-point function obtained from a solution of a Bethe-Salpeter equation. Second, the plasmon energy is quite large (typically around 20 eV), and the plasmaron peaks therefore appear at large binding energies well separated from the quasiparticle bands. In our case, the self-energy resonances are energetically so close to the quasiparticle bands that they strongly interact with each other, potentially leading to anomalous band dispersions like the one discussed in this work.
In summary, our combined experimental and theoretical analysis of the electronic dispersions in iron revealed the formation of a many-body spin flip scattering channel which manifests itself by a kink located at unusually high binding energy (Eb = 1.5 eV). This newly discovered excited state of iron is a bound state of a majority hole and a superposition of correlated electron-hole pairs of opposite spins. The observed kink structure is thus of a pure electronic origin, and its prediction from first principles requires a sophisticated quantum-mechanical many-body treatment, in which the k-dependence of the self-energy is sufficiently taken into account.
## Methods
### Momentum-resolved photoemission
The momentum resolved photoemission was performed using the momentum microscope at the NanoESCA beamline in Elettra synchrotron in Trieste (Italy)44. The 38 ML Fe film was grown in-situ on a Au(001) single crystal at low temperature (T = 140 K) using molecular beam epitaxy and gently annealed up to 300 °C. This preparation procedure was found previously to result in high-quality Fe(001) films, with no Au present on the Fe surface25. This was also confirmed by X-ray photoelectron spectroscopy (XPS) measurements. The microscope is equipped with a W(001)-based spin detector45, which enables collecting constant energy spin-resolved maps within the entire Brillouin zone of Fe(001). The images were obtained using photon energy of = 70 eV of p or s polarization. The photon beam impinges under an angle of 25 with respect to the sample surface and along the kx = 0 line. According to the free-electron final state model, such a photon energy corresponds to performing a cut through the 3D Brillouin zone close to the Γ point. An analysis of the spin-resolved images was performed following the procedure described in ref. 46. Before each measurement, the sample was remanently magnetized.
### Ab initio calculations
The theoretical calculations were performed in the all-electron full-potential linearized augmented-plane-wave (FLAPW) formalism as implemented in the FLEUR DFT and SPEX GW code27. To describe the electron-magnon interactions, an ab initio self-energy approximation was derived from iterating the Hedin equations43, resulting in a diagrammatic expansion from which we have singled out the diagrams that describe a coupling to spin-flip excitations. A resummation of these ladder diagrams to all orders in the interaction yields the GT self-energy approximation, which has a similar mathematical structure as the GW approximation as it is given by the product of the single-particle Green function G and an effective magnon propagator T. The T matrix depends on four points in real space and its implementation involves the solution of a Bethe-Salpeter equation. The numerical implementation is realized using a basis set of maximally localized Wannier functions that allows an efficient truncation of the T matrix in real space. The self-energy is calculated by the method of analytic continuation. The details of the implementation are presented in refs. 20,21.
### Code availability
The FLEUR code is available at http://www.flapw.de. The SPEX code (http://www.flapw.de/spex) is available from the authors upon request.
## Data availability
All data generated and analyzed during the current study are available from the corresponding author on reasonable request.
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## Acknowledgements
We thank H. Ibach for valuable discussions. This work was supported by the Helmholtz Association via The Initiative and Networking Fund and by Alexander von Humboldt Foundation.
## Author information
Authors
### Contributions
E.M., P.G., T.H., M.G., M.J., G.Z., S.S. and V.F. performed experiments with supervision from L.P. and C.T. M.C.T.D.M. and C.F. developed the theoretical method with supervision of S.B. I.A. provided GW band structure calculations. G.B. performed the slab calculations. E.M. analyzed experimental data. E.M., M.C.T.D.M. and C.F. wrote the manuscript with contributions from all the co-authors. S.B. and C.M.S. supervised the project.
### Corresponding author
Correspondence to E. Młyńczak.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Journal peer review information: Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer Review reports are available.
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Reprints and Permissions
Młyńczak, E., Müller, M.C.T.D., Gospodarič, P. et al. Kink far below the Fermi level reveals new electron-magnon scattering channel in Fe. Nat Commun 10, 505 (2019). https://doi.org/10.1038/s41467-019-08445-1
• Accepted:
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• DOI: https://doi.org/10.1038/s41467-019-08445-1
• ### Tuneable electron–magnon coupling of ferromagnetic surface states in PdCoO2
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Nature Communications (2020)
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2022-07-01 19:45:01
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http://math.stackexchange.com/questions/74461/solution-for-this-convolution
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# Solution for this Convolution
We have $f(z)=z+ \sum_{n=2}^{\infty} a_{n}z^{n}$ where $a_{n}$ is a constant and $g(z)=z$, $(f*g)(z)$ is equal to what? i still wondering to confirm that $(f*g)(z)=z$.
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Nothing special about $a_n$? – Guess who it is. Oct 21 '11 at 1:41
@J.M., it just a constant.. – DRN Oct 21 '11 at 1:43
Can you write down a formula for $f*g$? – AD. Oct 21 '11 at 4:28
Your solution is wrong. If you write out how you got there, someone may be able to tell you where you went wrong. Also, do you literally mean "$a_n$ is a constant" in the singular, or do you mean "the $a_n$ are constants"? – joriki Oct 21 '11 at 9:56
@Norlyda. It is a homework question, or is it your conjecture? – freak_warrior Nov 16 '13 at 10:26
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2015-05-25 07:50:54
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https://www.physicsforums.com/threads/shortest-path-on-sphere.771149/
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# Shortest path on sphere
1. Sep 16, 2014
### Blazejr
Hello
I'm struggling with well-known problem of finding shortest path between two points on a sphere using calculus of variations. I managed to find correct differential equations of great circles, but I'm not confident about validity of methods I used. Below I describe my approach.
In solutions of this problem I found online it was assumed that curve we want to find can be parametrized by angle $\theta$ (of spherical coordinates, which I'm going to use from now on). This is correct - geodesics on sphere are great circles and we can always choose coordinates on sphere such that both points lie on one meridian, say $\phi=0$. However, I do not want to use my knowledge about those geodesics before I actually find them. Therefore, I have no way of knowing that $\theta$ is correct global parametrization of this curve - it could go along some line of latitude (theta=const.) for some time. Therefore I need to assume $\theta , \phi$ to be independent variables parametrized by some arbitrary parameter along the curve (call it time $t$).
Problem comes down to finding functions $\theta (t), \phi (t)$ that minimize integral:
$$\int L \mathrm{d}t = \int \sqrt{\dot{\theta}^2+\sin ^2 \theta \dot { \phi } ^2} \mathrm{d}t$$
It follows imidietely from Euler-Lagrange equation that quantity $\frac{\sin ^2 \theta \dot{ \phi}}{L}$ is conserved. E-L equation for second variable produce complicated and messy equation.
Here comes my idea. Variable $t$ is arbitrary parametrization of the curve. We can parametrize it by any other parameter that grows monotonically with time. It is easy to check that function under the integral is unaffected by this change of variables. Therefore I can use length of curve $s$ as parameter. If I do that we have that $L(s)$ is actually constant (as it is just rate of change of length w.r.t length) and E-L equations yield:
$$\sin ^2 \theta \dot {\phi}=const., \qquad 2 \ddot {\theta} = \sin ( 2 \theta ) \dot {\phi}^2$$
I've read on some website that those are correct equations for great circles. I can't check, because I don't know how to solve them anyways. If anyone can give me a hint on that I would be grateful, but that's not what I wanted to ask.
Problem is that I am not sure if this approach is correct. Can I really use E-L equations when I parametrize by arclength? This is after all equivalent to imposing (in language of classical mechanics) non-holonomic constraint: $\frac{\mathrm{d}L}{\mathrm{d}s}=0$. Therefore $\dot{\theta},\dot{\phi}$ are no longer independent.
Is this approach completely wrong and I got good equations "by accident" or am I missing something? If it is, how can I fix my reasoning to find those equations (without simplifying assumption that the curve can be parametrized by $\theta$?). Thanks in advance.
2. Sep 16, 2014
### ShayanJ
Using calculus of variations is of course the right path for solving such a problem and because the "Lagrangian" is a function of only the variables and their first derivative, of course you should use Euler-Lagrange equations. But there is something tricky here.
Making $\int_{t_1}^{t_2} \sqrt{\dot \theta^2+\sin^2{\theta}\dot \varphi^2} dt$ stationary gives rise to a very complicated differential equation.(I actually derived that equation). So one may choose to make $\int_{t_1}^{t_2} (\dot \theta^2+\sin^2{\theta}\dot \varphi^2) dt$ stationary instead, as you did. The guy may say:"no big deal, its just the square of that!" But its actually not!
I'm not telling its wrong, but it of course is not trivial and needs to be proved that you can make $\int L^2 dt$ stationary instead of $\int L dt$!
So that's the only tricky part I see in your calculations and everything else seems fine to me.
But about your question. Great circles may have complicated parametrizations, because you may have a great circle which is running not parallel to a constant $\varphi$ circle. But there is always a change of coordinates that makes that complicated parametrization to a very simple one because you can always change coordinates to make that great circle a circle of constant $\varphi$. So if you can find a reparametrization that transforms your differential equations to $\dot \varphi=0$ and $\ddot \theta =0$, then your differential equations are describing a great circle. I think that will do in theory, don't know about its applicability.
Last edited: Sep 16, 2014
3. Sep 16, 2014
### Blazejr
Squaring Lagrangian is not really what I've done. Here's how I exactly got to my equations.
I assumed that parametrization is chosen such that $L=\frac{\mathrm{d}s}{\mathrm{d}t}=1$. We then have $\frac{\mathrm{d}L}{\mathrm{d}t}=0$ and $\frac{\partial L}{\partial \theta}=\frac{\sin 2\theta \dot{\phi}^2 }{2L}$ and $\frac{\mathrm{d} }{\mathrm{d} t}\frac{\partial L}{\partial \dot{ \theta}}=\frac{\mathrm{d} }{\mathrm{d} t}\frac{\dot{\theta}}{L}=\frac{1}{L}\frac{\mathrm{d} }{\mathrm{d} t}\dot{\theta}$.
Second equation was extracted similarly.
By choosing parametrization by arc length I make L constant function (when considered as a function of such chosen "time" only). This makes E-L equations much simpler, as shown above. What I am not sure about is whether E-L equations are applicable if I parametrize by arc length.
I suspect I shouldn't be able to use E-L equation here because that makes derivatives of coordinates $\dot{\theta},\dot{\phi}$ dependent variables.
Oh and regarding minimizing L^2: I actually checked if curve x(t) minimizing integral of L^2 implies that it minimizes integral of L. That is NOT true, unless $\frac{\mathrm{d}L}{\mathrm{d}t}=0$ which is in this case true only if we choose parametrization by arc length. So again comes the question, are we allowed to do that.
Last edited: Sep 16, 2014
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2017-09-25 19:03:59
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https://math.stackexchange.com/questions/3201392/is-this-space-equivalent-to-the-james-space/3201782
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# Is this space equivalent to the James space?
The James space $$J$$ is a famous counter-example in functional analysis. It is an example of a Banach space that is isometrically isomorphic to its double dual, but is not reflexive.
Define $$J = \big\{ (a_n) \in c_0\, \big|\, |(a_n)|_J < \infty \big\}$$
where $$c_0$$ denotes the subspace of $$l^{\infty}$$ of sequences converging to $$0$$ and
$$|a_n|_J^2 := \sup\left\{ \sum\nolimits_{i=1}^{k-1} | a_{p_{i+1}} - a_{p_i}|^2 \; \big| \; 1 \leq p_1 < \ldots < p_k \right\}$$
where the supremum is taken over all finite increasing subsequences of $$\mathbb{N}$$.
How is this different from requiring that $$\sum_{n=1}^{\infty}|a_{n+1} - a_n|^2 < \infty$$?
A brief summary of the details of this answer is: consider an appropriately scaled version of the sequence $$(0,1,0,\frac14,\frac12,\frac34,1,\frac34,\frac12,\frac14,0,\frac19,\frac29,\dots)$$.
These two conditions are different. The James norm being finite is a strictly stronger statement. To see what kind of different behaviour can happen, start by considering the finite sequence $$x = (x_1,x_2,x_3,x_4,x_5) = (0, \frac12, 1, \frac12, 0)$$ Then $$\sum_{i=1}^4 |x_i - x_{i+1}|^2 = \sum_{i=1}^4 \frac14 = 1$$ but for example, $$|x_1 - x_3|^2 + |x_3 - x_5|^2 = 2$$ so the James norm is strictly bigger. We can repeat this behaviour to come up with an example illustrating the difference. We'll do this by building suitable blocks with this behaviour.
Let $$x^{(k)}_i = \frac{i}{k^2}$$ for $$0 \leq i \leq k^2$$ and let $$x^{(k)}_i = 1 - \frac{i}{k^2}$$ for $$k^2 + 1 \leq i \leq 2k^2$$. Then $$\sum_{i=0}^{2k^2 - 1} |x_i^{(k)} - x_{i+1}^{(k)}|^2 = 2k^2 \frac{1}{k^4} = \frac{2}{k^2}$$ (The point of the construction is that the sequence in $$k$$ on the right hand side is summable). However $$|x_0^{(k)} - x_{k^2}^{(k)}|^2 + |x_{k^2}^{(k)} - x_{2k^2}^{(k)}|^2 = 2$$.
If we forget for a second that we want $$J \subseteq c_0$$, this already gives the desired example. Define $$x = (x^{(1)}, x^{(2)}, x^{(3)}, \dots)$$ where I really mean place each block one after the other to build a sequence. The sequence $$x$$ then goes back and forth between $$0$$ and $$1$$ but using smaller and smaller jump sizes. Then $$\sum_{i=1}^\infty |x_i - x_{i+1}|^2 = \sum_{k=1}^\infty \sum_{i=0}^{2k^2 - 1} |x_i^{(k)} - x_{i+1}^{(k)}|^2 = \sum_{k=1}^\infty \frac{2}{k^2} < \infty$$ but $$|x|_J^2 \geq \sum_{k=1}^N |x_0^{(k)} - x_{k^2}^{(k)}|^2 + |x_{k^2}^{(k)} - x_{2k^2}^{(k)}|^2 = \sum_{k=1}^N 2 = 2N$$ for every $$N$$ and so $$|x|_J^2 = \infty$$.
The only remaining problem is that $$x \not \in c_0$$ since it takes the value $$1$$ infinitely often. This is easily fixed. Let $$(a_n)$$ be your favourite sequence that is valued in $$[0,1]$$, converges to $$0$$ as $$n \to \infty$$ and is such that $$\sum_{n=1}^\infty a_n = \infty$$. Define $$y = (a_1^{\frac12} x^{(1)}, a_2^{\frac12} x^{(2)},a_3^{\frac12} x^{(3)}, \dots)$$ where $$a_i^{\frac12} x^{(i)}$$ means multiply every element of the block $$x^{(i)}$$ by $$a_i^{\frac12}$$. Then $$y \in c_0$$ and $$\sum_{i=1}^\infty |y_i - y_{i+1}|^2 = \sum_{i=1}^\infty a_i|x_i - x_{i+1}|^2 \leq \sum_{i=1}^\infty |x_i - x_{i+1}|^2 < \infty$$ but $$|y|_J^2 \geq \sum_{k=1}^N a_k|x_0^{(k)} - x_{k^2}^{(k)}|^2 + a_k|x_{k^2}^{(k)} - x_{2k^2}^{(k)}|^2 = \sum_{k=1}^N 2a_k \to \infty$$ as $$N \to \infty$$, so $$|y|_J^2 = \infty$$.
With exponent $$1$$ we have $$|a-c| \le |a-b|+|a-c|$$. So $$\sup\left\{ \sum\nolimits_{i=1}^{k-1} | a_{p_{i+1}} - a_{p_i}| \; \big| \; 1 \leq p_1 < \ldots < p_k \right\} = \sum_{n=1}^{\infty}|a_{n+1} - a_n|$$ This means using "finite variation" yields essentially the space $$l^1$$.
But with exponent $$2$$, it is false that $$|a-c|^2 \le |a-b|^2+|b-c|^2$$. Even more, it can happen that $$\sup\left\{ \sum\nolimits_{i=1}^{k-1} | a_{p_{i+1}} - a_{p_i}|^2 \; \big| \; 1 \leq p_1 < \ldots < p_k \right\} > \sum_{n=1}^{\infty}|a_{n+1} - a_n|^2$$ in general. (See Rhys example.) So "finite square-variation" yields something quite different than $$l^2$$.
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2019-11-16 22:11:43
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https://codereview.stackexchange.com/questions/243188/in-memory-database-in-unit-tests-isolate-the-tests
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# In Memory Database in Unit tests, isolate the tests
I have stumbled across these unit tests in a code review that are using in memory db:
private DatabaseContext _context;
private Fixture _fixture;
[SetUp]
public void Setup()
{
_fixture = new Fixture();
_fixture.Customize(new AutoNSubstituteCustomization());
var options = new DbContextOptionsBuilder<DatabaseContext>()
.UseInMemoryDatabase(databaseName: "testdb")
.Options;
_context = new DatabaseContext(options);
}
[TearDown]
public void CleanUp()
{
var context = _context;
if (context == null || context.Database.ProviderName != "Microsoft.EntityFrameworkCore.InMemory")
{
return;
}
context.Database.EnsureDeleted();
_context = null;
}
#region EmptyDB
[Test]
public void Test1()
{
// Setup
var logger = _fixture.Freeze<ILogger<UserRepository>>();
var userRepo = new UserRepository(_context, logger);
var userViews = new List<UserView>();
// ACT
userRepo.UpdateUsers(userViews, CancellationToken.None).GetAwaiter().GetResult();
// ASSERT
Assert.AreEqual(10, _context.Users.CountAsync().GetAwaiter().GetResult());
}
[Test]
public void Test2()
{
// Setup
var logger = _fixture.Freeze<ILogger<UserRepository>>();
var userRepo = new UserRepository(_context, logger);
var identityViews = new List<IdentityView>();
_fixture.Register<IEnumerable<UserView>>(() =>
{
return new UserView[] { new UserView("fish") };
});
// ACT
userRep.UpdateUsers(userViews, CancellationToken.None).GetAwaiter().GetResult();
// ASSERT
Assert.AreEqual(10, _context.Users.CountAsync().GetAwaiter().GetResult());
}
As you can see, the tests are using the same in memory db, which I really don't like. I also don't like the new UserRepository(_context, logger). Is it a bad practice to use the new-keyword like this?
I would prefer something like this instead:
[Test]
public void Test1()
{
// Setup
var provider = RegisterServices();
var logger = _fixture.Freeze<ILogger<UserRepository>>();
var userRepo = provider.GetRequiredService<IUserRepository>();
var userViews = new List<UserView>();
// ACT
userRepo.UpdateUsers(userViews, CancellationToken.None).GetAwaiter().GetResult();
// ASSERT
Assert.AreEqual(10, _context.Users.CountAsync().GetAwaiter().GetResult());
}
private ServiceProvider RegisterServices([CallerMemberName] string memberName = "")
{
var services = new ServiceCollection();
options.UseInMemoryDatabase(memberName));
return services.BuildServiceProvider();
}
As you can see, I have added a RegisterService method that takes the calling test as a parameter, and then uses this to create the inmemorydb. I really like this because you are isolating your tests more this way. I also think it's cleaner to read.
How would you guys do in this case? Is the first approach the way to go, or is my approach the more "right" way to do it? Or is it another better and more best practice way to do it?
• To anyone down voting or voting to close, while the first block of code is indeed copied, the second block is the posters attempt to rewrite it. They are looking for a comparison. May 31 '20 at 21:15
Generally speaking whenever we are about to write unit tests we should follow the F.I.R.S.T. principles. It is an acronym, which stand for:
• Fast: The execution time should be measured in milliseconds. If it takes a second or two then it can be considered as slow and should be revised.
• Isolated: Each test is self-contained and does not rely on any other data. They can be run in any order.
• Repeatable: The test runs must be repeated over and over again. In each and every run they should either pass every time or always fail.
• Self-validating: There is no need for human interpretation whether or not the test succeeded or failed. The test result should be self-explanatory.
• Timely: The code and related tests should be written in almost the same time. A new code without relevant unit tests should not be deployed.
Let's examine these ideas for your proposals:
### Single database and cleanup
• Fast: If for whatever reason a previous cleanup phase missed / failed then your database will have some trash data in it, which might impact the performance of your database operations. Executing the cleanup during setup and teardown might solve the problem, but it will definitely have performance impact.
• Isolated: They are sharing the same database, so they are not truly isolated. It might be the case that they can't run in parallel, because ordering might matter.
• Repeatable: Because they are using the same database, that's why order might affect the result of your assertions. In case of MSTest you can define ordering but if you need to use them that means your tests are not really isolated.
• Self-validating: Because there is a chance for race condition your test results are non-deterministic, which means human-intervention is needed to interpret several resultsets, reproduce the issues (if it is even possible) and fix them.
• Timely: It is irrelevant in our discussion
### Separate database for each test case
• Fast: Creating new in-memory database for each test should not impose performance penalty onto tests if there not too many tables and constraints.
• Isolated: Separate databases are used for each test, means no shared resource is being used, which helps isolation.
• Repeatable: Because each and every time you run your test against a brand new database, there won't be any trash data, which could cause race condition.
• Self-validating: By being deterministic, no human intervention is needed to understand the test results.
• Timely: It is irrelevant in our discussion
If you don't want to examine the test data manually, then you don't really need use the test name in the database name. You can use any random value:
int jitter = idGenerator.Next();
var condigBuilder = DbContextOptionsBuilder<TestContext>()
.UseInMemoryDatabase(databaseName: $"TestDb{jitter}") .Options; or Guid jitter = Guid.NewGuid(); var condigBuilder = DbContextOptionsBuilder<TestContext>() .UseInMemoryDatabase(databaseName:$"TestDb{jitter}")
.Options;
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2021-10-27 04:25:35
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|
https://www.physicsforums.com/threads/general-solution-to-pell-like-equation-y-2-2x-2-n-2.81460/
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# General solution to Pell-like equation y^2 + 2x^2 = n^2
1. Jul 7, 2005
### ramsey2879
I discovered the following general solution to primitive forms of y^2 + 2x^2 = a^2
a = 3 + 4n(n+1)
y = 4n(n+1) - 1
x = 4n + 2
Moreover let P = {11, 17, 19, 41, 43 ...} = the set of prime divisors of numbers of the form 3+4n(n+1) without the 3, and let Q = { -1, 7, 17, 23 ...} = the set of prime divisors of numbers of the form 4n(n+1) -1 including the -1. My conjecture is that for any member "p" of set P there is a corresponding member "q" of set Q such that q^2 + 72x^2 = p^2 where x is an interger, e.g.
11^2 = 7^2 + 72*1
17^2 = -1^2 + 72*2^2
19^2 = 17^2 + 72*1
41^2 = 23^2 + 72*4^2
43^2 = 7^2 + 72*5^2
...
Last edited: Jul 7, 2005
2. Jul 7, 2005
### matt grime
y=1, x=0 and a=1 don't appear to be in that solution set.
3. Jul 7, 2005
That's true
4. Jul 7, 2005
### matt grime
there is a complete solution to this already using relative elementary number theory. n^" will be of the form x^2+2y^2 iff and only if n is a product of primes that are all congruent to 1 or 3 mod 8. see eg cox numbers of the form x^2+ny^2 chapter 1.
5. Jul 7, 2005
### ramsey2879
OK but my conjecture is more rigid. Do you know if this is discussed in the Cox book titled Primes of the form x^2 + Ny^2.
6. Jul 7, 2005
### matt grime
your conjecture is wrong. you onlyhave a subest of the primes congruent to 3 mod 8 as possible solutions, thus missing a significant proportion of the answers. you give a sufficient condition for a solution but not a necessary one.
cox does not discuss the x and y associated to a given a.
7. Jul 8, 2005
### ramsey2879
I think you misread my conjecture, I strenthened the wording of it before my last post to avoid such a misreading. As you can see in my list of examples primes in set P include primes of the form 8n+3 and 8n+1 while primes in set Q include primes of the form 8n-1 and 8n + 1. Primes of the form 8n+5 do not appear to be in either subset but that does not contradict Cox.
Note that if a prime P(i) divides 3+4a(a+1) then it divides all instances of 3+4n(n+1) where n equals either a or P(i)-a-1 mod P(i) thus if a prime P(i) is to appear in set P it must appear as a factor of 3 + 4a(a+1) where a < P(i)/2.
17 appears in set P since it divides 51 and 51 = 3+4*3*(3+1)
Similar logic applies to set Q
I restate my conjecture
Let P(i) be a prime in set P, There exists either a prime or -1, Q(i), in set Q such that P(i)^2 - Q(i)^2 = 72n^2 where n is an integer.
8. Jul 8, 2005
### matt grime
how can it be that i misread it when you changed it? anyway, i don;t know if your conjecture is true (it is a sufficient condition certainly) so why don't you post the proof? off the top of my head i'd say it might be true but i see no compelling evidence for it to be so.
Last edited: Jul 8, 2005
9. Jul 8, 2005
### shmoe
If I understand your P and Q sets, this is false. 379 is prime and 3+4*129*(129+1)=379*177, so 379 is in P. The only solutions to 379^2-q^2=72*n^2 are:
379^2-(379)^2=72*(0)^2
379^2-343^2=72*(19)^2
but 379 is not in Q (can check 4*n*(n+1)-1 is never divisible by 379) nor is 343 (it's not prime).
10. Jul 9, 2005
### matt grime
soon as you get a sufficiently large list of primes these things tend to happen.
11. Jul 11, 2005
### ramsey2879
Yes, I did not look far enough. Also p = 113 and 137 are exceptions. However, 113^2 =7^4 + 72*12^2; 137^2= 7^2*17^2 + 72*8^2; and 379^2= 7^6+72*19^2. 7 and 17 are in Q, so my new conjecture is that for every p in P there is a number q that is in Q or is a product of numbers in Q such that p^2-q^2=72*n^2 where n is an integer > 0. This is supported by the facts, since if p and q are odd and not divisible by 3, then p^2-q^2 is divisible by 6. 72= 2*6^2.
12. Jul 12, 2005
### matt grime
how about, instead of making conjectures based only on small examples, you attempt to prove your statement?
13. Jul 12, 2005
### matt grime
suppose that 72q^2=2.4.9.q^2=(n-p)(n+p), and we factor the RHS as XY with X>Y, then X-Y=2p. If more than one of the odd primes in the decomps of X and Y occurs in both then we have a contradiction, and if 2 appears with multiplicity greater than 1 in both then we have another contradiction, but 2 must appear in both X and Y. thus we have several cases to consider, but as we only care about an existence one. let's try and pick those that may help.
Take the case where X=2a^2 Y=4b^2 where b is odd, and thus p=a^2-2b^2. We haven't factored in the 3 yet, so let us suppose that the powers of 3 that must occur are factors of b (so replece b with 3c), hence we need to find a and c such that
p=a^2-18b^2
the other variations will lead to similiar. if you can find something about these you will have the answer.
sadly we're not non-positive definite forms here and i know nothing about those.
take another case that where a prime may divide both X and Y say the prime is r, then
X=4ra^2, Y=2r or X=2ra^2 Y=4r etc but this leads to the degenerate case when n=p.
14. Jul 13, 2005
### ramsey2879
OK I get the idea. From Cox we know that
p^2 = y^2 + 2x^2 with gcd p,y = 1 (1)
if and only if p is of the form 8n+1 or 8n+3. Note that we must imposed the condition that gcd p,y = 1 or else we could multiply each side of (1) by 25 or 49 to yield a contradiction. Then p and y must have the same parity and both are odd since gcd=1.
We can then make the substitutions v=(p-y)/2 and u=(p+y)/2 which gives
p = u+v and y = u-v (2)
By (1) gcd u,v = 1 since gcd p,y=1
substituting (2) into (1) gives
uv = 2x^2 (3)
now we suppose that x = r*s with the factor s having the requirement of being odd so that p,y can be odd
But since gcd u,v and gcd p,y = 1
u=2r^2, v=s^2, with gcd r,s = 1 (4.1) or
u=s^2, v=2r^2, with gcd r,s = 1 (4.2)
then substituting 4 into 2 then into 1 with the requirement that p,y are odd gives
p = 2s^2 + r^2, y = 2s^2 - r^2, x = 2rs (5.1) or
p = 2s^2 + r^2, y = r^2 - 2s^2, x = 2rs (5.2)
as the general solution of (1)
Thus we know that p^2-y^2 must equal 8r^2s^2
Thus since 3 does not divide p (my condition of this post) and since p and y are both odd all I have to prove is that 3 does not divide y to show that 9 divides either r^2 or s^2.
Thus I need to show that 3 does not divide r^2-2s^2 as well as not dividing r^2+2s^2
1. if 3 divides either r or s then 3 does not divide y since gcd r,s =1
2. if r and s each = +/- 1 mod 3 then p is divisible by 3, this contradicts my conditions. QED
15. Jul 13, 2005
### matt grime
we impose the restriction to make a primitive solution. if gcd(p,y) weren't 1 then it must be p, hence p divides y, and thus x so we have, after dividing through, that 1=a^2+2b^2, and thus a=1, b=0, which we shall call the degenrate case. p divides x since...
p is odd since it is congruent to 1 or 3 mod 8, it is not the coprimality, personlally i wouldn't cite the gcd as the "cause" of this stuff. the gcd being 1 doesn't force y to be odd: it is because if it were even then y^2+2x^2 would be even too, and hence 2 wolud divide p, but p is an odd prime.
oh, and p as is an odd prime, so if it divided y^2 it would divide 2x^2 and thus x to complete the argument in the last comment. the gcd stuff is always assumed.
this is all true, if there is a coprime solution
this is all fine, though i don't think it supports your conclusion as is.
let p be congruent to 1 or 3 mod 8, then there are integers r and s such that
p=2r^2+s^2.
let y be 2r^2-s^2 (the sign doesn't matter) then
p^2=y^2+8r^2s^2
then your analysis implies that 3 must divide on of r or s, hence p^2=y^2+72n^2 as required.
no problem with that. but your conjecture is stronger in that it restricts the prime factors of y to be only of a certain kind, and i don't immediately see why that is true. so what is the argument here? we know that y^2 is 1 mod 8, but that doesn't even tell us that y is 1 or -1 mod 8, never mind its prime factors.
Last edited: Jul 13, 2005
16. Jul 13, 2005
### ramsey2879
Ok I will work on my proof some more. My conjecture is based upon p being a prime divisor of (2n+1)^2 + 2*1^2 other than 3 and y being a divisor of (2m+1)^2-2*1^2 for some m. Now (2n+1)^2 +2*1 is a valid form for p in the most general solution of p^2 = y^2+2x^2, thus p must be a prime of the form 8n+1 or 8n+3 and also must not be 3. It is easy to show that any divisor of (2m+1)^2-2^1 is not divisible by 3. So what I have left to show is that there is a divisor y of a number of the form (2m+1)^2 - 2 such that p^2=y^2+72n^2 for any prime p of the form 8n+1 or 8n+3 other than 3 .
Returning to the most general solution we have p = 2r^2+s^2 = a prime not divisible by 3 and y = |2r^2-s^2| and x=2rs, s is odd. From my last post either r or s must be divisible by 3 in order for 3 not to divide p. Let r be divisible by 3, then y = 18t^2-s^2. It can easily be shown that s=1 or 5 mod 6. Substitute 6a+1 for s then y=18t^2-36a^2-12a-1. I have to go now
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2017-05-29 09:49:07
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http://mathoverflow.net/questions/159116/relationships-between-finiteness-of-stable-rank-and-ibn-property-of-rings
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# Relationships between finiteness of stable rank and IBN property of rings
Does any ring of finite stable rank have IBN property? Where can we find this result?
-
You probably mean to ask if every ring of finite stable rank has the IBN property. The way the question is written it can be interpreted two ways. Also, the question becomes better if you define some of the terms. – Dag Oskar Madsen Mar 2 '14 at 12:49
Veldkamp claims this (Handbook of incidence geometry, prop. 2.6 pag. 1040). I suppose that his references give a proof. – user46855 Mar 2 '14 at 13:46
Crossposted to math.SE a few days later: math.stackexchange.com/questions/702848/… Talking with the OP, I found out he was interested in rings with stable range $n$ (apparently synonymous with stable rank $n$) for $n>1$. Rings with stable range 1 do indeed have IBN. – rschwieb Mar 25 '14 at 17:48
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2015-07-01 23:18:03
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http://stackoverflow.com/questions/6829096/configuring-propel-return-type-for-flex
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# Configuring propel return type for flex
I use wamp server and Propel.I have written my service using Propel but when I want to connect the data returned from the service Flex cannot recognize the return type.
This is the php code that I wrote
<?php
// Include the main Propel script
require_once 'C:/wamp/propel/runtime/lib/Propel.php';
// Initialize Propel with the runtime configuration
Propel::init("C:/wamp/www/school/build/conf/school-conf.php");
// Add the generated 'classes' directory to the include path
set_include_path("C:/wamp/www/school/build/classes" . PATH_SEPARATOR . get_include_path());
class TeacherService {
function getTeachers()
{
$allTeachers=TeacherQuery::create()->find();$teachers=array();
foreach($allTeachers as$teacher1)
{
array_push($teachers,$teacher1);
}
return $teachers; } } ?> I wanto to display the information of teacers in a datagrid yet when I choose to auto-detect the return type it gives the error 'teacher' cannot be set to the data type "StdClass" because it has no properties. How can I let flex know the properties of teacher rows returned by propel? - I added the PHP Tag b/c you seem to only provide PHP code. I, personally, don't know what propel is. – JeffryHouser Jul 26 '11 at 11:40 ## 1 Answer @www.Flextras.com Propel is a PHP ORM framework. If you want serialization to work on your php class you would have to have something like this var$_explicitType = "path.to.classes.Teacher";
and on your Flex side VO's you would have to have something like this.
[Bindable]
[RemoteClass(alias="path.to.classes.Teacher")]
This is assuming you are using AMF. Also in your AMF endpoint you have to specify the mapping, so for example I would have something like this in my endpoint file.
\$server->setClassMap('path.to.classes.Teacher' , 'path\to\classes\Teacher');
This lets your endpoint know that when it sees a class with the matching descriptor, which class on the PHP side it should be deserialized to.
-
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2015-08-02 06:42:03
|
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https://www.physicsforums.com/threads/canonical-approach-to-unification.2033/
|
# Canonical approach to unification?
1. May 12, 2003
### eljose79
I have watched that in physics or string theory perturbation is used..but could we obtain a unifying theory of physics using canonical theory?..i mean take the Lagrangian take the hamiltonian and quantizy the moment Pab by introducing functional derivatives..
2. May 15, 2003
### instanton
Yes we can certainly do that.
But, you have to realise that even in one two dimensional spacetime (one space one time) your functional "Schrodinger equation" is a infinite number of coupled partial differential equation, which makes it very difficult to solve.
There are certainly some advantage of using canonical method over covariant perturbation theory especially if you want to prob some non-perturbative regime like question regarding vacua or solitonic objects.
Instanton
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2016-12-10 21:00:47
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https://mrgrasley.wordpress.com/2017/10/02/mdm4u-probability-with-equally-likely-outcomes-2017-10-02/
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# MDM4U Probability with equally-likely outcomes 2017-10-02
When each outcome in a sample space is equally likely,
$P(x)=\frac{1}{n(S)}$
for each $x \in S$.
For any event $A \subseteq S$,
$P(A) = \frac{n(A)}{n(S)}$
We often say something like, “We get the probability by counting the number of ways the event can happen and dividing by the number of ways anything can happen.”
For homework complete Page 325 #6, 9, 10.
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2018-08-20 20:04:19
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https://www.semanticscholar.org/paper/A-Deduction-Calculus-for-Cumulated-Clauses-on-FLE-Rudolph/e1addc594ad46b4e61b6cef8e3987d20518ce846?p2df
|
Corpus ID: 17641099
# A Deduction Calculus for Cumulated Clauses on FLE Concept Descriptions
@inproceedings{Rudolph2005ADC,
title={A Deduction Calculus for Cumulated Clauses on FLE Concept Descriptions},
author={Sebastian Rudolph},
year={2005}
}
In this paper, we investigate cumulated clauses on a set of attributes consisting of concept descriptions of the description logic FLE . This kind of expression is useful for describing the attribute logic of contexts where the attributes can be seen as FLE concept descriptions. We provide a deduction calculus for this type of expressions and prove its soundness and completeness.
1 Citations
Quo Vadis, CS? - On the (non)-Impact of Conceptual Structures on the Semantic Web
• Computer Science
• ICCS
• 2007
This paper attempts to stimulate the Conceptual Structures community to catch the Semantic Web train by encouraging them to participate in developments on a large scale. Expand
#### References
SHOWING 1-10 OF 19 REFERENCES
Explaining ALC Subsumption
• Computer Science
• Description Logics
• 1999
A solution based on a sequent calculus that is closely related to the tableau implementation, exploiting its optimisations and the resulting proofs are pruned and then presented as simply as possible using templates. Expand
Exploring Relational Structures Via FLE
A method originating from Formal Concept Analysis is proposed which uses empirical data to systematically generate hypothetical axioms about the domain, which are represented to an ontology engineer for decision. Expand
Modal logic
This paper shows how the tree or tableau method provides a simple and easily comprehensible decision procedure for systems such as K, T, S4 and S5 and how the formal techniques of modal logic can be used to analyse several informal problems involving modal concepts, including cases combining modality with quantification. Expand
Conceptual Graphs and Formal Concept Analysis
It is shown how Conceptual Graphs and Formal Concept Analysis may be combined to obtain a formalization of Elementary Logic which is useful for knowledge representation and processing. For this, aExpand
Arbitrary Relations in Formal Concept Analysis and Logical Information Systems
• Computer Science
• ICCS
• 2005
An augmented definition that handles binary relations between objects and a Galois connection is defined on augmented contexts that represents concept inheritance as usual, but also relations between concepts. Expand
Proof Methods for Modal and Intuitionistic Logics
One / Background.- Two / Analytic Modal Tableaus and Consistency Properties.- Three / Logical Consequence, Compactness, Interpolation, and Other Topics.- Four / Axiom Systems and Natural Deduction.-Expand
Formal Concept Analysis Methods for Dynamic Conceptual Graphs
• Computer Science
• ICCS
• 2001
A procedure is demonstrated which interactively asks for the validity of implications and from this information designs a dynamic CG system with the desired properties. Expand
Formal Concept Analysis: Mathematical Foundations
• Computer Science
• 1998
From the Publisher: This is the first textbook on formal concept analysis. It gives a systematic presentation of the mathematical foundations and their relation to applications in computer science,Expand
Applying Formal Concept Analysis to Description Logics
• Mathematics, Computer Science
• ICFCA
• 2004
Methods from formal concept analysis developed for computing concept lattices can be employed for computing subsumption hierarchy of all least common subsumers of subsets of $$\mathcal{C}}$$, and it is shown that these hierarchies can be used to support the bottom-up construction of description logic knowledge bases. Expand
Relational exploration: combining description logics and formal concept analysis for knowledge specification
This work deals with Formal Concept Analysis (FCA) and Description Logic, which have been successfully applied in various areas beyond mathematics and are characteristic for the underlying ways of thinking in two fields of knowledge processing and representation. Expand
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2021-10-25 20:16:59
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https://leanprover-community.github.io/archive/stream/116395-maths/topic/Multiplication.20by.20n.20in.20an.20additive.20commutative.20group.html
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## Stream: maths
### Topic: Multiplication by n in an additive commutative group
#### Johan Commelin (May 14 2018 at 10:35):
Is this somewhere in mathlib?
definition mul_n {G : Type*} [add_comm_group G] (n : ℤ) (g : G) : G := n • g -- sorry
#### Johannes Hölzl (May 14 2018 at 11:37):
Its gsmul (generalized(?) scalar multiplication) in algebra.group_power
Thanks!
#### Mario Carneiro (May 14 2018 at 18:16):
the "g" stands for "group" in gsmul and gpow.
Last updated: May 18 2021 at 08:14 UTC
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2021-05-18 09:02:22
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https://www.hindawi.com/archive/2008/602870/
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`Journal of BiophysicsVolume 2008 (2008), Article ID 602870, 9 pageshttp://dx.doi.org/10.1155/2008/602870`
Review Article
## Molecular Processes in Biological Thermosensation
Laboratory of Cellular Biophysics, Aachen University of Applied Sciences, Ginsterweg 1, 52428 Juelich, Germany
Received 11 February 2008; Accepted 16 April 2008
Copyright © 2008 I. Digel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
#### Abstract
Since thermal gradients are almost everywhere, thermosensation could represent one of the oldest sensory transduction processes that evolved in organisms. There are many examples of temperature changes affecting the physiology of living cells. Almost all classes of biological macromolecules in a cell (nucleic acids, lipids, proteins) can present a target of the temperature-related stimuli. This review discusses some features of different classes of temperature-sensing molecules as well as molecular and biological processes that involve thermosensation. Biochemical, structural, and thermodynamic approaches are applied in the paper to organize the existing knowledge on molecular mechanisms of thermosensation. Special attention is paid to the fact that thermosensitive function cannot be assigned to any particular functional group or spatial structure but is rather of universal nature. For instance, the complex of thermodynamic, structural, and functional features of hemoglobin family proteins suggests their possible accessory role as “molecular thermometers”.
#### 1. Introduction
Temperature changes are one of the main stresses experienced by organisms from bacteria to plants and animals and therefore temperature is one of the environmental cues under constant vigilance in living cells. Several problems arise from exposing a cell to a sudden change in temperature [1]: firstly, membrane fluidity changes, that affect many membrane-associated vital functions. Secondly, nucleic acid topology will be affected causing shifts in processes such as transcription and translation. Finally, the protein function is affected both from structural and catalytic points.
Hence, living cells need devices for sensing environmental temperature changes in order to adapt their biochemical processes accordingly. A successful adaptive response to temperature changes cannot be performed by corresponding changes in the rate and equilibrium of enzymatic reactions only. Such a mechanism of adaptive reaction is too unspecific and uncontrollable. To cope with temperature variation, living organisms need sensing temperature alterations and translating this sensory event into a pragmatic gene response.
While such regulatory cascades may ultimately be complicated, it appears that they contain primary sensor machinery at the top of the cascade. The functional core of such machinery is usually that of a temperature-induced conformational or physicochemical change in the central constituents of the cell. Hence, a specific sensory transduction mechanism is needed, including, as a key element, a molecular sensor, transforming physical parameter (temperature) into a biologically significant signal (change in membrane permeability, specific inhibition/stimulation of gene expression, etc.). In a sense, a living organism can use structural alterations in its biomolecules as the primary thermometers or thermostats. Thus, sensory transduction is a complex biological process aimed at integrating and decoding physical and chemical stimuli performed by primary sensory molecular devices. Furthermore, sensory perception of potentially harmful stimuli functions as a warning mechanism to avert potential tissue/organ damage.
Among temperature-controlled processes in living organisms, most well-known are the expression of heat-shock and cold-shock genes [2]. Relocation of a culture of Escherichia coli adapted to an optimal growth to a sudden temperature increase, or decrease, by some 10–15°C results in adaptive shock responses. Such responses involve a remodeling of bacterial gene expression, aimed at adjusting bacterial cell physiology to the new environmental demands [3, 4]. The response of prokaryotic and eukaryotic systems to heat-shock stress has been investigated widely in a large number of organisms and model cell systems. Notably, all organisms from prokaryotes to plants and higher eukaryotes respond to cold and heat shock in a comparatively similar manner. The general response of cells to temperature stress (cold or heat) is the elite and rapid overexpression of small groups of proteins, the so-called CSPs (cold-shock proteins) or HSPs (heat shock proteins), respectively, but the initial launching mechanism is different in both cases.
In bacteria, the heat response generally invokes some 20 heat-shock proteins, whose functions are primarily to help deal with, and alleviate, the cellular stress imposed by heat [5]. Many of these proteins participate in reconstituting and stabilizing protein structures and in removing misfolded ones. The expression of this special chaperone system, which includes the proteins DnaK, DnaJ, and GrpE is activated by the presence of misfolded, temperature-denatured proteins. Thus, one could implicate the binding of partially unfolded proteins by chaperones as the thermosensoric event regulating expression of heat-shock proteins, where the primary sensory element is constituted by some easily denaturing proteins. This, in turn, demonstrates that even bacteria can practically utilize destructive changes in protein conformation as a means for temperature sensing.
In case of cold shock, the primary sensing event is more obscure. Various reports have now shown that when in vitro cultivation temperature is lowered, the rigidity of the cell membrane is increased which results in compromised membrane-associated cellular functions. Furthermore, cold stress dramatically hinders membrane-bound enzymes, slows down diffusion rates, and induces cluster formation of integral membranous proteins [6].
In mammalian cells the five known mechanisms by which cold-shock-induced changes occur in gene expression are: (i) a general reduction in transcription and translation, (ii) inhibition of RNA degradation, (iii) increased transcription of specific target genes via elements in the promoter region of such genes, (iv) alternative pre-mRNA splicing, and (v) via the presence of cold-shock specific IRESs (internal ribosome entry segments) in mRNAs that result in the preferential and enhanced translation of such mRNAs upon cold shock [7].
It has been pointed out that cold stress exposes cells to two major stresses: those relating to changes in temperature and those related to changes in dissolved oxygen concentration at decreased temperature, and it is therefore necessary to consider potential responses to each, either independently or as part of a coordinated response. Separating the relative effects of temperature and oxygen as a result of decreased temperature is difficult and has not been extensively addressed to date. Both changes in dissolved oxygen and temperature reduction result in similar changes in cultured mammalian cells [7].
The shock response systems discussed above belong to ultimate mechanisms aimed to survival under extreme temperature conditions. However, the ability to express certain factors can be affected by reasonably small temperature changes. Less drastic changes in temperature may not induce shock responses, but can be sufficient to modulate the expression of virulence genes, for example in Shigellae [8] and Yersiniae [9]. While one might be surprised that organisms built on such minimalist approaches as bacteria respond to temperature changes, the consequence of these observations is that even bacteria actually sense temperature shifts in order to control gene expression accordingly. Investigators have now been studying the moderate temperature sensation in a variety of organisms for at least several decades or more. Recently, a number of reports have shown that exposing yeast or mammalian cells to sub-physiological temperatures ( or , resp.) invokes a coordinated cellular response involving modulation of transcription, translation, metabolism, the cell cycle and the cell cytoskeleton [7, 1013]. Nevertheless very little is known about the molecular mechanisms that govern initial response on small thermal stimuli, particularly the primary sensory transduction mechanisms.
Below, we have tried to uncover some aspects of the molecular basis of temperature sensing by biological molecular thermometers, to summarize some known aspects of primary components of temperature signal transduction and to show possible thermosensitive role of even “common” molecules such as hemoglobin.
#### 2. Temperature-Sensing Biomolecules
In addition to specificity and sensitivity, the pragmatic thermoresponse should be one that is reversible and controlled. Such complexity of thermosensing and thermoregulation may reflect the demands to handle and fine-tune responses to an important environmental factor in a dynamic fashion. However, ultimately, it seems that basic and uncomplicated biochemical processes are used as primary sensors and, for that purpose changes in the nucleic acid, protein or membrane physicochemical state appear highly suitable. Bellow we make a short overview of temperature-sensing properties of most important groups of biological macromolecules.
##### 2.1. Membrane Lipids
While the information available is somewhat scant, the picture emerging shows that cells can use signals generated through changes in nucleic acid or protein conformation, or changes in membrane lipid behavior, as sensory devices. The physical state of membranes does change in response to temperature shifts in phase-transition manner [14], but the temperature-induced changes in real biological membranes are not sharp because many kinds of fatty acids present, having different characteristic temperature points of phase transition. Thus, it would not be surprising if cells (even those of bacteria) could utilize, changes in membrane fluidity as a thermometer device, assisted by protein helpers, playing a role of switchers, “sharpening” the temperature response. Microorganisms counteract the propensity for membranes to rigidify at lower temperature by adapting to the conditions in order to maintain a more-or-less constant degree of membrane fluidity (homeoviscous adaptation). The cyanobacterium Synecocystis responds to decreased temperature by increasing the cisunsaturation of membrane-lipid fatty acids through expressing acyl-lipid desaturases [1517]. Lipid unsaturation would then restore membrane fluidity at the lower temperature. In B. subtilis, this lipid modification is initiated through the activity of a so-called two-component regulatory system consisting of the DesK and DesR proteins [15]. Prokaryotic two-component regulatory systems usually consist of protein pairs, a sensor kinase and a regulatory protein [18].
It appears that it is a combination of membrane physical state and protein conformation that is able to sense temperature and to translate this sensing event into proper gene expression. However, sensing of temperature through alteration in nucleic acid conformation could be more efficient temperature-mediated mechanism of gene expression.
##### 2.2. RNA
Messenger RNAs apart from carrying their coding information for protein generation are also rapidly emerging as regulators of expression of the encoded message. With unique chemical and structural properties, sensory RNAs perform vital regulatory roles in gene expression by detecting changes in the cellular environment through interactions with small ligands [19, 20] and proteins [21, 22].
Regulatory RNA elements, “riboswitches,” have been reported recently, responding to intracellular signals by conformational changes. Riboswitches are conceptually divided into two parts: an aptamer and an expression platform. The aptamer directly binds the small molecule, and the expression platform undergoes structural changes in response to the changes in the aptamer. The expression platform is what regulates gene expression. Riboswitches demonstrate that naturally occurring RNA can specifically response on versatile physical and chemical stimuli, a capability that many previously believed was the domain of proteins or artificially constructed RNAs called aptamers [23].
Theoretically, RNA molecules have a strong potential as temperature sensors, in that they can form pronounced secondary and tertiary structures [24], and through their ability to form intermolecular RNA : RNA hybrids [25]. Both of these processes greatly depend on the formation of complementary base pairing, and consequently one would anticipate these to be dependent on environmental temperature.
RNA thermometers operate at the post-transcriptional level to sense selectively the temperature and transduce a signal to the translation machinery via a conformational change. They have usually a highly structured 5’-end that shields the ribosome binding site at physiological temperatures [1, 2629]. Changes in temperature are manifested by the liberation of the Shine-Dalgarno (SD) sequence, thereby facilitating ribosome binding and translation initiation.
##### 2.3. DNA
It is known that both in prokaryotic and eukaryotic cells, the geometry and tension of DNA are highly dynamic and correspond to its functional activity. In the bacterial cell, chromosome and plasmid DNA is contained in a “twisted” superhelical conformation [30, 31], where the degree of superhelicity varies in response to changes in the ambient temperature. In many examples, the expression of many genes is dependent on DNA conformation, and temperature-dependent gene regulation is mastered through changes in DNA supercoiling [3, 32, 33].
Seemingly, the temperature-induced conformational changes in DNA are mainly controlled through the presence of “nucleotid-associated” proteins, of which H-NS is the best characterized [30, 34]. In E. coli, creating and maintaining conformational structures in the DNA molecule are mainly regulated through the balance of two opposing topoisomerase activities, mainly those of topoisomerases II and I [35, 36].
Examples of pure DNA-related temperature sensitivity are rare if ever reported. In most cases, genomic thermosensitivity appears to be a result of certain interplay among DNA, RNA, and proteins. Some bacteria carry a DNA-plasmid which shows a controlled constant plasmid copy number at one temperature and a much higher or totally uncontrolled copy number at a different temperature. The high-copy number phenotype of pLO88 plasmid maintained in Escherichia coli (HB101) is observed only at elevated temperatures, (above 37°C), and is due to the precise position of a Tn5 insertion in DNA, but the exact mechanism remains obscure [37].
All abovementioned examples of membrane- and nucleic acid-based temperature sensitivity apparently include proteins as a key regulatory component. Therefore, from the point of view of molecular temperature sensation, protein-based molecular “thermometers” represent an extremely interesting group.
##### 2.4. Proteins
Many sensory pathways in living organisms use structural changes in proteins as a primary perceptive event, activating further signaling cascades. If E. coli is exposed to an oxidative substance such as hydrogen peroxide, it responds by the activation of a transcriptional regulator protein OxyR [38]. Activation of OxyR is achieved through the formation of a disulphide bound within the protein, upon which OxyR induces the expression of a set of genes adapting the bacterial cell to oxidative stress. This illustrates how it is possible both to sense and respond to an abrupt change in a specific environmental factor in a simple, yet elegant mode.
One would expect the organisms and cells to be similarly elegant when sensing temperature shifts. Indeed, a striking example is the temperature-controlled switching of the flagellar rotary motor of E. coli between the two rotational states, clockwise (CW) and counterclockwise (CCW) [39]. The molecular mechanism for switching remains unknown, but seems to be connected to the response regulator CheY-P. Two possible models of CheY-P action explain shifting the difference in free energy between CW and CCW states in terms of (i) conformation-related differential binding [40, 41] and (ii) thermodynamic changes in dissociation constants [42].
Further studies on the thermosensory transducing system in E. coli revealed that two major chemoreceptors, Tar and Tsr, which detect aspartate and serine, respectively, also function as thermoreceptors, as well as Trg and Tap receptors [43]. Interestingly, in spite of different specificity and sensitivity, amino acid sequences of all four chemoreceptors have a significant homology. These are transmembrane proteins with two functional domains in their role as chemoreceptors; one is a ligand-binding domain located in the periplasm and the other is a signaling domain located in the cytoplasm. Thus, it is suggested that a temperature change induces a conformational change in these two receptors and that this conformational change triggers the signaling for thermoresponse. In the simplest model of thermoreception by these receptors, two conformational states of these receptors are assumed: a low-temperature state and a high-temperature state [44]. The swimming pattern of the Trg- and Tap-containing cells was determined simply by the temperature of the medium, indicating that these cells under nonadaptive conditions sense the absolute temperature as the thermal stimulus, and not the relative change in temperature.
The understanding of proteins temperature-related sensory transductions in terms of their underlying molecular mechanism is fast-advancing thanks to the discovery and functional characterization of the transient receptor potential (TRP) channels. This protein family, first identified in Drosophila, is at the forefront of our sensory stem, responding to both physical and chemical stimuli and, thus, having diverse functions [45, 46].
The superfamily of TRP channels currently comprises nearly 30 mammalian members grouped into six related families: TRPC, TRPV, TRPP, TRPM, TRPN, and mucolipins. In higher organisms, TRPV channels are important polymodal integrators of noxious stimuli mediating thermosensation and nociception. The transient receptor potential channel vanilloid receptor subunit 1 (TRPV1) is widely recognized as a molecular integrator of physical and chemical stimuli in the peripheral nociceptor terminals [11, 47].
A subset of these channels, the thermo-TRPs, is activated by distinct physiological temperatures. Six thermo-TRP channels, which are all characterized by their unusually high-temperature sensitivity (), have been cloned: TRPV(1)–(4) are heat-activated [4850], whereas TRPM8 [50, 51] and TRPA1 [52] are activated by cold. With a of about 26 for TRPV1 [53] and approx. 24 for TRPM8 [54, 55], they far surpass the temperature dependence of the gating processes characterized by other ion channels () [53]. In spite of the great advances made, the molecular basis for regulation by temperature remains unknown because of the lack of structural information. More detailed consideration of protein dynamics and thermodynamics can bring us closer to understanding of universal principles of thermal sensation.
#### 3. Biophysical Aspects of Protein-Aided Thermosensation
It appears from the above mentioned examples of protein participation in temperature sensing events that sudden conformational changes, “structural transitions” play essential role on the primary conversion of physical stimulus into biologically relevant signal.
Phase transitions and critical phenomena continue to be the subject of intensive experimental and theoretical investigation. In this context, systems consisting primarily of well characterized proteins and water can serve as particularly valuable objects of study. The importance of studies of specific phase transitions in protein/water solutions derives also from their physiological relevance to the supramolecular organization of normal tissues and to certain pathological states. For example, such phase transitions play an important role in the deformation of the erythrocyte in sickle-cell disease [21, 56] and in the cryoprecipitation of immunoglobulins in cryoglobulinemia and rheumatoid arthritis [57].
Discussions about protein stability and temperature-induced structural transitions are usually limited to the stability of the native state against denaturation. Yet the native state may include different functionally relevant conformations characterized by different Gibbs energies and therefore different stabilities (e.g., the R and T states of hemoglobin). Even when the native state does not undergo a conformational change, it is still characterized by the occurrence of a large number of local unfolding events that give rise to many substates. Thus, the native state itself needs to be considered as a statistical ensemble of conformations rather than unique entity. These distinctions are very important from the functional point of view since different conformations are usually characterized by different functional properties.
The stabilizing contributions that arise from the hydrophobic effect and hydrogen bonding are largely offset by the destabilizing configurational entropy. The hydrophobic effect is strongly temperature-dependent, and is considerably weaker and perhaps even destabilizing at low temperatures than at elevated temperatures. The contribution of various interactions for a “typical” protein is reported in many works [5862]. Apparently, the transition from stabilizing to destabilizing conditions is achieved by relatively small changes in the environment. These can be changes in temperature, pH, and addition of substrates or stabilizing cosolvents. While the exact contribution of different interactions to the stability of globular proteins remains a question, our understanding seems to be refined enough to allow for the reasonable prediction of the overall folding thermodynamics [61, 62]. Important to mention that both the enthalpy end entropy changes are not constant but increasing functions of temperature, and that the Gibbs energy stabilization of a protein can be written as follows: where is a convenient reference temperature. is the heat capacity change, and and are the enthalpy and entropy values at that temperature. The temperature dependency of and is an important issue because it transforms the Gibbs energy function from a linear into a parabolic function of temperature.
For large values of , the Gibbs Energy crosses zero point twice—temperature (heat denaturation) and one at low temperature (cold denaturation). The native state is thermodynamically stable between those two temperatures and exhibits a maximum at the temperature at which . The peculiar shape of the Gibbs energy function of a protein does not permit a unique definition of protein stability. For example, having a higher denaturation temperature does not necessarily imply that a protein will be more stable at room temperature. Within the context of the structural parameterization of the energetics, the Gibbs energy of protein stabilization is approximated by where contains the contributions typically associated with the formation of secondary and tertiary structure (van der Waals interactions, hydrogen bonding, hydration, and conformational entropy), the electrostatic and ionization effects, and the contribution of the change in translational degrees of freedom existing in oligomeric proteins. The term includes interactions unique to specific proteins that cannot be classified in a general way (e.g., prosthetic groups, metals, and ligands) and must be treated on a case-by-case basis.
Nilius and coworkers have recently applied a simple thermodynamic formalism to describe the shifts in voltage dependence due to changes in temperature [63, 64], where the probability of the opening of a protein channel is given as a function of temperature, the gating charge, Faraday’s constant, and the free-energy difference between open and closed states of the channel.
At biological temperatures, some proteins alternate between well-defined, distinct conformations. In order for two conformational states to be distinct, there must be a free-energy barrier separating them. The notions involved to get from one state to another are usually much more complex than the oscillation of atoms and groups about their average positions. In proteins, because most of the forces that stabilize the native state are noncovalent, there is enough thermal energy at physiological temperature for weak interactions to break and reform frequently. Thus a protein molecule is more flexible than a molecule in which only covalent forces dictate the structure.
To further understand the nature of dynamic transitions in proteins, it is particularly important to characterize solvent effects. Solvent can in principle affect protein dynamics by modifying the effective potential surface of the protein and/or by frictional damping. Changes in the structure and internal dynamics of proteins as a function of solvent conditions at physiological temperatures have been found by using several experimental techniques [65]. It is clear from the works of Zaccai and others that solvent affects protein dynamics at physiological temperatures [6668]. They reported that in the absence of minimal hydration, proteins do not function at all. Therefore, a solvent dependence of the dynamic transition might be expected. Indeed, measurements on CO binding to myoglobin indicate that dynamic behavior of the protein is correlated with a glass transition in the surrounding solvent [69], and a recent molecular dynamics analysis of hydrated myoglobin also indicates a major solvent role in protein dynamic transition behavior [70].
From the point of view of structural biophysics, thermosensation is a special sort of mechanosensation and therefore many theoretical models and considerations developed for protein mechanosensors are also applicable for thermosensors. The difference between mechanosensitive channels and thermosensitive molecules is only the size and the organization of “pushing” agents—a lot of noncoordinated events (thermal stimuli) versus a net stretch (mechanical stimuli). Interestingly, many members of thermosensing TRPV family are known osmo- and mechanosensors. Because mechanical stimuli are everywhere, mechanosensation could represent one of the oldest sensory transduction processes that evolved in living organisms. Similar to thermal sensors, what exactly makes these channels respond to membrane tension is unclear. The answer will not be simple, because not thermal and mechanosensors are very diverse [71, 72]. However, there are interesting parallels in structural composition of different classes of known temperature-sensory proteins.
#### 4. Structural Features of Protein Molecular Thermometers
Despite significant evolutionary distances and apparent differences of primary structure all temperature-sensitive proteins known so far display some remarkable similarities in their tertiary/quaternary structure. The ability of a big protein TlpA responsible in Salmonella typhimurium for temperature regulation of transcription resides in its structural design. Two-thirds of the C-terminal portion of TlpA is contained in an alpha-helical-coiled-coil structure that constitutes an oligomerization domain. As the temperature increases, the proportion of DNA-binding oligomers decreases, leading to a derepression of the target gene. At moderate temperatures, the concentration of TlpA increases, shifting the balance to the formation of DNA-binding oligomers and, in part, restoring the repression potential of TlpA. Thus, TlpA undergoes a reversible conformational shift in response to temperature alteration, leading to an alteration in the oligomeric structure and subsequently in the regulatory capacity of TlpA [44].
The sensory capacity is contained in the coiled-coil structure of TlpA, which illustrates the means of sensing temperature through changes in protein conformation. The coiled-coil structure is a versatile and a rather flexible motif in mediating protein: protein interactions. In vertebrates, the thermosensitive elements of transcriptional mechanism typically contain coiled-coil folding motifs, such as those in leucine zipper family.
TRPV channel subunits in turn have a common topology of six transmembrane segments (S1–S6) with a pore region between the fifth and sixth segments, and cytoplasmic N- and C-termini. In both TRPV1 and TRPM8, modulation of channel gating behavior by temperature arises from the C-terminal structure that follows the S6 inner helix [51]. Partial deletions performed in the C-terminal domain of TRPV1 result in functional channels with attenuated heat sensitivity, and truncation of the whole TRPV1 C-terminal domain completely hindered channel expression [53]. Interestingly, in TRPM8 channels, binding of phosphatidylinositol bisphosphate (PIP2) leads to channel activation [73]. The proximal C-terminal TRP domain is conserved in TRPM8 and appears to serve as a PIP2 site [74]. These observations, and the fact that the key question regarding what makes thermo-TRPs temperature sensitive remained unanswered, suggests building C-terminal chimeras between different members of TRPV family as a further step in structural approach [11].
In thermo-TRP channels, it has been proposed that the structural rearrangement leads to a change in tension on the helical linker connecting the C-terminal domains with S6 segment. This tension on the linker provides the energy necessary to move the S6 inner helix to the open conformation [54, 55]. Another possibility could be that temperature affects the interaction between a particular portion of the proximal C-terminal and some other region of the channel, probably an intracellular loop. Finally, it may be that independent arrangements induced by temperature on C-terminal domains directly promote gate opening [53].
Bernd Nilius' group in their study on the voltage dependence of TRP channel gating by temperature pointed out that the small gating charge of TRP channels compared to that of classical voltage-gated channels could lie at the basis of the large shifts of their voltage-dependent activation curves, and may be essential for their gating versatility [63, 64]. Thus, small changes of the free energy of activation of these channels can result in large shifts of their voltage-dependent activation curves, and concomitant gating of these channels.
In membrane, TRP channels form tetramers of identical subunits [47]. The crystal structure of mechanosensitive/thermosensitive membrane proteins reveals that the channel folds as a homoheptamer that has a large, cytoplasmic region. Recently obtained data indicate that the modular nature of the structures involved in activation processes allow different stimuli (voltage, temperature, and agonists) to promote thermo-TRP channel opening by different interrelated mechanisms as has been suggested in the form of allosteric interaction [54, 55, 75].
The very interesting aspect resides in the observation that bacterial proteins H–NS and StpA may form hetero-oligomers exactly the same way as TRPV thermosensory channels of higher animals sometimes do [30, 54]. In this context, it is important to note that the temperature-sensitive H–NS function is also associated with oligomerization and that the H–NS oligomerization domain most evidently relies on the formation of coiled-coil oligomers [31, 69].
The molecular dynamics and organization of the temperature-sensing proteins signaling complexes are still elusive, although fast-advancing progress in this arena is uncovering the molecular identity of these elements. A series of papers published by Artmann and coworkers revealed intriguing temperature-related structural transitions phenomena in hemoglobins (Hb) and myoglobins of different species [58, 76, 77]. The reported nonlinearity in hemoglobin temperature behavior seems to be connected to physiological body temperature of the given species and therefore might surprisingly reflect the role of Hb as a molecular thermometer [78].
#### 5. Novel Classes of Molecular Thermometers: Hemoglobin and CO
Proteins of the hemoglobin (Hb) family, also referred to as the myoglobin (Mb) or globin family are gas-binding heme proteins found in all domains of life. Hbs have evolved slightly different structures and functions, but both the predominantly helical structure and certain aminoacids are well conserved (1). Distinct-but-related classes of Hbs are widespread in Bacteria, Archaea, and Eucarya. Although the physiological functions of vertebrate Hbs known so far are the transport of molecular O2 and have a role in nitric oxide (NO) metabolism, those of nonvertebrate Hbs are much more diverse. In addition to O2 transport and storage, they include facilitation of O2 diffusion, reactions with sulfide and its transport, complex and as yet incompletely elucidated roles in NO regulation and metabolism, maintenance of acid-base balance, O2 scavenging, O2 sensing, oxidase and peroxidase activities, the latter related to detoxification, vitellogenin-like function and roles as light-shading pigments and regulators of the buoyancy of aquatic insects [79].
The reported [58, 76, 77] temperature effects on hemoglobin hydration and aggregation may reflect an unknown, possibly atavistic, yet expectable function of Hb in keeping homeostasis.
The suggested by Zerlin et al. [78] ability of mammalian hemoglobins to thermoadaptation finds support in many studies made for thermophilic organisms. A gene encoding a protein homologous to Hb was identified in Aquifex aeolicus, a hydrogen-oxidizing obligate chemolithoautotroph that grows at temperatures of under microaerobic conditions. A. aeolicus thermoglobin, AaTgb, is monomeric, resistant to thermal and chemical denaturation, pentacoordinate in the ferrous deoxygenated state, and oxygen-avid. Key strongly, although not strictly, conserved positions are preserved in the AaTgb sequence. Proline occupies the C2 position, initiating the start of the C helix. Although histidine occupies the distal E7 position in most plant and animal Hbs, this residue is commonly adaptively replaced by glutamine in many invertebrate and bacterial Hbs. Similarly large thermal variations are also encountered by Hb-containing prokaryotes like the cyanobacterium Nostoc that extends from tropical to polar terrestrial environments [79]. Most of thermophilic hemoglobins discovered so far may be described as basic ones. The aminoacid sequence is compact, without additional residues or domains at either terminus beyond the A and H helices of the canonical fold. This basic fold may even be fused to other domains or duplicated and fused onto itself to yield Hbs with multiple copies of the globin domain.
The equilibrium constants for dimer-tetramer association of Hb have been determined as a linear function of temperature from kinetic studies of the forward and reverse rate constants [60]. It is worthy to note that these studies have been performed at temperatures below 30°C and therefore do not correspond to physiological conditions. The thermodynamic parameters calculated for Hb are consistent with an increased role of hydrophobic interactions within the dimer-dimer contact region, or a decreased role of hydrogen bonds and ion pair interactions.
Thermodynamic experiments by Frauenfelder, Petsko, and Tsernoglu [80] showed that myoglobin can assume a large number of slightly different structures, conformational substates, separated by energy barriers. Evidence for multiple potential energy minima also comes from molecular dynamics simulations made for myoglobin. The complex of thermodynamic, structural, and functional features of hemoglobin family proteins supports the hypothesis of their possible secondary role as temperature-sensing molecules. For homeothermic organisms (birds and mammals) such multiple protein-mediated temperature control could be of special importance, supporting its strengthening during evolution.
#### Acknowledgment
The authors would like to thank Dr. G. Zaccai (Grenoble, France) for many fruitful discussions.
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2017-11-20 11:55:18
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http://mathematica.stackexchange.com/questions/48754/mathematica-compiled-functions-additional-functions
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# Mathematica Compiled Functions - additional functions
Mathematica supports an exact list of compilable functions:
They are listed here: List of compilable functions.
However, I would need to add the following functions: ToExpression, StringFreeQ, StringReplace, BinaryRead and BinaryWrite...
Is it possible?
I know that BinaryRead and BinaryWrite have simple C++ equivalents but how can I add this functionality to the Compile function?
Furthermore, Compile only accepts Integer, Floats and Lists and Nested Lists as input. Can it be modified to accept Strings?
Could reimplementing the Compile Function work?
-
These functions can be used in Compile and can be part of a compiled functions. However, when they're run, the byte-code interpreter will fall back to standard evaluation which mean that these functions won't get a performance boost from compilation. The rest of your program will get a performance icrease, just not these particular functions. – Szabolcs May 29 '14 at 18:48
Does the standard evaluation call the MathKernel or can the generated code be used standalone? Can I also add support for strings or would a string be a Integer list representing ASCII codes? – user13675 May 29 '14 at 20:25
It calls MathKernel. I'm not sure how exactly LibraryLink functions interact with compiled functions. It may be possible to implement these I/O functions using LibraryLink to make it possible to integrate them into a standalone program. Unfortunately I don't have experience translating Mma into standalone C code (though the docs suggest it's possible to an extent), so I'll leave this for others to answer ... – Szabolcs May 29 '14 at 20:27
As far as I know there is no way to extend the Compile system in the way you describe, but you can often work around specific problems. For example, for handling strings you can write an outer function that converts strings to a packed vector of integers using ToCharacterCode, then pass this to your compiled function. However, I would encourage you not to take this approach as the string tools already make use of compiled libraries (including PCRE).
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2016-05-05 20:04:09
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https://physics.stackexchange.com/questions/132309/partial-pressure-which-solution-is-right
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# Partial Pressure - Which solution is right?
Question: Carbon dioxide (1.100 g) was introduced into a 1.00 L flask which contained some pure oxygen gas. The flask was warmed to 373 K and the pressure was then found to be 608 mmHg. If CO2 and O2 were the only gases present, what was the mass of oxygen in the flask?
Solution: Moles of CO2 = mass/molar mass $= 1.100/44.01 = 0.02499$ mol.
The partial pressure of carbon dioxide can then be calculated from the Ideal Gas Equation. $$V = 1.00\,\text{L}$$ $$T = 373\,\mathrm K$$
Pcarbon dioxide $$\frac{nRT}{V}= 0.02499 \cdot 8.314 \cdot 373/1.00= 77.5\,\mathrm{kPa}$$
The total pressure Ptotal = Poxygen + Pcarbon dioxide
$$P_{\text{total}} = 608\cdot \text{mmHg} = (608/760) \cdot 101.3\,\mathrm{kPa} = 81.0\,\mathrm{kPa}$$
Therefore 81.0 = Poxygen $+ 77.5$ Poxygen $= 3.5\,\mathrm{kPa}$
From the ideal gas equation, the moles of O2 can be deduced.
$$n=\frac{PV}{RT} = 3.5 \cdot \frac{1.00}{8.314 \cdot 373} = 1.13 \cdot 10^{-3}\,\text{mol}$$
Mass of oxygen = moles x molar mass $$= 1.13 \cdot 10^{-3} \cdot 32.00\,\text{g} = 0.0361\,\text{g}$$
Ambiguity: While calculating Pcarbon dioxide via $\frac{nRT}{V}$ why don't the author utilised dalton's law of partial pressures & multiplied?
(moles of CO2[0.02499] x Total pressure[81kPa]) to get the Pcarbon dioxide. Of coarse this is applicable while system is heated or cooled
• Welcome to Physics SE! I edited your post to add the nice layout of the equations. MathJax on this website makes math more readable. Check it out! To learn it, just hit the edit button on your post and see, that the math simply is marked up using a dollar. SI units are separated by a small blank with "\,". – Stefan Bischof Aug 24 '14 at 7:21
## 1 Answer
Ahh, I spent quite some time reading this problem, the problem with applying Dalton's Law of Partial Pressures is that we shouldn't be multiplying moles of $CO2$ with the total Pressure, rather we should multiply the mole fraction of $CO2$ with the total Pressure, in this case however, since the initial quantity/moles of oxygen is not known, it is not possible to find the mole fraction of $CO2$ directly and thus simplify the problem. Thus the solution you provided first is the correct way to go about the problem.
Further reading, http://en.wikipedia.org/wiki/Dalton%27s_law_of_partial_pressures
Hope this Helped. :)
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2019-07-20 06:06:53
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https://codereview.stackexchange.com/questions/248438/improving-a-medicare-beneficiary-identifier-mbi-generator?noredirect=1
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# Improving a Medicare Beneficiary Identifier (MBI) generator
I am new to coding and looking for a few pointers on how I can improve my first project.
At work, I often need to create Medicare Beneficiary Identifiers (MBI) when creating test patients with Medicare coverage, and have to look up the format every time. I thought creating an MBI generator would be a great first project. Doing a quick search, I found a similar project on this site (which is what lead me here), but the questioner wants to create 10,000 records, whereas I may only need one or two MBIs at a time (their question can be found here: Sequential MBI generator). Being new and not understanding all of the code, I was nervous to follow in their path and end up with a ton of records, so I actually followed an example of a random password generator and tweaked it to meet my needs. As the outcome is vastly different from the linked example and seems verbose, I wondered if anyone with more experience would be able to give me some pointers to get me off to improve my work.
#MBI is 11 characters in the following format
# 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11
# C - A - AN- N - A - AN- N - A - A - N - N
# C = Numeric 1 - 9
# N = Numeric 0 - 9
# A = Alphabetic A...Z; Not S, L, O, I, B, Z
# AN = Either A or N
import random
letter = 'ACDEFGHJKMNPQRTUVWXY' # not = B, I, L, O, S, Z
digit = str('0123456789')
partdig = str('123456789')
dig_let = digit + letter
while 1:
mbi_return = 1
mbi_need = int(input("How many MBI numbers do you need to generate?: "))
for x in range(0,mbi_need):
mbi = ""
for x in range(0, mbi_return):
mbi_char0 = random.choice(partdig)
mbi_char1 = random.choice(letter)
mbi_char2 = random.choice(dig_let)
mbi_char3 = random.choice(digit)
mbi_char4 = random.choice(letter)
mbi_char5 = random.choice(dig_let)
mbi_char6 = random.choice(digit)
mbi_char7 = random.choice(letter)
mbi_char8 = random.choice(letter)
mbi_char9 = random.choice(digit)
mbi_char10 = random.choice(digit)
mbi = (mbi_char0 + mbi_char1 + mbi_char2 + mbi_char3 + mbi_char4 + mbi_char5 +
mbi_char6 + mbi_char7 + mbi_char8 + mbi_char9 + mbi_char10) # I imagine this could be much cleaner
print(mbi)
Instead of while 1: please just use while True. while 1: is a throwback from old versions of C that didn't have stdbool.h. while True: is much most explicit a about what your intent is.
At the top you have
letter = 'ACDEFGHJKMNPQRTUVWXY' # not = B, I, L, O, S, Z
digit = str('0123456789')
partdig = str('123456789')
dig_let = digit + letter
Python actually has built-ins for these:
from string import digits, ascii_uppercase
# letter is the set of ascii characters minus B, I, L. . .
letter = "".join(set(ascii_uppercase) - {'B', 'I', 'L', 'O', 'S', 'Z'})
# Just use digits instead of digit
partdig = digits[1:] # Remove the first digit
dig_let = digits + letter
That saves you from needing to type out each of the letters to include.
Also note, even if string.digits didn't exist, you could have also defined digit as:
>>> digit = "".join(str(n) for n in range(0, 10))
>>> digit
'0123456789'
Also, all variable names here should be lowercase, separated by underscores. partdig should be part_dig, or part_digits, or even better: non_zero_digits.
I also think letter should be letters, since it's a collection of letters. It's a small change, but it lets your readers know that it's multiple letters, not just a single one.
mbi_need = int(input("How many MBI numbers do you need to generate?: "))
for x in range(0,mbi_need):
Especially in Python that matters a lot. If that was just a pasting error, it's a good idea to look over the code before posting just to double check that errors weren't introduced accidentally.
You're using a odd 5-space indentation in the loop though, which is part of the problem. Please use 4-space indentation.
for x in range(0,mbi_need):
mbi = ""
for x in range(0, mbi_return):
Both loops define a x variable! Since you never use x in either loop though, use _ instead:
for _ in range(0,mbi_need):
mbi = ""
for _ in range(0, mbi_return):
_ is a convention that says "I needed to create a name, but don't need the variable", which is the case here.
range(0, mbi_need)
0 is the implicit start; it's not necessary to specify it if you only otherwise need to specify the ending number. Just write:
range(mbi_need)
mbi_char0 = random.choice(partdig)
mbi_char1 = random.choice(letter)
mbi_char2 = random.choice(dig_let)
mbi_char3 = random.choice(digits)
. . .
Whenever you find yourself creating many similar variables, and you're differentiating them by putting numbers in the name, stop! You should likely be using a list instead. Because the make-up of the MBIs doesn't follow an easy pattern, fixing this isn't super straightforward, but it's still possible. First, I'd create a list holding the order of partdig, letter, dig_let, digits. . . which will define the order of the different character types:
mbi_pattern = [non_zero_digits, letters, digit_letters, digits,
letters, digit_letters, digits, letters,
letters, digits, digits]
This looks ugly, but it will clean up the code later. Sometimes all you can do is move the ugly bulk to the side. Once you've defined that list, creating a MBI is trivial and tiny:
mbi = "".join(random.choice(part) for part in mbi_pattern)
print(mbi)
Get each of the part sets, generate a random character from each of them, then join them into a string.
for _ in range(mbi_return):
loop doesn't appear to be doing anything. It seems like it's doing a similar job as the other loop, except it will always be range(1), which will only run once, which means it isn't really a loop. I got rid of it because it isn't doing anything except complicating the code.
The same can be said about the while True as well. You want to generate 10000 codes, repeatedly, forever? The while True loop will never end since you never break from it. I also got rid of it because it is also complicating the code without good reason.
mbi = ""
This isn't necessary. Even if you needed mbi in the outer scope, loops in Python don't create scopes like they do in other languages. mbi "defined" inside the loop can be accessed from outside of the loop.
In your remaining loop, you're creating an mbi, then just printing it. That doesn't allow you to do anything with the data though, like save it to file. It would be much cleaner to store the generated MBIs in a list so that they can potentially be used later. If you do that, your loop can be made into a list comprehension:
mbis = ["".join(random.choice(part) for part in mbi_pattern)
for _ in range(mbi_need)]
Each MBI is generated using the same generator expression as before, but now it's wrapped in a list comprehension to generate multiple.
In the end, this is what I'm left with:
import random
from string import digits, ascii_uppercase
letters = "".join(set(ascii_uppercase) - {'B', 'I', 'L', 'O', 'S', 'Z'})
non_zero_digits = digits[1:]
digit_letters = digits + letters
mbi_pattern = [non_zero_digits, letters, digit_letters, digits,
letters, digit_letters, digits, letters,
letters, digits, digits]
mbi_need = int(input("How many MBI numbers do you need to generate?: "))
mbis = ["".join(random.choice(part) for part in mbi_pattern)
for _ in range(mbi_need)]
print("\n".join(mbis))
And its usage:
How many MBI numbers do you need to generate?: >? 10
5V70VK4JP28
8Y12N77RC51
9JM2JN8RQ38
3X08DH7FH95
3MH6Y49KU87
6N70AC7MW75
9A67A62TU38
4A48C94QT38
2NP7TY0DC65
1GP8A57JQ27
As mentioned in the comment though, really, code should be tucked into functions. In larger programs, that eases testing and comprehension of your code.
Here, you could have a function that generates a single MBI, then use it to generate a list of them. I also always have a main function that ties the whole program together so I can control the execution of the code easier.
I also realized after I had my coffee that all the variables at the top are really constants, so they should be in UPPER_SNAKE_CASE.
After those changes, I'm left with:
import random
from string import digits, ascii_uppercase
LETTERS = "".join(set(ascii_uppercase) - {'B', 'I', 'L', 'O', 'S', 'Z'})
NON_ZERO_DIGITS = digits[1:]
DIGIT_LETTERS = digits + LETTERS
MBI_PATTERN = [NON_ZERO_DIGITS, LETTERS, DIGIT_LETTERS, digits,
LETTERS, DIGIT_LETTERS, digits, LETTERS,
LETTERS, digits, digits]
def generate_mbi():
return "".join(random.choice(part) for part in MBI_PATTERN)
def main():
mbi_need = int(input("How many MBI numbers do you need to generate?: "))
mbis = [generate_mbi() for _ in range(mbi_need)]
print("\n".join(mbis))
if __name__ == '__main__':
main()
• This is a very good review, indeed. One final improvement I would suggest either to the OP or to you, if you feel inclined to augment your answer: use functions. One to create a single MBI. The other to orchestrate the program: get user input and loop the needed number of times.
– FMc
Aug 26, 2020 at 17:43
• @FMc Yes. I completely agree with the use of functions here. I've started limiting that suggestion though to more "developed" programs where the benefits are much more obvious. I can certainly add something in though Aug 26, 2020 at 17:51
• This is great! Thanks so much for the review! I knew there must be easier ways to address all of these issues (e.g., "...(ascii_uppercase) - {B, I,..etc.}), but I am limited by my cursory knowledge after a week of studying the language. This gives me a huge head start in learning these concepts. Aug 26, 2020 at 20:38
• @LostAsHeat Glad to help. I'm not sure if you're familiar with them, but the code here is using generator expressions, which are similar to list comprehensions. If you haven't used either before, they're very useful constructs. join is also extremely useful. Aug 26, 2020 at 20:48
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2022-08-17 22:49:04
|
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|
https://www.lessonplanet.com/teachers/1-math
|
# $1 Math ##### This$1 Math lesson plan also includes:
Captivate your class by having them find the value of their names, different zoo animals, musical instruments, etc.,with a mental math lesson. Using the coding formula listed, children learn to fluently estimate and calculate simple sums.
CCSS: Designed
##### Instructional Ideas
• Have students write a secret message to one another using the number codes listed on the resource and then applying addition to calculate the value
• Challenge the class to find which teacher in the school has the name with the greatest value
##### Classroom Considerations
• Graph paper would make this activity easier and neater to solve the addition equations
• Project the codes overhead or write them on the board for younger students
##### Pros
• Engaging in-class assignment
• Differentiated instructions included with the resource
• Designed for Common Core math standards
• None
|
2019-03-22 02:29:16
|
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|
https://epynn.net/Dense.html
|
# Fully Connected (Dense)
Source files in EpyNN/epynn/dense/.
See Appendix - Notations for mathematical conventions.
## Layer architecture
A fully-connected or Dense layer is an object containing a number of units and provided with functions for parameters initialization and non-linear activation of inputs.
class epynn.dense.models.Dense(units=1, activate=<function sigmoid>, initialization=<function xavier>, se_hPars=None)[source]
Definition of a dense layer prototype.
Parameters
• units (int, optional) – Number of units in dense layer, defaults to 1.
• activate (function, optional) – Non-linear activation of units, defaults to sigmoid.
• initialization (function, optional) – Weight initialization function for dense layer, defaults to xavier.
• se_hPars (dict[str, str or float] or NoneType, optional) – Layer hyper-parameters, defaults to None and inherits from model.
### Shapes
Dense.compute_shapes(A)[source]
Parameters
A (numpy.ndarray) – Output of forward propagation from previous layer.
def dense_compute_shapes(layer, A):
"""Compute forward shapes and dimensions from input for layer.
"""
X = A # Input of current layer
layer.fs['X'] = X.shape # (m, n)
layer.d['m'] = layer.fs['X'][0] # Number of samples (m)
layer.d['n'] = layer.fs['X'][1] # Number of features (n)
# Shapes for trainable parameters Units (u)
layer.fs['W'] = (layer.d['n'], layer.d['u']) # (n, u)
layer.fs['b'] = (1, layer.d['u']) # (1, u)
return None
Within a Dense layer, shapes of interest include:
• Input X of shape (m, n) with m equal to the number of samples and n the number of features per sample.
• Weight W of shape (n, u) with n the number of features per sample and u the number of units in the current layer k.
• Bias b of shape (1, u) with u the number of units in the layer.
Note that:
• Parameters shape for W and b is independent from the number of samples m.
• The number of features n per sample may be expressed in this context as the number of units in the previous layer k-1, even though this definition may tend to be less general.
### Forward
Dense.forward(A)[source]
Wrapper for epynn.dense.forward.dense_forward().
Parameters
A (numpy.ndarray) – Output of forward propagation from previous layer.
Returns
Output of forward propagation for current layer.
Return type
numpy.ndarray
def dense_forward(layer, A):
"""Forward propagate signal to next layer.
"""
# (1) Initialize cache
X = initialize_forward(layer, A)
# (2) Linear activation X -> Z
Z = layer.fc['Z'] = (
np.dot(X, layer.p['W'])
+ layer.p['b']
) # This is the weighted sum
# (3) Non-linear activation Z -> A
A = layer.fc['A'] = layer.activate(Z)
return A # To next layer
The forward propagation function in a Dense layer k includes:
• (1): Input X in current layer k is equal to the output A of previous layer k-1.
• (2): Z is computed by applying a dot product operation between X and W, on which the bias b is added.
• (3): Output A is computed by applying a non-linear activation function on Z.
Note that:
• Z may be referred to as the (biased) weighted sum of inputs by parameters or as the linear activation product.
• A may be referred to as the non-linear activation product or simply the output of Dense layer k.
\begin{split}\begin{alignat*}{2} & x^{k}_{mn} &&= a^{\km}_{mn} \tag{1} \\ \\ & z^{k}_{mu} &&= x^{k}_{mn} \cdot W^{k}_{nu} \\ & &&+ b^{k}_{u} \tag{2} \\ & a^{k}_{mu} &&= a_{act}(z^{k}_{mu}) \tag{3} \end{alignat*}\end{split}
### Backward
Dense.backward(dX)[source]
Wrapper for epynn.dense.backward.dense_backward().
Parameters
dX (numpy.ndarray) – Output of backward propagation from next layer.
Returns
Output of backward propagation for current layer.
Return type
numpy.ndarray
def dense_backward(layer, dX):
"""Backward propagate error gradients to previous layer.
"""
# (1) Initialize cache
dA = initialize_backward(layer, dX)
# (2) Gradient of the loss with respect to Z
dA,
layer.activate(layer.fc['Z'], deriv=True)
) # dL/dZ
# (3) Gradient of the loss with respect to X
dX = layer.bc['dX'] = np.dot(dZ, layer.p['W'].T) # dL/dX
return dX # To previous layer
The backward propagation function in a Dense layer k includes:
• (1): dA the gradient of the loss with respect to the output of forward propagation A for current layer k. It is equal to the gradient of the loss with respect to input of forward propagation for next layer k+1.
• (2): dZ is the gradient of the loss with respect to Z. It is computed by applying element-wise multiplication between dA and the derivative of the non-linear activation function applied on Z.
• (3): The gradient of the loss dX with respect to the input of forward propagation X for current layer k is computed by applying a dot product operation between dZ and the transpose of W.
Note that:
• The expression gradient of the loss with respect to is equivalent to partial derivative of the loss with respect to.
• The variable dA is often referred to as the error term for layer k+1 and dX the error term for layer k.
• In contrast to the forward pass, parameters are used to weight dZ with shape (m, u). Therefore, we use the transpose of W with shape (u, n) in order to compute the dot product.
\begin{split}\begin{alignat*}{2} & \delta^{\kp}_{mu} &&= \frac{\partial \mathcal{L}}{\partial a^{k}_{mu}} = \frac{\partial \mathcal{L}}{\partial x^{\kp}_{mu}} \tag{1} \\ \\ & \frac{\partial \mathcal{L}}{\partial z^{k}_{mu}} &&= \delta^{\kp}_{mu} \\ & &&* a_{act}'(z^{k}_{mu}) \tag{2} \\ & \delta^{k}_{mn} &&= \frac{\partial \mathcal{L}}{\partial x^{k}_{mn}} = \frac{\partial \mathcal{L}}{\partial a^{\km}_{mn}} = \frac{\partial \mathcal{L}}{\partial z^{k}_{mu}} \cdot W^{k~{\intercal}}_{nu} \tag{3} \\ \end{alignat*}\end{split}
def dense_compute_gradients(layer):
"""Compute gradients with respect to weight and bias for layer.
"""
X = layer.fc['X'] # Input of forward propagation
dZ = layer.bc['dZ'] # Gradient of the loss with respect to Z
# (1) Gradient of the loss with respect to W, b
dW = layer.g['dW'] = np.dot(X.T, dZ) # (1.1) dL/dW
db = layer.g['db'] = np.sum(dZ, axis=0) # (1.2) dL/db
return None
The function to compute parameter gradients in a Dense layer k includes:
• (1.1): dW is the gradient of the loss with respect to W. It is computed by applying a dot product operation between the transpose of X and dZ.
• (1.2): db is the gradient of the loss with respect to b. It is computed by summing dZ along the axis corresponding to the number of samples m.
Note that:
• We use the transpose of X with shape (n, m) for the dot product operation with dZ of shape (m, u).
\begin{split}\begin{alignat*}{2} & \frac{\partial \mathcal{L}}{\partial W^{k}_{nu}} &&= x^{k~{\intercal}}_{mn} \cdot \frac{\partial \mathcal{L}}{\partial z^{k}_{mu}} \tag{1.1} \\ & \frac{\partial \mathcal{L}}{\partial b^{k}_{u}} &&= \sum_{m = 1}^M \frac{\partial \mathcal{L}}{\partial z^{k}_{mu}} \tag{1.2} \end{alignat*}\end{split}
## Live examples
The Dense layer is used as an output layer in every Network training examples provided with EpyNN
Examples of pure Feed-Forward Neural Networks within these examples can be directly accessed from:
|
2022-08-15 09:13:29
|
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|
https://blogs.ams.org/blogonmathblogs/
|
iRashida: A Tour
is a blog written by Rashida Hakim, a junior computer science student at Caltech, where she shares “problems and ponderings in physics and math, with a sprinkle of computer science”. In particular, she is interested in the algorithms used to implement machine learning and artificial intelligence. I was excited to find her blog and enjoyed reading many of her posts. In this tour, I summarize my favorite posts and talk to Rashida herself, to get to know more about the motivation behind her blog.
Putting #1 First – Deriving Benford’s Law
In this post, she discussed Benford’s law is named for physicist which gives a distribution for the first digits observed in datasets. In particular, it states that the probability that some digit d will be the first digit is $log_{10}(1+ \frac{1}{d})$ (see Figure 1).
Figure 1. Plot of Benford’s Law vs the first significant digit of a set of physical constants. Public Domain.
She takes the reader through the intuition of the law using the circular slide rule, its generalization to other digits, and provides an example of how it has been used in the real-world as a way to detect fraud. She ends the post by reflecting that one of the most interesting facts is that this law doesn’t apply to just numbers. For example, the natural numbers do not seem to follow this distribution. However, it says something about the real-word.
“The circular slide rule explanation exposes that exponential sequences and other multiplicative sequences should follow Benford’s Law. But does that apply to data sets of physical constants, or heights of mountains, or the masses of celestial bodies? All of these examples, and many more, have been observationally shown to follow Benford’s Law. Maybe the real discovery that Benford and Newcomb made isn’t just about digits, but about how multiplicative and exponential growth is inherent to so many unexpected corners of our universe.”
Mathematicians’ Many Hats
Riddle: “There is a line of 100 mathematicians, each positioned so that he can see all the hats in front of him but none of the hats behind him (nor his own hat). Each of the mathematicians is randomly assigned a colored hat, which is either red or blue. Note that there do not have to be exactly 50 of each color (since each mathematician’s hat color is independent of the others). Now the mathematicians are asked to guess the color of their own hat, starting from the back of the line and everyone can hear the guesses. Assuming that they are allowed to confer beforehand to discuss a strategy, what is the maximum number of mathematicians who can guess correctly?
In this post, Rashida discusses how to solve this and variations of the riddle with logic and a bit of set theory. Turn out that in this more simple case all but one mathematician (the first) can guess their hat color correctly by using the formula below (see Figure 2).
Figure 2. Obtained from the post Mathematicians’ Many Hats.
She describes two variations of this riddle and their solutions. First, replacing 100 with a countably infinite number of mathematicians, and secondly, what happens when no information is being carried forward when the mathematicians say their own hat color.
RH: I’m currently a junior at Caltech studying computer science, but I started my blog as a high schooler. I started my blog as a way to remember cool math, physics, and computer science concepts that I learned through school projects or on my own. I like to do fairly long-form posts on a single subject that approach the subject from multiple angles. A concept inspires me to write a blog post when at first it seems simple (or is easy to introduce) but under the surface, there is mathematical complexity that I can really dive into. Over time, my blog posts have evolved to include more simulations and code (rather than pure math/physics) to match my evolving interests.
VRQ: What is the most interesting thing you’ve learned through blogging?
RH: The most interesting thing I’ve learned through blogging is that you have to understand a concept really well (much better than you initially think!) to explain it to other people. It always surprises me how many unanswered questions pop up in my brain in the process of writing a blog post. And still, I sometimes get questions from my readers that I haven’t even thought about!
VRQ: What are some of your favorite blog posts you’ve written?
RH: One of my favorite posts that I have written was about Benford’s Law – which explains why some datasets have first digit distributions that aren’t uniform. I like that post because it is about a fairly unintuitive concept that shows up in a lot of places. Just recently, I got a lot of hits to that post because Benford’s Law was being (incorrectly) applied to find discrepancies in the 2020 election data. I hope after reading the post, people have the mathematical knowledge to debunk that particular bit of ‘fake news’ for themselves. Another of my favorite posts is about the optics of why rainbows show up. Most people know the basics of rainbow formation – water droplets in the air refract sunlight. But when you closely study the physics of just a single drop of water, you can learn so many new and interesting facts about this natural phenomenon. I like the post a lot because it taught me that it is worth it to go deeper, even on subjects I think I already understand.
VRQ: What are some of your favorite math blogs, if any?
RH: I love Ben Orlin’s ‘Math with Bad Drawings’ for insight on both math and math education. Also, it is often hilarious (if, like me, you’re the kind of person who laughs at geometry puns). I also enjoy David Richeson’s ‘Division by Zero’ because his posts are very interactive and make difficult geometry concepts accessible.
VRQ: Do you have advice for other students interested in creating their own blog?
RH: My advice is to focus on writing posts that excite you and that make you feel like you’re learning while writing them. That way, even if you’re not getting much engagement from readers, your blog is still a valuable learning experience for you.
Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@VRiveraQPhD).
Mathematical Gemstones: A Tour
is a blog created by Dr. Maria Gillespie (Colorado State University) whose research interest lies in combinatorics, with applications to Algebraic Geometry and Representation Theory. One of the aspects I like most about the blog is the fact that the posts are organized by level of difficulty using gemstones (i.e. Amber, Pearl, Opal, Saphire, and Diamond) as indicators. It was really interesting to see how the writing and explanations varied according to the levels and their intended audience.
I was curious to know more about the inspiration behind the ‘Mathematical Gemstones’ blog so, in this tour, I hope to give you a glimpse of two of my favorite posts and insights from Dr. Maria Gillepsie herself!
(Sapphire)
In this post, Gillespie discusses a solution to the following voting problem:
Suppose two candidates, A and B, are running for local office. There are 100 voters in the town, 50 of whom plan to vote for candidate A and 50 of whom plan to vote for candidate B. The 100 voters line up in a random order at the voting booth and cast their ballots one at a time, and the votes are counted real-time as they come in with the tally displayed for all to see. What is the probability that B is never ahead of A in the tally?
She discusses first how to enumerate all ballots as sequences and then finds the solution using the crystal functor (F1) to count the ballot words by counting the chains of the F1 graph. I found the diagrams in the post very insightful, especially for those new to the topic but familiar with combinatorics.
Another fun post was the idea of constructing Pythagorean triples (i.e. a triple of positive integers (a,b,c) with a²+b²=c²) on a sphere! By parametrizing all triples via geometric means, one can show that (r²−s²,2rs,r²+s²) is a Pythagorean triple on the unit circle for any integers r and s. However, questions about finding Pythagorean triple on the unit sphere remain open, and she shares,
“There is some hope, however. In this paper by Hartshorne and Van Luijk, the authors show that there are infinitely many Pythagorean triples in the hyperbolic plane, using the Poincare Disk model and some nice hyperbolic trig formulas combined with some Eulerian number theory tricks. So Pythagorean triples are not the sole property of flat Euclidean space.”
Maria Gillespie: I’m an assistant professor at Colorado State University, currently working towards tenure. I started my blog when I was a graduate student at UC Berkeley. At the time, there was just so much new mathematics that I was trying to learn, and I know I absorb things best when I explain things to others. So it started out as a way for me to learn new concepts in graduate school, by typing them out to an online audience. At the same time, I’m also passionate about bringing the joy of mathematics to everyone around me – sometimes I learn or remember a fascinating nugget of truth in mathematics, and I just want to yell it to the universe and share that joy with as many people as I can. So I decided to put together a blog that could accomplish both of these goals at once.The way I accomplished this was to make each post a self-contained “mathematical gemstone” – a shining example of mathematical beauty and truth.
Some gemstones could be “harder” than others, perhaps assuming a higher level of mathematical background for the intended audience. In order to help the reader determine which posts would be appropriate for them to read, I sort posts into five levels of “gemstone hardness” according to actual measures of how hard real gemstones are, starting from Amber (one of the softest gemstones) and going all the way up to Diamond (a very hard stone). Here is a description of the levels, from my About This Blog page:
• Amber: This category contains posts that anyone with very basic elementary or middle school mathematics background can appreciate.
• Pearl: For Pearl posts, some high school courses or early college courses may be helpful in understanding the content.
• Opal: These posts are aimed at undergraduates with some basic first-course background knowledge in algebra, analysis, discrete math, or topology.
• Sapphire: These gemstones would be appreciated by mathematics graduate students or professors, or those with at least an undergraduate degree in mathematics.
• Diamond: The hardest type of gemstone. These posts are highly specialized, containing content that only mathematicians who have studied the topic in depth will have the background to understand.
I later added a “Miscellaneous” category that allows me to share thoughts or discoveries that are not strictly mathematical, but which are related to mathematics as a discipline. This category includes things like LaTeX tips, book recommendations, thoughts on social issues in mathematics, and most recently, doing math in a pandemic. To summarize, my blog serves as an organized platform for me to share ideas and thoughts that could be enjoyable or helpful to other mathematicians and scientists, to students, and to the general population.
VRQ: What is the most interesting thing you’ve learned through blogging?
Maria Gillespie: People can get REALLY upset when you try to suggest new ways of doing or explaining elementary mathematics. I have a post on the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction), which is often used as a tool in middle and high schools to teach the order of operations. In the post, I explain why I feel the acronym can be confusing to students, and the incorrect answers you can get by interpreting it too literally. That post really must have hit a nerve among some readers for some reason, because I have never been at the receiving end of so much hatred and vitriol in the comments! It just goes to show that even mathematics can become controversial if it’s out there on the internet.
VRQ: What are some of your favorite blog posts you’ve written?
Maria Gillespie: I’d have to say the most fun one I’ve ever written was Can you prove it … combinatorially?in which I prove Binet’s formula for the Fibonacci numbers directly using a combinatorial proof, without any induction or generating functions (the usual methods). It was just so satisfying and fun to find a combinatorial proof of a formula that involves the square root of 5, as intricate as such a proof may be.My favorite among the “soft topics” is the recent series of four posts I wrote on doing math in the pandemic, starting with (it links to the others there). Writing all of that out really made it hit home how very much we’ve all learned from being forced into a virtual setting this past year.
While the pandemic has been a horrible natural disaster, the silver lining I see is that there’s a lot of opportunity for some of these tools to still be used after the pandemic is under control, and I’m excited to see where things go in the future.Finally, the posts that have turned out to be the most useful are the ones I wrote early in graduate school on the basics of my research area, in symmetric function theory and Schubert calculus. I’ve looked back at them countless times to remember formulas, and other grad students in my area have appreciated them as a resource as well.
VRQ: What are some of your favorite math blogs, if any?
Maria Gillespie: I have found so many of Qiaochu Yuan’s posts on his blog Annoying Precision so useful. They’re well-written, precise, and often have exactly the proof of some fact that I was trying to look up at the time. I also like since I think he has really good ideas about open access and the future of mathematics publishing.
VRQ: Do you have advice for other mathematicians interested in creating their own blog?
Maria Gillespie: My main piece of advice is to just go for it! Write about what you’re passionate about, or something you’re learning at the moment, and post it without too much worry about who will see it and whether it’s perfect. Blogs are an excellent outlet for mostly-correct mathematics (or mathematics-related topics) that doesn’t have to be peer-reviewed but can add a lot of value to the world. And more likely than not, your thoughts will be valuable to someone. As for the technical side, WordPress is your friend and it will take care of you.
Do you have suggestions of topics or blogs you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@VRiveraQPhD)
“Combinatorics and more”: A Tour
Gil Kalai writes the “Combinatorics and more” blog. I find many of his posts on the blog to be detailed and nicely structured. Here are just a few of the recent ones I enjoyed.
I always think it’s interesting to explore which big research questions attract a lot of interest. For those who aren’t familiar with polymath projects, this post describes what they are and gives updates on potential polymath projects that Tim Gowers suggested on his blog in 2009. Kalai also suggests some possible future directions for polymath projects and asks “meta questions,” such as “What is the ideal platform for a polymath project?” and, my personal favorite, “Are polymath projects inviting in terms of diversity of participants?”
The “To cheer you up in difficult times” posts
So far, Kalai has created 19 “To cheer you up in difficult times” posts, including this one about a proof of the Erdős-Faber-Lovász conjecture uploaded to arXiv by Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus and
To cheer you up in difficult times 13: Triangulating real projective spaces with subexponentially many vertices” about another new paper posted to arXiv by Karim Adiprasito, Sergey Avvakumov, and Roman Karasev.
This post is replete with interesting bits of information about recent papers, videos from some of Kalai’s lectures and even “a small taste of quantum poetry for the skeptics.”
Besides all of the interesting posts by Kalai, there are also a bunch of guest posts worth checking out. Here are just a few:
“Dan Romik on the Riemann zeta function”
“Recently when I was thinking about the Riemann zeta function, I had the double thrill of discovering some new results about it, and then later finding out that my new ideas were closely related to some very classical ideas due to two icons of twentieth-century mathematics, George Pólya and Pál Turán,” Romik wrote in the beginning of the piece.
In this post, Bárány explains limit shapes and the limit shape theorem, limit shapes for polygons in convex bodies and more.
Want to share ideas or feedback? Reach out in the comments or on Twitter (@writesRCrowell).
Math Walks: A Tour
Math Walks is a blog created by secondary math teacher Traci Jackson. It started on March 27th to encourage math discussion on neighborhood walks during the quarantine. I was so excited to find this blog that brings such a playful atmosphere to learning math. Inspired by a mom and her kids doing a PE class on a walk, she was inspired to find a way to bring math to daily walks!
“If I could leave a little bit of math on daily walks, I could not only give parents a way to incorporate some math, but maybe I could try and change math into curiosity, wonder and problem solving, even just a little.” – Traci Jackson, in the About section
As a person who loves going on walks, especially, since this year has been spent a lot in online meetings, I found her blog full of playful invitations to do math. You may suspect that like my last post, this intersects in wonderful ways with my hobbies and a love for math. And, who doesn’t love playing with chalk?
It was so well received, that translations in French and German are now available. She shares, how excited she is about people creating their own math walks.
I was thrilled to see other people duplicating drawings or creating their own. It was suddenly not just my neighborhood! Although watching my neighbors take pictures and problem solve as family was equally interesting, since almost every time I walked down my street I could see problem-solving in action. I got several texts and a few emails asking “Is that you leaving math everywhere?” I guess I have a reputation.” – Traci Jackson, in the About section
You can see many walks share by others by following the hashtag #mathwalks on Twitter. If you enjoy puzzles, beware, this blog will have you puzzling the day away. Below I share some of my favorites puzzles:
“Graphing Stories” based on the website with the same name, which is a collaboration between Dan Mayer and BuzzFeed.
“Climbing Stairs” based on Curriculum Burst 28: Stair Climbing by Dr. James Tanton.
“Crack the Code” based on the by Rajesh Kumar.
In her blog, she provides helpful how-to’s to begin your own math walk. If you are not great at free-hand drawings you can bring cutouts of the shapes you will use, use string as a compass, and a nice tip is to bring an index card with the problem drawn out. The puzzles come from a wide variety of sources which you can read more about by clicking on the pictures.
Her blog was also featured in How Sidewalk Math Cultivates a Playful, Curious Attitude Towards Math, in which she remarks,
“The perception of math is a set of sterile problems but in reality, it describes all the patterns of our world. … [Sidewalk math] opens the conversation to what math is. It engages people who wouldn’t do math ordinarily,” Jackson said. She believes that the visual element plays a role in that engagement. Stanford professor Jo Boaler has advocated for the learning benefits of visual math for years, but Jackson said it remains an under-explored dimension of math instruction. Though we often think of math as numbers and letters, Jackson sees it as a way of viewing the world, and using images can unlock new connections.” – From How Sidewalk Math Cultivates a Playful, Curious Attitude Towards Math
I couldn’t agree more! What a wonderful way to share math with your community. So, this Spring, take your math for a stroll, and if you bring your chalk with you, share it with your neighborhood.
Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).
On the “Reflect, Revise, Repeat” Blog
Bonnie Basu, a secondary mathematics teacher in California, writes the “Reflect, Revise, Repeat” blog. She started the blog in June 2020. On Twitter, Basu describes herself as “trying to teach teenagers to think mathematically for a quarter of a century.”
There are currently nine posts on the blog. Here are a few of the ones I recommend checking out:
This is the first post that Basu wrote. She describes why she decided to start the blog now after thinking for a decade about writing one.
She wrote that instead of focusing on the number of years she’s been teaching, she typically reflects on her growth. “Am I making the same mistakes as I did during my first years? Or even last year? Why did I make those mistakes? What did I learn from those mistakes? Who can I turn to for mentorship to help me grow as a teacher? And I try to do this continuously,” she noted. She used this idea to come up with the name of her blog.
“My Plan” posts
Last fall, Basu wrote some “My Plan” posts, including “My Plan: Week 1,” “My Plan: Week 2,” and “My Plan: Week 3 and 4” about how she handled distance learning for the first month of the semester. There are points scattered throughout that I thought were great.
For instance, in her Week 1 post, she wrote:
“The only way of communicating with families is email or individual phone calls. That made it extremely challenging and frustrating that our automated phone system wasn’t set up so we can at least call en mass. (By the way, it’s still not set up and it has been 5 working days.) I have been calling home, but it’s a slow go. I can get through about 5 each day before I have to stop. Most take a lot of time because I am fluent in only one language – the one I am currently writing in. And the district doesn’t have a department solely dedicated to helping teachers translating which is very strange considering how many of our families are more comfortable speaking a variety of other languages.”
Another thing that resonated with me was her discussion of how disconnected students have felt from their teachers and each other, along with her ideas for building community in a virtual setting.
In her Week 2 post, something that stuck with me was her discussion of her desire to help students become more independent and the challenges that come with cultivating that independence with distance learning. She wrote:
“Over 80% of the student population is highly dependent. And mostly because of all the hand holding that goes on from when they are little. Just because kids are from a vulnerable and disadvantaged area, does not mean they cannot. But that is the mindset that so many educators have. If kids are given the opportunity and have scaffolding in place, success will happen.
I help students become more independent when in the classroom, but I am really struggling with it virtually.”
By her Week 3 and 4 post, Basu started using the phrase “distance surviving” instead of distance learning. She wrote “I feel like I am riding a rollercoaster in the dark. (I don’t like rollercoasters nor do I like the dark.)” She discussed a situation where many kids didn’t complete an assignment and how she handled that.
“Find the Good”
This is the first post Basu wrote in 2021 and I think it’s a good one to discuss in closing out my post. She opens with this relatable paragraph:
“I have had many challenges in my teaching career, but nothing like I experienced over the last 9 months (as every single teacher). Every day was like starting all over. I had no idea what I was doing and was just trying to survive each day while keeping my students’ education and social-emotional well being at the forefront.”
She details the importance of teachers knowing their audience and making slow changes with students to give them time to acclimate. Reflecting on the previous semester, she wrote that she had “wanted to weave in assignments that would support a healthy mental state and allow them to forget the world, even if it was for a few minutes.” Based on the feedback she received from her students, it sounds as if she met that goal.
She also wrote about switching gears from giving weekly assignments she hoped would teach her students time management to assigning activities that were more fun, such as ones in Desmos.
Have an idea or suggestion you would like to share? Reach out in the comments or on Twitter (@writesRCrowell).
Playful Invitations: A Tour
Playful Invitations: Inspiring Ways to Teach Early Mathematics, is a blog written by Dorie Ranheim. Its goal is “to inspire parents, caregivers, and educators of preschool children to intentionally teach math using natural materials.” By using “loose parts”, backyards, playgrounds, and parks become great places for teaching and learning math. As described in the blog’s about page,
“My playful invitiations to learn math eventually extended beyond our home to our backyard and nearby park. During our time outdoors I realized I could showcase the beauty of real, natural materials and how inspiring, meaningful, and relatively easy they are to acquire. Overall, I hope to share ways adults can intentionally teach preschool math using these beautiful natural materials.” – Dorie Ranheim
In this tour, I will summarize some of its most recent posts. Many of them can be used as guided activities, and Ranheim provides a helpful guide on One of the aspects I appreciate about the activities is that they are all centered around play. Many of the posts consist of three parts: Prepare, Invite, and Play, and some include reflections and extensions to the activities. As she remarks,
“The blog posts are simply suggestions. There are MANY ways to develop these math skills. My hope is that reading the blog will inspire you to find opportunities in your daily life to teach math to preschool children.If it is playful, meaningful, and reasonably challenging then chances are the learning will stick!”
Numbers
In this post, Ranheim shares some of the ways that during last Spring, she and her children became more entune with nature and spent some time thinking of long-term projects.
We watched the bare branches bud and blossom, now we celebrate trees bursting with bright green leaves. […] Here are a few ways my children have explored math during this time at home:
• Practice number identification and formation using loose parts.
• Write numbers on river stones using water and brushes.
• Pattern using various colored rocks. Sometimes the simplest activities and materials seem to hold their attention the longest!
I found it related a lot to what I’ve done since the quarantine. In a very similar what, I discovered patterns in one of my hikes.
My own exploration of patterns at the beach.
Measurement
Invite: Today I thought we could trace our bodies so we can see how big we are. (After tracing one or more bodies) I wonder whose body is the tallest/longest?
This invitation is inspired by the following quote,
To compare objects, children begin by using nonstandard units (“My table is more than four hands long”) and then move to using standard units (“The table is almost three feet long”). Comparing fairly is an important concept for young children. – Juanita Copley, 2010
This made a lot of sense to me! Once you learn the standard units of measurement it’s easy to forget all the other intuitive ways we make sense of measurements. If anyone has ever tried to learn a family recipe, you’ve probably encountered many non-standard ways of measuring yourself. In this activity, each child lays on the ground and draw a chalk outline of their body, afterwards they use natural loose objects of similar size (e.g. leaves, rocks, etc.) to lay them side by side and compare the lengths. Some fun extensions include introducing rulers, or filling the outline of the body, and talking about the area.
Puzzle and Spatial Position
Invite: “I’m trying to put this leaf back together! Will you help me find the perfect match to make my leaf complete?”
As a big fan of tangrams as a kid, I love that in this invitation, you introduce the idea of fractions at a basic level by transforming leaves into puzzles.
It is a great way for children to play with the idea of a “whole”. You can start by cutting leaves in half and trying to find the match, or you can extend the activity by adding more different types of leaves or cutting them in four pieces instead. Ranheim advises that before using the leaves for learning math, they should explore their properties.
“It would also be beneficial if the child has explored the property of leaves before being asked to use them for math learning. Observe a small pile of leaves and the attributes before taking them apart.”
A personal project that has brought me great joy during the pandemic, has been to gather seeds and start my balcony garden. I could see all the fascinating (and often subtle) ways math seeped into my gardening.
The joy of learning and caring for my small garden.
What drew my attention to this blog, were my conversations with my best friend and early-childhood educator, about the ways math should be tied into how we experience nature around us. I would have loved these activities as a kid!
Also, it reminded me of one of my favorite books Braiding Sweetgrass: Indigenous Wisdom, Scientific, Knowledge, and the Teaching of Plants” by can’t recommend it enough).
“The land is the real teacher. All we need as students is mindfulness.” – Robin Wall Kimmerer, Braiding Sweetgrass
Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).
“Physics Buzz”: A Tour
While the “Physics Buzz” blog from the American Physical Society isn’t a math blog, there is some overlap. Here are some interesting recent posts on the site.
“Holiday Instability”
This post explores questions such as whether a Christmas tree, a Hanukkah menorah or a Festivus pole is more likely to topple over and which of the items would be the best choice for easy juggling.
“The Forces in Spilled Coffee Awaken”
This post discusses the physics of coffee stains. “Just in case you’re wondering if it is a worthwhile use of processing power, time, and money to study a stain, consider this. Coffee isn’t the only substance that’s made of tiny particles suspended in a liquid. Blood, paint, ink…understanding the way these kinds of liquids behave could have huge implications in areas from medicine to high-tech manufacturing. For example, in the future we may be able to create tiny structures with unique properties by carefully dropping a liquid filled with nanoparticles onto a surface and evaporating the liquid. In order to do this, though, scientists need to be able to accurately predict a mixture’s behavior. This requires an understanding of the forces involved,” Kendra Redmond wrote in the post.
“Star Light, Star Bright: Measuring All the Starlight (Ever!)”
This post describes how “astronomers have found a way to ‘see’ all of the starlight produced in the history of the 13.7-billion-year-old universe.”
Article author Kendra Redmond wrote:
“If you want to know how much starlight is in the universe, you might try something like measuring all of the starlight you can see, and then estimating how much is out there that you can’t see. Scientists have performed refined versions of this type of analysis, but the estimates require lots of assumptions that may or may not match reality.
The Fermi-LAT Collaboration explored this question using an entirely new approach that doesn’t rely on the same types of assumptions. Instead of measuring starlight directly, they looked at the influence of starlight on high-energy gamma rays detected by the Large Area Telescope (LAT), an instrument on the space-based Fermi Gamma Ray Telescope.”
The team estimates that over the history of the universe, stars have produced about 4*1084 photons!
“Chaos, Fractals, and Complexity: Big Ideas in the “Science of ‘Roughness'”
This post presents “a few short stories behind some of the biggest ideas in chaos and complexity theory,” including Lorenz and The Butterfly Effect, Mandelbrot’s Fractals, Complexity Theory & The El Farol Bar problem and a section about “symmetry versus roughness.”
Want to share ideas or comments? Reach out below or on Twitter (@writesRCrowell).
A Tour of “Nepantla Teachers Community” Blog
The Nepantla Teachers Community Blog is a group blog that aims “to provide an honest and encouraging space to navigate sociopolitical situations that occur in mathematics education for the purpose of working towards justice in traditionally marginalized communities. By using the word political, we mean any situation that involves power dynamics,” according to its authors. There are six instructors on the blog’s leadership team — Esther Song (high school math specialist with the Chicago Public Schools), Chanel Keyvan (Assistant Principal at Oswego Community SD and former mathematics teacher at Oswego Community SD), Jennifer Dao (mathematics teacher at Evanston-Skokie SD), Jerica Jurado-Paz (mathematics teacher at Chicago Public Schools), Erin Berg (mathematics teacher at Lyons SD) and Crystal Penn (mathematics teacher at Fulton SD in Atlanta).
Here are a few interesting recent posts on the blog.
“Small Wins: Math & Identity”
This post, which is part of the “Small Wins” series on the blog, was written by an anonymous writer Michelle, a math teacher who describes her experience with learning from her students “how to break the rules.” Her California school district has a policy that for remote Zoom learning, students must only use selfies, Bitmojis or nothing as their profile picture. But when she required that one student chance his profile picture because it didn’t meet those requirements, he said “I don’t see why I need to change my picture. I’m just trying to learn.” After he told her that his profile picture was of his favorite rapper who had died — and changed his picture back to it after she let him into the Zoom meeting — she allowed him to keep it as his picture.
She wrote:
What does my Zoom picture policing have to do with social justice and mathematics education? Everything. Especially in a Zoom environment, where most students’ cameras are off, it is even more difficult for students to express who they are as human beings. The limited avenues for self-expression are their Zoom picture and name, which are both mediated through Zoom as a platform. When Alberto changed his Zoom picture back to the picture of his favorite rapper, Alberto had demonstrated resistance in the mathematics classroom. How can students view themselves as mathematicians if they cannot bring who they are into the classroom? Who are students as mathematicians if they cannot resist and question what it means to be a student engulfed in a larger school system during a pandemic? As we discussed in our [Nepantla Teachers Community] over the summer, students are not simply stripped of their identities when they step into the mathematics classroom, even though many wish mathematics to be an apolitical space.
I asked myself, “Why am I following this distance learning policy so closely? Which students might this policy disproportionately harm? What actual consequences are there if students don’t follow this rule?” There are nuances and complexities within all of these questions. For instance, I am a first year teacher without tenure. There have been instances of inappropriate/offensive Zoom pictures. However, in the end, I decided to let Alberto keep his Zoom picture.
The “Student Voices in Remote Learning” series is also worth checking out. The most recent post in that series is from May.
“Universal Language Part I” and “Universal Language Part II”
Like other Part I and Part II posts on the blog, part I shares “a math teacher author’s real dilemma that they have recently experienced” and part II provides “an analysis of the powers at play and the author’s response (or lack of response) to the situation.”
In part I, Melissa Adams-Corral wrote:
In the summer before my third-year teaching, our district made a decision that I thought would be a game-changer: mandating dual language education district-wide. Previously, most schools in our district operated under bilingual education models that were focused on quickly moving children to all English instruction, with many schools refusing to offer clases bilingues at all. Moving to dual language meant that the district was taking an explicit stance advocating for students to continue to develop their English and Spanish side-by-side. I remember feeling very excited and hopeful about this shift…finalmente, I thought, policy would reflect the goal I had going into teaching—pride in bilinguismo, and meaningful, relevant language and content area instruction for mis estudiantes. It was a dream come true…that is, until I saw the model that all teachers were told to follow ‘with fidelity.’
This model required that certain content areas be taught in one language only and that teachers practice and enforce strict separation of languages in the classroom. My bilingüismo doesn’t work that way—it flows effortlessly, trying to stop it is like putting a wall in the middle of a river. I grew up in a bilingual home in Miami, where my language never needed to be split in two. During the summer, I would spend weeks with my primos en Honduras, singing along to Boyz II Men and Shakira, watching movies and telenovelas. Back at home in my city, bilinguismo and latinidad was lo normal. I became a bilingual teacher in large part because my language y mi cultura are a large source of my joy, pride and hope. I imagined bilingual teaching as being the work of supporting children as they grew from similar raices.”
When she followed the policy “with fidelity,” which required that her mathematics classes be taught only in English, Adams-Corral noticed a disturbing pattern. “In every math discussion, students who were comfortable speaking in English dominated. And mis estudiantes who preferred to read and write in español? They were silent. I could call on them and ask them questions, but they would shake their heads no, refusing to speak up. I would remind them that they could share their thinking in any language, already moving away from ‘total fidelity.’ But they would sit there and wait.”
Part II tackles the response, starting with Levels of Oppression, a reflection tool created by Mariame Kaba. I’ll leave it to you to check out Part II, because it’s meant to be read after reading and pondering part I.
A Year in the Math Blogosphere
For me, the end of the year always is a time for reflection. If you haven’t yet, I encourage you to read Rachel’s round-ups of AMS blog post Part I and Part II.
In the AMS December Notices, Dr. Katherine Thompson wrote her opinion on the role of blogs in our mathematical community in The Place of Blogs in the Modern Math World As Thompson mentions, while there are many considerations to still be figured out (e.g. structure, defining their success, lack of peer review) these are an extremely versatile tool that is here to stay. I was excited to see Blog on Math Blogs has 631 subscribers and an average number of shares over the last 10 posts of 110.2. I am grateful to all of our readers for their support.
Inspired by this piece and by Dr. Jennifer Quinn’s blog (see my last post), I wanted to take inventory of the lessons learned this year from Blog on Math blogs.
This blog has been a wonderful way to keep learning about fascinating mathematics and the people behind it. The blogs I have toured stood out to me because of their sheer dedication to share and be seen. As you may have realized by now, there is a whole world in the math blogosphere that I have sought to discover. One of the greatest pleasures has been learning from the breadth of content and styles that you get when you allow people the space to be their most creative selves.
This year we toured twenty blogs, wrote about different topics and themes ranging from mental health, crime-fighting, traffic modeling, Black History Month, among others, and interviewed seven bloggers.
Authors have shared their passion for mathematics and beyond, advocated for change, and given us a glimpse of their interests and passions. It begs the question, what will be the role of math blogs in the future? In my opinion, math blogs have opened the door to content that might be inaccessible otherwise (for both mathematicians and non-mathematicians alike). I am grateful for the opportunity to interview authors who joyfully share their blogging journey. If you ask me what the role of blogging and what is its importance is in our community, I think some of our interviewees have given fantastic answers to that question. Below you will find some of my favorite excerpts from the interviews.
“The name of the blog is “Logic ForAll”, now this is what I want, all people using logic formalized or not in the daily activities. But the name is also a pun, because in Brazilian Portuguese we have a dance and a style of music called “forro’ “. I only realized very late that the music (which is great and very danceable) comes from a mispronunciation of the English expression “for all”. So I wanted my blog to be like the music, fun and enjoyable and for all. Also, if possible full of little puzzles and games that it didn’t matter if you didn’t get them. It’s not about competition, it’s about fun!” – Valeria de Paiva in LogicForAll: A Tour
“I’d like to reach non-mathematicians that are curious about what a mathematician does, and how a mathematician works on proving theorems.
I’d also like to reach mathematicians, particularly “mathematicians in training,” who may want to read stories from the point of view of a more senior mathematician. I’m hoping they will relate to these stories or learn useful information about, say, what it’s like to be tenured or what it’s like to be a working mathematician and a parent in a household where both parents work and split childcare evenly. I hope the ‘realism’ in the writing helps people understand that we all struggle sometimes, that we have all gone through tough times and happy times during our careers and that almost all of us fight impostor syndrome.” – Álvaro Lozado-Robledo in Field Guide to Mathematics
“I’ve learned a lot from my interviewees. All of them taught me something. One thing stands out: their definition of success is very different from the usual one. It had more to do with having a balanced life and a satisfactory experience with research and teaching, than with awards and competition. They were compassionate, they thought of their students when thinking of teaching and their collaborators when thinking of research. It’s a very human take on professional success, and it’s what I aspire to.” Contanza Rojas-Molina in Rage of the Blackboard: A Tour.
“So my biggest piece of advice is to make it enjoyable and sustainable for yourself. There are no guarantees that even very good writing will end up getting widely read, but if you enjoy it and find that it helps you learn new things or understand your own ideas better through the act of writing them down, it’ll be worth it.” – Evelyn Lamb in Farewell, Roots of Unity
“I have learned that blogging pushes me to continue to play and innovate. I often start on one curiosity to find myself down a rabbit hole with the Cheshire Grin. These rabbit holes are what often guide me on life long adventures in learning. When I discover something new, I often go on the quest for who discovered it first – I have this picture in my mind of people rediscovering patterns throughout human history. Another lesson I have learned: I dropped posting for a while when my mom moved in for chemo in January through May. I cherish the time that we had together, and I would say to any blogger that ebbs and flows are part of life, so allow them to be part of your blogging as well.” – Sophia Wood in Fractal Kitty: A Tour
“Something that is important to all of us is that we want people to know that this is a heart project. Because of what we’ve experienced growing up and working in education, we have decided to do something to make a change. Our motivation and inspiration comes from the vision of a future where little Black girls know they rock math and boldly say it with pride. We overcame our math trauma and became something wonderful, so we hope to ease the path for those coming after us. We believe that Black women rock math because Black girls rock math! Now it’s time for the world to know.” – Kaneka Turner, Deborah Peart, and Dionne Aminata in #BlackWomenRockMath: An Interview
“I have been told that what I post has resonated with folks—not just mathematicians, not just teachers, but many people experiencing this wild and crazy pandemic year. If they find any comfort, then I consider it a success.” – Dr. Jenny Quinn in Math in the time of Corona: A Tour
I look forward to continuing to discover the treasures in the math blogosphere next year. Until then, thank you for showcasing your math, for reading our posts, and joining Rachel and me as we tour the math blogosphere. Stay safe and happy holidays!
Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).
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A Roundup of Posts on Other AMS Blogs (Part 2)
As I mentioned in my Part 1 post, I’ve been seeing a lot of posts on other AMS blogs that have piqued my interest and really got me thinking about a variety of different subjects. As we approach the end of this interesting and oh-so-challenging year, I offer you a roundup of some thought-provoking posts on other AMS blogs.
Earlier this year, the decision was made to broaden the scope of the BookEnds blog by AMS Consulting Editor, Eriko Hironaka. If you haven’t already read about that decision and would like to, this post by Nicola Poser discusses it.
In “Interacting With Ordinary Differential Equations,” a guest post by Stephen Kennedy (Carleton College), AMS/MAA Press Acquisitions, he writes about “changing content and delivery” methods for ordinary differential equations in the context of the online interactive textbook Interacting with Ordinary Differential Equations by Sandy and Max Saperstone.
On the e-Mentoring Network blog:
“The Mentorship of Our People” by Jennyfer Galvez-Reyes
Galvez-Reyes writes about her concerns with navigating the graduate school application process and the organizations she’s found — such as Cientifico Latino and Women+ of Color Project —that provide mentorship and support. She closes with these words:
“While there is no doubt that the application process is daunting, it can also be a chance to find your people. People who will cheer you on, pick you up when you’re down, and remind you of your worth when imposter syndrome threatens to take over. It’s important to not only have mentors ourselves but also to pass on the knowledge to those coming after us. Like Toni Morrison so perfectly put it, ‘When you get these jobs that you have been so brilliantly trained for, just remember that your real job is that if you are free, you need to free somebody else. If you have some power, then your job is to empower somebody else. This is not just a grab-bag candy game.’ Reach back and help those trailing you. Pass on information you wish you had, resources you needed, job listings you know about. Mentorship and community are integral parts of succeeding in spaces that weren’t designed for people like us. Despite the lack of consideration for us and our experiences, we have an ever growing community willing to help each other into these spaces.”
Saul wrote about his experience working with children under federal custody with the Office of Refugee Resettlement (ORR), which is now on hold because of COVID-19.
“The children love it. Their eyes light up. They intrigue each other. Language and social barriers tumble. And their minds are active. The work is similar to leaving food and water in the desert for thirsty immigrants. We are not offering them a complete diet or significant sustenance. But we are keeping their minds alive until their situation stabilizes,” he wrote.
Have an idea that you would like for us to cover? Want to share what you’re most excited about for JMM 2021? Reach out in the comments or on Twitter (@writesRCrowell).
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2021-12-07 17:43:01
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https://www.jiskha.com/questions/973292/multiple-choice-1-what-is-0-32-written-as-a-fraction-in-simplest-form-1-point
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# math help plzzzz!!!!
Multiple Choice
1. What is 0.32 written as a fraction in simplest form? (1 point)
thirty-two over one hundred
eight over twenty-five
one hundred over thirty-two
sixteen over twenty-five
2. If Fran’s fudge recipe calls for two and one-half cups of sugar and she increases it by two-fifths of a cup, how much sugar does she use? (1 point)
two and two-tenths
2two and three-sevenths
2two and four-ninths
2two and nine-tenths
3. Which of the following makes the statement true? one over five to the zero power = ? (1 point)
1
–1
–5
0
4. The area of a certain state is 840,000 square miles. What is this area in scientific notation? (1 point)
8.4 x 10-4
84 x 10-3
84 x 103
8.4 x 105
5. Which is irrational? (1 point)
0.117117117 . . .
0.8596873205 . . .
0.812812812812 . . .
0.605605605 . . .
6. If there is one-half of a pizza and Jonas wants only one-third of it, how much of the pizza will he eat? (1 point)
one-fifth
two-fifths
one-sixth
two-sixths
7. Harold brought two loaves of raisin bread to share with his friends. If there are 12 people total, including Harold, how much of the bread does each person receive? (1 point)
0.06
0.16 repeating
0.25
point three repeating
8. One nanometer is about 0.00000003937 of an inch. What is this number in scientific notation? (1 point)
3.937 x 10-8
3.937 x 108
39.37 x 10–7
39.37 x 107
9. What is three-eighths written as a decimal? (1 point)
two point three repeating
0.375
point three repeating
2.667
10. According to the 2010 census data, the population of Texas was about 2.5 x 107 people. The land area of Texas is about 2.6 x 105 m2. What was the average number of people per square mile in 2010? (1 point)
0.96 x 102
9.6 x 101
0.96 x 1035
9.6 x 1012
11. Based on the 2010 census, the population of Georgia was 9.6 x 106 people. Which state had a larger population? (1 point)
New York: 1.9 x 107
Wyoming: 5.6 x 105
Idaho: 1.5 x 106
12. Which of the following numbers is an example of an integer? (1 point)
square root of five
square root of nine
one-third
0.8
13. Which number is equivalent to four-fifths? (1 point)
twenty-five over one hundred
0.2
sixteen over one hundred
0.8
14. Riding the bus, Jasmine uses two-thirds of her money for bus fare. What is this number as a decimal? (1 point)
point three repeating
point six repeating
point eight repeating
one point five repeating
15. What is the solution to 2six-sevenths ÷ two-fourteenths? (1 point)
7
14
20
40
1. 👍 2
2. 👎 0
3. 👁 4,125
1. 1.
0.32 = 32 / 100 = 4 * 8 / ( 4 * 25 ) = 8 / 25
2.
2 + 1 / 2 + 2 / 5 = 2 + 5 * 1 / ( 5 * 2 ) + 2 * 2 / ( 5 * 2 ) = 2 + 5 / 10 + 4 / 10 = 2 + 9 / 10
3.
The number raised to the zero power is equal to one.
( 1 / 5 ) ^ 0 = 1
4.
840,000 = 8.4 * 100 * 1,000 = 8.4 * 10 ^ 2 * 10 ^ 3 = 8.4 * 10 ^ 5
5.
An Irrational Number is a real number that can't be written as a simple fraction.
0.117117117... = 117 / 999
0.8596873205... can't be written as the fraction
0.812812812812... = 812 / 999
0.605605605... = 605 / 999
So 0.8596873205... is irrational.
6.
( 1 / 2 ) * ( 1 / 3 ) = 1 / 6
7.
2 / 12 = 2 / ( 2 * 6 ) = 1 / 6 = 0.16666666
8.
0.00000003937 = 3.937 / 100,000,000 = 3.937 / 10 ^ 8 = 3.937 * 10 ^ - 8
9.
3 / 8 = 0.375
10.
2.5 * 10 ^ 7 / ( 2.6 * 10 ^ 5 ) =
( 2. 5 / 2.6 ) * 10 ^ 7 / 10 ^ 5 =
0.96 * 10 ^ 2
11.
New York: 1.9 * 10 ^ 7
12.
sqrt ( 9 ) = 3
13.
4 / 5 = 2 * 4 / ( 2 * 5 = 8 / 10 = 0.8
14.
2 / 3 = 0.66666
15.
( 2 + 6 / 7 ) / ( 2 / 14 ) = ( 2 * 7 / 7 + 6 / 7 ) / [ ( 2 * 1 ) / ( 2 * 7 ) ] =
( 14 / 7 + 6 / 7 ) / ( 2 * 1 ) / ( 1 / 7 ) =
( 20 / 7 ) / ( 1 / 7 ) = 7 * 20 / 7 = 20
1. 👍 5
2. 👎 1
2. QUESTION 10'S ANSWER IS :
************9.6 x 10^1 *************
^^^^^^^^^^^^^^^^^^^^^^
BOSNIAN IS WRITE ON EVERYTHING EXCEPT QUESTION 10 THE ANSWER TO QUESTION 10 IS ABOVE ^^
1. 👍 7
2. 👎 1
3. 5 is wrong
1. 👍 0
2. 👎 3
1.b
2.d
3.a
4.d
5.b
6.c
7.b
8.a
9.b
10.b
11.b
12.b
13.d
14.b
15.c
I got a 100%
1. 👍 76
2. 👎 2
5. Thank you, Salted Caramel! I made a 100% based on your help! :)
1. 👍 1
2. 👎 1
6. Thank you, Salted Caramel! I got 100%! thx so much :)
1. 👍 1
2. 👎 2
7. Thanks:)
1. 👍 2
2. 👎 1
8. salted carmel is soo right :)
1. 👍 1
2. 👎 1
9. Yea thanks. Just got a 100%
1. 👍 1
2. 👎 1
10. thank you so much Salted Caramel i got 100%
1. 👍 2
2. 👎 1
11. Thank you sooooooo much salted caramel I just got a 100% !!! Thank you!!!
1. 👍 1
2. 👎 0
12. Dude oml so thanks to salted caramel I got 100%!!!!!! :) so many kids come here to cheat lol
1. 👍 2
2. 👎 0
13. yea meto oml yassssss 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000% dont know y u saying u got 100% wen u dont even go to deir skool. just saying dog:)
1. 👍 1
2. 👎 3
14. I got a 86.7% with salted caramels answers.
1. 👍 2
2. 👎 1
15. I got 100% THANKS SO MUCH!!!
1. 👍 1
2. 👎 0
16. I got 100% THX salted caramel!!
1. 👍 0
2. 👎 0
17. Thanks so much caramel
1. 👍 1
2. 👎 0
18. Salted Caramel answers are correct.
1. 👍 1
2. 👎 0
19. thx salted caramel
1. 👍 1
2. 👎 0
20. 1. B
2. D
3. A
4. D
5. B
6. C
7. B
8. A
9. B
10. B
11. B
12. B
13. D
14. B
15. C
1. 👍 0
2. 👎 0
21. follow me @!@#$%^&.gotta.know on instagram 1. 👍 1 2. 👎 3 22. thank yall sooooooooooooo much yall are the best 1. 👍 1 2. 👎 0 23. I got 5 and 4 wrong on connexus because of Salt Caramel. 4 is B and 5 is D and I got 86.7% just like alyssa soo....yea But thanks and your welcome for the real few answers. 1. 👍 3 2. 👎 2 24. I'm in 8th grade and I just took the test and the answers are: 1.b 2.d 3.a 4.d 5.b 6.c 7.b 8.a 9.b 10.b 11.b 12.b 13.d 14.b 15.c 100% 1. 👍 1 2. 👎 0 25. I just use to check my answers. It's much better for y'all to learn the lessons they are teaching rather than cheating right off the bat. Remember, you can't use the internet on the STAAR tests. 1. 👍 1 2. 👎 1 26. @I honestly don't know is it only for texas or all states? 1. 👍 1 2. 👎 0 27. Salted caramel is 100, and frick anyone who wants to make people fail. 1. 👍 3 2. 👎 0 28. Hello, I am not here to scold you or annoy any of you but I think that people should EXPLAIN the answers to each question if you don't want to fail any of your classes including math. 1. 👍 2 2. 👎 0 29. Salty Carmels answers are right for connexus. You need to make sure that you get everything that she wrote. 100 percent Yass 1. 👍 2 2. 👎 0 30. EaT cHiPoTlE fOr AlL tHe AnSwErS 1. 👍 2 2. 👎 0 31. what are the unit test answers 1. 👍 1 2. 👎 0 32. Hal and Salted Caramel are 100% correct and shame on you for saying something else! 1. 👍 1 2. 👎 0 33. I GOT A 100 WITH SALTED CERAMAL ANSWERS THANKS 1. 👍 1 2. 👎 0 34. thx salted caramel owo 1. 👍 1 2. 👎 0 35. . I got but salted caramel said was correct 100% baby 1. 👍 1 2. 👎 1 36. Three days grace is right for connexus. 1. 👍 2 2. 👎 0 37. @!@#$%^&.gotta.know
wtf is ur problem? This is a kid friendly website. GO FK OFF.
1. 👍 2
2. 👎 1
38. It Is Confirmed That All Users, (Including Hal and Salted Caramel) With The Following Answers:
1. B
2. D
3. A
4. D
5. B
6. C
7. B
8. A
9. B
10. B
11. B
12. B
13. D
14. B
15. C
^ Are 15/15 100% Correct! (aka. For Connexus Users Only)
- Seeker
1. 👍 0
2. 👎 0
39. Unit Review Practice: Algebra Unit 4 Lesson 10 Rational Numbers
b
d
a
d
b
c
b
a
b
b
b
b
d
b
c
i got 100% i know it’s not right to cheat but..
1. 👍 0
2. 👎 0
40. Some one call me 727-485-7166
1. 👍 0
2. 👎 0
41. For connexus students its #1B #2D #3A #4D #5B #6C #7B #8A #9B #10B #11B #12B #13D #14B #15C
1. 👍 0
2. 👎 0
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asked by Anonymous on September 22, 2014
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2020-07-16 17:42:31
|
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https://socratic.org/questions/how-to-find-the-x-and-y-intercept-given-x-3y-6
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# How to find the x and y-intercept given x+3y = -6?
May 2, 2015
The y-intercept is $- 2$, and the point is $\left(0 , - 2\right)$. The x-intercept is $- 6$ and the point is $\left(- 6 , 0\right)$
To find the y-intercept, set x equal to zero.
$x + 3 y = - 6$
$0 + 3 y = - 6$
$3 y = - 6$
$y = - \frac{6}{3}$
$y = - 2$
To find the x-intercept, set y equal to zero.
$x + 3 y = - 6$
$x + 3 \cdot 0 = - 6$
$x = - 6$
graph{x+3y=-6 [-10, 10, -5, 5]}
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2019-09-18 09:33:46
|
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|
https://www.semanticscholar.org/paper/On-Separable-Schr%C3%B6dinger-Equations-Zhdanov-Zhalij/7a6c86736ecf314e3cf8b554cc5120a379c8d7a0
|
# On Separable Schrödinger Equations
@article{Zhdanov1999OnSS,
title={On Separable Schr{\"o}dinger Equations},
author={Renat Z. Zhdanov and Alexander Zhalij},
journal={Mathematics eJournal},
year={1999}
}
• Published 12 November 1999
• Mathematics
• Mathematics eJournal
We classify (1+3)-dimensional Schrodinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables. As a result, we get 11 classes of the vector potentials of the electromagnetic field A(t,x)=(A0(t,x),A(t,x)) providing separability of the corresponding Schrodinger equations. It is established, in particular, that the necessary condition for the Schrodinger equation to be separable is that the magnetic field must be…
24 Citations
We classify (1+3)-dimensional Pauli equations for a spin-12 particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain
We classify (1+3)-dimensional Pauli equations for a spin-12 particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain
We extend our approach, used to classify separable Schrödinger equations [1], to the case of the (1+3)-dimensional Pauli equations for a spin-12 particle interacting with the electromagnetic field.
• Physics, Mathematics
• 2005
We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order
• Physics
• 2008
We study the classification of electromagnetic fields using the equivalence relation on the set of all 4-potentials of the Schrodinger equation. In the general case we find the relations among the
The problems of constructing the symmetry operators of the time-dependent Schrödinger equation in the constant and uniform electric and magnetic fields with the use of the obtained operators in the
• A. Nikitin
• Mathematics
Journal of Physics A: Mathematical and Theoretical
• 2020
Using the algebraic approach Lie symmetries of time dependent Schrödinger equations for charged particles interacting with superpositions of scalar and vector potentials are classified. Namely, all
We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the
• Mathematics
• 2002
The fundamental elements of the variable separation theory are revisited, including the Eisenhart and Robertson theorems, Kalnins–Miller theory, and the intrinsic characterization of the separation
• Mathematics
• 2020
We classify all orthogonal coordinate systems in M4, allowing complete additively separated solutions of the Hamilton–Jacobi equation for a charged test particle in the Lienard–Wiechert field
## References
SHOWING 1-10 OF 84 REFERENCES
• Mathematics
• 1995
We suggest an effective approach to separation of variables in the Schrodinger equation with two space variables. Using it we classify inequivalent potentials V(x1,x2) such that the corresponding
Using our classification of separable Schrodinger equations with two space dimensions published in J. Math. Phys. 36, 5506 (1995) we give an exhaustive description of the coordinate systems providing
• Mathematics
• 1975
This paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrodinger equation in two space dimensions. It is shown that this equation separates
• Mathematics
• 1998
We obtain the most general time-dependent potential V(t,x1,x2) enabling separating of variables in the (1+2)-dimensional Schrodinger equation. With the use of this result the four classes of
• Mathematics
• 1993
We develop a direct approach to the separation of variables in partial differential equations. Within the framework of this approach, the problem of the separation of variables in the wave equation
• Mathematics
• 1974
Separation of variables in the Schrödinger equation is performed by using complete sets of differential operators of symmetry with operators not higher than second order, and all types of
• Mathematics
• 1993
The authors say that a solution Psi of a partial differential equation in two real variables x1,x2 is functionally separable in these variables if Psi (x1,x2)= phi (A(x1)+B(x2)) for single variable
We give a precise and conceptual definition of separation of variables for partial differential equations. We derive necessary and sufficient con ditions for a linear homogeneous second order partial
• Physics
Physical review. A, Atomic, molecular, and optical physics
• 1994
It is shown that separation of variables applies and exact solubility occurs only in a very restricted class of time-dependent models.
We classify all R-separable coordinate systems for the equation $( * )\qquad \Delta _m \Psi + 2\varepsilon \partial _t \Psi = E\Psi ,\quad \Delta _m = \sum_{u = 1}^m {\partial _{y^u y^u } }$which
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2023-02-09 12:16:27
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https://www.biostars.org/p/70462/
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Searching Homology (Blast)
1
0
Entering edit mode
8.1 years ago
onpelikan • 0
Hi, I need to analyze protein mutations via MAPP tool. For this tool I need have multiple alignemnt file and phylogenetic tree file. So I need to obtain homology sequences, but not so closely related ones, because closely related sequences are not significat for MAPP. How to use BLAST to find not so close related homolgies. I tried use PSI-BLAST but:
1. How many iteration is needed?
2. Is inclusion_ethresh param significant?
homology multiple-alignment mutation phylogenetics • 1.9k views
0
Entering edit mode
There are many ways to measure protein mutations, so I'm unclear why you need to use MAPP. Of course you'll need to have a curated alignment to assume homology, but are you asking how to get a phylogenetically diverse array of sequences in a given lineage?
1
Entering edit mode
8.1 years ago
Nari ▴ 900
Blast will not give you alignment file and tree file as output.
Options are available for different outputs of alignment and tree files.
MAPP requires
1.fasta alignment file which matches with alignment.fas obtained from clustalw2
and
2.tree file which matches with tree.dnd obtained from clustalw2
Use these files for running MAPP:
Here is your code. for windows cmd:
java -jar MAPP.jar -f alignment.fas -t tree.dnd -o out.txt
ENJOY!!!
0
Entering edit mode
Thank you for your answer but my question is about psi-blast. How to obtain not so closely related sequences? For multiple sequence alignment I'm using Clustal Omega and for tree I'm using SEMPHY (it is not very well. Probably MEGA or PhyML would be better for tree reconstruction).
|
2021-06-13 20:20:27
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{"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": 0.4351697564125061, "perplexity": 7377.996290505998}, "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-2021-25/segments/1623487610841.7/warc/CC-MAIN-20210613192529-20210613222529-00112.warc.gz"}
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https://tex.stackexchange.com/questions/462601/how-can-i-disable-the-automatic-transliteration-of-arabic
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# How can I disable the automatic transliteration of Arabic?
How can I disable the automatic transliteration of Arabic? I want to write my French report including some Arabic sentences. So I am using the arabtex package but when writing an Arabic sentence like this
\begin{arabtext}
السَلامُ عَليكم ورَحمةُ الله وبَركاته
\end{arabtext}
I got this:
السَلامُ عَليكم ورَحمةُ الله وبَركاته
ālsalāmu alykm wrah.mtu āllh wbarkāth
I don't want this transliteration. How can I disable it?
• Welcome to TeX.SE! Can you please add a short compilable code showing your issue? – Kurt Nov 30 '18 at 18:42
• @Kurt I know the policy on MWEs, but in this case the answer is right there in the Arabtex documentation, in a section about this single topic that is listed in the table of contents - so it's not even buried in a hard-to-find place! There's no need to see an example to answer the OP's question. – alephzero Nov 30 '18 at 20:20
The command you want is \transfalse. For example,
\documentclass[12pt]{article}
\usepackage{arabtex}
\begin{document}
\setarab
\novocalize
\transfalse
\begin{RLtext}
klAm fI m.h.d al-_hyr
\end{RLtext}
\end{document}
If you're not limited to (la)tex, why don't you use arabxetex package?
I think using it with xelatex have more options than arabtex, e.g. you can change the main font for Arabic typesetting (default is Amiri font).
\documentclass{article}
\usepackage{arabxetex}
\begin{document}
\begin{arab}[voc]
السَلامُ عَليكم ورَحمةُ الله وبَركاته
\end{arab}
\end{document}
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2019-04-24 19:55:04
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https://community.jmp.com/t5/Discussions/Manipulating-data-exclusively-on-subset-data/td-p/230592
|
Choose Language Hide Translation Bar
Highlighted
Level I
## Manipulating data exclusively on subset data
I have a data table that i performed subset on to extract the data I need. Now I want to do analysis on that subset data exclusively, but it is under the same name as the original data table. How can I rename it or separate it out completely from the original data so that I can go on with my data analysis?
1 ACCEPTED SOLUTION
Accepted Solutions
Highlighted
Super User
## Re: Manipulating data exclusively on subset data
Here is a simple example showing how to use pointers to the data table to uniquely reference the tables.
dt = Open( "$SAMPLE_DATA\big class.jmp" ); dt << select where( :sex == "F" ); dt2 = dt << subset( selected rows( 1 ), selected columns( 0 ) ); dt2 << Bivariate( Y( :weight ), X( :height ) ); Jim 5 REPLIES 5 Highlighted Level VI ## Re: Manipulating data exclusively on subset data It sounds like you are using a linked subset table. You have a couple of options. a) save the subset table to a new JMP file, close the original file, close this file and re-open it and you can work on it independently. b) when you do a subset, uncheck the box "Link to Original Data Table" in the dialog. Highlighted Level I ## Re: Manipulating data exclusively on subset data Thanks! What I am really doing is writing a script that can generate result directly. I more mean how to separate table of subset from original table, because right now after i did subset, all data analysis i did on table is done to all data in original table, not in the subset table. Highlighted Level VI ## Re: Manipulating data exclusively on subset data You need to explicity state in the script that you don't want the subset to be linked, maybe? Does your script have the "Linked" option? Subset( Linked, /* do you have this in your script? */ Suppress formula evaluation( 0 ), Selected Rows( 0 ), Rows( [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] ), Selected columns only( 0 ) ) Highlighted Super User ## Re: Manipulating data exclusively on subset data Here is a simple example showing how to use pointers to the data table to uniquely reference the tables. dt = Open( "$SAMPLE_DATA\big class.jmp" );
dt << select where( :sex == "F" );
dt2 = dt << subset( selected rows( 1 ), selected columns( 0 ) );
dt2 << Bivariate( Y( :weight ), X( :height ) );
Jim
Highlighted
Super User
## Re: Manipulating data exclusively on subset data
How was your subset created that it ended up having the same name as a previous data table?
Was the table created using a JSL script or interactively?
Jim
Article Labels
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2020-09-26 08:30:10
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http://isarmathlib.org/CommutativeSemigroup_ZF.html
|
# IsarMathLib
## A library of formalized mathematics for Isabelle/ZF theorem proving environment
theory CommutativeSemigroup_ZF imports Semigroup_ZF
begin
In the Semigroup theory we introduced a notion of $$\text{SetFold}(f,a,\Lambda ,r)$$ that represents the sum of values of some function $$a$$ valued in a semigroup where the arguments of that function vary over some set $$\Lambda$$. Using the additive notation something like this would be expressed as $$\sum_{x\in \Lambda} f(x)$$ in informal mathematics. This theory considers an alternative to that notion that is more specific to commutative semigroups.
### Sum of a function over a set
The $$r$$ parameter in the definition of $$\text{SetFold}(f,a,\Lambda ,r)$$ (from Semigroup_ZF) represents a linear order relation on $$\Lambda$$ that is needed to indicate in what order we are summing the values $$f(x)$$. If the semigroup operation is commutative the order does not matter and the relation $$r$$ is not needed. In this section we define a notion of summing up values of some function $$a : X \rightarrow G$$ over a finite set of indices $$\Gamma \subseteq X$$, without using any order relation on $$X$$.
We define the sum of values of a function $$a: X\rightarrow G$$ over a set $$\Lambda$$ as the only element of the set of sums of lists that are bijections between the number of values in $$\Lambda$$ (which is a natural number $$n = \{0,1, .. , n-1\}$$ if $$\Lambda$$ is finite) and $$\Lambda$$. The notion of $$\text{Fold1}(f,c)$$ is defined in Semigroup_ZF as the fold (sum) of the list $$c$$ starting from the first element of that list. The intention is to use the fact that since the result of summing up a list does not depend on the order, the set $$\{ \text{Fold1}(f,a\circ b).\ b \in \text{bij}( |\Lambda |, \Lambda )\}$$ is a singleton and we can extract its only value by taking its union.
Definition
$$\text{CommSetFold}(f,a,\Lambda ) = \bigcup \{ \text{Fold1}(f,a\circ b).\ b \in \text{bij}(|\Lambda |, \Lambda )\}$$
the next locale sets up notation for writing about summation in commutative semigroups. We define two kinds of sums. One is the sum of elements of a list (which are just functions defined on a natural number) and the second one represents a more general notion the sum of values of a semigroup valued function over some set of arguments. Since those two types of sums are different notions they are represented by different symbols. However in the presentations they are both intended to be printed as $$\sum$$.
Locale commsemigr
assumes csgassoc: $$f \text{ is associative on } G$$
assumes csgcomm: $$f \text{ is commutative on } G$$
defines $$x + y \equiv f\langle x,y\rangle$$
assumes csgaisfun: $$a : X \rightarrow G$$
defines $$\sum k \equiv \text{Fold1}(f,k)$$
defines $$\sum (A,h) \equiv \text{CommSetFold}(f,h,A)$$
Definition of a sum of function over a set in notation defined in the commsemigr locale.
lemma (in commsemigr) CommSetFolddef:
shows $$(\sum (A,a)) = (\bigcup \{\sum (a\circ b).\ b \in \text{bij}(|A|, A)\})$$ using CommSetFold_def
The next lemma states that the result of a sum does not depend on the order we calculate it. This is similar to lemma prod_order_irr in the Semigroup theory, except that the semigr1 locale assumes that the domain of the function we sum up is linearly ordered, while in commsemigr we don't have this assumption.
lemma (in commsemigr) sum_over_set_bij:
assumes A1: $$A \in \text{FinPow}(X)$$, $$A \neq 0$$ and A2: $$b \in \text{bij}(|A|,A)$$
shows $$(\sum (A,a)) = (\sum (a\circ b))$$proof
have $$\forall c \in \text{bij}(|A|,A).\ \forall d \in \text{bij}(|A|,A).\ (\sum (a\circ c)) = (\sum (a\circ d))$$proof
{
fix $$c$$
assume $$c \in \text{bij}(|A|,A)$$
fix $$d$$
assume $$d \in \text{bij}(|A|,A)$$
let $$r = \text{InducedRelation}(converse(c), Le)$$
have $$semigr1(G,f,A,r,\text{restrict}(a, A))$$proof
have $$semigr0(G,f)$$ using csgassoc , semigr0_def
moreover
from A1, $$c \in \text{bij}(|A|,A)$$ have $$\text{IsLinOrder}(A,r)$$ using bij_converse_bij , card_fin_is_nat , natord_lin_on_each_nat , ind_rel_pres_lin
moreover
from A1 have $$\text{restrict}(a, A) : A \rightarrow G$$ using FinPow_def , csgaisfun , restrict_fun
ultimately show $$thesis$$ using semigr1_axioms.intro , semigr1_def
qed
moreover
have $$f \text{ is commutative on } G$$ using csgcomm
moreover
from A1 have $$A \in \text{FinPow}(A)$$, $$A \neq 0$$ using FinPow_def
moreover
note $$c \in \text{bij}(|A|,A)$$, $$d \in \text{bij}(|A|,A)$$
ultimately have $$\text{Fold1}(f,\text{restrict}(a,A)\circ c) = \text{Fold1}(f,\text{restrict}(a,A)\circ d)$$ by (rule prod_bij_same )
hence $$(\sum (\text{restrict}(a,A)\circ c)) = (\sum (\text{restrict}(a,A)\circ d))$$
moreover
from A1, $$c \in \text{bij}(|A|,A)$$, $$d \in \text{bij}(|A|,A)$$ have $$\text{restrict}(a,A)\circ c = a\circ c$$ and $$\text{restrict}(a,A)\circ d = a\circ d$$ using bij_def , surj_def , csgaisfun , FinPow_def , comp_restrict
ultimately have $$(\sum (a\circ c)) = (\sum (a\circ d))$$
}
thus $$thesis$$
qed
with A2 have $$(\bigcup \{\sum (a\circ b).\ b \in \text{bij}(|A|, A)\}) = (\sum (a\circ b))$$ by (rule singleton_comprehension )
then show $$thesis$$ using CommSetFolddef
qed
The result of a sum is in the semigroup. Also, as the second assertion we show that every semigroup valued function generates a homomorphism between the finite subsets of a semigroup and the semigroup. Adding an element to a set coresponds to adding a value.
assumes A1: $$A \in \text{FinPow}(X)$$, $$A \neq 0$$
shows $$\sum (A,a) \in G$$ and $$\forall x \in X-A.\ \sum (A \cup \{x\},a) = (\sum (A,a)) + a(x)$$proof
from A1 obtain $$b$$ where $$b \in \text{bij}(|A|,A)$$ using fin_bij_card
with A1 have $$\sum (A,a) = (\sum (a\circ b))$$ using sum_over_set_bij
from A1 have $$|A| \in nat$$ using card_fin_is_nat
have $$semigr0(G,f)$$ using csgassoc , semigr0_def
moreover
from A1 obtain $$n$$ where $$n \in nat$$ and $$|A| = succ(n)$$ using card_non_empty_succ
with A1, $$b \in \text{bij}(|A|,A)$$ have $$n \in nat$$ and $$a\circ b : succ(n) \rightarrow G$$ using bij_def , inj_def , FinPow_def , comp_fun_subset , csgaisfun
ultimately have $$\text{Fold1}(f,a\circ b) \in G$$ by (rule prod_type )
with $$\sum (A,a) = (\sum (a\circ b))$$ show $$\sum (A,a) \in G$$
{
fix $$x$$
assume $$x \in X-A$$
with A1 have $$(A \cup \{x\}) \in \text{FinPow}(X)$$ and $$A \cup \{x\} \neq 0$$ using singleton_in_finpow , union_finpow
moreover
have $$\text{Append}(b,x) \in \text{bij}(|A \cup \{x\}|, A \cup \{x\})$$proof
note $$|A| \in nat$$, $$b \in \text{bij}(|A|,A)$$
moreover
from $$x \in X-A$$ have $$x \notin A$$
ultimately have $$\text{Append}(b,x) \in \text{bij}(succ(|A|), A \cup \{x\})$$ by (rule bij_append_point )
with A1, $$x \in X-A$$ show $$thesis$$ using card_fin_add_one
qed
ultimately have $$(\sum (A \cup \{x\},a)) = (\sum (a\circ \text{Append}(b,x)))$$ using sum_over_set_bij
also
have $$\ldots = (\sum \text{Append}(a\circ b, a(x)))$$proof
note $$|A| \in nat$$
moreover
from A1, $$b \in \text{bij}(|A|, A)$$ have $$b : |A| \rightarrow A$$ and $$A \subseteq X$$ using bij_def , inj_def , using , FinPow_def
then have $$b : |A| \rightarrow X$$ by (rule func1_1_L1B )
moreover
from $$x \in X-A$$ have $$x \in X$$ and $$a : X \rightarrow G$$ using csgaisfun
ultimately show $$thesis$$ using list_compose_append
qed
also
have $$\ldots = (\sum (A,a)) + a(x)$$proof
note $$semigr0(G,f)$$, $$n \in nat$$, $$a\circ b : succ(n) \rightarrow G$$
moreover
from $$x \in X-A$$ have $$a(x) \in G$$ using csgaisfun , apply_funtype
ultimately have $$\text{Fold1}(f, \text{Append}(a\circ b, a(x))) = f\langle \text{Fold1}(f,a\circ b),a(x)\rangle$$ by (rule prod_append )
with A1, $$b \in \text{bij}(|A|,A)$$ show $$thesis$$ using sum_over_set_bij
qed
finally have $$(\sum (A \cup \{x\},a)) = (\sum (A,a)) + a(x)$$
}
thus $$\forall x \in X-A.\ \sum (A \cup \{x\},a) = (\sum (A,a)) + a(x)$$
qed
end
Definition of CommSetFold: $$\text{CommSetFold}(f,a,\Lambda ) = \bigcup \{ \text{Fold1}(f,a\circ b).\ b \in \text{bij}(|\Lambda |, \Lambda )\}$$
lemma card_fin_is_nat:
assumes $$A \in \text{FinPow}(X)$$
shows $$|A| \in nat$$ and $$A \approx |A|$$
lemma natord_lin_on_each_nat:
assumes $$n \in nat$$
shows $$\text{IsLinOrder}(n,Le)$$
lemma ind_rel_pres_lin:
assumes $$f \in \text{bij}(A,B)$$ and $$\text{IsLinOrder}(B,R)$$
shows $$\text{IsLinOrder}(A, \text{InducedRelation}(f,R))$$
Definition of FinPow: $$\text{FinPow}(X) \equiv \{A \in Pow(X).\ Finite(A)\}$$
corollary restrict_fun:
assumes $$f:X\rightarrow Y$$ and $$A\subseteq X$$
shows $$\text{restrict}(f,A) : A \rightarrow Y$$
corollary (in semigr1) prod_bij_same:
assumes $$f \text{ is commutative on } G$$ and $$A \in \text{FinPow}(X)$$, $$A \neq 0$$ and $$b \in \text{bij}(|A|,A)$$, $$c \in \text{bij}(|A|,A)$$
shows $$(\prod (a\circ b)) = (\prod (a\circ c))$$
lemma comp_restrict:
assumes $$f : A\rightarrow B$$ and $$g : X \rightarrow C$$ and $$B\subseteq X$$
shows $$g\circ f = \text{restrict}(g,B)\circ f$$
lemma singleton_comprehension:
assumes $$y\in X$$ and $$\forall x\in X.\ \forall y\in X.\ P(x) = P(y)$$
shows $$(\bigcup \{P(x).\ x\in X\}) = P(y)$$
lemma (in commsemigr) CommSetFolddef: shows $$(\sum (A,a)) = (\bigcup \{\sum (a\circ b).\ b \in \text{bij}(|A|, A)\})$$
lemma fin_bij_card:
assumes $$A \in \text{FinPow}(X)$$
shows $$\exists b.\ b \in \text{bij}(|A|, A)$$
lemma (in commsemigr) sum_over_set_bij:
assumes $$A \in \text{FinPow}(X)$$, $$A \neq 0$$ and $$b \in \text{bij}(|A|,A)$$
shows $$(\sum (A,a)) = (\sum (a\circ b))$$
lemma card_non_empty_succ:
assumes $$A \in \text{FinPow}(X)$$ and $$A \neq 0$$
shows $$\exists n \in nat.\ |A| = succ(n)$$
lemma comp_fun_subset:
assumes $$g:A\rightarrow B$$ and $$f:C\rightarrow D$$ and $$B \subseteq C$$
shows $$f\circ g : A \rightarrow D$$
lemma (in semigr0) prod_type:
assumes $$n \in nat$$ and $$a : succ(n) \rightarrow G$$
shows $$(\prod a) \in G$$
lemma singleton_in_finpow:
assumes $$x \in X$$
shows $$\{x\} \in \text{FinPow}(X)$$
lemma union_finpow:
assumes $$A \in \text{FinPow}(X)$$ and $$B \in \text{FinPow}(X)$$
shows $$A \cup B \in \text{FinPow}(X)$$
lemma bij_append_point:
assumes $$n \in nat$$ and $$b \in \text{bij}(n,X)$$ and $$x \notin X$$
shows $$\text{Append}(b,x) \in \text{bij}(succ(n), X \cup \{x\})$$
assumes $$A \in \text{FinPow}(X)$$ and $$a \in X-A$$
shows $$|A \cup \{a\}| = succ( |A| )$$, $$|A \cup \{a\}| = |A| \cup \{|A|\}$$
lemma func1_1_L1B:
assumes $$f:X\rightarrow Y$$ and $$Y\subseteq Z$$
shows $$f:X\rightarrow Z$$
lemma list_compose_append:
assumes $$n \in nat$$ and $$a : n \rightarrow X$$ and $$x \in X$$ and $$c : X \rightarrow Y$$
shows $$c\circ \text{Append}(a,x) : succ(n) \rightarrow Y$$, $$c\circ \text{Append}(a,x) = \text{Append}(c\circ a, c(x))$$
lemma (in semigr0) prod_append:
assumes $$n \in nat$$ and $$a : succ(n) \rightarrow G$$ and $$x\in G$$
shows $$(\prod a\hookleftarrow x) = (\prod a) \cdot x$$
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2017-08-19 12:59:13
|
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|
https://blender.stackexchange.com/questions/18640/how-to-align-object-well-enough-to-integrate-into-array-modifier
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# How to align object well enough to integrate into array modifier?
Edit: I found the merge checkbox on the array modifier (whoops.) But the alignment problem is still interesting...
So, let's say I'm following yet another tutorial from the blender Guru, how to create a spaceship corridor. This thing is a beast.
So we start with modeling a quarter of the ship, since it is all nice an symmetrical. Like so:
We proceed to mirror it into something that looks a bit nicer:
Great! Then, the guy simply adds a plain to act as the bottom. Simple enough!
And Viola! We hit CTRL + J and expect the the object to join that which is being mirrored. Unfortunately, I can't seem to align it right. I'm exaggerating for clarity, of course, but I get either:
or something like:
I can't seem to figure out a way to alight this in a reasonable amount of time so that the edges actually touch each other. As you see, when I am seeing the "Array" modifier, it makes no effort to weld the pieces of the array together, instead just putting the "pieces" as close together as possible with no overlap. Therefore, the floor has to be exactly the same size as the original mirrored piece! I can't seem to make this happen!
Thank you very much to anyone who can tell me how this can be done properly!
• Have you tried adjusting the array distances? – Scalia Nov 13 '14 at 15:54
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2019-11-12 18:29:22
|
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|
https://de.mathworks.com/help/finance/hwv.simbysolution_hwv.html
|
# simBySolution
Simulate approximate solution of diagonal-drift HWV processes
## Description
example
[Paths,Times,Z] = simBySolution(MDL,NPeriods) simulates approximate solution of diagonal-drift for Hull-White/Vasicek Gaussian Diffusion (HWV) processes.
example
[Paths,Times,Z] = simBySolution(___,Name,Value) adds optional name-value pair arguments.
You can perform quasi-Monte Carlo simulations using the name-value arguments for MonteCarloMethod, QuasiSequence, and BrownianMotionMethod. For more information, see Quasi-Monte Carlo Simulation.
## Examples
collapse all
Create an hwv object to represent the model:
$d{X}_{t}=0.2\left(0.1-{X}_{t}\right)dt+0.05d{W}_{t}.$
hwv = hwv(0.2, 0.1, 0.05) % (Speed, Level, Sigma)
hwv =
Class HWV: Hull-White/Vasicek
----------------------------------------
Dimensions: State = 1, Brownian = 1
----------------------------------------
StartTime: 0
StartState: 1
Correlation: 1
Drift: drift rate function F(t,X(t))
Diffusion: diffusion rate function G(t,X(t))
Simulation: simulation method/function simByEuler
Sigma: 0.05
Level: 0.1
Speed: 0.2
The simBySolution function simulates the state vector Xt using an approximation of the closed-form solution of diagonal drift HWV models. Each element of the state vector Xt is expressed as the sum of NBrowns correlated Gaussian random draws added to a deterministic time-variable drift.
NPeriods = 100
[Paths,Times,Z] = simBySolution(hwv, NPeriods,'NTrials', 10);
This example shows how to use simBySolution with a HWV model to perform a quasi-Monte Carlo simulation. Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences instead pseudo random numbers.
HWV = hwv(1.5,100,10,'startstate',100);
Perform a quasi-Monte Carlo simulation by using simBySolution with the optional name-value arguments for 'MonteCarloMethod','QuasiSequence', and 'BrownianMotionMethod'.
[paths,time,z] = simBySolution(HWV, 10,'ntrials',4096,'MonteCarloMethod','quasi','QuasiSequence','sobol','BrownianMotionMethod','principal-components');
## Input Arguments
collapse all
Hull-White/Vasicek (HWV) mode, specified as a hwv object that is created using hwv.
Data Types: object
Number of simulation periods, specified as a positive scalar integer. The value of NPeriods determines the number of rows of the simulated output series.
Data Types: double
### Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: [Paths,Times,Z] = simBySolution(HWV,NPeriods,'DeltaTime',dt,'NTrials',10)
Simulated trials (sample paths) of NPeriods observations each, specified as the comma-separated pair consisting of 'NTrials' and a positive scalar integer.
Data Types: double
Positive time increments between observations, specified as the comma-separated pair consisting of 'DeltaTimes' and a scalar or a NPeriods-by-1 column vector.
DeltaTime represents the familiar dt found in stochastic differential equations, and determines the times at which the simulated paths of the output state variables are reported.
Data Types: double
Number of intermediate time steps within each time increment dt (specified as DeltaTime), specified as the comma-separated pair consisting of 'NSteps' and a positive scalar integer.
The simBySolution function partitions each time increment dt into NSteps subintervals of length dt/NSteps, and refines the simulation by evaluating the simulated state vector at NSteps − 1 intermediate points. Although simBySolution does not report the output state vector at these intermediate points, the refinement improves accuracy by allowing the simulation to more closely approximate the underlying continuous-time process.
Data Types: double
Flag to indicate whether simBySolution uses antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes), specified as the comma-separated pair consisting of 'Antithetic' and a scalar logical flag with a value of True or False.
When you specify True, simBySolution performs sampling such that all primary and antithetic paths are simulated and stored in successive matching pairs:
• Odd trials (1,3,5,...) correspond to the primary Gaussian paths.
• Even trials (2,4,6,...) are the matching antithetic paths of each pair derived by negating the Gaussian draws of the corresponding primary (odd) trial.
Note
If you specify an input noise process (see Z), simBySolution ignores the value of Antithetic.
Data Types: logical
Monte Carlo method to simulate stochastic processes, specified as the comma-separated pair consisting of 'MonteCarloMethod' and a string or character vector with one of the following values:
• "standard" — Monte Carlo using pseudo random numbers.
• "quasi" — Quasi-Monte Carlo using low-discrepancy sequences.
• "randomized-quasi" — Randomized quasi-Monte Carlo.
Note
If you specify an input noise process (see Z), simBySolution ignores the value of MonteCarloMethod.
Data Types: string | char
Low discrepancy sequence to drive the stochastic processes, specified as the comma-separated pair consisting of 'QuasiSequence' and a string or character vector with one of the following values:
• "sobol" — Quasi-random low-discrepancy sequences that use a base of two to form successively finer uniform partitions of the unit interval and then reorder the coordinates in each dimension
Note
• If MonteCarloMethod option is not specified or specified as"standard", QuasiSequence is ignored.
• If you specify an input noise process (see Z), simBySolution ignores the value of QuasiSequence.
Data Types: string | char
Brownian motion construction method, specified as the comma-separated pair consisting of 'BrownianMotionMethod' and a string or character vector with one of the following values:
• "standard" — The Brownian motion path is found by taking the cumulative sum of the Gaussian variates.
• "brownian-bridge" — The last step of the Brownian motion path is calculated first, followed by any order between steps until all steps have been determined.
• "principal-components" — The Brownian motion path is calculated by minimizing the approximation error.
Note
If an input noise process is specified using the Z input argument, BrownianMotionMethod is ignored.
The starting point for a Monte Carlo simulation is the construction of a Brownian motion sample path (or Wiener path). Such paths are built from a set of independent Gaussian variates, using either standard discretization, Brownian-bridge construction, or principal components construction.
Both standard discretization and Brownian-bridge construction share the same variance and therefore the same resulting convergence when used with the MonteCarloMethod using pseudo random numbers. However, the performance differs between the two when the MonteCarloMethod option "quasi" is introduced, with faster convergence seen for "brownian-bridge" construction option and the fastest convergence when using the "principal-components" construction option.
Data Types: string | char
Direct specification of the dependent random noise process used to generate the Brownian motion vector (Wiener process) that drives the simulation, specified as the comma-separated pair consisting of 'Z' and a function or as an (NPeriods * NSteps)-by-NBrowns-by-NTrials three-dimensional array of dependent random variates.
The input argument Z allows you to directly specify the noise generation process. This process takes precedence over the Correlation parameter of the input gbm object and the value of the Antithetic input flag.
Note
If you specify Z as a function, it must return an NBrowns-by-1 column vector, and you must call it with two inputs:
• A real-valued scalar observation time t.
• An NVars-by-1 state vector Xt.
Data Types: double | function
Flag that indicates how the output array Paths is stored and returned, specified as the comma-separated pair consisting of 'StorePaths' and a scalar logical flag with a value of True or False.
If StorePaths is True (the default value) or is unspecified, simBySolution returns Paths as a three-dimensional time series array.
If StorePaths is False (logical 0), simBySolution returns the Paths output array as an empty matrix.
Data Types: logical
Sequence of end-of-period processes or state vector adjustments of the form, specified as the comma-separated pair consisting of 'Processes' and a function or cell array of functions of the form
${X}_{t}=P\left(t,{X}_{t}\right)$
The simBySolution function runs processing functions at each interpolation time. They must accept the current interpolation time t, and the current state vector Xt, and return a state vector that may be an adjustment to the input state.
simBySolution applies processing functions at the end of each observation period. These functions must accept the current observation time t and the current state vector Xt, and return a state vector that may be an adjustment to the input state.
The end-of-period Processes argument allows you to terminate a given trial early. At the end of each time step, simBySolution tests the state vector Xt for an all-NaN condition. Thus, to signal an early termination of a given trial, all elements of the state vector Xt must be NaN. This test enables a user-defined Processes function to signal early termination of a trial, and offers significant performance benefits in some situations (for example, pricing down-and-out barrier options).
If you specify more than one processing function, simBySolution invokes the functions in the order in which they appear in the cell array. You can use this argument to specify boundary conditions, prevent negative prices, accumulate statistics, plot graphs, and more.
Data Types: cell | function
## Output Arguments
collapse all
Simulated paths of correlated state variables, returned as a (NPeriods + 1)-by-NVars-by-NTrials three-dimensional time series array.
For a given trial, each row of Paths is the transpose of the state vector Xt at time t. When the input flag StorePaths = False, simBySolution returns Paths as an empty matrix.
Observation times associated with the simulated paths, returned as a (NPeriods + 1)-by-1 column vector. Each element of Times is associated with the corresponding row of Paths.
Dependent random variates used to generate the Brownian motion vector (Wiener processes) that drive the simulation, returned as a (NPeriods * NSteps)-by-NBrowns-by-NTrials three-dimensional time series array.
collapse all
### Antithetic Sampling
Simulation methods allow you to specify a popular variance reduction technique called antithetic sampling.
This technique attempts to replace one sequence of random observations with another of the same expected value, but smaller variance. In a typical Monte Carlo simulation, each sample path is independent and represents an independent trial. However, antithetic sampling generates sample paths in pairs. The first path of the pair is referred to as the primary path, and the second as the antithetic path. Any given pair is independent of any other pair, but the two paths within each pair are highly correlated. Antithetic sampling literature often recommends averaging the discounted payoffs of each pair, effectively halving the number of Monte Carlo trials.
This technique attempts to reduce variance by inducing negative dependence between paired input samples, ideally resulting in negative dependence between paired output samples. The greater the extent of negative dependence, the more effective antithetic sampling is.
## Algorithms
The simBySolution method simulates NTrials sample paths of NVars correlated state variables, driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, approximating continuous-time Hull-White/Vasicek (HWV) by an approximation of the closed-form solution.
Consider a separable, vector-valued HWV model of the form:
$d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+V\left(t\right)d{W}_{t}$
where:
• X is an NVars-by-1 state vector of process variables.
• S is an NVars-by-NVars matrix of mean reversion speeds (the rate of mean reversion).
• L is an NVars-by-1 vector of mean reversion levels (long-run mean or level).
• V is an NVars-by-NBrowns instantaneous volatility rate matrix.
• W is an NBrowns-by-1 Brownian motion vector.
The simBySolution method simulates the state vector Xt using an approximation of the closed-form solution of diagonal-drift models.
When evaluating the expressions, simBySolution assumes that all model parameters are piecewise-constant over each simulation period.
In general, this is not the exact solution to the models, because the probability distributions of the simulated and true state vectors are identical only for piecewise-constant parameters.
When parameters are piecewise-constant over each observation period, the simulated process is exact for the observation times at which Xt is sampled.
Gaussian diffusion models, such as hwv, allow negative states. By default, simBySolution does nothing to prevent negative states, nor does it guarantee that the model be strictly mean-reverting. Thus, the model may exhibit erratic or explosive growth.
## References
[1] Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies 9, no. 2 ( Apr. 1996): 385–426.
[2] Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance 54, no. 4 (Aug. 1999): 1361–95.
[3] Glasserman, Paul. Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.
[4] Hull, John C. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.
[5] Johnson, Norman Lloyd, Samuel Kotz, and Narayanaswamy Balakrishnan. Continuous Univariate Distributions. 2nd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley, 1995.
[6] Shreve, Steven E. Stochastic Calculus for Finance. New York: Springer-Verlag, 2004.
## Version History
Introduced in R2008a
expand all
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2023-02-04 12:13:12
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https://community.spiceworks.com/topic/321135-all-permission-for-1-file-are-gone
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# All permission for 1 file are gone
## File Sharing
WorkDocs Alternative
So here's the deal. On our network share drive, there is one file that no one can access. It was, up until at least last Friday, accessible by several people, and seemed to be working alright. Then all of a sudden, people who work with this file start calling me saying that they can't get it open. They just get a message that says "Access Denied. Contact Administrator". I tried to open it, and I got the same message. This is a problem, because I am the administrator.
I tried to check on the permissions for the file, but it says that I don't have permissions to view or edit the permissions. I tried to take ownership of the file with several accounts, mine, our general IT account, every other IT person's account, as well as the CEO and everyone who had access to the file. No luck. I got a message that said that I didn't have permissions to take ownership of the file.
Not really sure how, but it appears that no one has permissions to even view the permissions on this file. Any thoughts would be welcome.
Popular Topics in File Sharing
Which of the following retains the information it's storing when the system power is turned off?
• GPU
• RAM
• CPU
• ROM
88% of IT pros got this right.
## 18 Replies
· · ·
Ghost Chili
OP
##### ErikN Apr 4, 2013 at 5:18 UTC
This page has a black background so it has to be right: http://wangpidong.blogspot.com/2010/06/how-to-delete-files-without-permission.html
I've dealt with similar issues in the past but it's been so long that I only remember the process was similar to what I found in the link.
0
· · ·
Mace
OP
##### Chamele0n Apr 4, 2013 at 8:06 UTC
Is this shared file an office file, like excel or word?
0
· · ·
Jalapeno
OP
##### Jared7787 Apr 4, 2013 at 8:10 UTC
@Chamele0n: Yeah, it is an excel file.
@Erik6041: I'll give that a shot. Thanks.
0
· · ·
Mace
OP
##### Chamele0n Apr 4, 2013 at 8:14 UTC
Jared7787 wrote:
@Chamele0n: Yeah, it is an excel file.
@Erik6041: I'll give that a shot. Thanks.
Do you have the shared workbook check box checked? So that multiple users can edit the file at the same time? And do you have DFS that replicates this file to other Namespace servers?
0
· · ·
Jalapeno
OP
##### Jared7787 Apr 4, 2013 at 8:26 UTC
The last time I had this file open (about a month ago) the shared workbook box was checked. But now it is not a matter of multiple users not being able to work on it at the same time, but no one can open it at all. As for DFS replication, no, we do not.
0
· · ·
Mace
OP
##### Chamele0n Apr 4, 2013 at 8:29 UTC
Jared7787 wrote:
The last time I had this file open (about a month ago) the shared workbook box was checked. But now it is not a matter of multiple users not being able to work on it at the same time, but no one can open it at all. As for DFS replication, no, we do not.
I was about to say if you had DFS enabled and shared workbook, it does cause permission loss issues. That doesn't seem to be what caused yours. But if multiple people try to save somewhere close to the same time. Or even if VSS has the file locked while the use tries to save the permissions will disappear. But you should be able to change the permissions back. Probably not from the network share location, you will have to find the file on the server that is hosting it and login as administrator there, then tack ownership and re-apply appropriate permissions.
0
· · ·
Mace
OP
##### Limey Apr 5, 2013 at 1:05 UTC
Copy it off to a FAT32 drive like a USB stick, then copy it to your desktop.
Or restore from Thursday's backup.
0
· · ·
Jalapeno
OP
##### Jared7787 Apr 5, 2013 at 1:49 UTC
So I tried going on to the server last night, both as the domain admin and as the local admin, and still couldn't access the file or take ownership. I could not copy the file to a USB stick, got the same permissions error. I do have backups from last Thursday, and I checked this file in the backups and permissions were correct. The problem there is that I couldn't override the file on the server with the backup copy, got the permissions error there, too. If I can't find a fix I'll just have to make a name change to the copy from the backup and leave this untouchable copy there until I figure something out.
0
· · ·
Mace
OP
##### Limey Apr 5, 2013 at 2:03 UTC
I'd drop to the command line and try to delete it. Failing that, boot from Linux live cd and delete from there.
0
· · ·
Ghost Chili
OP
##### ErikN Apr 5, 2013 at 2:17 UTC
What results did you get from the command line takeown and cacls attempts?
0
· · ·
Jalapeno
OP
##### Jared7787 Apr 5, 2013 at 2:25 UTC
Eric6041, I got an Invalid argument/option error when I tried to do the takeown. I think it may be because there is a space in the file name, because I tested it out on another file and it worked (granted I already had permissions for the other file, but I didn't have ownership of it and was able to take ownership).
Limey, I've got a Linux disk that I could use, and I may have to do that. Wasn't able to delete it from the command line, though.
Thanks everyone for the assistance.
0
· · ·
Ghost Chili
OP
##### ErikN Apr 5, 2013 at 2:35 UTC
You should be able to enclose the file name in quotes if the space was the issue.
2
· · ·
Mace
OP
##### Chamele0n Apr 5, 2013 at 3:20 UTC
Erik6041 wrote:
You should be able to enclose the file name in quotes if the space was the issue.
Erik is 100% correct, you should enclose any file name with spaces in it in quotes (e.g. "C:\Share\Folder\File.xls")
0
· · ·
Jalapeno
OP
##### Jared7787 Apr 5, 2013 at 3:30 UTC
Alright, tried it in quotes and at least it got the file, but I got an Error: Access is denied message. Tried it as the domain admin and the local admin.
0
· · ·
Mace
OP
##### Chamele0n Apr 5, 2013 at 3:35 UTC
Try to open command prompt with "run as Administrator" and then type
Del /F "C:\Path\to\file.xls"
Just to see if forcing it while in elevated command prompt works. If that fails you may be forced to boot into a linux live CD to remove the file. This will require some downtime for your server obviously.
You could also try the free unlocker tool. http://www.emptyloop.com/unlocker I have had to use it once before.
0
· · ·
Jalapeno
OP
##### Jared7787 Apr 5, 2013 at 3:51 UTC
Yeah, I've used the unlocker before. Tried it here, but no go. Same problem with the Del command while cmd was running as Administrator. I guess I'll have to schedule some down time for that server and Linux boot it.
0
· · ·
Jalapeno
OP
##### Jared7787 Apr 11, 2013 at 1:54 UTC
I loaded from a linux boot and was able to delete the file. Thanks everyone for the help.
0
· · ·
Mace
OP
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2017-09-23 20:18:54
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https://itectec.com/askdifferent/greek-letter-shortcuts-in-keynote-6-5/
|
# Greek letter shortcuts in Keynote 6.5
keynote
I recently switched to Keynote for writing scientific talks. I use tons of greek letters and need a fast way to type them in. I have seen on other questions and forums that people just change the auto-correct settings to replace things like \lambda with the Greek letter. However, in Keynote 6.5 I cant seem to find the auto-correct settings to do this. Its looks like they moved it out of the preferences menu. Anyone know where this is, or have another solution?
Using the Keyboard System Preference, in combination with the Greek keyboard, I was able to get this to work.
Open System Preferences > Keyboard
Select the Keyboard tab and check Show Keyboard & Character Viewers in menu bar. The flag of your default language should now appear to the right of the Date/Time menu item.
Select the Input Sources tab > Add Greek keyboard
Select the Text tab > select then add \lambda to the Replace field
From the Character Viewer menu bar item, choose the Greek keyboard > Select the With field and type the 'l' key. This should insert the λ character in this field.
From the Character ViewSwitch back to the default keyboard.
Now, whenever you type \lambda you will see this replacement pop-up:
Repeat with other characters.
|
2021-11-27 14:36:18
|
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|
http://www.r-bloggers.com/why-pictures-are-so-important-when-modeling-data/
|
# Why pictures are so important when modeling data?
October 31, 2012
By
(This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers)
(bis repetita) Consider the following regression summary,
Call:
lm(formula = y1 ~ x1)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.0001 1.1247 2.667 0.02573 *
x1 0.5001 0.1179 4.241 0.00217 **
---
Signif. codes: 0***0.001**0.01*0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.237 on 9 degrees of freedom
Multiple R-squared: 0.6665, Adjusted R-squared: 0.6295
F-statistic: 17.99 on 1 and 9 DF, p-value: 0.00217
obtained from
> data(anscombe)
> reg1=lm(y1~x1,data=anscombe)
Can we say something if we look (only) at that output ? The intercept is significatively non-null, as well as the slope, the $R^2$ is large (66%). It looks like we do have a nice model here. And in a perfect world, we might hope that data are coming from this kind of dataset,
But it might be possible to have completely different kinds of patterns. Actually, four differents sets of data are coming from Anscombe (1973). And that all those datasets are somehow equivalent: the $X$'s have the same mean, and the same variance
> apply(anscombe[,1:4],2,mean)
x1 x2 x3 x4
9 9 9 9
> apply(anscombe[,1:4],2,var)
x1 x2 x3 x4
11 11 11 11
and so are the $Y$'s
> apply(anscombe[,5:8],2,mean)
y1 y2 y3 y4
7.500909 7.500909 7.500000 7.500909
> apply(anscombe[,5:8],2,var)
y1 y2 y3 y4
4.127269 4.127629 4.122620 4.123249
Further, observe also that the correlation between the $X$'s and the $Y$'s is the same
> cor(anscombe)[1:4,5:8]
y1 y2 y3 y4
x1 0.8164205 0.8162365 0.8162867 -0.3140467
x2 0.8164205 0.8162365 0.8162867 -0.3140467
x3 0.8164205 0.8162365 0.8162867 -0.3140467
x4 -0.5290927 -0.7184365 -0.3446610 0.8165214
> diag(cor(anscombe)[1:4,5:8])
[1] 0.8164205 0.8162365 0.8162867 0.8165214
which yields the same regression line (intercept and slope)
> cbind(coef(reg1),coef(reg2),coef(reg3),coef(reg4))
[,1] [,2] [,3] [,4]
(Intercept) 3.0000909 3.000909 3.0024545 3.0017273
x1 0.5000909 0.500000 0.4997273 0.4999091
But there is more. Much more. For instance, we always have the standard deviation for residuals
> c(summary(reg1)$sigma,summary(reg2)$sigma,
+ summary(reg3)$sigma,summary(reg4)$sigma)
[1] 1.236603 1.237214 1.236311 1.235695
Thus, all regressions here have the same R2
> c(summary(reg1)$r.squared,summary(reg2)$r.squared,
+ summary(reg3)$r.squared,summary(reg4)$r.squared)
[1] 0.6665425 0.6662420 0.6663240 0.6667073
Finally, Fisher's F statistics is also (almost) the same.
+ c(summary(reg1)$fstatistic[1],summary(reg2)$fstatistic[1],
+ summary(reg3)$fstatistic[1],summary(reg4)$fstatistic[1])
value value value value
17.98994 17.96565 17.97228 18.00329
Thus, with the following datasets, we have the same prediction (and the same confidence intervals). Consider for instance the second dataset (the first one being mentioned above),
> reg2=lm(y2~x2,data=anscombe)
The output is here exactly the same as the one we had above
> summary(reg2)
Call:
lm(formula = y2 ~ x2, data = anscombe)
Residuals:
Min 1Q Median 3Q Max
-1.9009 -0.7609 0.1291 0.9491 1.2691
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.001 1.125 2.667 0.02576 *
x2 0.500 0.118 4.239 0.00218 **
---
Signif. codes: 0***0.001**0.01*0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.237 on 9 degrees of freedom
Multiple R-squared: 0.6662, Adjusted R-squared: 0.6292
F-statistic: 17.97 on 1 and 9 DF, p-value: 0.002179
Here, the perfect model is the one obtained with a quadratic regression.
> reg2b=lm(y2~x2+I(x2^2),data=anscombe)
> summary(reg2b)
Call:
lm(formula = y2 ~ x2 + I(x2^2), data = anscombe)
Residuals:
Min 1Q Median 3Q Max
-0.0013287 -0.0011888 -0.0006294 0.0008741 0.0023776
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.9957343 0.0043299 -1385 <2e-16 ***
x2 2.7808392 0.0010401 2674 <2e-16 ***
I(x2^2) -0.1267133 0.0000571 -2219 <2e-16 ***
---
Signif. codes: 0***0.001**0.01*0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.001672 on 8 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 7.378e+06 on 2 and 8 DF, p-value: < 2.2e-16
Consider now the third one
> reg3=lm(y3~x3,data=anscombe)
i.e.
> summary(reg3)
Call:
lm(formula = y3 ~ x3, data = anscombe)
Residuals:
Min 1Q Median 3Q Max
-1.1586 -0.6146 -0.2303 0.1540 3.2411
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.0025 1.1245 2.670 0.02562 *
x3 0.4997 0.1179 4.239 0.00218 **
---
Signif. codes: 0***0.001**0.01*0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.236 on 9 degrees of freedom
Multiple R-squared: 0.6663, Adjusted R-squared: 0.6292
F-statistic: 17.97 on 1 and 9 DF, p-value: 0.002176
This time, the linear model could have been perfect. The problem is one outlier. If we remove it, we have
> reg3b=lm(y3~x3,data=anscombe[-3,])
> summary(reg3b)
Call:
lm(formula = y3 ~ x3, data = anscombe[-3, ])
Residuals:
Min 1Q Median 3Q Max
-0.0041558 -0.0022240 0.0000649 0.0018182 0.0050649
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.0056494 0.0029242 1370 <2e-16 ***
x3 0.3453896 0.0003206 1077 <2e-16 ***
---
Signif. codes: 0***0.001**0.01*0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.003082 on 8 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 1.161e+06 on 1 and 8 DF, p-value: < 2.2e-16
Finally consider
> reg4=lm(y4~x4,data=anscombe)
This time, there is an other kind of outlier, in $X$'s, but again, the regression is exactly the same,
> summary(reg4)
Call:
lm(formula = y4 ~ x4, data = anscombe)
Residuals:
Min 1Q Median 3Q Max
-1.751 -0.831 0.000 0.809 1.839
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.0017 1.1239 2.671 0.02559 *
x4 0.4999 0.1178 4.243 0.00216 **
---
Signif. codes: 0***0.001**0.01*0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.236 on 9 degrees of freedom
Multiple R-squared: 0.6667, Adjusted R-squared: 0.6297
F-statistic: 18 on 1 and 9 DF, p-value: 0.002165
The graph is here
So clearly, looking at the summary of a regression does not tell us anything... This is why we do spend some time on diagnostic, looking at graphs with the errors (the graphs above could be obtained only with one explanatory variable, while errors can be studied in any dimension): everything can be seen on thise graphs. E.g. for the first dataset,
or the second one
the third one
or the fourth one,
R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...
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2013-12-05 15:42:59
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http://self.gutenberg.org/Articles/Teleparallelism
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#jsDisabledContent { display:none; } My Account | Register | Help
# Teleparallelism
Article Id: WHEBN0000903686
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Title: Teleparallelism Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:
### Teleparallelism
Teleparallelism (also called teleparallel gravity), was an attempt by Einstein[1] to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field.
## Contents
• Teleparallel spacetimes 1
• New teleparallel gravity theory 2
• Møller Tetrad Theory of Gravitation 2.1
• New translation teleparallel gauge theory of gravity 3
• Non-gravitational contexts 4
• References 6
• Books 7
## Teleparallel spacetimes
The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set \{\mathrm X_1, \dots,\mathrm X_4\} of four vector fields defined on all of M\, such that for every p\in M\, the set \{\mathrm X_1(p), \dots,\mathrm X_4(p)\} is a basis of T_pM\,, where T_pM\, denotes the fiber over p\, of the tangent vector bundle TM\,. Hence, the four-dimensional spacetime manifold M\, must be a parallelizable manifold. The tetrad field was introduced to allow the distant comparison of the direction of tangent vectors at different points of the manifold, hence the name distant parallelism. His attempt failed because there was no Schwarzschild solution in his simplified field equation.
In fact, one can define the connection of the parallelization (also called Weitzenböck connection) \{\mathrm X_{i}\} to be the linear connection \nabla\, on M\, such that [2]
\nabla_{v}(f^{i}\mathrm X_{i})=(vf^{i})\mathrm X_{i}(p)\,,
where v\in T_pM\, and f^{i}\, are (global) functions on M\,; thus f^{i}X_{i}\, is a global vector field on M\,. In other words, the coefficients of Weitzenböck connection \nabla\, with respect to \{X_{i}\} are all identically zero, implicitly defined by:
\nabla_{\mathrm{X}_i} \mathrm{X}_j = 0 \, ,
hence W^k{}_{ij}=\omega^k(\nabla_{\mathrm{X}_i} \mathrm{X}_j)\equiv0 \, , for the connection coefficients (also called Weitzenböck coefficients) —in this global basis. Here \omega^k\, is the dual global basis (or co-frame) defined by \omega^i(\mathrm{X}_j)=\delta^i_j\,.
This is what usually happens in Rn, in any affine space or Lie group (for example the 'curved' sphere S3 but 'Weitzenböck flat' manifold).
Using the transformation law of a connection, or equivalently the \nabla properties, we have the following result.
Proposition. In a natural basis, associated with local coordinates (U, x^{\mu}), i.e., in the holonomic frame \partial_{\mu}, the (local) connection coefficients of the Weitzenböck connection are given by:
\Gamma^{\beta}{}_{\mu\nu}= h^{\beta}_{i} \partial_{\nu}h^{i}_{\mu}\,,
where \mathrm X_{i} = h^{\mu}_{i}\partial_{\mu}\quad {i,\mu} = 1,2, \dots n are the local expressions of a global object, that is, the given tetrad.
Weitzenböck connection has vanishing curvature, but —in general— non-vanishing torsion.
Given the frame field \{X_{i}\}, one can also define a metric by conceiving of the frame field as an orthonormal vector field. One would then obtain a pseudo-Riemannian metric tensor field g\, of signature (3,1) by
g(X_{i},X_{j})=\eta_{ij}\,,
where
\eta_{ij}={\mathrm {diag}}(-1,-1,-1,1)\,.
The corresponding underlying spacetime is called, in this case, a Weitzenböck spacetime.[3]
It is worth noting to see that these 'parallel vector fields' give rise to the metric tensor as a by-product.
## New teleparallel gravity theory
New teleparallel gravity theory (or new general relativity) is a theory of gravitation on Weitzenböck space-time, and attributes gravitation to the torsion tensor formed of the parallel vector fields.
In the New teleparallel gravity theory the fundamental assumptions are as follows: (A) Underlying space-time is the Weitzenböck space-time, which has a quadruplet of parallel vector fields as the fundamental structure. These parallel vector fields give rise to the metric tensor as a by-product. All physical laws are expressed by equations that are covariant or form invariant under the group of general coordinate transformations. (B) The equivalence principle is valid only in classical physics. (C) Gravitational field equations are derivable from the action principle. (D) The field equations are partial differential equations in the field variables of not higher than the second order.
In 1961 Møller[4] revived Einstein’s idea, and Pellegrini and Plebanski[5] found a Lagrangian formulation for absolute parallelism.
### Møller Tetrad Theory of Gravitation
In 1961, Møller[6][7] showed that a tetrad description of gravitational fields allows a more rational treatment of the energy-momentum complex than in a theory based on the metric tensor alone. The advantage of using tetrads as gravitational variables was connected with the fact that this allowed to construct expressions for the energy-momentum complex which had more satisfactory transformation properties than in a purely metric formulation.
## New translation teleparallel gauge theory of gravity
In 1967, quite independently, Hayashi and Nakano[8] revived Einstein’s idea, and Pellegrini—Plebanski[9] started to formulate the gauge theory of the space-time translation group. Hayashi pointed out the connection between the gauge theory of space-time translation group and absolute parallelism. The first fiber bundle formulation was provided by Cho.[10] This model was later studied by Schweizer et al.,[11] Nitsch and Hehl, Meyer, and more recent advances can be found in Aldrovandi and Pereira, Gronwald, Itin, Maluf and da Rocha-Neto, Muench, Obukhov and Pereira, and Schucking and Surowitz.
Nowadays, people study teleparallelism purely as a theory of gravity [12] without trying to unify it with electromagnetism. In this theory, the gravitational field turns out to be fully represented by the translational gauge potential B^a{\!}_\mu, as it should be for a gauge theory for the translation group.
If this choice is made, then there is no longer any Lorentz gauge symmetry because the internal Minkowski space fiber—over each point of the spacetime manifold—belongs to a fiber bundle with the abelian R4 as structure group. However, a translational gauge symmetry may be introduced thus: Instead of seeing tetrads as fundamental, we introduce a fundamental R4 translational gauge symmetry instead (which acts upon the internal Minkowski space fibers affinely so that this fiber is once again made local) with a connection B and a "coordinate field" x taking on values in the Minkowski space fiber.
More precisely, let \pi\colon{\mathcal M}\to M be the Minkowski fiber bundle over the spacetime manifold M. For each point p\in M, the fiber {\mathcal M}_p is an affine space. In a fiber chart (V,\psi)\,, coordinates are usually denoted by \psi = (x^\mu,x^a)\,, where x^{\mu}\, are coordinates on spacetime manifold M, and xa are coordinates in the fiber {\mathcal M}_p\,.
Using the abstract index notation, let abc, ... refer to {\mathcal M}_p and μν, ... refer to the tangent bundle TM. In any particular gauge, the value of xa at the point p is given by the section
x^\mu \to (x^\mu,x^a = \xi^a(p)).
D_\mu \xi^a \equiv (d \xi^a)_\mu + B^a{\!}_\mu = \partial_\mu \xi^a + B^a{\!}_\mu
is defined with respect to the connection form B, a 1-form assuming values in the Lie algebra of the translational abelian group R4. Here, d is the exterior derivative of the ath component of x, which is a scalar field (so this isn't a pure abstract index notation). Under a gauge transformation by the translation field αa,
x^a\rightarrow x^a+\alpha^a
and
B^a{\!}_\mu\rightarrow B^a{\!}_\mu - \partial_{\mu}\alpha^a
and so, the covariant derivative of x^a=\xi^a(p) is gauge invariant. This is identified with the translational (co-)tetrad
h^a{\!}_\mu = \partial_{\mu}\xi^a + B^a{\!}_\mu
which is a one-form which takes on values in the Lie algebra of the translational abelian group R4, whence is gauge invariant.[13] But what does this mean? x^a=\xi^a(p) is a local section of the (pure translational) affine internal bundle {\mathcal M} \to M, another important structure in addition to the translational gauge field B^a{\!}_\mu. Geometrically, this field determines the “origin” of the affine spaces; it is known as Cartan’s “radius vector”. In the gauge-theoretic framework, the 1-form
h^a = h^a{\!}_\mu dx^{\mu} = (\partial_{\mu}\xi^a + B^a{\!}_\mu)dx^{\mu}
arises as the nonlinear translational gauge field with \xi^a interpreted as the Goldstone field describing the spontaneous breaking of the translational symmetry.
A crude analogy: Think of {\mathcal M}_p as the computer screen and the internal displacement as the position of the mouse pointer. Think of a curved mousepad as spacetime and the position of the mouse as the position. Keeping the orientation of the mouse fixed, if we move the mouse about the curved mousepad, the position of the mouse pointer (internal displacement) also changes and this change is path dependent; i.e., it doesn't only depend upon the initial and final position of the mouse. The change in the internal displacement as we move the mouse about a closed path on the mousepad is the torsion.
Another crude analogy: Think of a crystal with line defects (edge dislocations and screw dislocations but not disclinations). The parallel transport of a point of {\mathcal M} along a path is given by counting the number of (up/down, forward/backwards and left/right) crystal bonds transversed. The Burgers vector corresponds to the torsion. Disinclinations correspond to curvature, which is why they are left out.
The torsion, i.e., the translational field strength of Teleparallel Gravity (or the translational "curvature"),
T^a{\!}_{\mu\nu} \equiv (DB^a)_{\mu\nu} = D_\mu B^a{\!}_\nu - D_\nu B^a{\!}_\mu,
is gauge invariant.
Of course, we can always choose the gauge where xa is zero everywhere (a problem though; {\mathcal M}_p is an affine space and also a fiber and so, we have to define the "origin" on a point by point basis, but this can always be done arbitrarily) and this leads us back to the theory where the tetrad is fundamental.
Teleparallelism refers to any theory of gravitation based upon this framework. There is a particular choice of the action which makes it exactly equivalent [14] to general relativity, but there are also other choices of the action which aren't equivalent to GR. In some of these theories, there is no equivalence between inertial and gravitational masses.
Unlike GR, gravity is not due to the curvature of spacetime. It is due to the torsion.
## Non-gravitational contexts
There exists a close analogy of geometry of spacetime with the structure of defects in crystal.[15][16] Dislocations are represented by torsion, disclinations by curvature. These defects are not independent of each other. A dislocation is equivalent to a disclination-antidisclination pair, a disclination is equivalent to a string of dislocations. This is the basic reason why Einstein's theory based purely on curvature can be rewritten as a teleparallel theory based only on torsion. There exists, moreover, infinitely many ways of rewriting Einstein's theory, depending on how much of the curvature one wants to reexpress in terms of torsion, the teleparallel theory being merely one specific version of these.[17]
A further application of teleparallelism occurs in quantum field theory, namely, two-dimensional non-linear sigma models with target space on simple geometric manifolds, whose renormalization behavior is controlled by a Ricci flow, which includes torsion. This torsion modifies the Ricci tensor and hence leads to an infrared fixed point for the coupling, on account of teleparallelism ("geometrostasis").[18]
## References
1. ^ A. Einstein (1928). "Riemann-Geometrie mit Aufrechterhaltung des Begriffes des Fernparallelismus". Preussische Akademie der Wissenschaften, Phys.-math. Klasse, Sitzungsberichte 1928: 217–221.
2. ^ Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds, p. 223
3. ^ On the History of Unified Field Theories
4. ^ C. Møller (1961). "Conservation laws and absolute parallelism in general relativity". K. Dan. Vidensk. Selsk. Mat. Fys. Skr. 1 (10): 1–50.
5. ^ C. Pellegrini & J. Plebanski (1963). "Tetrad fields and gravitational fields". Mat. Fys. Skr. Dan. Vid. Selsk. 2 (4): 1–39.
6. ^ Møller, Christian (1961). "Conservation laws and absolute parallelism in general relativity". Mat. Fys. Dan. Vid. Selsk. 1 (10): 1–50.
7. ^ Møller, Christian (1961). "Further remarks on the localization of the energy in the general theory of relativity". Ann. Phys. 12 (1): 118–133.
8. ^ K. Hayashi & T. Nakano (1967). "Extended Translation Invariance and Associated Gauge Fields". Prog. Theor. Phys. 38 (2): 491–507.
9. ^ C. Pellegrini & J. Plebanski (1963). "Tetrad fields and gravitational fields". Mat. Fys. Skr. Dan. Vid. Selsk. 2 (4): 1–39.
10. ^ Y.M. Cho (1976). "Einstein Lagrangian as the translational Yang–Mills Lagrangian". Physical Review D 14: 2521.
11. ^ M. Schweizer; N. Straumann & A. Wipf (1980). "Postnewtonian generation of gravitational waves in a theory of gravity with torsion". Gen. Rel. Grav. 12: 951–961.
12. ^ Arcos, H.I.; J.G. Pereira (January 2005). "Torsion Gravity: a Reappraisal". Int.J.Mod.Phys. D 13 (10): 2193–2240.
13. ^ F.W. Hehl; J.D. McCrea; E.W. Mielke & Y. Ne’eman (1995). "Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance". Phys. Rep. 258: 1–171.
14. ^ Y.M. Cho (1976). "Einstein Lagrangian as the translational Yang–Mills Lagrangian". Physical Review D 14: 2521.
15. ^
16. ^
17. ^
18. ^ Braaten, E.; Curtright, T. L.; Zachos, C. K. (1985). "Torsion and geometrostasis in nonlinear sigma models". Nuclear Physics B 260 (3–4): 630.
## Books
• Aldrovandi, R.; Pereira, J.G. (2012), Teleparallel Gravity: An Introduction, Springer: Dordrecht,
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2019-10-21 05:17:47
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https://www.physicsforums.com/threads/potential-energy.228767/
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# Potential energy
A proton is brought near a positively charged sphere. As it is brought closer its potential energy...
A. increases
B. decreases
C. remains the same
D. cannot determine
Does it decrease (B) because same forces repel causing a decrease in PE?
Last edited:
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2019-12-09 04:58:42
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https://www.homebuiltairplanes.com/forums/threads/traveling-with-a-twin-engine-2-stroke-airplane.34728/page-2
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# Traveling with a twin engine 2 stroke airplane?
### Help Support HomeBuiltAirplanes.com:
#### blane.c
##### Well-Known Member
HBA Supporter
Extra engines Lazair
#### Armilite
##### Well-Known Member
As you know if you have a total of 50hp with two engines it is unlikely to maintain altitude on a single engine (with any appreciable weight) because of losses due to drag will make the effective power available closer to 37% than 50% so on single engine you will have effectively more like 18hp than 25hp. Also you likely don't cruise @ full power so being on one engine will be a judgement call, if in order to maintain some kind of glide path better than complete engine loss you decide to reduce power on the one remaining due to engine temps going up regards power setting and airspeed.
My crazy thinking is three engines are better. Usually with three engines one will be in the center and will not adversely affect yaw and therefor not be included to any appreciable degree in drag losses due to the failure of either engine mounted on the side. So worst case if you lose a side mounted engine on an aircraft with (3) 17hp engines (51hp total) then you would have effective power of 37% of the two side mounted engines plus the full power of the center engine so 37% of 34hp = 12.5hp effective + 17hp center engine equals 29.5hp effective which is over 10hp better than a twin with a engine loss and much more room to reduce some power a bit to maintain temps.
==============================
With a Triple, your Cost & Weight goes up. Even most Commerical made Small Twins aren't being produced today. I haven't seen a Triple made since WWII.
The Lazair is a unique Plane account it was really more of a Motor Glider than an Ultralight and flew on (1) 9.5hp Engine, originally used (2) 5.5hp = 11hp.
Most Twins ever made don't share that Power to Weight Ratio!
A 2 Stroke or 4 Stroke can be built to be more Durable. You Can't Fix Stupid people who run out of Gas, forget to put Oil in the Engine/Gas, use cheap Low Octane Gas, don't change the Plugs, Fuel Filter, etc.
Just like when upgrading these Honda/Clones Singles, 99% only do the $400 worth of upgrade Mods, but yet the weakest Point on a 4 Stroke is the Valve Train. I would spend the extra$180-$200 and use the HD Billet Needle Bearing Rocker Arms. I would also use these Engine Coatings. This adds to the total Engine Cost, but you can do them yourself fairly cheap. #### Speedboat100 ##### Well-Known Member I am curious what the responses would be if a new cheaper design came out utilizing small twin 2 strokes....yea or nay? Cri Cri is such. It can also be fitted with jet engines. #### Toobuilder ##### Well-Known Member HBA Supporter Log Member I am curious what the responses would be if a new cheaper design came out utilizing small twin 2 strokes....yea or nay? A response would require some significant requirements definition, but even for talking points : "Cheaper" than what? What is the cost theshold you are trying to met? What is the mission of the aircraft -speed, range, payload, seating capacity, etc? What FOS are you trying to meet with a twin? Is the proposed configuration any safer or more reliable than the existing products? #### blane.c ##### Well-Known Member HBA Supporter ============================== With a Triple, your Cost & Weight goes up. Even most Commerical made Small Twins aren't being produced today. I haven't seen a Triple made since WWII. The Lazair is a unique Plane account it was really more of a Motor Glider than an Ultralight and flew on (1) 9.5hp Engine, originally used (2) 5.5hp = 11hp. Most Twins ever made don't share that Power to Weight Ratio! A 2 Stroke or 4 Stroke can be built to be more Durable. You Can't Fix Stupid people who run out of Gas, forget to put Oil in the Engine/Gas, use cheap Low Octane Gas, don't change the Plugs, Fuel Filter, etc. Just like when upgrading these Honda/Clones Singles, 99% only do the$400 worth of upgrade Mods, but yet the weakest Point on a 4 Stroke is the Valve Train. I would spend the extra $180-$200 and use the HD Billet Needle Bearing Rocker Arms. I would also use these Engine Coatings. This adds to the total Engine Cost, but you can do them yourself fairly cheap.
If you want the most efficient airplane and don't care about anything else then single engine is the way to go.
Twins are not produced because "they are inefficient, so is a triple", but it is inefficiency of a bygone era's opinion. There is a different type of efficiency gained with a three engine arrangement and that is the ability to make it to a runway if you have a loss of engine power or much better odds of it. I often see in the evening news a ga airplane accident and most of them are single engine. It isn't just because there are more single engine aircraft it is because a single engine aircraft has much less chance of making it to a runway than a multi-engine when there are engine problems.
I argue that a triple isn't a lot more expensive than a twin because the engines can be smaller than the twins engines and their smaller size offsets part of the expense. I also argue that a engine out in a triple is much easier to deal with than an engine out in a twin, because (unless you go to extreme sizes of engine on the twin) you will have more remaining power available with an engine loss with the triple and also less adverse yaw to control.
HBA Supporter
#### Toobuilder
HBA Supporter
Log Member
If thats true, then the Rotax costs $135 bucks per HP. Multiply that by 180 and you are right in Lycoming 0-360 country. #### Wanttaja ##### Well-Known Member Well, most engine setups end up being upwards$7-10k so was just thinking about cost vs reliability.
Don't think "reliability" is the term you're looking for. Adding a second engine only doubles the chance of an engine failure.
You're undoubtedly looking at the ability to continue the flight to a safe destination, vs. having to perform an immediate power-off landing. One of the OWTs about flying is that you're actually MORE likely to get killed after engine failure on a twin-engine aircraft. The way the story goes, pilots fly "power-off" landings all the time, but the gymnastics needed to keep a twin going with a fan out is something practiced only BFRs. A VMC roll at low altitude is very likely to kill you.
Again, though, just a story...haven't seen any sort of statistics on it.
Light twin aircraft rarely have a surfeit of power. Do so research on the Champion Lancer, a twin-engine version of the Aeronca Champ. Thing couldn't get out of its way with one engine out.
To maximize safety, too, you're going to want to have featherable propellers on the engines...kind of rare, for two-stroke engines. That was the downfall of the Lancer, two O-200s with fixed props. If your plane can't climb on a single engine, you're just stretching the glide, which may or may not really be an advantage. Though maybe two-stroke engines don't windmill....
Two-stroke engine issues are a factor in homebuilt aircraft accidents about twice as often as certified aircraft engines. If safety is your goal, you'd be better off with a single, more reliable powerplant instead of two two-stroke engines.
Ron Wanttaja
#### Hephaestus
##### Well-Known Member
With a Triple, your Cost & Weight goes up. Even most Commerical made Small Twins aren't being produced today. I haven't seen a Triple made since WWII.
Britten-Norman would dispute the lack of triples.
Caught a ride on one down in ? St Kitts? years ago.
#### n3puppy
##### Well-Known Member
Britten-Norman would dispute the lack of triples.
Caught a ride on one down in ? St Kitts? years ago.
So would de Havilland (DHA-3 Drover)
There are more in this thread
THREE ENGINE AIRCRAFT
#### Armilite
##### Well-Known Member
If you want the most efficient airplane and don't care about anything else then single engine is the way to go.
Twins are not produced because "they are inefficient, so is a triple", but it is inefficiency of a bygone era's opinion. There is a different type of efficiency gained with a three engine arrangement and that is the ability to make it to a runway if you have a loss of engine power or much better odds of it. I often see in the evening news a ga airplane accident and most of them are single engine. It isn't just because there are more single engine aircraft it is because a single engine aircraft has much less chance of making it to a runway than a multi-engine when there are engine problems.
I argue that a triple isn't a lot more expensive than a twin because the engines can be smaller than the twins engines and their smaller size offsets part of the expense. I also argue that a engine out in a triple is much easier to deal with than an engine out in a twin, because (unless you go to extreme sizes of engine on the twin) you will have more remaining power available with an engine loss with the triple and also less adverse yaw to control.
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First, you have to look at 40+ Years of Prior Accident Data! It's about 50/50 between Ultralights/Kitplanes and Small Commerical made Airplanes. The Top three leading causes for many years are:
1. Running out of Gas! (Human Error)
2. Flying in Bad Weather! (Human Error)
3. Flying with a known Mechanical problem. (Human Error)
There Top 10 today.
10) Thunderstorms Or Windshear (Human Error)
Weather is obviously one of the most hazardous parts of flying. This photo below is a Cessna 210 that flew into a level 6 thunderstorm. The pilot at the controls was Scott Crossfield, an accomplished Naval test pilot, and the first pilot to fly twice the speed of sound. Before he departed, he received a weather briefing, however he didn't get weather updates during his flight. The airplane broke apart in-flight, with wreckage found at three different locations.
9) Midair Collisions (Human Error)
Most midairs happen near airports, and in this accident, a Cessna 172 entered the traffic pattern and collided with a helicopter. Unfortunately, the 172 didn't make radio calls prior to entering the pattern, and the helicopter was unaware of them. The helicopter was able to land safely, but the 172 entered a spin, impacting the ground.
8) Systems Failure
This Cessna 335's attitude indicator failed in poor weather. The pilot became spatially disoriented and crashed.
7) Fuel Exhaustion Or Contamination (Human Error)
This Cessna 172 ran out of fuel in flight. The aircraft had just completed an STC (supplemental type certificate) to increase the engine's horsepower. However, new fuel burn rates weren't placed in the flight manual, and the pilot didn't plan for the increased fuel burn rate.
6) Flight In IMC (Human Error)
This King Air 200 was on a localizer approach, but the pilots were using a GPS to navigate to the IAF. The pilots inadvertently swapped the initial approach fix with the missed approach point on the GPS, using manually entered fixes. With no glideslope, and incorrect DME data, the plane flew approximately 5 miles past the missed approach point at the MDA altitude. As the pilots executed a missed approach, they impacted the top of a mountain.
5) Unknown/Undetermined
Sometimes the NTSB and FAA don't have enough information to determine the cause of an accident. In this crash, the NTSB and FAA believe the aircraft flew into a severe downdraft in mountainous terrain, based on radar data. Carb Ice is also hard to detect.
4) Low Altitude Operations (Human Error)
This P-3 air tanker was on a fire bombing run. The flight had an FAA examiner on board performing a checkride. As the P-3 descended over a hill, the left wingtip hit the ground, and the aircraft impacted terrain.
3) Powerplant Failure
In this crash, the aircraft had a right engine cylinder failure. The pilot feathered the prop but didn't have enough single-engine performance to maintain altitude. The pilot elected to ditch the aircraft in the water. Fortunately, the pilot and all the passengers survived.
2) Controlled Flight Into Terrain (Human Error)
This King Air 200 was on a medivac flight. The pilot was cleared for a visual approach into Bozeman, MT at night. Unfortunately, the pilot identified the wrong airport, overflew Bozeman, and impacted terrain.
1) Loss Of Control In Flight (Human Error)
In this accident, the pilot lost their right engine immediately after takeoff. The pilot lost directional control, rolled inverted, and impacted the runway.
You say "efficient" but I think you mean more Durabil or Safe. Let's say you have a Cessna 310, (4) Passengers and Full Fuel and then takeoff, get maybe 50 miles, and have an Engine Failure. Now is it going to make it back to the Airport?
General characteristics
• Crew: one
• Capacity: four passengers
• Length: 27 ft 0 in (8.23 m)
• Wingspan: 35 ft 0 in (10.67 m)
• Height: 10 ft 6 in (3.20 m)
• Wing area: 175 sq ft (16.3 m2) [75]
• Empty weight: 2,850 lb (1,293 kg)
• Gross weight: 4,600 lb (2,087 kg)
• Fuel capacity: 100 US gal (83 imp gal; 380 L)[75]
• Powerplant: 2 × Continental O-470-B horizontally opposed piston engines, 240 hp (180 kW) each. 480 hp total.
Off a different website:
5 MOST COMMON CAUSES OF AVIATION ACCIDENTS
Aviation accidents often take years to fully investigate, and so the exact cause of a plane crash may not be discovered for some time. However, there are several causes that have been found to be contributing factors in most aviation accidents:
1. Pilot error. The most common cause of aviation accidents is pilot error, which accounts for approximately half of all plane crashes. Flying a plane is among the most complex and difficult jobs available, despite modern innovations that automate many features of air travel. A pilot must monitor dozens of readouts and gauges throughout the flight, many of which seem downright mystic to non-pilots. Any miscalculation or misreading can result in a deadly crash. However, pilots should not be blamed for every accident.
2. Mechanical defects. Planes are massive feats of engineering and are made up of hundreds of separate systems. A defect or failure in any one of these systems can lead to a dangerous situation. These can include manufacturing defects, inadequate repairs or equipment replacements, and old or worn out parts.
3. Weather problems. Just as driving becomes more dangerous in bad weather, so does flying. Heavy rainstorms, fog, and snow can make it more difficult for airplanes to maneuver and can lead to deadly accidents. Visibility issues, high winds and skidding during takeoff and landing are the most dangerous weather-related threats to aircraft.
4. Air traffic controller error. Pilots rely on information and support from air traffic controllers while they are in the air. Air traffic controllers must coordinate with many different planes at once, and often must take factors such as weather and fuel into consideration when scheduling takeoffs and landings. Any error made by an air traffic controller has the potential to result in an aviation accident, possibly involving more than one aircraft.
5. Other causes. There are many other factors that can contribute to a plane crash, including sabotage and poor runway maintenance. One of the most surprisingly common factors in aviation accidents is birds. If a large bird collides with a windscreen or an engine, it can cause damage that may contribute to a plane crash.
Notice, Engine Failure isn't really a problem, it's Pilot/Human Error for not Maintaining their Engine. I would say 99% of all Engine Failures are Human related.
#### EzyBuildWing
##### Well-Known Member
Aircam with twin Rotax 4-stroke pushers can can easily climb-out on one engine at max gross.
Plenty of Youtube vids demonstrating Aircam climb-out on one.
How about a Moyes Dragonfly powered with THREE pusher Polini-Thor motors..... 36HP each, 18 kg, and 105kg thrust with a 1.5m dia prop?.......would give the pilot a substantially better chance of making a power-on decent/landing if one(or two) engines quit.
Hey, if 3 engines all quit together, then go for the ballistic-chute option!
HBA Supporter
BS
#### Wanttaja
##### Well-Known Member
Notice, Engine Failure isn't really a problem, it's Pilot/Human Error for not Maintaining their Engine. I would say 99% of all Engine Failures are Human related.
Since engines and aircraft are created by humans, any failure is thus human-related. Humans designed them, humans built components, humans assemble them, and humans operate them. On that basis, 100% of all accidents, not just engine failures, are "human related." Guess I should just delete my databases and relax.
I think it was Robert Heinlein who said, "When you get down to it, all deaths can be attributed to heart failure."
Ron Wanttaja
#### BBerson
##### Light Plane Philosopher
HBA Supporter
Just need two hearts.
#### Wanttaja
##### Well-Known Member
Just need two hearts.
Obviously, only Time Lords should fly.
Ron Wanttaja
#### BBerson
##### Light Plane Philosopher
HBA Supporter
Oh, instead of traveling with twins it's traveling in time.
I had to look that one up on wiki:
Time Lords and human beings look alike,[38][39] however they differ in many respects. Physiological differences from humans include two hearts which normally beat at 170 beats per minute,[40] three brain stems [S10E06 Extremis], an internal body temperature of 15 degrees Celsius (59 degrees Fahrenheit)[citation needed] and a "respiratory bypass system" that allows them to survive strangulation
HBA Supporter
#### Toobuilder
##### Well-Known Member
HBA Supporter
Log Member
Based on the graphic in post #34, Im never going to set my azz in a Glassair III... Way too dangerous!
#### TFF
##### Well-Known Member
Is that 104% chance of engine failure?
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2021-01-19 22:24:56
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http://interstat.statjournals.net/YEAR/1996/abstracts/9611002.php
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## Semiparametric EM-estimation of censored linear regression models for durations
### by Alfred Hamerle and Michael Moller .
Abstract: This paper investigates the sensitivity of maximum quasi likelihood estimators of the covariate effects in duration models in the presence of misspecification due to neglected heterogeneity or misspecification of the hazard function. We consider linear models for $r\left(T\right)$ where $T$ is duration and $r$ is a known, strictly increasing function. This class of models is also referred to as location-scale models. In the absence of censoring, Gould and Lawless (1988) have shown that maximum likelihood estimators of the regression parameters are consistent and asymptotically normally distributed under the assumption that the location-scale structure of the model is of the correct form. In the presence of censoring, however, model misspecification leads to inconsistent estimates of the regression coefficients for most of the censoring mechanisms that are widely used in practice. We propose a semiparametric EM-estimator, following ideas of Ritov (1990), and Buckley and James (1979). This estimator is robust against misspecification and is highly recommended if there is heavy censoring and if there may be specification errors. We present the results of simulation experiments illustrating the performance of the proposed estimator.
Key Words: Censored linear regression models, Accelerated failure time models, misspecified models semiparametric EM-estimation, simulation study
Authors:
>Alfred Hamerle, martin.spiess@wiwi.uni-regensburg.de
Michael Moller
Editor: Jugal K. Ghorai , jugal@csd.uwm.edu
READING THE ARTICLE: You can read the article in portable document (.pdf) format (151239 bytes.)
NOTE: The content of this article is the intellectual property of the authors, who retains all rights to future publication.
This page has been accessed 2445 times since April 1, 2007.
Return to the Home Page.
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2018-08-15 07:32:38
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http://teste.abramge.com.br/ntuwq/monte-carlo-simulation-example-in-operation-research.php
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##### Monte carlo simulation example in operation research
8. Theory of Thus operations research can Uniform Distribution with examples, manufacturing. Forecasts generated by both the time Research Article Reliability Assessment of Active Distribution System Using Monte Carlo Simulation Method ShaoyunGe, 1 LiXu, 1 HongLiu, 1 andMingxinZhao 2 Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin , China China Electric Power Research Institute, Beijing , China Correspondence should be addressed to Hong Dec 04, 2017 · “Forced Monte Carlo simulation strategy for the design of maintenance plans with multiple inspections. Simulations with Continuous Random Variables. for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. SIPmath development was led by Dr. g. The paper combined Monte Carlo simulation with traditional M/M/1 model. Machine Learning, 50, 5–43, 2003 c 2003 Kluwer Academic Publishers. Uniformly scatter some objects of uniform size (grains of rice or sand) over the square. Jan 08, 2014 · analysis and a Monte Carlo simulation method to construct a robust forecast for the shell usage consumption. Richard M. This sample use double precision hardware if a GTX 200 class GPU is present. A Business Planning Example using Monte Carlo SimulationImagine you are the marketing manager for a firm that is planning to introduce a new product. Statistical Analysis in Simulations. We have verified that electrons evolve much fa ster than photons. , Nature Physics, 2011, illustrating the examples in Figure 4 of Fieremans and Lee, NeuroImage 2018 with more details in supplmentary information. . Andrieu@br SIMULATION. No longer the approach of “last resort”! Industrial Engineering - EMJ Not a very old technique World War II “Monte Carlo” simulation: originated with the work on the atomic bomb. analysis, a Monte Carlo simulation was developed for use in further study of the situation and evaluation of potentially useful vehicle rear lighting systems. But for now, let's discuss the pros and cons of PRNG. There are a wide range of applications for simulation; for example, players in the electricity market can use simulation to decide whether or not an investme nt can be expected to be profitable, and authorities For example, Monte Carlo simulation is exploited for optimal DG allocation and sizing in distribution systems considering multi-level load models, in order to minimize the costs of active and reactive losses and improve the voltage profile and reliability [25]; or to assess the the distribution, Monte Carlo simulation utilizes the “S-shaped” cumulative probability curve. As a statistical approach, a typical Monte Carlo simulation provides an ensemble-averaged result of light propagation [that is, it ignores coherent effects (21)] and requires launching a large number of The methods are: (1) analytical method, (2) non-sequential Monte Carlo simulation, and (3) sequential Monte Carlo simulation. In these cases the number of extractions from the frequency distributions characterizing the model is inadequate or limited to just one, so it is necessary to in simulation and analytical method. Use 9. By using repeated random sampling to create a probability distribution for a variable, a Monte Carlo simulation can provide answers to questions that might otherwise be impossible to answer. Averaging over such rollouts can provide an effective position evaluation, achieving superhuman performance in backgammon 8 and Scrabble9, and weak amateur level play in Go 10. The phrase "Monte Carlo" derives from the well-known gambling city on the Mediterranean in Monaco. tract No. 2 TERMINOLOGIES In this section, we discuss a few terms which are used in Mar 18, 2019 · The Monte Carlo Simulation shows that the probability of the money lasting through retirement decreases to 87%. This dissertation is about how Monte Carlo simulation can be used to analyse electricity markets. 1 Further investigation led me to the Monte Carlo method page of Wikipedia 2 where I saw an example of approximating pi using this simulation. 1. Abonazel: A Monte Carlo Simulation Study using R 5. First the concept of using Monte Carlo methods to give solutions to PERT Bard, J. For a long time, Excel has stood out as one of the leading tools to create data models and simulation. Three-component system is taken up and analysis is performed with the consideration of repair actions. As noted by Mooney (1997), Monte Carlo simulation "offers an alternative to analytical mathematics for understanding a statistic's sampling distribution and evaluating its behavior in random samples" (p. Apr 09, 2018 · A Monte Carlo simulation consists of a large number (hundreds of thousands or millions are typically necessary to capture all the potential variability of the outcomes) of “trials” in which a new set of simulated variables (ε in our example) are selected based on defined distributions (a normal distribution is a frequently utilized Monte Carlo simulation was developed as part of the atomic program. We also discuss various application areas for Monte Carlo simulation in section7 and software for performing Monte Carlo simulation in section8, before concluding in section9. Journal of the Operational Research Society A farewell to the use of antithetic variates in Monte Carlo simulation simpler interpretation of the AV role is presented, showing AV as solely a procedure for input sample means compensation, In this paper the results of a Monte Carlo simulation of PERT networks are given. As an example of the use of such techniques this paper de scribes briefly a Monte Carlo simulation of the Air Force Eastern Test Range data reduction computer system. ac. It controls the sequence of demands realized for the simulation. I don't usually find myself in upscale company there, ha ha. The application of simulation involves specific steps in order for the simulation study to be successful. Monte Carlo Simulation: Project Appraisal 8. Indeed, an High quality decision support through, primarily but not limited to, discrete event and Monte Carlo simulation, linear programming, adhoc Excel analysis, and other operations research methods. Desired (but perhaps not mandatory) requirements might include: Should be able to easily handle unit conversions; Should be able to support distributed processing (for Monte Carlo simulation). 4. We have confirmed that our algorithm is quite promising using real world timetable data. SIMULATION WITH Operations Research is an applied science and is concerned with quantitative decision problems, generally involving the allocation and control of limited resources. Time series analysis is reviewed in Section 3. In this paper, we shall focus on dynamical interpreta- Applications of Monte Carlo Author: Herman Kahn Subject: A discussion of some of the ideas and techniques of the Monte Carlo method (applying probability theory and statistics to applied mathematics) that have proved useful in the solution of various problems. Given a random i. Used to simulate bombing raids. The code re-implements 2d Monte Carlo simulations originally developed in Fieremans, et al. , Zan, J. ingly complex systems. All three methods are utilized to calculate the pdf of a sample SF the Monte Carlo simulation of sea ice load on a GPU by using CUDA programming over CPU implementations. These simulations help you see the The thesis provides an introduction to Monte Carlo simulation in the financial markets. 2 Monte Carlo Simulation. Making statements based on opinion; back them up with references or personal experience. , 10. Sep 15, 2008 · Posted by Palisade September 15, 2008 Leave a comment on Monte Carlo Meets Simulation Latest to be touched by a greening of consciousness is the Formula One race crowd. In this webinar, we will use an example to demonstrate how to analyze and visualize your model's behavior across its design space using Monte Carlo simulations. It was named after the famous Casino de accounting research using deterministic simulations as the primary research tool. operation of power system which considers random events like outages of elements, dependent events and component behaviour, queuing of failed component, load variation, variations in energy source and different operating conditions. Scenarios (such as best, worst, or most likely case) for each Oct 10, 2019 · In finance, we use Monte Carlo simulations to define potential risk. Probability density functions (PDFs) explain the range of potential values of a given variable and the likelihood that different [11], economics [22] and operations research [6, 24]. Created Date: 4/27/2006 3:46:10 PM Historical meteorological data has been used to model and deploy a set of renewable energy distributed generators which maximize reliability in a 37-bus primary-distribution network. The following briefly describes the basic steps in the simulation process [6, 7]: Problem Definition Monte Carlo approximations of this type are routinely used in Bayesian optimal design problems. 5 Nov 2017 MONTE CARLO SIMULATION IN OPERATIONS RESEARCH BY GOURAV In this video you are going to learn how to solve Simulation problem using Monte Carlo method of simulation. In this section, I introduce an application of Monte Carlo methods in revenue management. Monte Carlo simulation is a rather down-market term (pardon my snobbery). 50 as tails, is a Monte Carlo simulation of the behavior of repeatedly tossing a coin. Monte Carlo method ? Draw a square on the ground, then inscribe a circle within it. Douglas Hubbard's The Failure of Risk Management: Why It's Broken dealing with numbers like operations, manufacturing, or engineering, if applicable. It then calculates results over and over, each time using a different set of random values from the probability functions. I use two helper function , get_continuation_function to create the TF operators. This site features information about discrete event system modeling and simulation. By Dr. The name ‘the Monte Carlo method’ was first used by Metropolisand Ulam (1949), and 4 yearslater Metropoliset al (1953)publishedtheir Markov The Monte Carlo simulation is a Monte Carlo Method. To simulate microscopic nucleation, which concerns the formation of clusters and collisions between clusters and monomers, a kinetic approach that may be applied to the direct simulation Monte Carlo (DSMC) method, is developed. Monte Carlo Method = a computer simulation that performs Monto Carlo experiments aimed to compute the above probability We will illustrate the Monto Carlo Method with a simple experiment to find Pi Software for risk and decision analysis, including @RISK and the DecisionTools Suite. If you have a background in operations research, and you want to reduce costs and control risks by building and solving optimization and/or simulation models, you've come to the right place. For example, with $$n=4$$ and $$N=10^5$$ the average probability in 25 full Monte Carlo experiments is 0. Operational Monte-Carlo-Verfahren: Additional Physical Format: Online version: Kohlas, Jürg, 1939-Monte Carlo Simulation im Operations Research. Large-scale simulation of systems such as Ising model requires a large amount of high per-formance computing resources, which are usually available in multi-core computing architectures based on distributed shared memory, or distributed clusters (a. It provides much better energy-per-operation than a GPU implementation, at a comparable performance level. Given the security code name “Monte-Carlo”. Sam Savage. Monte Carlo methods use stochastic simulations, meaning that they use random numbers and probability statistics to examine a system. , NMR Biomed, 2010 and Novikov, et al. An example is presented to illustrate Monte Carlo simulation. Deterministic simulation models have the alternatives clearly known. Basic Terminology. MONTE CAB-LO TECHNIQUES Paul F. However, its widespread use is hindered by the high computational cost. human computers I Early Monte Carlo Meetings This research studies the details of photon transport in an LMR system using the Monte Carlo technique and explains both qualitatively and quantitatively the reasons for the image contrast and features due to lateral migration. sample x 1;x 2; ;x N For example the program doesn't work if the seed is 0. Oct 08, 2013 · • Darker and Kac define monte carlo method as combination of probability methods & sampling techniques providing solution to complicated partial or integral differential equation. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. MonteCarlo Parallized Monte Carlo Simulation Description MonteCarlo runs a Monte Carlo simulation study for a correctly specified function and the desired parameter grids. Why Optimization Should Be Top of Mind for Operational Planning… A simple example of MC. com’s operation research and decision modeling tutors can assist you with tutoring for Monte Carlo simulation using XLSim and SIPmath. The inputs to this. Monte Carlo simulation involves the use of a computer to represent the operation of a complex financial system. Additionally, the R-indices are introduced for conquering the overparameterized problem in the optimization process. The seed in F13 is used for Monte Carlo simulation. This will be a great example of applying Excel and math to a business question. Operations research, systems engineering, management science, complex systems and Using a Monte Carlo simulation and the prediction of the repeatability and Previous research on this topic has examined mainly how firms undertook analyses Sample selection, Standard item could refer to international standard. Keywords: Radiation transport, electron-photon showers, Monte Carlo simulation, sampling algorithms, constructive quadric geometry. F to optimize the operation costs of prosumers. Two of the main virtues of simulation are flexibility and simplicity. d. Monte-Carlo Simulation The Monte-Carlo simulation method uses random numbers for generating some data by which a problem can be solved. It is used to model the probability of various outcomes in a project (or process) that cannot easily be estimated because of the intervention of random variables. V. Monte Carlo simulation usually requires several (perhaps many) runs at Stochastic (Monte Carlo): Operations of grocery store with randomly modeled Practical Side: Role of Sponsor and Management in Designing/Executing Simulation Study However, simplification can lead to seriously incorrect results; Example: Botev suggested the multilevel Monte Carlo example in Section 5. 5 2 2. Therefore, using Monte Carlo simulation method, taking the PPP project of township sewage treatment plant in Hunan Cili County as an example, to carry out the risk analysis and evaluation Mar 16, 2015 · Simulation is one of the most widely used techniques in operations research and management science… 39. Traditional NPV analysis identifies the expected cash flows and discount them according to their systematic risk. The proposed technic: The full steps to create a Monte Carlo simulation study In this section, we proved the completed algorithm of Monte Carlo simulation study. For example, you know the next product update is planned for June 15th and you want to know how many new features will be ready by then. At the start of simulation, the first random number 21 generates a demand of 25 cakes as shown in table 2. MBDoE for Parameter Estimation. However, with increasing complexity, the Monte Carlo method win: • Monte Carlo method: • Numerical (e. In this section we Adaptive optimal operation of a parallel robotic liquid handling station. Bases of Monte Carlo simulation are briefly described. Scientist a t the Los Alamos National Laboratory originally used it to model the random diffusion of neutrons. F. Just as in roulette we get random numbers produced by a roulette wheel when it is spun, so in Monte Carlo simulation we make use of random numbers generated by a computer. Using Monte Carlo simulation for your Kanban process keeps your forecasts grounded in reality. Keywords: Average arrival, Average service, M/M/1 and M/M/C queueing model, Monte Carlo Simulation, Analytical method, Queue length. LEBOWITZ This thesis project involves writing a Monte Carlo simulation of radiation transport to be used for benchmarking the Intel iPSC/2, iPSC/860, and Touchstone Delta Machine. Individual samples were often very simple to program 2. Example Suppose you are facing an investment which cost \$100 today, but generates cash flows for the next two years. A New Algorithm for Monte Carlo Simulation of king Spin Systems* A. B. To set up the estimate, randomly located points are generated within a 2×2 square which has a circle inscribed within it– think of a game of darts. way the optimal sample size came from studying Monte Carlo simulation models so it is necessary to replicate simulation runs many times in order to obtain a for project management, models for the description of repetitive operations some probability distribution that describes the operation of some aspect of a Monte Carlo simulation is the only type of simulation that will be addressed in this As an example of discrete-event simulation from queuing theory, consider a bank Research suggests that the demand for the chair during the first eight weeks Simulation Using Monte Carlo Technique Based on Queuing. There is a need for an efficient algorithm able to predict the life of power electronics component. future directions for our research. The uncertainty associated with a value of some quantity is widely recognized throughout scientific disciplines as a quantitative measure of the Apr 02, 2019 · The codes use Monte Carlo methods to estimate π. There have four steps be concerned when using this method to map the risks. Distributions of fifty-seven input parameters were defined as uniform or log uniform (Table 1) and then were used in the Monte Carlo particles produced by the operation of attitude control system (ACS) jets. monte carlo simulation explained monte carlo simulation technique monte carlo simulation method monte carlo simulation example Considerable confusion exists over the best terminology to use. Overview of Monte Carlo simulation. Symbols and numerical values of constants frequently used in the text (Mohr and Taylor, 2005) Quantity Symbol Value Jun 06, 2020 · Monte-Carlo methods are effective, for example, for estimating the solution of multi-dimensional boundary value problems at a point. Subramanian Monte Carlo Simulation Select numbers randomly from a probability distribution Use these values to observe how a model performs over time Random numbers each have an equal likelihood of being selected at random Dec 06, 2007 · What is claimed is: 1. L. It is important to know the possible expected output at the end of simulation. If you like our video then subscribe our video then subscribe our channel. This section is based on the excellent textbook on the topic by Talluri and van Ryzin (2006) 4 and the original paper by Talluri and van Ryzin (1999) 5. Monte Carlo simulation methods are widely used in medical physics research and are starting to be implemented in clinical applications such as radiation therapy planning systems. monte carlo simulation in operation research. Simulation is an important tool used by engineers to design and implement advanced communication systems that deliver optimal performance. Step B. Queuing problems (e. The numerical example illustrates the feasibility of the model. Small memory was not a big constraint for these methods 3. 6. Subset Simulation by Take the application of Monte Carlo simulation to risk assessment as an exam-ple. The Monte Carlo simu - lation method written in Microsoft Excel VBA is presented in Section 4. 2). Figure 4: Monte Carlo simulation method scheme [ Back to Monte Carlo Simulation Basics] A deterministic model is a model that gives you the same exact results for a particular set of inputs, no matter how many times you re-calculate it. Actually, a great Operational Research (OR) Synonyms: • Operations Research; Systems Analysis Definition: • The discipline of applying advanced analytical methods to help make better decisions. In the first example Monte Carlo procedure. I. Simulation models of these systems will play a major role in meeting this challenge expeditiously and economically. When Monte Carlo simulations are applied, their predictions of failures, cost overruns and schedule overruns are routinely A Monte Carlo algorithm is often a numerical Monte Carlo method used to find solutions to mathematical problems (which may have many variables) that cannot easily be solved, for example, by integral calculus, or other numerical methods. Learn Monte Carlo simulation and Optimization. In my workplace, I usually refer to Monte Carlo simulation, because many people wouldn't have a clue what I was talking about if I said stochastic simulation. 5 3-4 -2 0 2 4 P*(x) Figur e a The function P x exp Ho wtodra w samples from this densit y b The function Introducing Uncertainty in a ModelTo turn the spreadsheet model on the previous page into a risk analysis model, we need to replace the fixed average Sales Volume, Selling Price, and Unit Cost amounts with variable amounts that reflect their uncertainty. Using simulation to calculate the NPV of a project Marius Holtan Onward Inc. Build accurate predictive models in 3 key areas: Finance, Operation and Project Management: Download: Project Planning & Estimating Skills: Incorporate cutting edge simulation and optimization techniques into your project plans and portfolios. P. The entire monte Carlo method has three following steps. A simulation of Markov branching processes allows one to construct estimates of the solution of certain non-linear equations, for example, the Boltzmann equation in the theory of rarefied gases [3] . Thanks for contributing an answer to Operations Research Stack Exchange! Please be sure to answer the question. In the fact based simulation, durations between events are decided from analytical results of train traffic record data. Syst. GraduateTutor. A characteristic feature of Monte Carlo simulation is the generation of a large number of random samples from specified probability distribution(s) to represent the operation of risk in the system. As an example, a mutual fund manager may use the method to manage assets and liabilities to try and establish any downward risk – the risk that liabilities will outgrow the assets leading to a loss. An option useful here is the number of observations in the simulation. Kroese, T. murakami@kek. 50 as heads and greater than 0. Taimre, Z. Monte Carlo simulation ‘represents the uncertain relationships to Once a simulation is built and what-if scenarios can be run, the desire to keep testing more and more scenarios often grows. Nov 20, 2019 · The Monte Carlo simulation is one example of a stochastic model; it can simulate how a portfolio may perform based on the probability distributions of individual stock returns. Comparison of different scenario reduction • The heart of a Monte Carlo analysis is to obtain an estimate of a mean value (a. In the simulation of heteroge-neous aerosols Monte Carlo methods are also attractive because they do not require any a priori assumptions about the enclosure distribution in each droplet. This Monte Carlo example is based on a simple EO model created many years ago. finity, andis used in manyapplications: for example,in pricing financial instruments one typically requires the average price over all possible future outcomes, while in Monte-Carlo integration it is necessary to estimate the average occupancyover the integration domain. kobe-u. Analytical methods used (examples): • Linear Programming • Network Analysis • Meta Heuristics • Queuing Theory • Game Theory • Simulation 1. Problem with Monte Carlo Simulation Hatice Tekiner 1, David W. supermarket checkouts) were also modelled using Monte-Carlo simulation. Going back to the traffic simulation example, the initial goal of the simulation might be to determine whether to replace 4-way-stops with roundabout intersections in a particular section of town. As a statistical approach, a typical Monte Carlo simulation provides an ensemble-averaged result of light propagation [that is, it ignores coherent effects (21)] and requires launching a large number of MC simulation. Simulation Languages. In addition, a detailed discussion is presented of the characteristics of the Monte Carlo procedure. The Procedures of using Monte Carlo simulation problem-solving operations. by. Monte Carlo simulation results show that the offset standard Jan 09, 2004 · The hands-on, example-rich guide to modeling and simulating advanced communications systems. AF 49(638)-1700--monitored by the Directorate of Operaiondl Requirements and Development Plans, Deputy Chief of Staff, Research and Development, Hq USAF. We can now easily estimate this same probabilty using Monte Carlo simulation. We present a Monte Carlo algorithm that generates points randomly and uniformly on a set of arbitrary surfaces. simulation. a data-centers) with homogeneous 1 It recommends an approach to evaluating measurement uncertainty based on the propagation of distributions using Monte Carlo simulation. The The only way to perform Monte Carlo simulations in such models that I have been able to think of is to write a loop where I change the desired input variable to a random draw from the appropriate distribution and recalculate the workbook for each run through the loop and store the result from each run in an output sheet. Monte Carlo simulation is a computerized mathematical technique to generate random sample data based on some known distribution for numerical experiments. Let U denote the unavailability or failure probability of a system and x i be the zero–one indicator variable that can be obtained using a Monte Carlo simulation method: x i = 1 if the sampled system state is a failed one x i = 0 if the sampled system state is a monte carlo simulation is used to give solutions of deterministic problems whereas stochastic simulation is used for stochastic problems. When the appropriate physical and The simulation model of the reservoir behavior was used, which allows to evaluate the results of solutions and helps to reduce, for example, the cost of dam construction, the risk of poor design of reservoir volumes, future operational risk of failures and reduce water shortages during the operation of water reservoirs. 1. Each point on this curve is determined by dividing the area under the curve to the left of the “x” value of interest by the total area of the curve (which happens to be the square root of π). 29 Jun 2020 Example of Monte Carlo Simulations: The Asset Price Modeling. Several of the chapters are polished enough to place here. So a Monte Carlo simulation uses essentially random inputs (within realistic limits) to model the system Monte Carlo simulation has been used to model uncertainties since the Manhattan atomic bomb project by blasting randomly generated inputs through mathematical models. Instead, practitioners use techniques such as Monte Carlo analysis. Monte Carlo is used in corporate finance to model components of project cash flow , which are impacted by uncertainty. Bloustein School of Planning and Public Policy Abstract A new approach to the electricity generation expansion problem is proposed to minimize We have developeda Monte Carlo simulator of QCLs that treats electrons and photons on the same footing. The probability of each possible outcome. We explain our algorithm through an application in regression framework, especially; we will use the I want to introduce Monte Carlo methods for a group of 16-18-years-old high school students. Table 1: AP1000 PWR startup physics simulation results Monte Carlo VERA VERA vs Must support Monte Carlo simulation. The term ‘Monte Carlo’ is presently somewhat fashionable, the term ‘simulation’ is to be preferred, because it does not suggest that the technique is limited to what is familiar to statisticians as a sampling experiment. OPERATIONS RESEARCH CENTER Monte-Carlo simulation and Section 5 describes the statistical analysis used. ing the simulator to be run in a Monte Carlo-like fashion or to be used in complex large scale simulations that utilizes dynamic programming. Estimating integrals by Monte Carlo simulations. Similarly, we can calculate the next demand for others. In the later one, correlations and regularities in Simulation is normally used to assess the current, or predict the future, performance of a business process. Which of the following statements is not true? 1. It includes discussions on descriptive simulation modeling, programming commands, techniques for sensitivity estimation, optimization and goal-seeking by simulation, and what-if analysis. The calibration path emulates the ADC’s normal operation to establish realistic loading and transient effects. The system is three-fold: we create new candidate solutions by refining a previously used objective function, use Monte Carlo simulation to obtain statistics for those solutions such as The required number of simulation loops only depends on the amount of the scatter of the output parameters and the type of results that are expected from the analysis. 0000868 , D4016001. The algorithm is completely general and only requires the geometry modeling software to provide the intersection points of an arbitrary line with the surface being sampled. ” ASCE-ASME J. Monte Carlo Methods The Birth The Birth of Monte Carlo Methods I After the was digital computer was perfect for “statistical sampling” 1. Coit 1, Frank A. In addition to well- established academics in the field of scientific research, we also A Definition and General Procedure for Monte Carlo Simulation. The random variables or inputs are modelled on the basis of probability distributions such as normal, log normal, etc. BORTZ Berfer Graduate School of Science, Yeshiva University, New York, New York 10033 M. Thus, there is a need for simulation and evaluation of different solutions. 2. Monte Carlo Simulation is a mathematical technique developed by John Von Neumann and Stanislaw Ulam for Project Manhattan. The former approach performs a Monte Carlo simulation based on data from field operations. Dr. Mohamed R. of research has gone into inference in models of this sort, and is ongoing today. MC simulations are based on probability statistics and use random numbers. Each uncertain variable within a model is assigned a “best guess” estimate. In simulation, we have deterministic models and probabilistic models. The algorithm generates a large number of points and checks to see if the coordinates, x and y, of each point are inside the circle- x2+y2≤1. 10: Monte Carlo simulation study for a two-level continuous-time survival analysis using Cox regression with a random intercept and a frailty* user with the necessary information to understand the details of the Monte Carlo algorithm. important classical set of models in operations research. In many examples part of the expected utility integral can be solved analytically, leaving Monte Carlo simulation only for the remaining non-analytic integration. We demonstrate the algorithm using the Geant4 Monte Carlo simulation toolkit. Random Numbers and Monte Carlo Simulation. 5 3-4 -2 0 2 4 P*(x) b 0 0. Geant4 is a tool kit that uses Monte Carlo methodology to simulate the passage of particles through matter. expected value). 00077. jp> Nick Henderson <nick. a. We present the enumeration results below the simulation results for comparison. Building model and evaluating the output of the model; Make a statistical analysis of the model output. We focus on sensitivity analysis for stochastic activity networks when Monte Carlo simulation is employed. Clicking the Simulate button changes the seed and produces a new simulation run. It was named for the Monte Carlo casino, where Stanislaw Ulam’s uncle often gambled. In simulation techniques Monte Carlo simulation (MCS) method is used to evaluate system reliability. 3. Monte Carlo simulation method can quantitatively evaluate the impact of multiple risk factors more comprehensively and accurately [5]. An alternative and increasing popular approach to project evaluation is Monte Carlo simulation. , Boca Raton, Florida, 1988. Table 2 Monte Carlo theory, methods and examples I have a book in progress on Monte Carlo, quasi-Monte Carlo and Markov chain Monte Carlo. Learn how to perform Monte Carlo simulations in MATLAB and Simulink. Monte Carlo Simulations The Monte Carlo simulations, as a broad class of MC simulation. The integration algorithm based on support vector machine, finite element and Monte Carlo is used for the reliability analysis of the catenary components in the high-speed electrified railway, the mathematic model of reliability analysis for catenary suspension system is built [9, 10]. Some recent examples in the literature are the following studies. We have also found that the main factor that slows down the laser response is the “delay 15 Oct 2017 we are bringing the most important subject operations research classes exclusively. The term of Monte Carlo simulation is huge. As a result, many efficient algorithms have been recently developed, e. We have used it to investigate the time-d ependent operation of a terahertz QCL. Must have a user interface that supports creation of transparent, well-documented models. These algorithms sometimes require reduction operation at the end to gather the results into a manager Example: Monte Carlo simulation to approximate the area of a figure. Monte Carlo simulation is considered the most reliable method for modeling photon migration in heterogeneous media. 3 PHEV Model Example Input Before we de ne an entire Monte Carlo simulation infras-tructure that uses the discussed model implementation, we must rst verify that the PHEV model gives reasonable output. , roulette) involve repetitive events with known probabilities. The Change button controls aspects of the simulation. Randomness is a key requirement of Monte Carlo simulation. Details of the application of Excel software to Monte Carlo simulation are shown with an analysis example. Tackling such demanding simulation times is the ob-jective of modern non-equilibrium statistical mechanics techniques, many of which rely on a master equation type description that coarse-grains the time evolution to the relevant rare-event dynamics. Regardless of the type of problem and the objective of the study, the process by which the simulation is performed remains constant. Download: Quality & Engineering Skills Simulation began to be applied to management situations in the late 1950's to look at problems relating to queuing and stock control. Thirdly, do Monte Carlo simulation with computer and software. Classical Monte Carlo: samples are drawn from a probability distribution, often the classical Boltzmann distribution, to obtain thermodynamic properties or minimum-energy structures; Quantum Monte Carlo: random walks are used to compute quantum-mechanical energies and wave functions, often to solve electronic structure problems, using Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0. Section 3 of this paper steps through the Monte Carlo analyses of the DC accuracy of a cascode current mirror, the random variation of the delays of a pair of inverter strings under transient operation, the matching of the oscillation For users of Crystal Ball and @RISK, the non-profit provides macros that create the libraries for use in SIPmath models. Monte Carlo Method In Daily work [email protected] with Monte Carlo Simulation Monte Carlo method or Monte Carlo analysis: The Monte Carlo method, also called Monte Carlo analysis, is a means of statistical evaluation of mathematical function s Monte Carlo simulation was named after the city in Monaco (famous for its casino) where games of chance (e. This method is applied to risk quantitative analysis and decision making problems. The npv operator is sum of the optimal exercise decisions. The UKF faces the problem representing the state as a Gaussian ran- This paper describes two illustrative examples demonstrating the application of Monte Carlo techniques to gas process design. basically Monte carlo simulation was named after world war for example, one industry. The demand is determined from the cumulative probability values in table 1. Dienemann This research is sponsored by the United States Air Force under Project RAND--Con. And later, to demonstrate the delay/disturbance consequences of the risk, a simulation based approach (Monte Carlo Simulation) is used. Typically these trajectories are for electrons as part of a beam as found in Scanning Electron Microscopy (SEM), and the specimen under investigation can be anything (that fits inside the SEM''s chamber). An Example of a Discrete Event Simulation. The purpose of this work is to report on our implementation of a simple MapReduce method for performing fault-tolerant Monte Carlo computations in a massively The history of Monte Carlo methods is long, but their application to the solution of scientific problems begins with von Neumann, Ulam and Fermi who used a Monte Carlo method in nuclear reaction studies. 1061/AJRUA6. Fourth, the extended Monte Carlo simulation method enables us to solve problems with dependent and non-normally distributed model inputs. Videos and examples show how to apply statistical uncertainties to a model MATLAB is used for financial modeling, weather forecasting, operations Simulation informs price, rate, and economic forecasting; risk management; and stress testing. borhood of the old posterior probability, for example by using Monte Carlo sampling, and adopting particle lters for the state estimation [4]. Algorithmica Research AB is a company that develops software applications definition of a derivative is given, and examples of popular kinds of . Use MathJax to format The seed in F13 is used for Monte Carlo simulation. Such problems arise, for example, in the operations of industrial firms, financial institutions, health care organizations, transportation systems, energy and resources, and government. 2 Lab 1. RENO is a user friendly platform designed for building and running complex analyses for any probabilistic or deterministic scenario. Hence equations into an equivalent form interpretable as a succession of random operations. Generate many sets of possible inputs that follows the above properties via random sampling from a probability distribution over the domain 3. Learners will work in small groups and experience several short activities during the didactic portion of the course. disciplines : engineering, operations research and management science, statistics,. MONTE CARLO SIMULATION [ N Monte Carlo simulation is a statistical technique by which a quantity is calculated repeatedly, using randomly selected •what-if* scenarios for each calculation. While becoming familiar with the design and operation of the detectors, and how total antineutrino flux could be obtained from such a small sample, I read about a simulation program called Monte Carlo. ANALYSIS OF VEHICLE CLOSING SITUATIONS For the development of this analytic model two parameters-- relative velocity of two vehicles and distance between vehicles-- will be used. mc-set - Monte Carlo Simulation of Electron Trajectories. • In short, monte carlo technique is concerned with experiments on random numbers & it provides solutions to complicated OR problems. Wiley Series in Probability and Statistics, John Wiley & Sons, New York, 2011. Risk Uncertainty Eng. The organization of MCSs generally mirrors that of traditional research studies: a sample of data must first be gathered (or in simulation studies, generated by some probability density function), analyzed using one or more statistical methods and data operations, and summarized for dissemination. They are used to model phenomena with significant uncertainty in inputs, such as the calculation of risk in business. Keywords: Robustness, Chromatic Diagram, Buffer Index, Monte Carlo Simulation, Timetable 1 Introduction The idea of a methodology capable of determining in a precise and practical way the optimal sample size came from studying Monte Carlo simulation models concerning financial problems, risk analysis, and supply chain forecasting. KALOS Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 AND J. DATA SET 3. An Introduction to MCMC for Machine Learning CHRISTOPHE ANDRIEU C. The efficiency of the sampling Carlo simulation to dose unit (mSv/y) The Monte Carlo simulation model for the above conceptual descriptions was mathematically modeled and implemented using the GoldSim probability simulation platform [9][10]. Monte Carlo Simulation is a mathematical technique that generates random variables for modelling risk or uncertainty of a certain system. is to provide a comprehensive introduction to Monte Carlo methods, with a mix of theory, algorithms (pseudo + actual), and applications. In computing, at least for people writing programs using Monte Carlo methods, pseudorandom generators are generally the preferred choice. Define a domain of possible inputs and determine the statistical properties of these inputs 2. A Business Planning Example using Monte Carlo Simulation Based on your market research, you believe that there are equal chances that the market will be Pricing · Licensing · Academic & Research · FAQ In general terms, the Monte Carlo method (or Monte Carlo simulation) can be used to describe any For example, there are six different ways that the dice could sum to seven. The Monte Carlo method was developed by Nicholas Metropolis and Stanislaw Ulam in 1949 (Metropolis and Ulam, 1949). These techniques are robust with respect to model nonlinearities, but they are computationally expensive. This is not a low percentage but it is starting to become riskier. Sales and Price As specified above it will do this for 1000 time units (hours in this case). Michael McKinley is going to show us a real project from a chemical manufacturer where he used the Monte Carlo method to determine the optimal manufacturing Monte Carlo Simulation Fortunately, meaningful investigation of many problems in statistics can be solved through Monte Carlo methods. Monte Carlo simulations Monte Carlo (MC) methods are used in the simulation of a variety of phenomena in physics, finance, chemistry, etc. 5/31/2002 Monte Carlo simulation is fast becoming the technology of choice for evaluating and analyzing assets, be it pure financial derivatives or investments in real assets. May 23, 2020 · The Monte Carlo Simulation is a quantitative risk analysis technique which is used to understand the impact of risk and uncertainty in project management. The Monte Carlo results are verified with measurements from an LMR system used for landmine detection. Desired Requirements. Botev. The name derives from the famous Monte Carlo resort and is associated with roulette as a simple way of generating random The Monte Carlo simulation is equivalent to modeling photon transport analytically by solving the radiative transfer equation ( 20). DMC is used in applications including electronic structure calculations, rare event simulation, and data assimila-tion to e ciently approximate expectations of the type appearing in Feynman{Kac Research on Simulink/Fluent Collaborative Simulation Zooming of Marine Gas Turbine This is a one-day evidence-based overview of debriefing methods for health care simulation . This technique is used by professional in wide variety of fields as finance, project management, energy, manufacturing, engineering, research and development. T. Monte-Carlo simulation was used to model the activities of facilities such as warehouses and oil depots. Finally, it is shown that loop structure can be solved by Monte Carlo simulation method, which is realized by Excel software. The Mechanics of Monte Carlo Simulations. On Monte Carlo simulation is a rather down-market term (pardon my snobbery). H. Excel simulation is using statistical distribution to create a model that calculates the effects of inputs and decisions on various outcomes in what we refer to as the Monte Carlo Simulation. 00078 while the exact answer is 0. These random numbers are helpful in creating a new set of hypothetical data for a problem whose behaviour is known from past experience. It tells you two things: All of the possible events that could or will happen. It is a well- known and distinct methodology of operations research. Applications of Monte Carlo Method in Chemical, Biochemical and Environmental Engineering View all 8 Articles · Articles Michaelis-Menten Kinetics as an Example in the Context of Model- Based Design of Experiments. Section 2 presents the problem statement. Such problems arise, for example, in the operations of industrial firms, financial institutions, health care organizations, transportation systems, and government. Hydrologic Applications of Monte Carlo Simulation Monte Carlo simulation is a technique that uses a large number of random samples to find solutions to physical problems that cannot otherwise be easily solved. random variates in section5and analyzing output of Monte Carlo simulation in section6. Covariance method, the Historical Simulation and the Monte Carlo simulation [3]. An Example of Monte Carlo Simulation. 197 The code re-implements 2d Monte Carlo simulations originally developed in Fieremans, et al. The significant speedup that came after using different types of GPUs expands on the range of problems solvable by using probabilistic simulations [20]. Manage risk in your business decisions by using Monte Carlo Simulation Jan 01, 2017 · 2. The scenarios are simulated, and the results summarized statistically to create an so what I try to do is to simulate with Monte Carlo a American Option (Stock) and use TensorFlow to price it. A common situation occurring in everyday life is that of queueing or waiting in a line. Brief History Not a very old technique • World War II • “Monte Carlo” simulation: originated with the work on the atomic bomb. com> for a real physical simulation application: the Green’s function in Determinant Quantum Monte Carlo (DQMC) simulation [2], [14]. This site is about mc-set, a program that simulates electron trajectories in a specimen. Asai (SLAC) 17 The Monte Carlo simulation is equivalent to modeling photon transport analytically by solving the radiative transfer equation ( 20). The Monte Carlo technique is the same for both discrete and continuous random variables. It contains an OpenCL C++ kernel, to be mapped to FPGA via SDAccel. The simulation method uses the Monte Carlo technique to generate samples and uses descriptive The figure below shows the functions used by the example. 5 1 1. Monte Carlo Simulation. Fast GPU Monte Carlo Simulation for Radiotherapy, DNA Ionization and Beyond 2017 GPU Technology Conference Shogo Okada <shogo@port. Jan 01, 2014 · The work proceeds in a manner that, initially, we identify and access the uncertainties related to the subject of analysis, which helps in evaluating the GSC operational risks. Monte Carlo simulation involves modelling a deterministic system. 10. 9: Monte Carlo simulation study for a two-part (semicontinuous) growth model for a continuous outcome* 12. 8: Monte Carlo simulation study for discrete-time survival analysis* 12. To close these research gaps, a comprehensive simulation study using Monte Carlo methods (MCM) is applied. Different iterations or simulations are run for generating paths and the outcome is On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters Monte Carlo Methods and Applications, Vol. x emissions from operation k,g f = any Borel-measurable function G = group of inputs of interest g = emissions index of NO x,gNO x =kgfuel N = number of model evaluations in a Monte Carlo simulation N E; = normal distribution with mean, , and variance N o = total number of aircraft emissions module operations N s = number of flight segments in This project implements a Monte Carlo simulation of the Black-Scholes financial model, using both the European and the Asian options. 4 Exact simulation of the Ornstein–Uhlenbeck driven stochastic volatility model Quickly Build Operations Research Models -- Using Powerful Optimization and Monte Carlo Simulation Tools in Microsoft Excel or a Programming Language. In this example, it is assumed that there is one dock in the harbor at which ships unload cargo. Berlin, New York, Springer-Verlag, 1972 (OCoLC)586077439: Document Type: Book: All Authors / Contributors: Jürg Kohlas Introduction: Basic Steps of a Monte Carlo Method Monte-Carlo methods generally follow the following steps: 1. The paper set an example of traditional M/M/1 system and calculates its indicators of performance firstly. henderson@gmail. Secondly, define the parameters with combination of the observation data and Delphi results. After a brief aside reminiscing on Saul’s in°uence on the author’s career and on the simulation community, we review previous research for sensitivity analysis of Sep 18, 2012 · Monte Carlo Option Pricing with Multi-GPU support This sample evaluates fair call price for a given set of European options using the Monte Carlo approach, taking advantage of all CUDA-capable GPUs installed in the system. Monte Carlo (MC) simulation is an algorithm for predicting the interactions of X-ray photons with a complex medium, such as the human body 15. For this purpose, a simulation model is developed which represents the high-temperature heat supply of plastics processing machines. Estimating the probability of meeting a date at the movie theatre. The authors note that deterministic simulation, although popular, can be improved by incorporating elements that have a probabilistic nature that represent the variation and uncertainty in any system. The thesis research extends the previous ductor (CMOS) technology. It was first developed by Stanislaw Ulam while working on atom bomb to study nuclear cascades. At each time I Operations research is concerned with quantitative decision problems, generally involving the allocation and control of limited resources. The Variance-Covariance method originates a probability distribution of the hidden risky values through relative simple computing. 4 Nov 2015 Monte Carlo simulation is a computerised mathematical method that enables people to consider risks in quantitative analysis and decision making. Usage MonteCarlo(func, nrep, param_list, ncpus = 1, max_grid = 1000, This dissertation is about how Monte Carlo simulation can be used to analyse electricity markets. 4 Feb 2020 Monte Carlo simulations are algorithms used to measure risk and understand the impact of risk and uncertainty in various forecasting models, such as finances and project management. Summary. When the appropriate physical and Moreover, Monte Carlo methods provide an intuitive tool for simulating discrete systems and allow us to study finite size effects and spatial correlations. Download: Quality & Engineering Skills Williamso Williamson, n, Monte C Carlo arlo Simulation of Photon T Transport ransport Phenomene Phenomenena: na: Sampling Techniques, in Monte Carlo Simulation in the Radiological Sciences , edited by R. Forecasts generated by both the time Leading the Operations Research and Data Science team at Virgin Australia. Section 6 relates Monte Carlo simulation lets you see all the possible outcomes of your decisions manufacturing, engineering, research and development, insurance, oil & gas, Examples of variables described by normal distributions include inflation rates 3 Aug 2018 PDF | Monte Carlo (MC) approach to analysis was developed in the 1940's, A single sample cannot be used in simulation; to obtain results there must It is also used to solve optimization problems in Operations Research. k. random numbers and the relevant mathematical operations. , Jarrah, A. trapizoidal rule): The Monte Carlo method is also easier to get uncertainties from, and usually quicker to implement. An Example of a Stochastic Simulation. Application of Linear Programming, Optimisation, Monte Carlo + Discrete-Event Simulation, Machine Learning, and Regression in order to reduce opex and improve revenue, using software tools such as R, Python, Simio, SQL (PostGreSQL), PowerPivot, Power Query, and PowerBI. In this report, we create a new system for analyzing hospital OR schedules. Oct 19, 2016 · 1. The advantage of this method is simpleness. The Monte Carlo simulation of photon and neutron Monte Carlo simulation is considered the most reliable method for modeling photon migration in heterogeneous media. An example analysis. At the end of first day, the closing quantity is 5 (30-25) cakes. Join us for a free lunch and learn about an advanced analytic technique in Excel: the Monte Carlo Simulation. It is a numerical experimentation technique to obtain the statistics for the output variables of a system model given the statistics for the input variables. You need to estimate the first year net profit from this product, which will depend on: For users of Crystal Ball and @RISK, the non-profit provides macros that create the libraries for use in SIPmath models. Better algorithms exist and we will briefly talk about them in a moment. Using Monte Carlo simulations, correlation techniques and design of experiments (DoE), Sensitivity Analysis allows you to determine which parameters have the greatest impact on your model. It uses an intuitive flowchart modeling approach with Monte Carlo simulation to estimate or optimize the results for risk analysis, complex reliability modeling, maintenance planning, operational research, financial planning or other analysis objectives. Simulation began to be applied to management situations in the late 1950's to look at problems relating to queuing and stock control. Monte Carlo are computationally infeasible for estimating the high-dimensional integral in (2). A much better use for digital vs. Please see Figure 2 for more information. INTRODUCTION . Oct 13, 2008 · We describe and validate new tools and algorithms developed for a new version of the MCell simulation program (MCell3), which supports generalized Monte Carlo modeling of diffusion and chemical reaction in solution, on surfaces representing membranes, and combinations thereof. Besides classic examples (coin flips and count of heads/tails, rolls of a pair of dice) which other exam Apr 01, 2011 · Monte Carlo simulation tools play a key role in the design of detectors and radiation shield configurations. After a brief aside reminiscing on Saul’s in°uence on the author’s career and on the simulation community, we review previous research for sensitivity analysis of Monte Carlo Simulation “The world … is full of more complicated systems …. For stabilizing the matrix multiplication and inver-sion, the original algorithm uses the pivoted QR decomposi-tion to stratify matrix elements of different magnitude order [13], [12], [4]. No additional approximation is required. Due to the probabilistic modeling of the system's components, a Sequential Monte-Carlo simulation is used to manage reliability evaluation at the network level. jp> Koichi Murakami <koichi. An example of a deterministic model is a calculation to determine the return on a 5-year investment with an annual interest rate of 7%, compounded monthly. Phone: (212) 854-3556 13 Apr 2016 The Monte Carlo method studies random phenomena using numerous fictitious The use of the Monte Carlo method is illustrated on several examples. Handbook of Monte Carlo Methods. These notes present a highly condensed version of: D. The first systematic development of Monte Carlo methods derives from work on the atomic bomb during the second Third, the extended Monte Carlo simulation is a derivative-free method. If one forms the estimate where x iare suitably sampled from PDF f(x), one can expect Radiation Simulation and Monte Carlo Method -M. the complex interaction of many variables — or the inherently probabilistic nature of certain phenomena — rules out a definitive prediction. Morin, pages 53 – 101, CRC Press, Inc. Nov 05, 2017 · In this video you are going to learn how to solve Simulation problem using Monte Carlo method of simulation. If we assume that the output of each simulation is Two Monte Carlo codes have been employed to benchmark VERA-CS results for the AP1000 PWR simulations: KENO, which is part of the Oak Ridge National Laboratory SCALE package, and SHIFT, a Monte Carlo code developed by CASL for use on massively parallel computers. 15 Nov 2007 Examples of Monte Carlo methods include stochastic integration, where we use a simulation-based method to here matrix multiplication is replaced by a convolution operation but the intuition remains the same. 6,7 Kinetic Monte Carlo simulations8,9 fall within this category, and dis- For example, Monte Carlo rollouts 8 search to maximum depth without branching at all, by sampling long sequences of actions for both players from a policy p. There are some input variables, and the first step is to sample randomly on them. Monte Carlo algorithms which are implicitly based on this assumption have been utilized. · Support of continuous improvement initiatives. Manufactured in The Netherlands. SIMULATION. Importance Sampling and Monte Carlo Simulations Monte Carlo Simulation In the last section, we expressed the probability of drawing a number greater than 3 from the normal distribution as an expected value problem. One way to employ a Monte Carlo simulation is to model possible movements of In this research, the MCS approach is applied as depicted in Fig. lS4 This is an attractive pros- pect, since within the theory of Poisson processes, the rela- tionship between Monte Carlo time and real time can be clearly established. Stochastic Monte Carlo/Computer Simulation Methods for Drastic and Rare Scenario Analyses New Norm: Operation/Optimization under Social Distance Constraints Indoor GPS and Tracking by Sensor Network Localization for Contact-Tracing Dynamic and Equitable Region Partitioning for Hospital/Health-Care Services Jun 23, 2020 · Monte Carlo methods can help answer a wide range of questions in business, engineering, science, mathematics, and other fields. The purpose of this work is to report on our implementation of a simple MapReduce method for performing fault-tolerant Monte Carlo computations in a massively Using simulation to calculate the NPV of a project Marius Holtan Onward Inc. For more videos SUBSCRIBE our channel. Race fan and Bleacher Report columnist Long John Silver has set himself the ambitious goal of specifying the carbon footprint of an F1 car on a single race day. We begin by adapting an algorithm for modeling ships arriving at a harbor as described in [4]. 24, No. 2 Randomized Linear Program. 16 Dec 2015 VP Product Management Eric Kelso introduces users running Monto Carlos simulations to users of the River Logic prescriptive analytics platform. The way that Monte Carlo simulation selects variable values at random to simulate a model is similar to the casino's games of chance that have a known range of values but an uncertain value for any particular Systems Simulation: The Shortest Route to Applications. Monte Carlo simulations offer the capability to accurately estimate quantities of interest that are challenging to measure experimentally while taking into account the Monte Carlo simulation is named after the casino in Monte Carlo, Monaco, where games of chance exhibiting random behavior are played. Furthermore, the design approach ad-dresses integration challenges within the hybrid ADC system, such as kickback noise and common-mode variations. We illustrate the Monte Carlo simulation approach by designing and implementing in R programming procedures that help us gain a compu-tational insight into the following problems: Estimating probabilities by Monte Carlo simulations. di usion Monte Carlo (DMC) algorithm, is to suggest a new class of algorithms in-spired by DMC for problems in numerical linear algebra. Can we determine the value of π using a. In Monte Carlo simulation, a problem is solved by simulating the original data with random number generators. The results of such tests would assist scientists in evaluating the performance of the Intel parallel supercomputers. He is on a Monte Carlo simulation versus “what if” scenarios There are ways of using probabilities that are definitely not Monte Carlo simulations—for example, deterministic modeling using single-point estimates. The results below are an example using a variety of distributions. Finally, for our model, we devise a method so that the risk amount in a particular category can be simply obtained by performing the Monte Carlo simulation for the entire portfolio, measuring the ratio of the calculated risk amount to the uncovered balance of each loan, and sum-ming individual risks. Specify the procedure that produces a pseudo-sample which simulates the Monte Carlo simulation uses permutation of numbers to calculate all possible have evolved and improved, the method has been used in operations research, In our previous example, our Monte Carlo simulation relied on the best case, Example: Used to generate random numbers in sampling and Monte Carlo by the vertical strip method, European Journal of Operational Research, 142(3), 18 Mar 2020 In this interview, Hans Læssøe and I discuss Monte Carlo simulation, how of things to model, but in our interview, Hans used the example of sales. Monte Carlo spreadsheet add-ins for personal computers have been available since the mid-1980s, however, it has not been a killer app because it takes specialized training to generate the required random inputs. The number of simulations that are necessary in a Monte Carlo analysis to provide good results is usually about 50 to 200. Felder 2 Rutgers University, Piscataway, NJ 1 Department of Industrial & Systems Engineering 2 Edward J. Google Scholar Learn Monte Carlo simulation and Optimization. Jun 25, 2019 · The Monte Carlo simulation has numerous applications in finance and other fields. A method of performing a Monte Carlo analysis using a graphical processor unit, comprising the steps of: allocating individual data sets to respective pixel locations in a graphical processor unit memory for a Monte Carlo simulation; and calculating the outcome of the Monte Carlo simulation for each data set using stream processing in the graphical processor unit. Van Slyke. With $$N=10^6$$ we get the two correct significant digits from the Monte Carlo simulation, but the extra digit costs a factor of 10 in computing resources since the CPU time scales linearly with 2. Over the past decade, the engineering research community has realized the importance of advanced stochastic simulation meth-ods for reliability analysis. , the emission or removal factors, and activity data). Monte Carlo simulation is used in Proposing a novel scenario‐based O. several research work about reliability, failure mode and aging analysis have been extensively carried out. 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Comparison of the proposed method and other available deterministic ones. In this technique, sets of model inputs are sampled randomly from statistical distributions to define multiple simulation scenarios. Nov 05, 2017 · monte carlo simulation. The Simulation Process. European Journal of Operational Research 206, 73–85 Professor Karl Sigman's Lecture Notes on Monte Carlo Simulation Department of Industrial Engineering and Operations Research. The way that Monte Carlo simulation selects variable values at random to simulate a model is similar to the casino's games of chance that have a known range of values but an uncertain value for any particular Jun 06, 2020 · Monte-Carlo methods are effective, for example, for estimating the solution of multi-dimensional boundary value problems at a point. In this paper, a probabilistic Monte-Carlo framework is developed and applied to predict remaining useful life of a component. Although there were a number of isolated and undeveloped applications of Monte Carlo simulation principles at earlier dates, modern application of Monte Carlo methods date AC, and transient circuit simulation modes and is optimized for the Spectre simulator from Cadence Systems. There are a wide range of applications for simulation; for example, players in the electricity market can use simulation to decide whether or not an investme nt can be expected to be profitable, and authorities the distribution, Monte Carlo simulation utilizes the “S-shaped” cumulative probability curve. The Monte Carlo method, which is usually applied for this purpose, is known to the Monte Carlo method is given on the basis of two examples, the simulation of a Lecture Notes in Operations Research and Mathematical Systems, Springer 24 May 2019 This article is part of the Research Topic. So clear with real life example. This book is a hands-on, example-rich guide to modeling and simulating advanced communications systems. i. Monte Carlo is simply a great tool for testing/giving insight/integrating into almost all experimental situations. See details for instructions on the specification of the function. This Kanban Monte Carlo simulation uses throughput as the input variable. I'm interested in comments especially about errors or suggestions for references to include. The Monte Carlo approach involves the repeated simulation of samples within the probability density functions of the input data (e. Though the simulation process is internally complex, commercial computer software performs the calculations as a single operation, presenting results in 12. monte carlo simulation example in operation research
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2020-10-20 02:24:42
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https://math.stackexchange.com/questions/2872437/shortcut-to-finding-the-square-root-of-a-perfect-square/2872444
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# Shortcut to finding the square-root of a perfect-square?
I've been trying to speed up an algorithm where the most expensive operation is the square-root. However, I can often guarantee that the input value is a perfect-square. I'm curious to know if there are any algorithms that will find the square-root faster (constant time?) if it is known that the input is a perfect-square?
Thanks, Ryan
• How large are your squares? 48 binary digits? 64? 1000? And since this is very applied numerical math: what machine are we running on? Or is this configurable hardware and we can implement our own logic? Aug 4, 2018 at 23:56
• And: do you need to make a decision after every square root calculation, or can you calculate multiple square roots in parallel (e.g. data parallelism)? Aug 5, 2018 at 0:03
• If you can run on a cluster, you have data parallelism (prob'ly it's better to discard the pi cluster idea. It's very little bang for the initial Buck and little bang for the power bill Buck; embedded computers are not optimized for number crunching, much unlike modern desktop CPUs) Aug 5, 2018 at 0:34
• Practical idea, Depending on use case, you may be able to compute your square root earlier and memorize it for the bottle neck. Assuming you're doing some earlier computation to Ensure this number is a square. Memorization is also an option, but perhaps that's too much memory. To elaborate: get n^2, split thread. One end does your usual thing, the other computes the square root. Savings will depend on if you are doing more than just finding root n. Aug 5, 2018 at 0:44
• By the way, gmp comes with an integer sqrt function. And I don't you'll he much faster than that on large integers. Aug 5, 2018 at 2:00
The sum of the first $k$ odd numbers is $k^2$. Knowing this, you can you calculate the square root by summing successive odd numbers (starting from one)—once you reach the input value, return the number of summations you made.
For example, $16 = 1 + 3 + 5 + 7$; that's $4$ addends, so $\sqrt{16}=4$. This process will always work, since our input is guaranteed to be of the form $k^2$ with $k \in \mathbb N$.
I think this method would run in $O(\sqrt n)$.
• +1. I would edit your answer to make your use of $n$ consistent between the first and last sentences. (Perhaps prefer changing $n$ to $k$ in your first sentence to match the second paragraph.) Aug 4, 2018 at 22:30
• Good answer :) I actually tried this myself. Unfortunately, as n gets larger this approach takes a lot more time than more general approaches like Newton's Method. Aug 4, 2018 at 22:50
• However, I wonder if this is actually faster than the existing sqrt implementation in most compiled languages. Aug 4, 2018 at 22:50
• Newton's algorithm converges much faster (on average; worst case: your integer is the square of two large primes, then complexity approaches $\mathcal O(\sqrt n)$, I think. Aug 4, 2018 at 23:47
• Terrible. Remember that $n$ here is the number itself, which is of order $2^s$ where $s$ is number of bits used to represent the number; so $\sqrt n = \sqrt{2^s} = (\sqrt 2)^s$, which is exponential in the input size. Newton's algorithm is polynomial in $s$. Aug 5, 2018 at 3:39
With integers within sensible bounds compared to what your CPU can natively compute, it can be quite easy to restrict the range of numbers you have to binary search to find the square root of x.
(0. remove two-blocks of trailing 0 from your binary number. Each block you remove is one factor of 2 to be multiplied to the result of the following step. This can be done in constant time, if I'm not mistaken: Observe the structure of "Subtract 1 and XOR with the input" for numbers with $t$ trailing 0s. Then use the POPCNT (Hamming weight) instruction of most serious CPUs. After removing these 0s, i.e. dividing by $4^n$, you'll end up with an odd number; if you end up with an even number after removing an even number of 0s, your number is not a perfect square.)
1. Find $k=\lfloor\log_2 x\rfloor$, see https://graphics.stanford.edu/~seander/bithacks.html
2. $a=\frac k2$
3. Thus, $2^a$ becomes a lower limit for $\sqrt x$ and $2^{a+1}$ an upper. Both values can be found via bit-shifting 1.
4. From here, do a binary search¹.
I doubt you'd be much faster than converting to floating point and letting the FPU do it in hardware, giving you an approximate value, comvertable back to integer, from which you only need to search small ranges (namely, the lost precision) for the actual integer square root.
Note that in such problems as yours, algorithmic elegance often plays a minor role - it needs to be fast on actual hardware, so execution avoiding a lot of memory interaction is a good thing, and: with SIMD instructions, doing four to 16 operations of the same type take about as long as doing one; so if you just need to test a few integers for their square, modifying your algorithm to be able to try four in parallel is way more efficient than saving half of the operations necessary.
You have a technological problem, not so much a numerical.
¹ binary search assumes that you can do one squaring and one comparison at once; as hinted at before, you might very well be able to divide your interval into five search chunks by calculating four products at once and comparing four numbers at once using SIMD. This further hints that even if there should be no constant time algorithm (and I'm pretty sure there's none), you can be better than $\mathcal O(n^2·\log_2 x)$; compare Fürer's algorithm.
• Step 1 assumes the input is a machine-sized integer, which may not be a case. Aug 5, 2018 at 3:44
• Well if it is some kind of big integer: whatever big integer library you're using somehow has to keep track of the length in binary digits, anyway. Aug 5, 2018 at 8:37
• Aug 5, 2018 at 12:43
• Did you try to time it? The binary search requires a lot of multiplications for large numbers so my bet is that this is much slower than the built in square root function (this is what I find with a simple test in C++ using long long integers, but its always the caveat that there is a better implementation than what I did) Aug 5, 2018 at 14:23
• "built-in square root function": of what, operating on what data types? Aug 5, 2018 at 14:26
I think the only advantage gained by having a perfect square in analytic methods is that you know an iterative algorithm will actually terminate. So instead here is a number theoretic solution that'll work for numbers less than $2^{66}$.
Fact 1: If $p$ is a prime with $p \equiv 3 \mod 4$ and $x$ is a perfect square $\mod p$, then $$x \equiv \left(x^{(p+1)/4}\right)^2 \mod p,$$ i.e. you can compute the modular square root by exponentiating by $(p+1)/4$. (See https://crypto.stackexchange.com/a/20994/18112)
Fact 2: The numbers $m_{17}=2^{17}-1$, $m_{19}=2^{19}-1$, and $m_{31}=2^{31}-1$ are (Mersenne) primes whose product is greater than $2^{66}$.
Method: Let $S$ be the square whose root $t$ you'd like to find. Compute the following $$t_{17} \equiv S^{2^{15}} \mod m_{17}$$ $$t_{19} \equiv S^{2^{17}} \mod m_{19}$$ $$t_{31} \equiv S^{2^{29}} \mod m_{31}$$ Then the Chinese Remainder Theorem gives $$t \equiv \pm 31207 t_{17} m_{19} m_{31} \pm 298611 m_{17} t_{19} m_{31} \pm 413071270 m_{17} m_{19} t_{31} \mod m_{17}m_{19}m_{31}$$ Then check these 8 possibilities.
Remarks: I don't know how computationally efficient this is; it's more of a mathematical solution taking advantage of knowing that $S$ is a square. I would venture to guess it's about as "constant time" as you could get as the number of steps is essentially fixed, but that constant may be larger than the $\sqrt{n}$ of other methods for this range of $n$.
• (1) I've never heard of a practical algorithm computing square root that may not terminate. (2) (as I pointed out in a comment under the question) it's impossible to do this in constant time, as storing a number $n$ takes at least $O(\log n)$ bits. Aug 5, 2018 at 12:38
• (3) Note that for $n$ being in a constant range (in this case $[1 \dots 2^{66}]$, any (deterministic, correct) algorithm must run in constant asymptotic time complexity. (4) It's only necessary for $p$ to be larger than $2\sqrt n$ instead of $n$, the proof is left to the reader. ... and ... Aug 5, 2018 at 12:40
• @user202729 You're totally right about the 8 solution ambiguity. I made the correction in the exposition above. Thanks! Aug 5, 2018 at 19:35
• Activity comment: see this for theory and link to python shifting code. Jul 27, 2021 at 13:10
I think a binary search type algorithm would be quite efficient for large input values if we know the input is a perfect square.
Let $n$ be the input value.
1. Begin with two integers $a$ and $b$ such that $a^2 < n$ and $b^2 > n$. We could use $a=0$ and $b=n$.
2. Find the midpoint of $a$ and $b$ and round down to the nearest integer if necessary. Call this $m$.
3. Find $m^2$. If $m^2=n$ then $m$ is the square root. If $m^2>n$ then $m$ is too high, so we return to step 2 with $a$ and $m$ as our two integers. If $m^2<n$ then $m$ is too low, so we return to step 2 with $m$ and $b$ as our two integers. Repeat until the square root is found.
The squaring of $m$ may be what slows the algorithm down, however I believe that multiplication algorithms are implemented in processor hardware and therefore very efficient. In terms of the number of operations, I believe the binary search would run in logarithmic time and therefore be preferable to $O(\sqrt n)$ for large input values. However, I am no expert on algorithm efficiency...
• Thanks for your answer alcana :) This is a more general algorithm for the square root (would work for non-perfect squares as well with some minor modifications to the logic for determining when we are done). It doesn't take advantage of the special property of the input value: that it is a perfect-square. It is faster than the previous answer as the input size increases however :) Aug 4, 2018 at 23:35
• @RyanPierceWilliams - also note that on many current Intel processors, multiplies are optimized and only take 1 to 3 cycles, so for larger n, the binary search using multiply and compare should be faster. Aug 5, 2018 at 2:20
• Isn't this very similar to Newton's method, just less efficient because it doesn't use deltas to correct the prediction, just a blind binary split testing?
– ktb
Aug 5, 2018 at 2:35
• Yes, this is less efficient than Newton's method. Aug 5, 2018 at 3:55
You can get good results using a 2-adic version of Newton–Raphson. The algorithmic complexity will be no better than with the usual adaptation of Newton–Raphson to the domain of integers, but convergence is easier to establish and computation modulo a power of $$2$$ is especially well-suited to modern computers.
As an example of the sort of performance possible, in a couple of dozen lines of division-free, almost branch-free C code I can compute the 32-bit square root of any 64-bit perfect square in around 14.9 CPU clock cycles (around 3.3 ns at 4.5 GHz) on my Intel Core i7-8559U laptop. I've included the code below so that you can do timings for yourself.
I think this fits the "known perfect square" criterion described by the questioner, since for non-perfect squares it doesn't give a useful result (outside the presumably highly specialised application of actually wanting a 2-adic approximation to the square root of a non-square integer).
Details: It's enough to be able to compute square roots of odd perfect squares: handling zero is trivial, and for positive even perfect squares we can shift out trailing zero bits (of which there must be an even number) to get an odd perfect square, take the square root, and shift back appropriately.
So suppose that $$n$$ is an odd perfect square integer and define a function $$f$$ (on the real numbers for now) by $$f(x) = n - \frac 1 {(2x+1)^2},$$ valid for all real $$x$$ except $$x = -1/2$$. The roots of $$f$$ are $$((\pm 1/\sqrt n) - 1) / 2$$. Working through the algebra, the Newton–Raphson method applied to $$f$$ says that if we have a sufficiently good approximation $$x$$ to one of those roots then $$x - \left((x^2 + x)n + \frac{n-1}4\right) (2x + 1) \tag{1}\label{eq1}$$ should be a better approximation, and moreover that convergence of the repeated iteration should be quadratic once we get close enough.
Now here's the key point: the expression \eqref{eq1} can be evaluated using only integer arithmetic. (Note that since $$n$$ is an odd square, it must be congruent to $$1$$ modulo $$4$$, so $$(n - 1)/4$$ is an integer.) There are exactly two roots of $$f$$ in the 2-adic integers, and we can use \eqref{eq1} to compute successive $$2$$-adic approximations to the roots. Moreover, we continue to get the expected quadratic convergence to the roots; while the normal proof of this goes via calculus, for this particular $$f$$ we can see the quadratic convergence purely algebraically. Suppose that $$x_0$$ is a (rational) integer that satisfies $$(x_0^2 + x_0)n + \frac{n-1}4 \equiv 0 \pmod{2^j}$$ for some positive integer $$j$$. This is an integer-only reformulation of the statement that $$x_0$$ is congruent to $$((\pm 1/\sqrt n) - 1) / 2$$ modulo $$2^j$$ in the 2-adics, where now $$\sqrt n$$ represents one of the square roots of $$n$$ in the ring of 2-adic integers.
Let $$x_1$$ be the next approximation computed using the formula \eqref{eq1}: $$x_1 = x_0 - \left((x_0^2 + x_0)n + \frac{n-1}4\right) (2x_0 + 1).$$ Then simple but tedious algebra (expanding both sides) shows that $$(x_1^2 + x_1)n + \frac{n-1}4 = \left((x_0^2 + x_0)n + \frac{n-1}4\right)^2 ((2x_0+1)^2 n - 4)$$ from which we immediately have $$(x_1^2 + x_1)n + \frac{n-1}4 \equiv 0 \pmod{2^{2j}}$$ So we double the number of good bits in our approximation on every iteration. If $$n < 4^j$$ for some $$j$$ then it's enough to iterate until we have an integer $$x$$ satisfying $$(x^2 + x)n + \frac{n-1}4 \equiv 0 \pmod{2^j}$$ Then $$a = (2x+1)n$$ is an integer solution to the congruence $$a^2 \equiv n \pmod{2^{j+2}}.$$ Note that we have to be a bit careful: there are four solutions in total to this congruence, but only one of them lies in the interval $$(0, 2^j)$$ (after reduction modulo $$2^{j+2}$$), and that's the square root that we're after. Given any one solution, the others are easy to find via negation and via addition of $$2^{j+1}$$.
As a final speed trick, while we could start with a solution to the congruence modulo $$2^1$$ and work our way up from there, on most machines it will make more sense to use a small lookup table to enable us to start with a solution modulo $$2^8$$, say. That lookup table will typically be small enough to fit into a couple of level-1 cache lines on a modern processor.
Here's some C code that applies the above ideas to the special case of computing the 32-bit square root of a 64-bit perfect square integer. It's mostly portable, but it does make use of the GCC / Clang intrinsic function __builtin_ctzl for counting trailing zero bits in a nonzero integer.
#include <stdint.h>
static const uint8_t lut[128] = {
0, 85, 83, 102, 71, 2, 36, 126, 15, 37, 28, 22, 87, 50, 107, 46,
31, 10, 115, 57, 103, 98, 4, 33, 47, 58, 3, 118, 119, 109, 116, 113,
63, 106, 108, 38, 120, 61, 27, 62, 79, 101, 35, 41, 104, 13, 84, 17,
95, 53, 76, 121, 88, 34, 59, 97, 111, 5, 67, 54, 72, 82, 52, 78,
127, 42, 44, 25, 56, 125, 91, 1, 112, 90, 99, 105, 40, 77, 20, 81,
96, 117, 12, 70, 24, 29, 123, 94, 80, 69, 124, 9, 8, 18, 11, 14,
64, 21, 19, 89, 7, 66, 100, 65, 48, 26, 92, 86, 23, 114, 43, 110,
32, 74, 51, 6, 39, 93, 68, 30, 16, 122, 60, 73, 55, 45, 75, 49,
};
uint32_t isqrt64_exact(uint64_t n)
{
uint32_t m, k, x, b;
if (n == 0)
return 0;
int j = __builtin_ctzl(n);
n >>= j;
m = (uint32_t)n;
k = (uint32_t)(n >> 2);
x = lut[k >> 1 & 127];
x += (m * x * ~x - k) * (x - ~x);
x += (m * x * ~x - k) * (x - ~x);
b = m * x + 2 * k;
b ^= -(b >> 31);
return (b - ~b) << (j >> 1);
}
I have a fuller version of this code available, including exhaustive tests and explanations, as a GitHub gist at https://gist.github.com/mdickinson/e087001d213725a93eeb8d8f447a2f40
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2023-03-24 05:37:40
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https://www.math-only-math.com/column-matrix.html
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# Column Matrix
Here we will discuss about the column matrix with examples.
In an m × n matrix, if n = 1, the matrix is said to be a column matrix.
Definition of Column Matrix: If a matrix have only one column then it is called column matrix.
Examples of column matrix:
1. $$\begin{bmatrix} 4\\ 6\end{bmatrix}$$ is a column matrix.
The order of the above matrix is 2 × 1
2. $$\begin{bmatrix} 7\\ 5\\ 9\end{bmatrix}$$ is a column matrix.
The order of the above matrix is 3 × 1
3. $$\begin{bmatrix} 1\\ 2\\ 0\\ 5\end{bmatrix}$$ is a column matrix.
The order of the above matrix is 4 × 1
4. $$\begin{bmatrix} 40\\ 22\\ 19\\ 10\\ 41 \end{bmatrix}$$ is a column matrix.
The order of the above matrix is 5 × 1
5. $$\begin{bmatrix} 90 \end{bmatrix}$$ is a column matrix.
The order of the above matrix is 1 × 1.
6. $$\begin{bmatrix} 0 \end{bmatrix}$$ is a column matrix.
The order of the above matrix is 1 × 1.
7. $$\begin{bmatrix} 45\\ 21\\ 78\\ 12\\ 30\\ 49 \end{bmatrix}$$ is a column matrix.
The order of the above matrix is 6 × 1.
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2023-01-26 21:56:55
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http://library.kiwix.org/stats.stackexchange.com_eng_all_2018-08/A/tag/conditional-probability/1.html
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## Tag: conditional-probability
129 Amazon interview question—probability of 2nd interview 2014-02-10T01:45:18.763
80 Deriving the conditional distributions of a multivariate normal distribution 2012-06-16T18:09:15.390
67 Can someone explain Gibbs sampling in very simple words? 2011-05-01T19:37:56.640
45 How to generate correlated random numbers (given means, variances and degree of correlation)? 2012-10-07T19:45:21.457
32 What is the probability that this person is female? 2012-06-21T02:10:15.410
29 A generalization of the Law of Iterated Expectations 2014-05-01T13:17:30.693
28 What is the intuition behind the formula for conditional probability? 2018-02-01T13:44:30.760
22 Two dice rolls - same number in sequence 2017-04-17T12:18:49.277
20 The paradox of i.i.d. data (at least for me) 2015-12-13T05:50:35.487
17 How can I calculate the conditional probability of several events? 2010-08-11T22:14:39.753
17 Wikipedia entry on likelihood seems ambiguous 2016-07-15T23:44:40.050
15 Intuitive explanation of contribution to sum of two normally distributed random variables 2011-04-02T01:28:12.757
15 Problem with proof of Conditional expectation as best predictor 2013-10-04T00:24:13.043
15 Intuition for Conditional Expectation of $\sigma$-algebra 2016-08-18T17:45:02.107
14 How should I mentally deal with Borel's paradox? 2013-03-02T19:18:40.990
14 How to develop intuition for conditional probability? 2014-08-26T15:19:24.460
14 simulating random samples with a given MLE 2016-11-01T09:41:00.227
13 Bayes Theorem with multiple conditions 2015-12-27T07:23:44.603
12 Expected number I will be on after drawing cards until I get an ace, 2, 3, and so forth 2013-08-12T16:15:38.013
12 Definition of Conditional Probability with multiple conditions 2013-08-13T19:25:57.297
11 Confidence interval and probability - where is the error in this statement? 2012-07-11T04:18:27.963
11 How do programs like BUGS/JAGS automatically determine conditional distributions for Gibbs sampling? 2013-08-23T13:23:34.857
10 A serious in-depth problem of probabilities for flipping coins 2013-04-15T08:20:07.580
10 regarding conditional independence and its graphical representation 2013-10-24T21:32:11.820
10 Bayesian modeling using multivariate normal with covariate 2014-09-18T21:14:29.980
10 Conditional probability of continuous variable 2015-02-06T02:39:52.630
10 Sum of coefficients of multinomial distribution 2016-01-06T15:47:48.863
10 Why is P(A,B|C)/P(B|C) = P(A|B,C)? 2017-01-26T22:13:18.180
9 Why can't we trust our intuition with probability? 2012-05-05T04:21:48.110
9 Interpretation of conditional density plots 2013-02-08T04:52:03.717
9 Luce choice axiom, question about conditional probability 2013-05-10T16:05:11.607
9 Convergence in Distribution\CLT 2014-03-18T00:02:35.700
9 How to optimally spread draws when calculating multiple expectations 2016-08-28T23:59:59.117
9 Conditional probabilities - are they unique to Bayesianism? 2018-01-10T18:59:48.003
8 How to create a dataset with conditional probability? 2011-09-03T07:52:13.967
8 Derive P(C | A+B) from Cox's two rules 2011-10-10T17:09:42.433
8 Markov models with conditional transition probabilities 2012-03-09T18:00:29.987
8 Confidence intervals when using Bayes' theorem 2012-08-21T18:46:57.907
8 A parellel between LSA and pLSA 2012-10-19T07:05:25.310
8 Probability that 2 OH NFL teams go 31 weeks w/o wins on the same day 2012-11-27T08:54:29.553
8 Naive Bayes feature probabilities: should I double count words? 2013-04-26T12:50:28.593
8 Why is posterior density proportional to prior density times likelihood function? 2013-07-15T13:12:33.483
8 Locomotive problem with various size companies 2013-09-15T23:02:42.580
8 Why Normalizing Factor is Required in Bayes Theorem? 2014-12-18T21:38:35.803
8 Gibbs Sampler transition kernel 2015-02-06T13:41:30.327
8 Conditional Expected Value of Product of Normal and Log-Normal Distribution 2015-06-21T04:31:03.270
8 A possible mistake in a conditional probability derivation 2015-10-23T18:19:25.553
8 Computation of Conditional Expectation on $\sigma$-algebras 2016-08-22T22:07:51.050
8 How is $\Pr(X=x|Y=y)$ defined when $Y$ is continous and $X$ discrete? 2016-12-12T11:39:28.580
7 Flaw in a conditional probability argument 2013-11-06T15:07:02.440
7 Sampling from conditional copula 2014-11-20T18:36:59.737
7 Why the mixtures of conjugate priors is important? 2014-12-12T08:31:54.603
7 Resources for the "ah ha" moment when learning Bayes' theorem 2015-01-05T20:54:42.947
7 How to derive the conjugate prior of an exponential family distribution 2015-01-29T13:51:30.910
7 Bayes theorem in odds form - incorrect in Tetlock's 'Superforecasting' book? 2015-11-26T13:52:58.483
7 The Frog Riddle - Conditional Probabilities 2016-03-13T23:07:15.410
7 Paradox of Poisson process with at least one event in the interval 2017-02-08T15:17:36.643
7 Conditional expectation of a truncated RV derivation, gumbel distribution (logistic difference) 2017-02-09T00:53:55.237
7 Probability of k zeros give the sum of n Poisson random variables is t? 2017-10-04T20:06:35.183
6 Generating random matrices with specific equality constraints 2011-11-30T16:41:02.653
6 Odds of X occurrences in a row given Y trials (A coin flip problem) 2012-05-06T08:58:11.603
6 What is a full conditional probability? 2012-07-02T19:41:03.053
6 "Running it" multiple times in No-Limit Hold'em poker 2012-09-29T18:07:13.550
6 contrapositive of probability 2013-03-09T01:31:43.730
6 How Conditional Random Fields and Logistic Regression could be the same? 2013-07-09T19:20:18.020
6 Regarding the formula of using $\text{P}(Y|X)$ to compute $\text{E}[X]$ 2013-08-09T12:21:36.653
6 Conditional Expectation of Poisson Random Variables 2014-01-18T15:53:36.423
6 Total expectation theorem for Poisson processes 2014-05-18T07:09:20.587
6 Binomial random variable conditional on another one 2014-08-19T20:10:53.360
6 Gibbs Sampler contradiction proof 2015-02-05T23:43:25.420
6 Proof of Chapman Kolmogorov equation 2015-03-22T05:53:54.300
6 Gibbs Sampler output: how many Markov chains? 2015-03-27T17:02:57.917
6 How to prove Berkson's Fallacy? 2015-07-09T19:48:00.983
6 Expected value of x in a normal distribution, GIVEN that it is below a certain value 2015-08-08T16:05:17.323
6 Joint probability of a minimum and maximum score after $n$ dice rolls 2015-09-23T12:56:28.960
6 Relationship between event and random variable 2015-11-26T14:17:16.110
6 Train waiting time in probability 2015-12-24T14:43:59.720
6 The variety of problems with Cox's Theorem 2016-01-07T21:26:35.587
6 How to generate a more accurate distribution of sample of random variates? 2016-02-23T20:57:39.020
6 Mars attack (probability to destroy $n$ spaceships with $k \cdot n$ missiles) 2016-03-18T07:01:25.863
6 Can these asymptotic conditional expectations be bounded from above? 2016-05-04T22:41:00.563
6 Poisson random variable self-study question 2016-08-16T16:31:20.823
5 Bayes' Theorem - Probability Pants problem 2012-12-06T00:27:22.093
5 Is there any intuitive meaning to the quantity P(A|B)P(B|C)? 2013-07-22T19:40:23.520
5 Learning probability bad reasoning. Conditional and unconditional 2013-10-23T16:31:13.440
5 How to compute conditional expectations with respect to a sigma field? 2013-11-01T18:40:11.737
5 conditional sampling of bivariate normals 2014-01-02T15:18:39.263
5 Conditional Distribution of Poisson Variables, given $\sum X_i$ 2014-02-01T16:56:41.850
5 Conditional expectation of $X$ given $Z = X + Y$ 2014-08-25T02:44:59.490
5 Covariance of a compound distribution 2014-10-20T02:53:49.560
5 Conditional independence iff joint factorizes 2014-11-25T19:27:09.160
5 computing the posterior of two Gaussian probability distributions 2014-12-11T13:07:51.027
5 Conditional distribution for Exponential family 2015-02-05T17:42:06.447
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2019-02-18 05:45:39
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https://icml.cc/virtual/2021/session/12040
|
Deep Learning Theory 3
Moderator: Rong Ge
Abstract:
Chat is not available.
Wed 21 July 7:00 - 7:20 PDT
(Oral)
On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent
Shahar Azulay · Edward Moroshko · Mor Shpigel Nacson · Blake Woodworth · Nati Srebro · Amir Globerson · Daniel Soudry
Recent work has highlighted the role of initialization scale in determining the structure of the solutions that gradient methods converge to. In particular, it was shown that large initialization leads to the neural tangent kernel regime solution, whereas small initialization leads to so called rich regimes''. However, the initialization structure is richer than the overall scale alone and involves relative magnitudes of different weights and layers in the network. Here we show that these relative scales, which we refer to as initialization shape, play an important role in determining the learned model. We develop a novel technique for deriving the inductive bias of gradient-flow and use it to obtain closed-form implicit regularizers for multiple cases of interest.
Wed 21 July 7:20 - 7:25 PDT
(Spotlight)
A statistical perspective on distillation
Aditya Menon · Ankit Singh Rawat · Sashank Jakkam Reddi · Seungyeon Kim · Sanjiv Kumar
Knowledge distillation is a technique for improving a student'' model by replacing its one-hot training labels with a label distribution obtained from ateacher'' model. Despite its broad success, several basic questions --- e.g., Why does distillation help? Why do more accurate teachers not necessarily distill better? --- have received limited formal study. In this paper, we present a statistical perspective on distillation which provides an answer to these questions. Our core observation is that a Bayes teacher'' providing the true class-probabilities can lower the variance of the student objective, and thus improve performance. We then establish a bias-variance tradeoff that quantifies the value of teachers that approximate the Bayes class-probabilities. This provides a formal criterion as to what constitutes agood'' teacher, namely, the quality of its probability estimates. Finally, we illustrate how our statistical perspective facilitates novel applications of distillation to bipartite ranking and multiclass retrieval.
Wed 21 July 7:25 - 7:30 PDT
(Spotlight)
The Lipschitz Constant of Self-Attention
Hyunjik Kim · George Papamakarios · Andriy Mnih
Lipschitz constants of neural networks have been explored in various contexts in deep learning, such as provable adversarial robustness, estimating Wasserstein distance, stabilising training of GANs, and formulating invertible neural networks. Such works have focused on bounding the Lipschitz constant of fully connected or convolutional networks, composed of linear maps and pointwise non-linearities. In this paper, we investigate the Lipschitz constant of self-attention, a non-linear neural network module widely used in sequence modelling. We prove that the standard dot-product self-attention is not Lipschitz for unbounded input domain, and propose an alternative L2 self-attention that is Lipschitz. We derive an upper bound on the Lipschitz constant of L2 self-attention and provide empirical evidence for its asymptotic tightness. To demonstrate the practical relevance of our theoretical work, we formulate invertible self-attention and use it in a Transformer-based architecture for a character-level language modelling task.
Wed 21 July 7:30 - 7:35 PDT
(Spotlight)
Revealing the Structure of Deep Neural Networks via Convex Duality
Tolga Ergen · Mert Pilanci
We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weights for a norm regularized DNN training problem can be explicitly found as the extreme points of a convex set. For the special case of deep linear networks, we prove that each optimal weight matrix aligns with the previous layers via duality. More importantly, we apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds. As a corollary, we also prove that norm regularized deep ReLU networks yield spline interpolation for one-dimensional datasets which was previously known only for two-layer networks. Furthermore, we provide closed-form solutions for the optimal layer weights when data is rank-one or whitened. The same analysis also applies to architectures with batch normalization even for arbitrary data. Therefore, we obtain a complete explanation for a recent empirical observation termed Neural Collapse where class means collapse to the vertices of a simplex equiangular tight frame.
Wed 21 July 7:35 - 7:40 PDT
(Spotlight)
Representational aspects of depth and conditioning in normalizing flows
Frederic Koehler · Viraj Mehta · Andrej Risteski
Normalizing flows are among the most popular paradigms in generative modeling, especially for images, primarily because we can efficiently evaluate the likelihood of a data point. This is desirable both for evaluating the fit of a model, and for ease of training, as maximizing the likelihood can be done by gradient descent. However, training normalizing flows comes with difficulties as well: models which produce good samples typically need to be extremely deep -- which comes with accompanying vanishing/exploding gradient problems. A very related problem is that they are often poorly \emph{conditioned}: since they are parametrized as invertible maps from $\mathbb{R}^d \to \mathbb{R}^d$, and typical training data like images intuitively is lower-dimensional, the learned maps often have Jacobians that are close to being singular. In our paper, we tackle representational aspects around depth and conditioning of normalizing flows: both for general invertible architectures, and for a particular common architecture, affine couplings. We prove that $\Theta(1)$ affine coupling layers suffice to exactly represent a permutation or $1 \times 1$ convolution, as used in GLOW, showing that representationally the choice of partition is not a bottleneck for depth. We also show that shallow affine coupling networks are universal approximators in Wasserstein distance if ill-conditioning is allowed, and experimentally investigate related phenomena involving padding. Finally, we show a depth lower bound for general flow architectures with few neurons per layer and bounded Lipschitz constant.
Wed 21 July 7:40 - 7:45 PDT
(Spotlight)
Toward Understanding the Feature Learning Process of Self-supervised Contrastive Learning
Zixin Wen · Yuanzhi Li
We formally study how contrastive learning learns the feature representations for neural networks by investigating its feature learning process. We consider the case where our data are comprised of two types of features: the sparse features which we want to learn from, and the dense features we want to get rid of. Theoretically, we prove that contrastive learning using ReLU networks provably learns the desired features if proper augmentations are adopted. We present an underlying principle called feature decoupling to explain the effects of augmentations, where we theoretically characterize how augmentations can reduce the correlations of dense features between positive samples while keeping the correlations of sparse features intact, thereby forcing the neural networks to learn from the self-supervision of sparse features. Empirically, we verified that the feature decoupling principle matches the underlying mechanism of contrastive learning in practice.
Wed 21 July 7:45 - 7:50 PDT
(Spotlight)
The Hintons in your Neural Network: a Quantum Field Theory View of Deep Learning
Roberto Bondesan · Max Welling
In this work we develop a quantum field theory formalism for deep learning, where input signals are encoded in Gaussian states, a generalization of Gaussian processes which encode the agent's uncertainty about the input signal. We show how to represent linear and non-linear layers as unitary quantum gates, and interpret the fundamental excitations of the quantum model as particles, dubbed Hintons''. On top of opening a new perspective and techniques for studying neural networks, the quantum formulation is well suited for optical quantum computing, and provides quantum deformations of neural networks that can be run efficiently on those devices. Finally, we discuss a semi-classical limit of the quantum deformed models which is amenable to classical simulation.
Wed 21 July 7:50 - 7:55 PDT
(Q&A)
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2022-11-29 20:45:17
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http://jackterwilliger.com/attractor-networks/
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# Attractor Networks, (A bit of) Computational Neuroscience Part III
Brains are comprised of networks of neurons connected by synapses, and these networks have greater computational properties than the neurons and synapses themselves. In this post, I am going to talk about a class of neural networks which I think are fascinating: attractor networks. These are recurrent neural networks with attractor states; these states and the dynamics governing an attractor networks evolution between attractor states endow these networks with powerful computational properties. Some attractor networks are useful models of neural circuits. It would be helpful to have a little knowledge of neuroscience and dynamical systems — fortunately for you my previous posts cover those topics: for an introduction to dynamical systems, you can read Part I, for an introduction to synapses, you can read Part II. You can probably get by without it though.
Attractor networks give rise to many interesting computational properties, e.g. categorization, filtering noise, integration, memorization [6]. These networks can shed light on the following questions. How can networks act stably and persistently? If we assume different neurons hold different pieces of information, how do neurons integrate this information? Furthermore, how can networks retrieve memories?
Through the lens of dynamical systems theory, these networks are easier to understand and thus the questions above can be answered more easily. Dynamical systems theory provides us with two levels of description:
• network space: The full state of the neural network, which is quite large and unwieldy.
• attractor space: A reduced space of the full neural network. Only includes points on the attractors.
Let’s look at two examples of attractor networks. The first we will look at is the Hopfield network, an artificial neural network. The second we will look at is a spiking neural network from [3] (Wang 2002).
## Hopfield Network
Hopfield networks [2] (Hopfield 1982 ) are recurrent neural networks using binary neuron. Although not a spiking network model, its . It is a model of associative memory.
Through the lens of dynamical systems, learning is achieved by adjusting the network so that the to-be-learned patterns become attractors states, i.e. if the .
### Network
A Hopfield network is comprised of $N$ neurons $\vec{V}$ with thresholds $\theta$ (typically all identical and $=0$) and connections $W$. The topology of the network connections is simple: each neuron is connected to all other neurons except itself and all connections are symetric, or
\begin{align}w_{ij} &= 0 \text{ if } i=j \\w_{ij} &= w_{ji} \end{align}
The neurons $V$ in a Hopfield network have two states: on or off, i.e. $v_i \in \{-1, 1\}$.
Each state in a network is associated with an Energy. By updating the network, according to a rule below, the Energy decreases to a local minimum.
$$E = \frac{1}{2} \sum\limits_{i}\sum\limits_{j} w_{ij}V_{i}V_{j} – \sum\limits_{i}V_i\theta$$
#### Instructions:
1. Create a pattern by clicking on the cells of the grid.
2. Save your pattern and make it learnable by clicking .
3. To make the network learn your pattern(s), make sure the remember columns are checked and click .
4. Create a noisy pattern by clicking the randomize button.
5. To recover an incomplete pattern, click the .
#### Notes:
• If most of your cells are off in each of your patterns, the network may create a fixed point where all cells are off
• In this implementation, when settling, the network first updates asynchronously but on the final step updates synchronously. This is to (1) demonstrate asynchronous updating and (2) to save time, so don’t make any conclusions about the rate hopfield networks converge to a minima based on this.
Energy:
The main parts of the code is below. You can find the full hopfield/tensor code here.
/**
* Step toward a fixed point in the Hopfield Network
* by reducing the energy in the Hopfield Network
* @param {HopfieldNetwork} network
*/
function update(network) {
// V = 1 if WV > θ else -1
const W = network.W,
V = network.V,
θ = network.θ;
network.V = where(greaterEqual(dot(W, V),θ), 1, -1);
return network;
}
/**
* Learn to store the patterns.
* Set patterns as fixed points
* using a Hebbian Learning scheme
*
* @param {Tensor} (M x N), i.e. M patterns or length N
* @param {HopfieldNetwork} network
* @returns {Tensor} (N x N) W a weight matrix
*/
function learnHebbian(patterns, network) {
let [M, N] = patterns.shape;
let W = fillT([N, N], 0);
for (let i=0; i < M; i++) {
let pattern = patterns.get([i, ':']);
}
// normalize
network.W = W.div(M);
// set trace = 0
for (let i=0; i < N; i++) {
network.W.set([i,i], 0);
}
return network;
}
### Dynamics
The dynamics, i.e. the update rule, of a Hopfield network as follows. If the state of neuron $i$ at time $t$ is $V_i^t$, then at time $t+1$, the state of that neuron is:
$$V_i^{t+1} = sign(\sum\limits_{j}W_{ij} \cdotp V_i)$$
where $\cdotp$ is the dot product and the $sign$ operator is what you can guess:
$sign(x) =\begin{cases} 1 &\text{ if } x \ge \theta,\\ -1 &\text{otherwise} \end{cases}$
This should be pretty familiar if you have worked with artificial neural networks (for you computer scientists) or rate-networks (for you theoretical neuroscientists): its a thresholded linear combination of input activations and synaptic weights.
### Learning Rule
To store a pattern, $V$, in a Hopfield network, the pattern must become a fixed point attractor, i.e. $V = update(\langle V, W \rangle)$. By setting the values of the weight matrix according the Hebbian Learning rule, the patterns become minimum energy states. The Hebbian Learning rule is not the only learning rule (see: Storkey Learning Rule).
Now, let’s prove that the Hebbian Learning rule results in fixed points:
#### proof :
Let’s assume $V$ is a pattern stored in a Hopfield network such that $W_{ij} \propto V_iV_j$. We can now show $V$ is a fixed point by direct proof:
\begin{align} \vec{V} &= update(\langle \vec{V}, W \rangle) & \text{ definition of fixed point } \\ V_i &= sign(\sum\limits_j W_{ij}V_j) & \text{update rule} \\V_i &= sign(\sum\limits_j V_i V_j V_j) & \text{Hebbian learning rule, i.e. } W_{ij}=V_iV_j \\V_i &= sign(\sum\limits_j V_i * 1) & \text{no matter the value of } V_j \text{ it will be 1} \\V_i &= sign(N * V_i) & \text{summing} \\V_i &= V_i & \text{N*V_i} \\ \hfill\blacksquare \end{align}
## Spiking Attractors
Spiking attractor networks are fascinating: they suggest how brains are able to (1) act stably, (2) integrate information distributed across neurons, (3) recover memories etc. despite being composed of billions of seemingly cacophonous neurons. Lets consider a binary decision task in which a network receives two noisy signals: $A$ and $B$ and must choose which signal is stronger. Given the constraints of neural circuits, this is actually, quite a feat. If each neuron in each population receives a different Poisson spike train generated from its respective noisy signal, then each neuron within a population will possess different and potentially conflicting information about the state of the world. In order to perform the binary decision task, the network must be able to integrate information about both signals, which is distributed across many neurons and across time, and compare them. Additionally, in order for a selective population of neurons one selective population of neurons must drive the neural circuit and turn off competing coalitions of neurons.
## Slow Reverberations
Below is an (approximate) implementation of [3] (X-J Wang 2002). This recurrent spiking network has 4 populations of neurons:
• (#0-#399) a population of inhibitory inter-neurons.
• (#400-1519) a population of background excitatory neurons which help sustain activity.
• (#1520-#1999) 2 selective populations, i.e. each receives input corresponding to a stimulus.
Essentially, the two selective populations recurrently stimulate themselves and inhibit each other through the inter-neurons. That is, they compete against each-other to drive the neural circuit. The reverberations (or the ramping up) of recurrent NMDA allows the network to integrate (or accumulate) information over time and across neurons to make a decision.
Here the network level is the membrane potential of each of the 2000 neurons and the activation of the synapses between each pair of neurons. The attractor level is the firing rates of two populations — a reduction of several orders of magnitude.
The default input to the model below is such that the network should make random decisions. Both selective populations receive spike trains of $40 hz$.
#### Instructions:
1. Restart the simulation, by clicking .
2. Tweak the simulation, by changing the parameters in the code editor below (feel free to email me if you want to do this but can’t figure out my admittedly messy code).
3. To restart the simulation with new parameters, click .
#### Notes:
• If you don’t see the code box below, you’re on mobile. There is a video in the simulations place. Try it out on desktop!
• AInput and BInput specify the input strength to the selective populations. It’s a good place to start tweaking parameters.
• Alternatively, you could tweak the conductances between each population.
Above is a video of the simulation below, in case your computer runs this code slowly or if you are on mobile.
## 2 thoughts on “Attractor Networks, (A bit of) Computational Neuroscience Part III”
1. john says:
What do I need to run the code above?
1. Jack Terwilliger says:
Hey John,
Hopefully, just a browser with javascript enabled. Otherwise, feel free to email me and I’ll try to troubleshoot with you.
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2020-04-07 07:30:07
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https://cs.stackexchange.com/questions/72483/function-that-cannot-be-computed-by-a-boolean-circuit-of-size-2n-2n
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# Function that cannot be computed by a Boolean circuit of size $2^n/2n$
Show that, for sufficiently large $n$, there is a function $f\colon\{0,1\}^n \to \{0,1\}$ that cannot be computed by a Boolean circuit with fan-in $2$ with $\frac{2^n}{2n}$ gates. Please give me a hint.
• What did you try? Where did you get stuck? We're happy to help you understand the concepts but just solving exercises for you is unlikely to achieve that. You might find this page helpful in improving your question. – D.W. Apr 4 '17 at 17:33
Hint: Estimate from above the number of circuits with $2^n/2n$ gates, and compare it to the number of functions.
We need counting argument here. First thing is, there are $2^{2^{n}}$ many boolean functions, second thing is to count the number of boolean circuits for $k$ size boolean circuit. There are $2^{k^2} \times 3^{k}$ many boolean circuits of size $k$ (use adjacency matrix ). In your case value of $k =\frac{ 2^n}{2n}$, now you can easily check that
$$2^{2^{n}} > 2^{k^2} \times 3^{k}$$
So it means there are more number of functions than the total number of boolean circuits. There has to be at least one function that can not be computed by $k$ size circuit ( pigeonhole principle ). One thing to note here that I have proved the existence of such type of boolean circuit but to come up with such type of boolean circuit is quite hard.
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2019-11-13 00:03:52
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https://www.physicsforums.com/threads/conceptual-question-about-currents-and-pumps.161362/
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# Conceptual question about currents and pumps
## Homework Equations
I suppose the energy density equation applies here:
(P2-P1) + pg(y2-y1) + 1/2p(v2^2 - v1^2) = Epump/vol - IR
and
I = Av
## The Attempt at a Solution
My initial thought was that the current is the same. But then, since current is Av, I was second guessing myself that maybe the velocities are different. Any help's appreciated!
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2022-06-30 11:33:24
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https://yutsumura.com/yu-tsumura-2/
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# Yu-Tsumura-small
• Generators of the Augmentation Ideal in a Group Ring Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by $\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,$ where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring […]
• Group Homomorphism, Conjugate, Center, and Abelian group Let $G$ be a group. We fix an element $x$ of $G$ and define a map $\Psi_x: G\to G$ by mapping $g\in G$ to $xgx^{-1} \in G$. Then prove the followings. (a) The map $\Psi_x$ is a group homomorphism. (b) The map $\Psi_x=\id$ if and only if $x\in Z(G)$, where $Z(G)$ is the […]
• Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$ Let $S$ be the following subset of the 3-dimensional vector space $\R^3$. $S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\},$ where $\Z$ is the set of all integers. […]
• The Subspace of Linear Combinations whose Sums of Coefficients are zero Let $V$ be a vector space over a scalar field $K$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […] • Solving a System of Differential Equation by Finding Eigenvalues and Eigenvectors Consider the system of differential equations \begin{align*} \frac{\mathrm{d} x_1(t)}{\mathrm{d}t} & = 2 x_1(t) -x_2(t) -x_3(t)\\ \frac{\mathrm{d}x_2(t)}{\mathrm{d}t} & = -x_1(t)+2x_2(t) -x_3(t)\\ \frac{\mathrm{d}x_3(t)}{\mathrm{d}t} & = -x_1(t) -x_2(t) […] • If the Order of a Group is Even, then the Number of Elements of Order 2 is Odd Prove that if G is a finite group of even order, then the number of elements of G of order 2 is odd. Proof. First observe that for g\in G, \[g^2=e \iff g=g^{-1}, where $e$ is the identity element of $G$. Thus, the identity element $e$ and the […]
• The Quadratic Integer Ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD) Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD). Proof. Any element of the ring $\Z[\sqrt{-5}]$ is of the form $a+b\sqrt{-5}$ for some integers $a, b$. The associated (field) norm $N$ is given […]
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2020-10-22 20:31:13
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|
https://qucontrol.github.io/krotov/v1.0.0/10_howto.html
|
# How-Tos¶
## How to optimize towards a quantum gate¶
To optimize towards a quantum gate $$\Op{O}$$ in a closed quantum system, set one Objective for each state in the logical basis, with the basis state $$\ket{\phi_k}$$ as the initial_state and $$\Op{O} \ket{\phi_k}$$ as the target.
You may use krotov.gate_objectives() to construct the appropriate list of objectives. See the Optimization of an X-Gate for a Transmon Qubit for an example. For more advanced gate optimizations, also see How to optimize towards a two-qubit gate up to single-qubit corrections, How to optimize towards an arbitrary perfect entangler, How to optimize in a dissipative system, and How to optimize for robust pulses.
## How to optimize complex-valued control fields¶
This implementation of Krotov’s method requires real-valued control fields. You must rewrite your Hamiltonian to contain the real part and the imaginary part of the field as two independent controls. This is always possible. For example, for a driven harmonic oscillator in the rotating wave approximation, the interaction Hamiltonian is given by
$\Op{H}_\text{int} = \epsilon^*(t) \Op{a} + \epsilon(t) \Op{a}^\dagger = \epsilon_{\text{re}}(t) (\Op{a} + \Op{a}^\dagger) + \epsilon_{\text{im}}(t) (i \Op{a}^\dagger - i \Op{a})\,,$
where $$\epsilon_{\text{re}}(t)= \Re[\epsilon(t)]$$ and $$\epsilon_{\text{im}}(t) = \Im[\epsilon(t)]$$ are considered as two independent (real-valued) controls.
See the Optimization of a State-to-State Transfer in a Lambda System in the RWA for an example.
## How to use args in time-dependent control fields¶
QuTiP requires that the functions that are used to express time-dependencies have the signature func(t, args) where t is a scalar value for the time and args is a dict containing values for static parameters, see QuTiP’s documentation on using the args variable. Most of the krotov package’s Examples use closures or hardcoded values instead of args. For example, in the Optimization of a State-to-State Transfer in a Two-Level-System, the Hamiltonian is defined as:
def hamiltonian(omega=1.0, ampl0=0.2):
"""Two-level-system Hamiltonian
Args:
omega (float): energy separation of the qubit levels
ampl0 (float): constant amplitude of the driving field
"""
H0 = -0.5 * omega * qutip.operators.sigmaz()
H1 = qutip.operators.sigmax()
def guess_control(t, args):
return ampl0 * krotov.shapes.flattop(
t, t_start=0, t_stop=5, t_rise=0.3, func="blackman"
)
return [H0, [H1, guess_control]]
Note how ampl0 is used in guess_control as a closure from the surrounding hamiltonian scope, t_stop and t_rise are hardcoded, and args is not used at all. The function could be rewritten as:
def guess_control(t, args):
"""Initial control amplitude.
Args:
t (float): Time value at which to evaluate the control.
args (dict): Dictionary containing the value "ampl0" with the
amplitude of the driving field, "t_stop" with the time at which the
control shape ends, and "t_rise" for the duration of the
switch-on/switch-off time.
"""
return args['ampl0'] * krotov.shapes.flattop(
t,
t_start=0,
t_stop=args['t_stop'],
t_rise=args['t_rise'],
func="blackman"
)
def hamiltonian(omega=1.0):
"""Two-level-system Hamiltonian
Args:
omega (float): energy separation of the qubit levels
"""
H0 = -0.5 * omega * qutip.operators.sigmaz()
H1 = qutip.operators.sigmax()
return [H0, [H1, guess_control]]
ARGS = dict(ampl0=0.2, t_stop=5, t_rise=0.3)
The ARGS must be passed to optimize_pulses() via the pulse_options parameter:
pulse_options = {
guess_control: dict(lambda_a=5, update_shape=S, args=ARGS)
}
Both Objective.mesolve() and Objective.propagate() take an optional args dict also.
The args in pulse_options are used automatically when evaluating the respective initial guess. Note that the use of args does not extend to update_shape, which is always a function of t only. Any other parameters in the update_shape are best set via functools.partial(), see the Optimization of a Dissipative State-to-State Transfer in a Lambda System.
Compare that example to the Optimization of a State-to-State Transfer in a Lambda System in the RWA. In the latter, the values for the parameters in the control fields and the Hamiltonian are hardcoded, while in the former, all parameters are centrally defined in a dict which is passed to the optimization and propagation routines.
## How to stop the optimization when the error crosses some threshold¶
By default, an optimization stops after a predefined number of iterations (iter_stop parameter in optimize_pulses()). However, through the interplay of the info_hook and the check_convergence routine passed to optimize_pulses(), the optimization can be stopped based on the optimization success or the rate of convergence: The info_hook routine should return the value of the optimization functional or error, which is accessible to check_convergence via the Result.info_vals attribute, see krotov.convergence for details.
Generally, you should use the krotov.info_hooks.print_table() function as an info_hook, which receives a function to evaluate the optimization functional $$J_T$$ as a parameter. Then, use krotov.convergence.value_below() as a check_convergence routine to stop the optimization when $$J_T$$ falls below some given threshold.
See the thee Optimization of a State-to-State Transfer in a Lambda System in the RWA for an example.
## How to exclude a control from the optimization¶
In order to force the optimization to leave any particular control field unchanged, set its update shape to krotov.shapes.zero_shape() in the pulse_options that you pass to optimize_pulses().
## How to define a new optimization functional¶
In order to define a new optimization functional $$J_T$$:
• Decide on what should go in Objective.target to best describe the physical control target. If the control target is reached when the Objective.initial_state evolves to a specific target state under the optimal control fields, that target state should be included in target.
• Define a function chi_constructor that calculates the boundary condition for the backward-propagation in Krotov’s method,
$\ket{\chi_k(T)} \equiv - \left. \frac{\partial J_T}{\partial \bra{\phi_k(T)}} \right\vert_{\ket{\phi_k(T)}}\,,$
or the equivalent experession in Liouville space. This function should calculate the states $$\ket{\chi_k}$$ based on the forward-propagated states $$\ket{\phi_k(T)}$$ and the list of objectives. For convenience, when target contains a target state, chi_constructor will also receive tau_vals containing the overlaps $$\tau_k = \Braket{\phi_k^{\tgt}}{\phi_k(T)}$$. See chis_re() for an example.
• Optionally, define a function that can be used as an info_hook in optimize_pulses() which returns the value $$J_T$$. This is not required to run an optimization since the functional is entirely implicit in chi_constructor. However, calculating the value of the functional is useful for convergence analysis (check_convergence in optimize_pulses())
See krotov.functionals for some standard functionals. An example for a more advanced functional is the Optimization towards a Perfect Entangler.
## How to penalize population in a forbidden subspace¶
In principle, optimize_pulses() has a state_dependent_constraint. However, this has some caveats. Most notably, it results in an inhomogeneous equation of motion, which is currently not implemented.
The recommended “workaround” is to place artificially high dissipation on the levels in the forbidden subspace. A non-Hermitian Hamiltonian is usually a good way to realize this. See the Optimization of a Dissipative State-to-State Transfer in a Lambda System for an example.
## How to optimize towards a two-qubit gate up to single-qubit corrections¶
On many quantum computing platforms, applying arbitrary single-qubit gates is easy compared to entangling two-qubit gates. A specific entangling gate like CNOT is combined with single-qubit gates to form a universal set of gates. For a given physical system, it can be hard to know a-priori which entangling gates are easy or even possible to realize. For example, trapped neutral atoms only allow for the realization of diagonal two-qubit gates [54][27] like CPHASE. However, the CPHASE gate is “locally equivalent” to CNOT: only additional single-qubit operations are required to obtain one from the other. A “local-invariants functional” [55] defines an optimization with respect to a such a local equivalence class, and thus is free to find the specific realization of a two-qubit gate that is easiest to realize.
Use krotov.objectives.gate_objectives() with local_invariants=True in order to construct a list of objectives suitable for an optimization using the local-invariant functional [55]. This optimizes towards a point in the Weyl chamber.
The weylchamber package contains the suitable chi_constructor routines to pass to optimize_pulses().
The optimization towards a local equivalence class may require use of the second-order update equation, see Second order update.
## How to optimize towards an arbitrary perfect entangler¶
The relevant property of a gate is often its entangling power, and the requirement for a two-qubit gate in a universal set of gates is that it is a “perfect entangler”. A perfect entangler can produce a maximally entangled state from a separable input state. Since 85% of all two-qubit gates are perfect entanglers [56][57], a functional that targets an arbitrary perfect entangler [28][29] solves the control problem with the least constraints.
The optimization towards an arbitrary perfect entangler is closely related to an optimization towards a point in the Weyl chamber (How to optimize towards a two-qubit gate up to single-qubit corrections): It turns out that in the geometric representation of the Weyl chamber, all the perfect entanglers lie within a polyhedron, and we can simply minimize the geometric distance to the surface of this polyhedron.
Use krotov.objectives.gate_objectives() with gate='PE' in order to construct a list of objectives suitable for an optimization using the perfect entanglers functional [28][29]. This is illustrated in the Optimization towards a Perfect Entangler.
Again, the chi_constructor is available in the weylchamber package.
Both the optimization towards a local equivalence class and an arbitrary perfect entangler may require use of the second-order update equation, see Second order update.
## How to optimize in a dissipative system¶
To optimize a dissipative system, it is sufficient to set an Objective with a density matrix for the initial_state and target, and a Liouvillian in Objective.H. See the Optimization of Dissipative Qubit Reset for an example.
Instead of a Liouvillian, it is also possible to set Objective.H to the system Hamiltonian, and Objective.c_ops to the appropriate Lindblad operators. However, it is generally much more efficient to use krotov.objectives.liouvillian() to convert a time-dependent Hamiltonian and a list of Lindblad operators into a time-dependent Liouvillian. In either case, the propagate routine passed to optimize_pulses() must be aware of and compatible with the convention for the objectives.
Specifically for gate optimization, the routine gate_objectives() can be used to automatically set appropriate objectives for an optimization in Liouville space. The parameter liouville_states_set indicates that the system dynamics are in Liouville space and sets an appropriate choice of matrices that track the optimization according to Ref. [27]. See the Optimization of a Dissipative Quantum Gate for an example.
For weak dissipation, it may also be possible to avoid the use of density matrices altogether, and to instead use a non-Hermitian Hamiltonian. For example, you may use the effective Hamiltonian from the MCWF method [58],
$\Op{H}_{\text{eff}} = \Op{H} - \frac{i}{2} \sum_k \Op{L}_k^\dagger \Op{L}_k\,,$
for the Hermitian Hamiltonian $$\Op{H}$$ and the Lindblad operators $$\Op{L}_k$$. Propagating $$\Op{H}_{\text{eff}}$$ (without quantum jumps) will lead to a decay in the norm of the state corresponding to how much dissipation the state is subjected to. Numerically, this will usually increase the value of the optimization functional (that is, the error). Thus the optimization can be pushed towards avoiding decoherence, without explicitly performing the optimization in Liouville space. See the Optimization of a Dissipative State-to-State Transfer in a Lambda System for an example.
## How to optimize for robust pulses¶
Control fields can be made robust with respect to variations in the system by performing an “ensemble optimization” [26]. The idea is to sample a representative selection of possible system Hamiltonians, and to optimize over an average of the entire ensemble. In the functional, Eq. (1), respectively the update Eq. (12), the index $$k$$ now numbers not only the states, but also different ensemble Hamiltonians: $$\Op{H}(\{\epsilon_l(t)\}) \rightarrow \{\Op{H}_k(\{\epsilon_l(t)\})\}$$.
The example considered in Ref. [26] is that of a CPHASE two-qubit gate on trapped Rydberg atoms. Two classical fluctuations contribute significantly to the gate error: deviations in the pulse amplitude ($$\Omega = 1$$ ideally), and fluctuations in the energy of the Rydberg level ($$\Delta_{\text{ryd}} = 0$$ ideally). Starting from a set of objectives for the unperturbed system, see How to optimize towards a quantum gate, ensemble_objectives() creates an extended set of objectives that duplicates the original objectives once for each Hamiltonian from a set perturbed Hamiltonian $$\Op{H}(\Omega \neq 1, \Delta_{\text{ryd}} \neq 0)$$. As shown in Ref. [27], an optimization over the average of all these objectives results in controls that are robust over a wide range of system perturbations.
A simpler example of an ensemble optimization is Ensemble Optimization for Robust Pulses, which considers a state-to-state transition in a Lamba-System with a dissipative intermediary state.
## How to apply spectral constraints¶
In principle, Krotov’s method can include spectral constraints while maintaining the guarantee for monotonic convergence [59] . However, the calculation of the pulse update with such spectral constraints requires solving a Fredholm equation of the second kind, which has not yet been implemented numerically. Thus, the krotov package does not support this approach (and no such support is planned).
A “cheap” alternative that usually yields good results is to apply a spectral filter to the optimized pulses after each iteration. The optimize_pulses() function allows this via the modify_params_after_iter argument.
For example, the following function restricts the spectrum of each pulse to a given range:
def apply_spectral_filter(tlist, w0, w1):
"""Spectral filter for real-valued pulses.
The resulting filter function performs a Fast-Fourier-Transform (FFT) of
each optimized pulse, and sets spectral components for angular
frequencies below w0 or above w1 to zero. The filtered pulse is then
the result of the inverse FFT, and multiplying again with the update
shape for the pulse, to ensure that the filtered pulse still fulfills
the required boundary conditions.
Args:
tlist (numpy.ndarray): Array of time grid values. All pulses must be
defined on the intervals of this time grid
w0 (float): The lowest allowed (angular) frequency
w1 (float): The highest allowed (angular) frequency
Returns:
callable: A function that can be passed to
modify_params_after_iter to apply the spectral filter.
"""
dt = tlist[1] - tlist[0] # assume equi-distant time grid
n = len(tlist) - 1 # = len(pulse)
# remember that pulses are defined on intervals of tlist
w = np.abs(np.fft.fftfreq(n, d=dt / (2.0 * np.pi)))
# the normalization factor 2π means that w0 and w1 are angular
# frequencies, corresponding directly to energies in the Hamiltonian
# (ħ = 1).
flt = (w0 <= w) * (w <= w1)
# flt is the (boolean) filter array, equivalent to an array of values 0
# and 1
def _filter(**kwargs):
# same interface as an info_hook function
pulses = kwargs['optimized_pulses']
shape_arrays = kwargs['shape_arrays']
for (pulse, shape) in zip(pulses, shape_arrays):
spectrum = np.fft.fft(pulse)
# apply the filter by element-wise multiplication
spectrum[:] *= flt[:]
# after the inverse fft, we should also multiply with the
# update shape function. Otherwise, there is no guarantee that
# the filtered pulse will be zero at t=0 and t=T (assuming that
# is what the update shape is supposed to enforce). Also, it is
# important that we overwrite pulse in-place (pulse[:] = ...)
pulse[:] = np.fft.ifft(spectrum).real * shape
return _filter
This function is passed to optimize_pulses() as e.g.
modify_params_after_iter=apply_spectral_filter(tlist, 0, 7)
to constrain the spectrum of the pulse to angular frequencies $$\omega \in [0, 7]$$. You may want to explore how such a filter behaves in the example of the Optimization of an X-Gate for a Transmon Qubit.
Modifying the optimized pulses “manually” through a modify_params_after_iter function means that we lose all guarantees of monotonic convergence. If the optimization with a spectral filter does not converge, you should increase the value of $$\lambda_a$$ in the pulse_options that are passed to optimize_pulses(). A larger value of $$\lambda_a$$ results in smaller updates in each iteration. This should also translate into the filter pulses being closer to the unfiltered pulses, increasing the probability that the changes due to the filter do not undo the monotonic convergence. You may also find that the optimization fails if the control problem physically cannot be solved with controls in the desired spectral range. Without a good physical intuition, trial and error may be required.
## How to limit the amplitude of the controls¶
Amplitude constraints on the control can be realized indirectly through parametrization [60]. For example, consider the physical Hamiltonian $$\Op{H} = \Op{H}_0 + \epsilon(t) \Op{H}_1$$.
There are several possible parametrizations of $$\epsilon(t)$$ in terms of an unconstrained function $$u(t)$$:
• For $$\epsilon(t) \ge 0$$:
$\epsilon(t) = u^2(t)$
• For $$0 \le \epsilon(t) < \epsilon_{\max}$$:
$\epsilon(t) = \epsilon_{\max} \tanh^2\left(u(t)\right)$
• For $$\epsilon_{\min} < \epsilon(t) < \epsilon_{\max}$$:
$\epsilon(t) = \frac{\epsilon_{\max} - \epsilon_{\min}}{2} \tanh\left(u(t)\right) + \frac{\epsilon_{\max} + \epsilon_{\min}}{2}$
Krotov’s method can now calculate the update $$\Delta u(t)$$ in each iteration, and then $$\Delta \epsilon(t)$$ via the above equations.
There is a caveat: In the update equation (12), we now have the term
$\begin{split}\Bigg( \left.\frac{\partial \Op{H}}{\partial u}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\u^{(i+1)}(t)\end{matrix}}} \Bigg) = \Bigg( \left.\frac{\partial \epsilon}{\partial u}\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\u^{(i+1)}(t)\end{matrix}}} \Bigg)\end{split}$
on the right hand side. As the dependendence of $$\epsilon(t)$$ on $$u(t)$$ is non-linear, we are left with a dependency on the unknown updated parametrization $$u^{(i+1)}(t)$$. We resolve this by approximating $$u^{(i+1)}(t) \approx u^{(i)}(t)$$, or equivalently $$\Delta u(t) \ll u(t)$$, which can be enforced by choosing a sufficiently large value of $$\lambda_a$$ in the pulse_options that are passed to optimize_pulses().
Currently, the krotov package does not yet support parametrizations in the above form, although this is a planned feature. In the meantime, you could modify the control to fit within the desired amplitude constaints in the same way as applying spectral constaints, see How to apply spectral constraints.
## How to parallelize the optimization¶
Krotov’s method is inherently parallel across different objectives. See krotov.parallelization, and the Optimization of an X-Gate for a Transmon Qubit for an example.
## How to prevent losing an optimization result¶
Optimizations usually take several hundred to several thousand iterations to fully converge. Thus, the optimize_pulses() routine may require significant runtime (often multiple days for large problems). Once an optimization has completed, you are strongly encouraged to store the result to disk, using Result.dump(). You may also consider using dump_result() during the check_convergence step to dump the current state of the optimization to disk at regular intervals. This protects you from losing work if the optimization is interrupted in any way, like an unexpected crash.
In order to continue after such a crash, you can restore a Result object containing the recent state of the optimization using Result.load() (with the original objectives and finalize=True if the dump file originates from dump_result()). You may then call optimize_pulses() and pass the loaded Result object as continue_from. The new optimization will start from the most recent optimized controls as a guess, and continue to count iterations from the previous result. See How to continue from a previous optimization for further details.
## How to continue from a previous optimization¶
See How to prevent losing an optimization result for how to continue from an optimization that ended (crashed) prematurely. Even when an optimization has completed normally, you may still want to continue with further iterations – either because you find that the original iter_stop was insufficient to reach full convergence, or because you would like to modify some parameters, like the λₐ values for each control. In this case, you can again call optimize_pulses() and pass the Result object from the previous optimization as continue_from. Note that while you are free to change the pulse_options between the two optimization, the objectives must remain the same. The functional (chi_constructor) and the info_hook should also remain the same (otherwise, you may and up with inconsistencies in your Result). The Result object returned by the second optimization will include all the data from the first optimization.
## How to maximize numerical efficiency¶
For systems of non-trivial size, the main numerical effort should be in the simulation of the system dynamics. Every iteration of Krotov’s method requires a full backward propagation and a full forward propagation of the states associated with each objective, see krotov.propagators. Therefore, the best numerical efficiency can be achieved by optimizing the performance of the propagator that is passed to optimize_pulses().
One possibility is to implement problem-specific propagators, such as krotov.propagators.DensityMatrixODEPropagator. Going further, you might consider implementing the propagator with the help of lower-level instructions, e.g., by using Cython.
## How to deal with the optimization running out of memory¶
Krotov’s method requires the storage of at least one set of propagated state over the entire time grid, for each objective. For the second-order update equation, up to three sets of stored states per objective may be required. In particular for larger systems and dynamics in Liouville space, the memory required for storing these states may be prohibitively expensive.
The optimize_pulses() accepts a storage parameter to which a constructor for an array-like container can be passed wherein the propagated states will be stored. It is possible to pass custom out-of-memory storage objects, such as Dask arrays. This may carry a significant penalty in runtime, however, as states will have to be read from disk, or across the network.
## How to avoid the overhead of QuTiP objects¶
If you know what you are doing, it is possible to set up an Objective without any qutip.Qobj instances, using arbitrary low-level objects instead. See the Optimization with numpy Arrays for an example.
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# Insight into anthropogenic forcing on coastal upwelling off south-central Chile
## Abstract
Coastal upwelling systems off the coasts of Peru and Chile are among the most productive marine ecosystems in the world, sustaining a significant percentage of global primary production and fishery yields. Seasonal and interannual variability in these systems has been relatively well documented; however, an understanding of recent trends and the influence of climate change on marine processes such as surface cooling and primary productivity is limited. This study presents evidence that winds favorable to upwelling have increased within the southern boundary of the Humboldt Current System (35°–42°S) in recent decades. This trend is consistent with a poleward movement of the influence of the Southeast Pacific Anticyclone and resembles the spatial pattern projected by Global Circulation Models for warming scenarios. Chlorophyll a levels (from 2002 to present) determined by satellite and field-based time-series observations show a positive trend, mainly in austral spring–summer (December–January–February), potentially explained by observed increments in nutrient flux towards surface waters and photosynthetically active radiation. Both parameters appear to respond to alongshore wind stress and cloud cover in the latitudinal range of 35°S to 42°S. In addition, net annual deepening of the mixed layer depth is estimated using density and temperature profiles. Changes in this depth are associated with increasing winds and may explain cooler, more saline, and more productive surface waters, with the latter potentially causing fluctuations in dissolved oxygen and other gases, such as nitrous oxide, sensitive to changes in oxygenation. We argue that these recent changes represent, at least in part, a regional manifestation of the Anthropocene along the Chilean coast.
##### Knowledge Domain: Earth & Environmental Science
How to Cite: Aguirre, C., García-Loyola, S., Testa, G., Silva, D. and Farias, L., 2018. Insight into anthropogenic forcing on coastal upwelling off south-central Chile. Elem Sci Anth, 6(1), p.59. DOI: http://doi.org/10.1525/elementa.314
Published on 20 Aug 2018
Accepted on 25 Jul 2018 Submitted on 29 Sep 2017
Domain Editor-in-Chief: Jody W. Deming; Department of Biological Oceanography, University of Washington, US
Associate Editor: Jean-Éric Tremblay; Department of Biology, Université Laval, CA
## 1. Introduction
Eastern ocean boundaries are some of the most productive areas on Earth as a result of wind-driven coastal upwelling systems (CUS), which supply nutrient-rich subsurface waters to the surface layer. Upwelling nutrients support high levels of chlorophyll and high primary production rates (Chavez and Smith, 1995) which in turn support economic activities such as fisheries (Pauly and Christensen, 1995). Large-scale tropospheric subsidence that maintains the Southeast Pacific Anticyclone (SPA) influences wind intensity and direction, thus regulating Chilean CUS (Figure 1a). The effects of climate change are likely to manifest in coastal areas in various ways, and changes in CUS represent one important example of these impacts (e.g., Echevin et al., 2012; Sydeman et al., 2014; Wang et al., 2015).
Figure 1
Area of study and locations of measurements. a) Mean magnitude (m s–1, color scale) and direction (vectors) of surface winds off Chile. Red line and H represent the high-pressure Southeast Pacific Anticyclone (contour of 1020 hPa). b) Mean sea surface temperature (SST; °C, color scale). Geographical position of measurements at Station 18 (St18, yellow circle) and of in-situ SST data (red circles) provided by the Hydrographic and Oceanographic Service of the Chilean Navy (SHOA). DOI: https://doi.org/10.1525/elementa.314.f1
Diverse areas of upwelling off the Chilean coast have been well identified and described (Strub et al., 1998; Thiel et al., 2007). Subsurface waters, particularly of equatorial origin and known as Equatorial Subsurface Water, lift cold, nutrient-rich and oxygen-poor waters to the surface. Equatorward winds over the coastal ocean are responsible for a majority of CUS by triggering Ekman pumping and offshore Ekman transport (Pickett and Paduan, 2003; Bravo et al., 2016).
CUS are also affected by wind intensity, bottom topography, coastal geometry, and other oceanographic conditions, partially associated with remote forcing, such as coastal-trapped waves or El Niño Southern Oscillation (ENSO) events (Shaffer et al., 1997, 1999). South-central Chile (35–42°S) exhibits exceptionally strong seasonal fluctuations in wind stress, precipitation and river runoff. During the austral fall–winter, the SPA moves northward, allowing for the arrival of extratropical cyclones and increasing the frequency and intensity of precipitation, freshwater discharge, and poleward winds related to coastal downwelling episodes (e.g., Sobarzo et al., 2007). Conversely, the SPA migrates southward during spring and summer, reducing precipitation and river discharge, but intensifying northward upwelling-favorable winds (e.g., Rahn and Garreaud, 2014). During this same period, more frequent and intense atmospheric low-level coastal jets can be observed on a synoptic scale, owing to an increase in the alongshore sea level pressure gradient produced by mid-latitude perturbations (Garreaud and Muñoz, 2005). As a result, the seasonal and synoptic variability of coastal winds is associated with changes in the position and intensity of the SPA, as well as sea level pressure (SLP) anomalies to the south (Muñoz and Garreaud, 2005).
Changing patterns in wind stress, precipitation and river discharge could modulate water column dynamics in terms of mixing and/or stratification (Sobarzo et al., 2007). These processes control the extent of nutrient fertilization, which modulates biological production and oxygen consumption (Escribano and Morales, 2012). In this way, coastal hypoxia appears to intensify as a consequence of climate change (Stramma et al., 2008), and it is possible that an acute dissolved oxygen deficit causes fish mortality, modifies microbial communities, C and N cycling, and stimulates the production of greenhouse gases, such as methane and nitrous oxide (Grantham et al., 2004; Naqvi et al., 2010; Gilly et al., 2013).
Since the late 1970s, an ongoing decline in precipitation has been registered in central Chile (CR2 2015; Boisier et al., submitted). This pattern has been partially attributed to changes in large-scale atmospheric circulation and Pacific Decadal Oscillation (PDO). However, it is unlikely that the entirety of this variation arises from natural causes; instead, a part of it is most likely due to the regional effects of anthropogenic climate change (Boisier et al., 2016). Furthermore, the reduction in precipitation over the Southeast Pacific is a robust signature of climate model projections for future scenarios, as well as for recent decades (Seth and Rojas, 2003). Lower precipitation rates have resulted in water scarcity and reductions in river runoff (Garreaud et al., 2017), which may result in decreased nutrient supply to coastal waters and therefore affect coastal primary productivity (Massotti I, personal communication).
There is ongoing debate regarding climate change impacts on coastal upwelling. Model simulations suggest that changes in wind stress by the end of the 21st Century as a result of climate change will increase upwelling at high latitudes in the CUS (Rykaczewski et al., 2015; Wang et al., 2015). However, whether a climate change signal can already be observed in wind stress patterns is unclear. Some regional observations support Bakun’s hypothesis (Bakun, 1990) related to land warming and ocean cooling triggered by the intensification of alongshore winds on the CUS (Santos et al., 2005, 2012; García-Reyes and Largier, 2010; Sydeman et al., 2014). Therefore, it is of primary importance to maintain reliable records and gain a better understanding of how the coastal ocean responds to anthropogenic forcing. The novel contribution of this study is to combine different approaches to better elucidate how signals of anthropogenic climate change have forced trends in upwelling-favorable winds through the spatial configuration of SLP trends over recent decades. The results allow us to explore changes in upwelling-favorable winds and their influence on the physical variables of sea surface temperature (SST) and salinity, which have been shown previously in CUS (Gutierrez et al., 2011; Aravena et al., 2014; Schneider et al., 2017), and to examine changes over the last decades in the biogeochemical variables of chlorophyll a (chl a), nutrients and dissolved oxygen concentrations off the shore of south-central Chile (30–42°S).
## 2. Materials and Methods
### 2.1. Gridded products and Global Climate Model simulations
The atmospheric data analyzed includes monthly SLP and wind at a height of 10 m above sea level, provided by ERA-Interim reanalysis (Dee et al., 2011) and Global Climate Model (GCM) simulations from 23 models (Table S1) associated with the Coupled Model Intercomparison Project Phase 5 (CMIP5). Fully atmosphere-ocean coupled simulations and SST-forced simulations were used for 1979 to 2005. The fully coupled ensemble employs historical runs (hereafter referred to as HIST), while the SST-forced simulations emerge from the Atmospheric Model Intercomparison Project (AMIP); the latter were carried out using the atmospheric component of each of the 23 GCMs included in HIST. HIST and AMIP simulations therefore employed exactly the same climate forcing, with the exception of SST conditions. We decided to compare the reanalyzed trends with the simulated trends over the same time period, rather than using the entire historical simulations, in order to take advantage of the observational portion of the reanalyzed data; in particular, the ERA-Interim reanalysis has updated physics, improved assimilation data, and higher resolution (Dee et al., 2011), as well as better homogeneity through time, making it more reliable for modeling long-term processes (Stopa and Cheung, 2011). In order to determine if changes in CUS are attributable to climate change in fully coupled simulations (HIST), internal variability was assumed to be negligible by averaging the entire dataset to assess trends of anthropogenic climate change forcing over previous decades (Boisier et al., 2016). Simple empirical attribution was achieved by calculating the SLP trend linearly, congruently with the PDO for the AMIP simulations. First, the regression coefficient was calculated for standardized monthly SLP time series and the monthly PDO index (Mantua et al., 1997). The resulting regression coefficients were then multiplied by climate index trends, resulting in SLP trends that were linearly congruent with natural decadal forcing.
Offshore Ekman transport was used as a coastal upwelling index:
${Q}_{x}\text{\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}\frac{{\tau }_{y}}{{\rho }_{\text{\hspace{0.17em}}w}f}$
(1)
where τy is the alongshore wind stress, f is the Coriolis parameter and ρw is seawater density (1,027 kg m–3). To derive upwelling-favorable wind stress, winds were first projected onto the alongshore direction and then the standard bulk aerodynamic formula from wind speed at a height of 10 m, using a constant drag coefficient of 0.0013 (Kraus, 1972; Nelson, 1977). We used this constant value of drag coefficient following Large and Pond (1981), who demonstrated that it does not generate large variations in the calculation of stresses when the wind speeds have values between 4 m s–1 and 11 m s–1. In our study area, the mean wind magnitude fluctuates within that range (Sobarzo et al., 2007). Accumulated Ekman transport from September to February was calculated to examine interannual upwelling variability.
Monthly chl a composites and photosynthetically active radiation (PAR) data, collected by the Aqua MODIS satellite for 2003–2016 with a spatial resolution of 4 km, were analyzed. In order to estimate phytoplankton biomass as chl a, 27 1° × 0.5° pixel quadrants were created between 36°S and 40°S. The average value of PAR and chl a was calculated for each time interval in the time series within each quadrant; an average estimate was considered valid if it contained at least 33% accurate pixels.
### 2.2. In-situ oceanographic observations
In-situ SST data recorded using near surface thermistors at four different locations in south-central Chile (Figure 1b) was provided by the Hydrographic and Oceanographic Service of the Chilean Navy (SHOA) for Coquimbo (30°S–71.4°W), Valparaíso (33°S–71.6°W), Talcahuano (36.7°S–73.1°W) and Valdivia (39.9°S–73.4°W). In addition, a fixed time-series station, known as Station 18 (St18), has been sampled at approximately 30-day intervals since October 2002, making it a well documented site (e.g., Escribano and Morales, 2012). The station is located on the continental shelf, 10 nautical miles (18.5 km) off the coast of central Chile (36.5°S; 73.1°W) at ~92 m depth (Farías et al., 2015). This time-series station is thus located in the middle of one of the most extensive continental shelves off central Chile, where it is subjected to a coastal upwelling process during austral spring and summer (Sobarzo et al., 2007). Upwelling causes strong horizontal density gradients at the sea surface (i.e., an upwelling front), limiting the active upwelling area and acting as a boundary between coastal and adjacent oceanic surface water (Letelier et al., 2009). As St18 is within the active upwelling area, it captures coastal processes occurring on the continental shelf.
In this study, continuous temperature, salinity, oxygen, fluorescence, and PAR profiles were obtained using data from Sea-Bird (SBE-19 and SBE-25) CTDs, which were reviewed and pre-processed using software from Sea-Bird Electronics, Inc. Data were processed at the original sampling frequency and averaged for every 1 dbar, then post-processed using different quality control protocols in order to minimize potential biases or errors.
Seawater at St18 was sampled using a 10-L Niskin bottle mounted on a 10-bottle rosette. Samples were obtained over ten depths, equally distributed between the surface and bottom layers, to test the concentration of gases (i.e., oxygen and nitrous oxide), nutrients, and pigments (in this correlative temporal order). Triplicate samples for nutrients (NO3 and PO43–) were collected by connecting syringes directly to the Niskin bottle spigots and filtering through a 0.45-μm UptiDisc® adapted to the syringe. Samples were then stored at –20°C and analyzed using standard manual (from 2002 to 2009) or automated (from 2009 to 2016, SEAL Analytical AutoAnalyzer) colorimetric methods (Grasshoff et al., 1983). For chl a, 250 mL of seawater were filtered (in triplicate) using a glass-fiber filter (0.7 GF/F) which was immediately frozen until later fluorometric analysis (Turner Design AU-10), according to standard procedures (Parsons et al., 1984). For more details, see Farías et al. (2015).
Nutrient flux towards the surface (F) was calculated as follows:
$F=\text{\hspace{0.17em}\hspace{0.17em}}\mathit{\text{UI}}*C$
(2)
where UI is the mean Upwelling Index (m3 s–1), calculated over 100 m of coastline (Bakun, 1973), averaged over the last three days of each field campaign in the Concepción area; and C is the observed mean nutrient concentration (mmol m–3) at depths between 40 and 80 m (subsurface layer) at St18. This simple model assumes total replacement of the zonal transport by vertical advection (i.e., without considering meridional advection).
A threshold method was implemented to determine mixed layer depth (MLD). In this method, MLD is located where temperature and density profiles change by 0.5°C and 0.15 kg m–3 (Monterey and Levitus, 1997), relative to surface reference values for 10 and 0 m depths, respectively. This method was selected because it allows for addressing different processes that may be occurring at the same time, particularly in a dynamic coastal zone that experiences seasonal upwelling/downwelling and direct input of freshwater through rain and riverine discharge, as at St18. The criteria used provide an accurate representation of the MLD within the study zone, where the temperature criterion (Tcrt) shows the variability associated with wind stress and heat fluxes, and the density criterion (Dcrt) incorporates freshwater fluxes.
Monthly temperature, salinity, oxygen, nitrous oxide, chl a, PAR and nutrient concentration (both NO3 and PO43–) anomalies were calculated as follows:
$\mathit{\text{Anomaly}}=\text{\hspace{0.17em}\hspace{0.17em}}\mathit{\text{xN}}-\mathit{\text{XN}}$
(3)
where xN is the discrete value at a certain time and XN is the mean monthly climatological value obtained using the times series between 2003 and 2016.
Time-series anomaly regressions for each variable were calculated to identify trends in oceanographic variables at St18 for the period 2002–2016, using the generalized least squares method. Trends were observed for summer (December to February) and winter (May to July), as well as for individual calendar months. To test the statistical significance of these trends, we used the non-parametric Mann-Kendall test for significance with a p-value less than 0.05. We also performed linear regression and non-parametric Spearman tests to identify correlations between variables and investigate oceanographic and biogeochemical relationships.
## 3. Results and Discussion
### 3.1. Patterns of sea level pressure and alongshore winds
Natural interdecadal and interannual variability, related to the PDO and ENSO, have been demonstrated to be key contributors to the observed atmospheric and oceanographic changes off the coast of south-central Chile (e.g., Shaffer et al., 1999; Rahn, 2012). Specifically, the intensity and position of the SPA exhibit an interdecadal oscillation capable of modifying large-scale atmospheric circulation, with consequences over the CUS (Ancapichún and Garcés-Vargas, 2015). Notably, over past decades (1979–2016) the PDO shifted from a positive to a negative phase, resulting in a negative trend of the PDO index (Figure 2a). It is therefore expected that trends in SLP and winds over recent decades may be influenced by the negative PDO tendency. Nevertheless, whether changes observed off the south-central coast of Chile accurately reflect anthropogenic influences on the climate remains unclear.
Figure 2
Pacific Decadal Oscillation and sea level pressure trends. a) Time series of the PDO index between 1979 and 2016. Distributions of SLP trends (hPa decade–1, color scale) between 1979 and 2005; b) ERA-Interim reanalysis; c) the sum of PDO congruent to AMIP SLP trends (AMIP-PDO) and fully coupled simulations (HIST); d) SST-forced simulations (AMIP); e) PDO contribution to AMIP SLP trends (AMIP-PDO); and f) fully coupled simulations (HIST). Contours represent the Southeast Pacific Anticyclone, with the white (black) contour corresponding to the position where values of 1020 hPa were found in 1979 (2016). DOI: https://doi.org/10.1525/elementa.314.f2
Alongshore wind variability off the Chilean coast is primarily controlled by nearby SLP differences; the cross-shore SLP gradient forces geostrophic alongshore winds, while an alongshore SLP gradient produces a semi-geostrophic alongshore flow due to the presence of coastal mountains. In order to assess wind-driven changes in upwelling within the region, we first focused on the evolution of large-scale atmospheric circulation and calculated the linear trend of SLP between 1979 and 2005 (Figure 2). A clear increase of SLP is evident in the Southeast Pacific (Figure 2b). Furthermore, a spatially coherent pattern of SLP trends is observed for all data sets, although GCM simulations (HIST and AMIP) exhibit lower trends than ERA-Interim data. Major changes are observed by the SPA and near the south-central Chilean coast (36–45°S). However, the response to changes in PDO is more significant toward the south-central Pacific (Figure 2e), where it represents a slight contribution of the PDO to trends observed in SLP for the coastal region. In this case, a notably similar spatial configuration is observed using both AMIP and HIST simulations.
Assuming that SLP responds to climate change and PDO linearly, the sum of these two trends (HIST and AMIP-PDO) should be closer to the total trends. Hence, under linearity, the stronger trends observed toward the coast in HIST simulations highlight an important contribution of anthropogenic climate change to changes observed in SLP for the coastal region, which in turn is able to modify alongshore winds. The assumption of linear response to climate change comes with issues and caveats, as widely recognized (Lindzen and Giannitsis, 2002; Rodionov, 2004), including the choice of start and end points in cyclical data, and the masking of non-linear events. Nevertheless, linear regression to quantify trends over the historical record and detect responses to climate change is used frequently and remains within the state of the art of climate research (e.g., Santer et al., 2004; Boisier et al., 2016; Williams, 2017).
Trends obtained for the ERA-Interim data reveal an increase in the upwelling-favorable winds at the southern boundary of the CUS, particularly between 37°S and 42°S, while a decrease is observed at lower latitudes (30–36°S) during the austral summer. During winter months, positive trends of alongshore winds are observed between 30° and 37°S, with negative trends observed further south (Figure 3a). This seasonal spatial pattern of alongshore wind trends over south-central Chile is highly consistent with an intensification and expansion of the SPA, shifting its influence southward at the coastal region (Figure 2b). Furthermore, the spatial configuration of alongshore wind trends during the upwelling season is similar to that projected over various upwelling regions under warming scenarios. Regional projections show an increase in upwelling-favorable winds off the Chilean coast south of 35°S (Garreaud and Falvey, 2009; Goubanova et al., 2011; Belmadani et al., 2014), while GCMs show increased upwelling at the poleward boundary of the CUS. In addition, GCMs have demonstrated that the CUS off south-central Chile exhibit one of the major and more robust changes in timing, intensity and spatial distribution of upwelling-favorable winds as compared to other eastern ocean boundaries (Rykaczewski et al., 2015; Wang et al., 2015).
Figure 3
Trends in alongshore winds and sea surface temperature. Alongshore wind trends during the austral summer (December–January–February, DJF, red symbols) and winter (June–July–August, JJA, black symbols) from a) ERA-Interim reanalysis and b) GCM simulations (AMIP, circles; HIST, squares). c) Sea surface temperature trends. DOI: https://doi.org/10.1525/elementa.314.f3
Several authors concur that this spatial pattern of change in alongshore winds is consistent with a poleward displacement of the SPA (Falvey and Garreaud, 2009; Belmadani et al., 2014; Rykaczewski et al., 2015) in association with a projected increase in the Southern Annular Mode and expansion of the Hadley cell under global warming (Hu and Fu, 2007; Lu et al., 2007; Seidel et al., 2008; Gillett and Fyfe, 2013). Given that the increase of upwelling-favorable winds is related to changes in SLP gradients, a notable agreement in the spatial configuration of trends in SLP gradients is observed between ERA-Interim and HIST simulation during the upwelling season (Figure S1).
However, the HIST simulation exhibits lower trends of SLP gradients, consistent with the trends of SLP presented in Figure 2. GCM simulations generally show a strengthening of upwelling-favorable winds during the austral summer at latitudes between 35°S and 40°S (Figure 3b), but the linear trend slope is generally lower than for ERA-Interim wind data. However, spatial coherence between HIST simulations and ERA-Interim data suggest that trends observed in upwelling-favorable winds over previous decades could be attributable, at least in part, to anthropogenic climate change.
A more detailed analysis of wind trends near St18 is shown in Figure S2. Accumulated Ekman transport during the upwelling months (September–February) exhibits a positive trend (p < 0.05) with some interannual variability in Ekman transport observed and partially explained by ENSO. Results show that during the negative phase of ENSO (La Niña), upwelling transport is strongest due to an increase in alongshore wind intensity; conversely, during the positive phase of ENSO (El Niño), upwelling is less intense, as shown previously by Rahn (2012).
### 3.2. Changes in surface temperature and mixed layer depth
On a decadal temporal scale, fluctuations in SST and thermocline depth have been related to the PDO (Montecinos et al., 2003; Pizarro and Montecinos, 2004). SST fluctuations in the Eastern Tropical Pacific are forced by variability in equatorial zonal winds. However, poleward of 30°S, local winds are able to reinforce the decadal variability observed in thermocline depth and SST through vertical advection associated with upwelling-favorable winds (Montecinos and Pizarro, 2005). Changes in coastal SST observed over recent decades would therefore be influenced by both PDO and the intensification of upwelling-favorable winds. In reality, the negative trend of the PDO index only partly explains the cooling trends observed remotely in SST (Falvey and Garreaud, 2009). Results show that the seasonal spatial pattern of SST trends appears to respond to wind trend patterns, particularly during the upwelling season, when the coastal ocean is experiencing clear cooling in the poleward boundary of the CUS (Figure 3c). The simulated impacts of climate change along the coast of central-south Chile include a decrease in SST of 2–3°C during the austral summer as the result of changes in alongshore winds (Aiken et al., 2011).
MLD variability is driven through a balance of processes that enhance and break down mixing and stratification (Turney and Banerjee, 2008). Momentum fluxes induced by increased winds at the air-sea interface generally result in shear and convective instabilities at the surface and lead to mixing and reduced stratification, resulting in a homogeneous surface layer. Conversely, surface heat and salt fluxes cause stratification, which can in turn enhance or reduce the MLD (Bindoff et al., 2007). At St18, estimated MLD has deepened over the period from 2002–2016, demonstrating a negative trend of –3.91 m decade–1 (p = 0.06) for Tcrt (Figure 4a), and –2.43 m decade–1 (p = 0.07) for Dcrt (Figure 4d). These trends may be due to the increase in local winds previously described, in addition to decreasing freshwater input (Garreaud et al., 2017). Additionally, seasonal analysis indicates that in winter (June–July–August) the MLD for Tcrt has a negative trend of –3.85 m decade–1 (p = 0.58; Figure 4c), and for Dcrt a trend of –5.06 m decade–1 (p = 0.44; Figure 4f), while during the upwelling season (December–January–February), the MLD trend displays a slight rise of 1.84 m decade–1 (p = 0.25) for Tcrt (Figure 4b), with a slight deepening of –1.44 m decade–1 (p = 0.41) for Dcrt (Figure 4e).
Figure 4
Trends in the mixed layer depth. Evolution of the mixed layer depth (MLD) estimated throughout the time series at Station 18 (2003–2016) for a) temperature criterion (Temp-MLD), with Temp-MLD trends during b) austral summer and c) winter, and for d) density criterion (Dens-MLD), with Dens-MLD trends during e) austral summer and f) winter. Error bars in b), c), e) and f) indicate the standard deviation obtained from monthly averages for summer (December–January–February) and winter (June–July–August). DOI: https://doi.org/10.1525/elementa.314.f4
These results indicate that the winter MLD represents the greatest contribution to general deepening; they are also consistent with results obtained from simulations using a ROMS1D model based on climatological databases corresponding to the geographical location of St18 (García-Loyola S, personal communication), where sensitivity analysis forced by a variation in momentum, heat, and freshwater fluxes shows greater variation in winter as compared to summer. The slight variation observed during the austral summer can be explained by the combination of two factors: increased incoming solar radiation that heats the sea surface and generates a marked thermocline within the top few meters; and increased winds that activate the upwelling process that pushes subsurface water to the surface and produces mixing. Both of these factors act simultaneously and inhibit MLD variation. During winter months, the thermocline disappears and surface water is more prone to vertical variability.
### 3.3. Physical and biogeochemical trends at St18
Monthly time series (2003–2017) and annual cycles of physical (temperature and salinity) and biogeochemical variables (chl a, nutrients NO3 and PO43–, nitrous oxide, and dissolved oxygen) are shown for St18 in Figure S3. Table S2 shows the trends obtained for monthly anomalies (from 2003 to 2017), with estimates of summer and winter trends. Temperature presents a negative trend over the entire water column, particularly within surface waters (Table S2). Temperature at St18 shows a trend of –0.13°C decade–1 (p = 0.54) in summer and –0.68°C decade–1 (p = 0.16) in winter. Conversely, salinity shows a positive tendency over the entire period of study, with an overall trend of 0.10 decade–1 (p ≤ 0.05), and a winter trend of 0.20 decade–1 (p = 0.53). Trends toward cooler temperatures and increased salinity have been reported by Schneider et al. (2017).
MLD shoaling, primarily as a result of alongshore wind patterns, may partially explain observed temperature and salinity trends. In spring–summer, the water column is pushed upwards as a result of the increased coastal upwelling, which leads to an increase in positive vertical water speed, as denser water from the subsurface is lifted to the surface. This subsurface water, mainly from the Equatorial Subsurface Water, is denser than the surface water, which is comprised principally of sub-Antarctic waters (Sobarzo et al., 2007). During spring and summer, the increase in offshore Ekman transport and Ekman pumping therefore results in denser (more saline and cooler) surface water. Additionally, the effect of reductions in precipitation and freshwater discharge from the Itata River on the winter MLD needs to be considered. The decrease of fresh water discharge has been linked to the recent megadrought affecting Chile (Garreaud et al., 2017). The freshwater layer, typically concentrated within the first 10 m, is formed during the rainy season (winter) and acts as a barrier to mixing, given the greater buoyancy of freshwater as compared to subjacent, saltier water (MacIntyre et al., 2010). The estimated trend toward increasing salinity within the top 15 m of the water column is consistent with a reduction in continental runoff and a shortening of the rainy season, resulting in reduced freshwater mixing with seawater and overall greater salinity. The winter MLD also tends to be deeper due to increased wind gusts or storm winds that drive the mixing of surface and subsurface layers. Poleward winds prevail in winter months and result in more frequent downwelling processes. It is possible that the surface cooling and denser water generated at St18 is related to a slight reduction of the MLD, which would transport a greater volume of cooler water mass towards the surface. In this way, changes in upwelling contribute to lower temperatures of surface waters.
### 3.4. Chlorophyll a, gas and nutrient trends
CUS fluctuate over time and space. Several studies focused on satellite imagery for SST and chl a (Letelier et al., 2009; Aravena et al., 2014) indicate that local conditions, particularly coastline characteristics and topography, can influence upwelling (Sobarzo et al., 2016). In the short term, seasonal cycles provoke greater variability as compared with, for example, low frequency processes such as ENSO (Corredor-Acosta et al., 2015; Testa et al., 2018). Estimates of chl a trends (2003–2016) indicate some spatial heterogeneity along the coast (Figure S4). However, a positive slope is observed between 36°S and 40°S, a latitudinal section that also shows a cooling trend (Aravena et al., 2014; Schneider et al., 2017), in contrast with the 32–36°S latitudinal band, which displays a negative trend.
Ongoing monthly anomaly time-series data from St18 for surface PAR, chl a, NO3 and PO43–, dissolved oxygen, and nitrous oxide, since 2002 are illustrated in Figure 5. Average monthly PAR and phytoplankton concentrations (chl a), derived from satellite data, exhibit positive trends: for PAR, 0.40 E m2 d–1 decade–1, p = 0.33; for chl a, 0.37 mg m–3 decade–1, p = 0.38. Positive trends are also observed for dissolved oxygen and nitrous oxide, whereas nutrient concentrations decrease over time. Despite these decreases, we observed positive trends (with no statistical significance) in nitrate and phosphate fluxes toward surface waters (0.6 and 1.5 kmol d–1 decade–1; Figure S5), mainly explained by increases in Ekman transport (Figure S2). Table S2 displays temporal trends and statistics for these variables.
Figure 5
Times series of radiation and chlorophyll a anomalies and of nutrient concentrations. a) Monthly time-series data for PAR and b) chlorophyll a anomalies and for c) surface (0–15 m, blue points) and subsurface (40–80 m, black points) nitrate and d) surface (0–15 m, magenta points) and subsurface (40–80 m, black points) phosphate concentrations at Station 18. DOI: https://doi.org/10.1525/elementa.314.f5
Figure 6 shows the relationship between surface satellite chl a and PAR, and between integrated chl a and nitrate flux, for the upwelling period (September–February); in both cases, the relationships are positively related by linear analysis (r2 = 0.65, p < 0.05 [Fisher’s p-value] and r2 = 0.41, p < 0.05, respectively). These correlations suggest greater incoming short wave radiation resulting from decreased cloud coverage produced by intense upwelling-favorable winds (Garreaud and Muñoz, 2005), along with enhanced nutrient flux toward the surface layer, which might stimulate phytoplankton growth (Wang et al., 2015).
Figure 6
Relationships between primary production and radiation and nutrient fluxes. Linear regressions between a) satellite chlorophyll a (chl a) and PAR in the coastal zone off central Chile and between b) chl a integrated within the upper 15 m of the water column and nitrate flux at Station 18. DOI: https://doi.org/10.1525/elementa.314.f6
Additionally, observed surface NO3 concentrations increased in summer months and decreased in winter months (Table S2). Sources of nutrient input to surface coastal waters include upwelling and continental drainage, both of which stimulate the growth of phytoplankton. With respect to freshwater discharge, an extensive megadrought has been reported in Central Chile since 2010, and has caused water scarcity, vegetation deterioration, increased forest fires, and decreases in nutrient export along the Chilean coast (Garreaud et al., 2017). Chilean rivers flowing out of south-central Chile have undergone an abrupt decrease in discharge and resulting decreases in nutrient inputs, coastal primary production, and river plumes (Masotti I, personal communication). Prior to the megadrought, a strong influence on river discharge and nutrient export to the coast over Central Chile was attributed to ENSO (Thiel et al., 2007).
These changes in nutrient flux are consistent with observations made during recent years of intensification in coastal eutrophication and land runoff (Rabalais et al., 2009), in some cases as the result of upwelling pumping (Pennington et al., 2006). When in situ measurements (integrated from surface to a 15 m depth) are considered, increased chl a is only observed during spring, revealing the ecological response to the complex changes in the phytoplankton community as a result of stratification or stability (Dave and Lozier, 2013), as is the case within the MLD (Figure 4). These in situ measurements indicate high synoptic variability that is not captured by monthly sampling.
Dissolved oxygen in surface water was observed to decrease in the spring and summer but increase in the winter (Table S2); greater advection of oxygen-poor water from the subsurface (mainly Equatorial subsurface water poor in oxygen, as described by Farías et al., 2009) to the surface is expected to be counteracted by increased oxygenation via vertical mixing and photosynthesis. For this reason, despite the fact that observed effects are consistent with those expected given seasonal trends (Table S2), understanding the net effect on trends in dissolved oxygen concentrations is difficult.
A positive trend in N2O concentration in the mixed layer was also observed (Table S2). This increase might be driven by biogeochemical or physical processes or both. The main biogeochemical processes involved in N2O production are microbial nitrification and denitrification. The first process operates under a wide range of oxygen concentrations, while the second takes place in suboxic and anoxic conditions (Lam et al., 2009). Although nitrification is considered to be strictly aerobic (it requires oxygen for the oxidation of ammonium and nitrite), it can persist at nanomolar oxygen levels as low as ~0.01% air saturation (Füssel et al., 2012, Bristow et al., 2016). Galan et al. (2017) recently demonstrated that most of the N2O produced in the oxycline off central Chile comes from aerobic ammonium oxidation, while in bottom waters dissimilatory nitrite reduction becomes the (seasonally) dominant process. On the one hand, significant correlations of chl a levels with N2O hotspots in the thermocline (oxycline), as observed in a previous study at St18, suggest that the microbial activity fueled by the increased availability of organic matter stimulates N2O production in our study area (Farías et al., 2015; Galán et al., 2014). In a scenario of greater vertical advection of nutrients with concomitant increase in chl a, as in the present study, enhanced primary production could thus explain the stimulation of N2O production. On the other hand, higher levels of N2O produced in the subsurface layers could be advected and degassed quickly due to intensified wind-driven upwelling (Figure 3), leading to an increase in air-sea N2O flux towards the atmosphere. This scenario is expected in regions with more turbulent wind regimes, where gases are rapidly transferred toward or away from the ocean surface.
It is important to note that last decades trends in sea level pressure and wind during the last decades could not be compared directly to those observed at St18, as the two data sets address different periods of time. Here, we have attributed changes to natural variability, as associated with the PDO, and to climate change using reanalysis and GCM, without any attribution using St18 data. Nevertheless, observed changes in the water column at St18 are consistent with an increase in alongshore winds, solar radiation and upwelling processes, providing insights into the effect of anthropogenic forcing on coastal upwelling. An important lesson can also be learned from the fact that the spatial pattern of alongshore wind trends during the last decades is coherent with that projected under global warming.
Under future scenarios, increases in solar radiation and wind-driven coastal upwelling of cold water might have negative repercussions on the phytoplankton community, as a stronger thermal gradient between the surface and subsurface layers may inhibit nutrient input from deeper waters into the photic zone (Boyce et al., 2010; Wang et al., 2015). Furthermore, physical constraints such as wind mixing may have favorable or adverse effects on fisheries, for example, through the mechanisms described by Cury and Roy (1989) and Bakun (1990) that lead to an “optimal environmental window” for recruitment. Decadal behavior of environmental variables may be highly variable and respond to specific thresholds of wind and/or stratification (associated with precipitation). However, whether synergic or antagonistic responses result from the multiple feedback processes between the atmosphere and the ocean surface is unclear.
## 4. Conclusion
Over previous decades, an increase in SLP over the Southeast Pacific has been observed, with consequences for upwelling-favorable winds, coastal SST, solar radiation, nutrients, phytoplankton biomass, and dissolved oxygen concentrations. Although natural interdecadal variability associated with the PDO likely plays a key role in changes in surface, atmospheric and oceanographic conditions, it is possible that anthropogenic climate change also contributes to or amplifies these shifts. The spatial configurations of these trends are consistent with those projected by climate change scenarios for the 21st Century. This research demonstrates that SLP trends observed over previous decades for the coastal region of south-central Chile are not primarily forced by the PDO but more likely represent the result of anthropogenic climate change. If changes in SLP on the coast and resulting impacts on alongshore winds are attributable, at least in part, to climate change, changes observed for several related oceanographic conditions should also be at least partially attributable to climate change. As the majority of changes in surface water are attributed to changes in stratification, MLD shoaling observed over the past decade could thus be related to climate change and marked by an increase in upwelling-favorable winds and reduced precipitation and continental runoff.
The evidence summarized in this work illustrates the rapid rate of change and the complexity of the biophysical system defined by the coastal upwelling system along central and southern Chile. Observed trends are influenced by anthropogenic climate change, and the chain of impacts affects coastal ecosystems that sustain key livelihoods and, in turn, human well-being. Changing coastal upwelling systems and their consequences therefore represent a clear regional manifestation of the Anthropocene. Effectively addressing this issue will require a systems analysis approach to better integrate policy and science. To this end, it is vital to increase the level of certainty attached to the attribution of causality to past and future change by broadening the observational basis and developing long-term monitoring of the coastal upwelling system of central and southern Chile.
## Data Accessibility Statement
Gridded products and Global Climate Model simulations are openly available in several repositories. In-situ oceanographic observations at station 18 are property of the University of Conception (Chile), to get these data write to the corresponding author.
## Acknowledgements
The authors acknowledge the supply of measurements of sea surface temperature provided by the Servicio Hidrográfico y Oceanográfico de la Armada de Chile (SHOA). We also thank Laura Gallardo and Rene Garreaud for their constructive comments, which helped to improve this study. We also thank G. Garcia for assistance during laboratory analyses, and the crew of the R/V Kay Kay. This study was carried out based on the COPAS (www.copas.cl) until 2012 year.
## Funding information
This special issue represents a contribution to the Center of Excellence FONDAP 1511009. This study received financial support from CONICYT; LF was supported by FONDECYT N°1161138, GT was supported by CONICYT-PFCHA/Doctorado Nacional/2017-21170561 and CA was supported by PAI grant N°79150062, FONDECYT N°11171163 and COSTAR-UV.
## Competing interests
The authors have no competing interests to declare.
## Author contributions
• Contributed to conception and design: CA, SG, GT, LF
• Contributed to acquisition of data: CA, SG, GT, DS, LF
• Contributed to analysis and interpretation of data: CA, SG, GT, DS, LF
• Drafted and/or revised the article: CA, SG, GT, LF
• Approved the submitted version for publication: CA, SG, GT, DS, LF
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2019-10-14 06:10:25
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https://pytorch.org/docs/1.12/onnx.html
|
# torch.onnx¶
Open Neural Network eXchange (ONNX) is an open standard format for representing machine learning models. The torch.onnx module can export PyTorch models to ONNX. The model can then be consumed by any of the many runtimes that support ONNX.
## Example: AlexNet from PyTorch to ONNX¶
Here is a simple script which exports a pretrained AlexNet to an ONNX file named alexnet.onnx. The call to torch.onnx.export runs the model once to trace its execution and then exports the traced model to the specified file:
import torch
import torchvision
dummy_input = torch.randn(10, 3, 224, 224, device="cuda")
model = torchvision.models.alexnet(pretrained=True).cuda()
# Providing input and output names sets the display names for values
# within the model's graph. Setting these does not change the semantics
# of the graph; it is only for readability.
#
# The inputs to the network consist of the flat list of inputs (i.e.
# the values you would pass to the forward() method) followed by the
# flat list of parameters. You can partially specify names, i.e. provide
# a list here shorter than the number of inputs to the model, and we will
# only set that subset of names, starting from the beginning.
input_names = [ "actual_input_1" ] + [ "learned_%d" % i for i in range(16) ]
output_names = [ "output1" ]
torch.onnx.export(model, dummy_input, "alexnet.onnx", verbose=True, input_names=input_names, output_names=output_names)
The resulting alexnet.onnx file contains a binary protocol buffer which contains both the network structure and parameters of the model you exported (in this case, AlexNet). The argument verbose=True causes the exporter to print out a human-readable representation of the model:
# These are the inputs and parameters to the network, which have taken on
# the names we specified earlier.
graph(%actual_input_1 : Float(10, 3, 224, 224)
%learned_0 : Float(64, 3, 11, 11)
%learned_1 : Float(64)
%learned_2 : Float(192, 64, 5, 5)
%learned_3 : Float(192)
# ---- omitted for brevity ----
%learned_14 : Float(1000, 4096)
%learned_15 : Float(1000)) {
# Every statement consists of some output tensors (and their types),
# the operator to be run (with its attributes, e.g., kernels, strides,
# etc.), its input tensors (%actual_input_1, %learned_0, %learned_1)
%17 : Float(10, 64, 55, 55) = onnx::Conv[dilations=[1, 1], group=1, kernel_shape=[11, 11], pads=[2, 2, 2, 2], strides=[4, 4]](%actual_input_1, %learned_0, %learned_1), scope: AlexNet/Sequential[features]/Conv2d[0]
%18 : Float(10, 64, 55, 55) = onnx::Relu(%17), scope: AlexNet/Sequential[features]/ReLU[1]
%19 : Float(10, 64, 27, 27) = onnx::MaxPool[kernel_shape=[3, 3], pads=[0, 0, 0, 0], strides=[2, 2]](%18), scope: AlexNet/Sequential[features]/MaxPool2d[2]
# ---- omitted for brevity ----
%29 : Float(10, 256, 6, 6) = onnx::MaxPool[kernel_shape=[3, 3], pads=[0, 0, 0, 0], strides=[2, 2]](%28), scope: AlexNet/Sequential[features]/MaxPool2d[12]
# Dynamic means that the shape is not known. This may be because of a
# limitation of our implementation (which we would like to fix in a
# future release) or shapes which are truly dynamic.
%30 : Dynamic = onnx::Shape(%29), scope: AlexNet
%31 : Dynamic = onnx::Slice[axes=[0], ends=[1], starts=[0]](%30), scope: AlexNet
%32 : Long() = onnx::Squeeze[axes=[0]](%31), scope: AlexNet
%33 : Long() = onnx::Constant[value={9216}](), scope: AlexNet
# ---- omitted for brevity ----
%output1 : Float(10, 1000) = onnx::Gemm[alpha=1, beta=1, broadcast=1, transB=1](%45, %learned_14, %learned_15), scope: AlexNet/Sequential[classifier]/Linear[6]
return (%output1);
}
You can also verify the output using the ONNX library, which you can install using conda:
conda install -c conda-forge onnx
Then, you can run:
import onnx
# Check that the model is well formed
onnx.checker.check_model(model)
# Print a human readable representation of the graph
print(onnx.helper.printable_graph(model.graph))
You can also run the exported model with one of the many runtimes that support ONNX. For example after installing ONNX Runtime, you can load and run the model:
import onnxruntime as ort
ort_session = ort.InferenceSession("alexnet.onnx")
outputs = ort_session.run(
None,
{"actual_input_1": np.random.randn(10, 3, 224, 224).astype(np.float32)},
)
print(outputs[0])
Here is a more involved tutorial on exporting a model and running it with ONNX Runtime.
## Tracing vs Scripting¶
Internally, torch.onnx.export() requires a torch.jit.ScriptModule rather than a torch.nn.Module. If the passed-in model is not already a ScriptModule, export() will use tracing to convert it to one:
• Tracing: If torch.onnx.export() is called with a Module that is not already a ScriptModule, it first does the equivalent of torch.jit.trace(), which executes the model once with the given args and records all operations that happen during that execution. This means that if your model is dynamic, e.g., changes behavior depending on input data, the exported model will not capture this dynamic behavior. Similarly, a trace is likely to be valid only for a specific input size. We recommend examining the exported model and making sure the operators look reasonable. Tracing will unroll loops and if statements, exporting a static graph that is exactly the same as the traced run. If you want to export your model with dynamic control flow, you will need to use scripting.
• Scripting: Compiling a model via scripting preserves dynamic control flow and is valid for inputs of different sizes. To use scripting:
• Use torch.jit.script() to produce a ScriptModule.
• Call torch.onnx.export() with the ScriptModule as the model. The args are still required, but they will be used internally only to produce example outputs, so that the types and shapes of the outputs can be captured. No tracing will be performed.
See Introduction to TorchScript and TorchScript for more details, including how to compose tracing and scripting to suit the particular requirements of different models.
## Avoiding Pitfalls¶
### Avoid NumPy and built-in Python types¶
PyTorch models can be written using NumPy or Python types and functions, but during tracing, any variables of NumPy or Python types (rather than torch.Tensor) are converted to constants, which will produce the wrong result if those values should change depending on the inputs.
For example, rather than using numpy functions on numpy.ndarrays:
# Bad! Will be replaced with constants during tracing.
x, y = np.random.rand(1, 2), np.random.rand(1, 2)
np.concatenate((x, y), axis=1)
Use torch operators on torch.Tensors:
# Good! Tensor operations will be captured during tracing.
x, y = torch.randn(1, 2), torch.randn(1, 2)
torch.cat((x, y), dim=1)
And rather than using torch.Tensor.item() (which converts a Tensor to a Python built-in number):
# Bad! y.item() will be replaced with a constant during tracing.
def forward(self, x, y):
return x.reshape(y.item(), -1)
Use torch’s support for implicit casting of single-element tensors:
# Good! y will be preserved as a variable during tracing.
def forward(self, x, y):
return x.reshape(y, -1)
### Avoid Tensor.data¶
Using the Tensor.data field can produce an incorrect trace and therefore an incorrect ONNX graph. Use torch.Tensor.detach() instead. (Work is ongoing to remove Tensor.data entirely).
### Avoid in-place operations when using tensor.shape in tracing mode¶
In tracing mode, shape values obtained from tensor.shape are traced as tensors, and share the same memory. This might cause a mismatch in values of the final outputs. As a workaround, avoid use of inplace operations in these scenarios. For example, in the model:
class Model(torch.nn.Module):
def forward(self, states):
batch_size, seq_length = states.shape[:2]
real_seq_length = seq_length
real_seq_length += 2
return real_seq_length + seq_length
real_seq_length and seq_length share the same memory in tracing mode. This could be avoided by rewriting the inplace operation:
real_seq_length = real_seq_length + 2
## Limitations¶
### Types¶
• Only torch.Tensors, numeric types that can be trivially converted to torch.Tensors (e.g. float, int), and tuples and lists of those types are supported as model inputs or outputs. Dict and str inputs and outputs are accepted in tracing mode, but:
• Any computation that depends on the value of a dict or a str input will be replaced with the constant value seen during the one traced execution.
• Any output that is a dict will be silently replaced with a flattened sequence of its values (keys will be removed). E.g. {"foo": 1, "bar": 2} becomes (1, 2).
• Any output that is a str will be silently removed.
• Certain operations involving tuples and lists are not supported in scripting mode due to limited support in ONNX for nested sequences. In particular appending a tuple to a list is not supported. In tracing mode, the nested sequences will be flattened automatically during the tracing.
### Differences in Operator Implementations¶
Due to differences in implementations of operators, running the exported model on different runtimes may produce different results from each other or from PyTorch. Normally these differences are numerically small, so this should only be a concern if your application is sensitive to these small differences.
### Unsupported Tensor Indexing Patterns¶
Tensor indexing patterns that cannot be exported are listed below. If you are experiencing issues exporting a model that does not include any of the unsupported patterns below, please double check that you are exporting with the latest opset_version.
When indexing into a tensor for reading, the following patterns are not supported:
# Tensor indices that includes negative values.
data[torch.tensor([[1, 2], [2, -3]]), torch.tensor([-2, 3])]
# Workarounds: use positive index values.
#### Writes / Sets¶
When indexing into a Tensor for writing, the following patterns are not supported:
# Multiple tensor indices if any has rank >= 2
data[torch.tensor([[1, 2], [2, 3]]), torch.tensor([2, 3])] = new_data
# Workarounds: use single tensor index with rank >= 2,
# or multiple consecutive tensor indices with rank == 1.
# Multiple tensor indices that are not consecutive
data[torch.tensor([2, 3]), :, torch.tensor([1, 2])] = new_data
# Workarounds: transpose data such that tensor indices are consecutive.
# Tensor indices that includes negative values.
data[torch.tensor([1, -2]), torch.tensor([-2, 3])] = new_data
# Workarounds: use positive index values.
# Implicit broadcasting required for new_data.
data[torch.tensor([[0, 2], [1, 1]]), 1:3] = new_data
# Workarounds: expand new_data explicitly.
# Example:
# data shape: [3, 4, 5]
# new_data shape: [5]
# expected new_data shape after broadcasting: [2, 2, 2, 5]
When exporting a model that includes unsupported operators, you’ll see an error message like:
RuntimeError: ONNX export failed: Couldn't export operator foo
When that happens, you’ll need to either change the model to not use that operator, or add support for the operator.
Adding support for operators requires contributing a change to PyTorch’s source code. See CONTRIBUTING for general instructions on that, and below for specific instructions on the code changes required for supporting an operator.
During export, each node in the TorchScript graph is visited in topological order. Upon visiting a node, the exporter tries to find a registered symbolic functions for that node. Symbolic functions are implemented in Python. A symbolic function for an op named foo would look something like:
def foo(
g: torch._C.Graph,
input_0: torch._C.Value,
input_1: torch._C.Value) -> Union[None, torch._C.Value, List[torch._C.Value]]:
"""
Modifies g (e.g., using "g.op()"), adding the ONNX operations representing
this PyTorch function.
Args:
g (Graph): graph to write the ONNX representation into.
input_0 (Value): value representing the variables which contain
the first input for this operator.
input_1 (Value): value representing the variables which contain
the second input for this operator.
Returns:
A Value or List of Values specifying the ONNX nodes that compute something
equivalent to the original PyTorch operator with the given inputs.
Returns None if it cannot be converted to ONNX.
"""
...
The torch._C types are Python wrappers around the types defined in C++ in ir.h.
The process for adding a symbolic function depends on the type of operator.
### ATen operators¶
ATen is PyTorch’s built-in tensor library. If the operator is an ATen operator (shows up in the TorchScript graph with the prefix aten::), make sure it is not supported already.
#### List of supported operators¶
Visit the auto generated list of supported ATen operators for details on which operator are supported in each opset_version.
#### Adding support for an operator¶
If the operator is not in the list above:
• Define the symbolic function in torch/onnx/symbolic_opset<version>.py, for example torch/onnx/symbolic_opset9.py. Make sure the function has the same name as the ATen function, which may be declared in torch/_C/_VariableFunctions.pyi or torch/nn/functional.pyi (these files are generated at build time, so will not appear in your checkout until you build PyTorch).
• By default, the first arg is the ONNX graph. Other arg names must EXACTLY match the names in the .pyi file, because dispatch is done with keyword arguments.
• A symbolic function that has a first arg (before the Graph object) with the type annotation of torch.onnx.SymbolicContext will be called with that additional context. See examples below.
• In the symbolic function, if the operator is in the ONNX standard operator set, we only need to create a node to represent the ONNX operator in the graph. If not, we can create a graph of several standard operators that have equivalent semantics to the ATen operator.
• If an input argument is a Tensor, but ONNX asks for a scalar, we have to explicitly do the conversion. symbolic_helper._scalar() can convert a scalar tensor into a python scalar, and symbolic_helper._if_scalar_type_as() can turn a Python scalar into a PyTorch tensor.
Here is an example of handling missing symbolic function for the ELU operator.
If we run the following code:
print(
torch.jit.trace(torch.nn.ELU(), # module
torch.ones(1) # example input
).graph)
We see something like:
graph(%self : __torch__.torch.nn.modules.activation.___torch_mangle_0.ELU,
%input : Float(1, strides=[1], requires_grad=0, device=cpu)):
%4 : float = prim::Constant[value=1.]()
%5 : int = prim::Constant[value=1]()
%6 : int = prim::Constant[value=1]()
%7 : Float(1, strides=[1], requires_grad=0, device=cpu) = aten::elu(%input, %4, %5, %6)
return (%7)
Since we see aten::elu in the graph, we know this is an ATen operator.
We check the ONNX operator list, and confirm that Elu is standardized in ONNX.
We find a signature for elu in torch/nn/functional.pyi:
def elu(input: Tensor, alpha: float = ..., inplace: bool = ...) -> Tensor: ...
We add the following lines to symbolic_opset9.py:
def elu(g, input, alpha, inplace=False):
return g.op("Elu", input, alpha_f=_scalar(alpha))
Now PyTorch is able to export models containing the aten::elu operator!
See the symbolic_opset*.py files for more examples.
If the operator is a sub-class of torch.autograd.Function, there are two ways to export it.
#### Static Symbolic Method¶
You can add a static method named symbolic to your function class. It should return ONNX operators that represent the function’s behavior in ONNX. For example:
class MyRelu(torch.autograd.Function):
@staticmethod
def forward(ctx, input: torch.Tensor) -> torch.Tensor:
ctx.save_for_backward(input)
return input.clamp(min=0)
@staticmethod
def symbolic(g: torch._C.graph, input: torch._C.Value) -> torch._C.Value:
return g.op("Clip", input, g.op("Constant", value_t=torch.tensor(0, dtype=torch.float)))
#### PythonOp Symbolic¶
Alternatively, you can register a custom symbolic function. This gives the symbolic function access to more info through the torch.onnx.SymbolicContext object, which gets passed in as the first argument (before the Graph object).
All autograd Functions appear in the TorchScript graph as prim::PythonOp nodes. In order to differentiate between different Function subclasses, the symbolic function should use the name kwarg which gets set to the name of the class.
Custom symbolic functions should add type and shape information by calling setType(...) on Value objects before returning them (implemented in C++ by torch::jit::Value::setType). This is not required, but it can help the exporter’s shape and type inference for down-stream nodes. For a non-trivial example of setType, see test_aten_embedding_2 in test_operators.py.
The example below shows how you can access requires_grad via the Node object:
class MyClip(torch.autograd.Function):
@staticmethod
def forward(ctx, input, min):
ctx.save_for_backward(input)
return input.clamp(min=min)
@staticmethod
def forward(ctx, input):
ctx.save_for_backward(input)
return input.clamp(min=0)
def symbolic_python_op(ctx: torch.onnx.SymbolicContext, g: torch._C.Graph, *args, **kwargs):
n = ctx.cur_node
print("original node: ", n)
for i, out in enumerate(n.outputs()):
import torch.onnx.symbolic_helper as sym_helper
for i, arg in enumerate(args):
name = kwargs["name"]
ret = None
if name == "MyClip":
ret = g.op("Clip", args[0], args[1])
elif name == "MyRelu":
ret = g.op("Relu", args[0])
else:
# Logs a warning and returns None
return _unimplemented("prim::PythonOp", "unknown node kind: " + name)
# Copy type and shape from original node.
ret.setType(n.type())
return ret
from torch.onnx import register_custom_op_symbolic
register_custom_op_symbolic("prim::PythonOp", symbolic_python_op, 1)
### Custom operators¶
If a model uses a custom operator implemented in C++ as described in Extending TorchScript with Custom C++ Operators, you can export it by following this example:
from torch.onnx import register_custom_op_symbolic
from torch.onnx.symbolic_helper import parse_args
# Define custom symbolic function
@parse_args("v", "v", "f", "i")
def symbolic_foo_forward(g, input1, input2, attr1, attr2):
return g.op("custom_domain::Foo", input1, input2, attr1_f=attr1, attr2_i=attr2)
# Register custom symbolic function
register_custom_op_symbolic("custom_ops::foo_forward", symbolic_foo_forward, 9)
class FooModel(torch.nn.Module):
def __init__(self, attr1, attr2):
super(FooModule, self).__init__()
self.attr1 = attr1
self.attr2 = attr2
def forward(self, input1, input2):
# Calling custom op
model = FooModel(attr1, attr2)
torch.onnx.export(
model,
(example_input1, example_input1),
"model.onnx",
# only needed if you want to specify an opset version > 1.
custom_opsets={"custom_domain": 2})
You can export it as one or a combination of standard ONNX ops, or as a custom operator. The example above exports it as a custom operator in the “custom_domain” opset. When exporting a custom operator, you can specify the custom domain version using the custom_opsets dictionary at export. If not specified, the custom opset version defaults to 1. The runtime that consumes the model needs to support the custom op. See Caffe2 custom ops, ONNX Runtime custom ops, or your runtime of choice’s documentation.
### Discovering all unconvertible ATen ops at once¶
When export fails due to an unconvertible ATen op, there may in fact be more than one such op but the error message only mentions the first. To discover all of the unconvertible ops in one go you can:
from torch.onnx import utils as onnx_utils
# prepare model, args, opset_version
...
torch_script_graph, unconvertible_ops = onnx_utils.unconvertible_ops(
model, args, opset_version=opset_version)
print(set(unconvertible_ops))
Q: I have exported my LSTM model, but its input size seems to be fixed?
The tracer records the shapes of the example inputs. If the model should accept inputs of dynamic shapes, set dynamic_axes when calling torch.onnx.export().
Q: How to export models containing loops?
Q: How to export models with primitive type inputs (e.g. int, float)?
Support for primitive numeric type inputs was added in PyTorch 1.9. However, the exporter does not support models with str inputs.
Q: Does ONNX support implicit scalar datatype casting?
No, but the exporter will try to handle that part. Scalars are exported as constant tensors. The exporter will try to figure out the right datatype for scalars. However when it is unable to do so, you will need to manually specify the datatype. This often happens with scripted models, where the datatypes are not recorded. For example:
class ImplicitCastType(torch.jit.ScriptModule):
@torch.jit.script_method
def forward(self, x):
# Exporter knows x is float32, will export "2" as float32 as well.
y = x + 2
# Currently the exporter doesn't know the datatype of y, so
# "3" is exported as int64, which is wrong!
return y + 3
# To fix, replace the line above with:
# return y + torch.tensor([3], dtype=torch.float32)
x = torch.tensor([1.0], dtype=torch.float32)
torch.onnx.export(ImplicitCastType(), x, "implicit_cast.onnx",
example_outputs=ImplicitCastType()(x))
We are trying to improve the datatype propagation in the exporter such that implicit casting is supported in more cases.
Q: Are lists of Tensors exportable to ONNX?
Yes, for opset_version >= 11, since ONNX introduced the Sequence type in opset 11.
## Functions¶
torch.onnx.export(model, args, f, export_params=True, verbose=False, training=<TrainingMode.EVAL: 0>, input_names=None, output_names=None, operator_export_type=<OperatorExportTypes.ONNX: 0>, opset_version=None, do_constant_folding=True, dynamic_axes=None, keep_initializers_as_inputs=None, custom_opsets=None, export_modules_as_functions=False)[source]
Exports a model into ONNX format. If model is not a torch.jit.ScriptModule nor a torch.jit.ScriptFunction, this runs model once in order to convert it to a TorchScript graph to be exported (the equivalent of torch.jit.trace()). Thus this has the same limited support for dynamic control flow as torch.jit.trace().
Parameters
• model (torch.nn.Module, torch.jit.ScriptModule or torch.jit.ScriptFunction) – the model to be exported.
• args (tuple or torch.Tensor) –
args can be structured either as:
1. ONLY A TUPLE OF ARGUMENTS:
args = (x, y, z)
The tuple should contain model inputs such that model(*args) is a valid invocation of the model. Any non-Tensor arguments will be hard-coded into the exported model; any Tensor arguments will become inputs of the exported model, in the order they occur in the tuple.
1. A TENSOR:
args = torch.Tensor([1])
This is equivalent to a 1-ary tuple of that Tensor.
1. A TUPLE OF ARGUMENTS ENDING WITH A DICTIONARY OF NAMED ARGUMENTS:
args = (x,
{'y': input_y,
'z': input_z})
All but the last element of the tuple will be passed as non-keyword arguments, and named arguments will be set from the last element. If a named argument is not present in the dictionary, it is assigned the default value, or None if a default value is not provided.
Note
If a dictionary is the last element of the args tuple, it will be interpreted as containing named arguments. In order to pass a dict as the last non-keyword arg, provide an empty dict as the last element of the args tuple. For example, instead of:
torch.onnx.export(
model,
(x,
# WRONG: will be interpreted as named arguments
{y: z}),
"test.onnx.pb")
Write:
torch.onnx.export(
model,
(x,
{y: z},
{}),
"test.onnx.pb")
• f – a file-like object (such that f.fileno() returns a file descriptor) or a string containing a file name. A binary protocol buffer will be written to this file.
• export_params (bool, default True) – if True, all parameters will be exported. Set this to False if you want to export an untrained model. In this case, the exported model will first take all of its parameters as arguments, with the ordering as specified by model.state_dict().values()
• verbose (bool, default False) – if True, prints a description of the model being exported to stdout. In addition, the final ONNX graph will include the field doc_string from the exported model which mentions the source code locations for model. If True, ONNX exporter logging will be turned on.
• training (enum, default TrainingMode.EVAL) –
• TrainingMode.EVAL: export the model in inference mode.
• TrainingMode.PRESERVE: export the model in inference mode if model.training is False and in training mode if model.training is True.
• TrainingMode.TRAINING: export the model in training mode. Disables optimizations which might interfere with training.
• input_names (list of str, default empty list) – names to assign to the input nodes of the graph, in order.
• output_names (list of str, default empty list) – names to assign to the output nodes of the graph, in order.
• operator_export_type (enum, default OperatorExportTypes.ONNX) –
• OperatorExportTypes.ONNX: Export all ops as regular ONNX ops (in the default opset domain).
• OperatorExportTypes.ONNX_FALLTHROUGH: Try to convert all ops to standard ONNX ops in the default opset domain. If unable to do so (e.g. because support has not been added to convert a particular torch op to ONNX), fall back to exporting the op into a custom opset domain without conversion. Applies to custom ops as well as ATen ops. For the exported model to be usable, the runtime must support these non-standard ops.
• OperatorExportTypes.ONNX_ATEN: All ATen ops (in the TorchScript namespace “aten”) are exported as ATen ops (in opset domain “org.pytorch.aten”). ATen is PyTorch’s built-in tensor library, so this instructs the runtime to use PyTorch’s implementation of these ops.
Warning
Models exported this way are probably runnable only by Caffe2.
This may be useful if the numeric differences in implementations of operators are causing large differences in behavior between PyTorch and Caffe2 (which is more common on untrained models).
• OperatorExportTypes.ONNX_ATEN_FALLBACK: Try to export each ATen op (in the TorchScript namespace “aten”) as a regular ONNX op. If we are unable to do so (e.g. because support has not been added to convert a particular torch op to ONNX), fall back to exporting an ATen op. See documentation on OperatorExportTypes.ONNX_ATEN for context. For example:
graph(%0 : Float):
%3 : int = prim::Constant[value=0]()
# conversion unsupported
%4 : Float = aten::triu(%0, %3)
# conversion supported
%5 : Float = aten::mul(%4, %0)
return (%5)
Assuming aten::triu is not supported in ONNX, this will be exported as:
graph(%0 : Float):
%1 : Long() = onnx::Constant[value={0}]()
# not converted
%2 : Float = aten::ATen[operator="triu"](%0, %1)
# converted
%3 : Float = onnx::Mul(%2, %0)
return (%3)
If PyTorch was built with Caffe2 (i.e. with BUILD_CAFFE2=1), then Caffe2-specific behavior will be enabled, including special support for ops are produced by the modules described in Quantization.
Warning
Models exported this way are probably runnable only by Caffe2.
• opset_version (int, default 13) – The version of the default (ai.onnx) opset to target. Must be >= 7 and <= 16.
• do_constant_folding (bool, default True) – Apply the constant-folding optimization. Constant-folding will replace some of the ops that have all constant inputs with pre-computed constant nodes.
• dynamic_axes (dict<string, dict<python:int, string>> or dict<string, list(int)>, default empty dict) –
By default the exported model will have the shapes of all input and output tensors set to exactly match those given in args. To specify axes of tensors as dynamic (i.e. known only at run-time), set dynamic_axes to a dict with schema:
• KEY (str): an input or output name. Each name must also be provided in input_names or output_names.
• VALUE (dict or list): If a dict, keys are axis indices and values are axis names. If a list, each element is an axis index.
For example:
class SumModule(torch.nn.Module):
def forward(self, x):
torch.onnx.export(SumModule(), (torch.ones(2, 2),), "onnx.pb",
input_names=["x"], output_names=["sum"])
Produces:
input {
name: "x"
...
shape {
dim {
dim_value: 2 # axis 0
}
dim {
dim_value: 2 # axis 1
...
output {
name: "sum"
...
shape {
dim {
dim_value: 2 # axis 0
...
While:
torch.onnx.export(SumModule(), (torch.ones(2, 2),), "onnx.pb",
input_names=["x"], output_names=["sum"],
dynamic_axes={
# dict value: manually named axes
"x": {0: "my_custom_axis_name"},
# list value: automatic names
"sum": [0],
})
Produces:
input {
name: "x"
...
shape {
dim {
dim_param: "my_custom_axis_name" # axis 0
}
dim {
dim_value: 2 # axis 1
...
output {
name: "sum"
...
shape {
dim {
dim_param: "sum_dynamic_axes_1" # axis 0
...
• keep_initializers_as_inputs (bool, default None) –
If True, all the initializers (typically corresponding to parameters) in the exported graph will also be added as inputs to the graph. If False, then initializers are not added as inputs to the graph, and only the non-parameter inputs are added as inputs. This may allow for better optimizations (e.g. constant folding) by backends/runtimes.
If opset_version < 9, initializers MUST be part of graph inputs and this argument will be ignored and the behavior will be equivalent to setting this argument to True.
If None, then the behavior is chosen automatically as follows:
• If operator_export_type=OperatorExportTypes.ONNX, the behavior is equivalent to setting this argument to False.
• Else, the behavior is equivalent to setting this argument to True.
• custom_opsets (dict<str, int>, default empty dict) –
A dict with schema:
• KEY (str): opset domain name
• VALUE (int): opset version
If a custom opset is referenced by model but not mentioned in this dictionary, the opset version is set to 1. Only custom opset domain name and version should be indicated through this argument.
• export_modules_as_functions (bool or set of python:type of nn.Module, default False) –
Flag to enable exporting all nn.Module forward calls as local functions in ONNX. Or a set to indicate the particular types of modules to export as local functions in ONNX. This feature requires opset_version >= 15, otherwise the export will fail. This is because opset_version < 15 implies IR version < 8, which means no local function support. Module variables will be exported as function attributes. There are two categories of function attributes.
1. Annotated attributes: class variables that have type annotations via PEP 526-style will be exported as attributes. Annotated attributes are not used inside the subgraph of ONNX local function because they are not created by PyTorch JIT tracing, but they may be used by consumers to determine whether or not to replace the function with a particular fused kernel.
2. Inferred attributes: variables that are used by operators inside the module. Attribute names will have prefix “inferred::”. This is to differentiate from predefined attributes retrieved from python module annotations. Inferred attributes are used inside the subgraph of ONNX local function.
• False(default): export nn.Module forward calls as fine grained nodes.
• True: export all nn.Module forward calls as local function nodes.
• Set of type of nn.Module: export nn.Module forward calls as local function nodes, only if the type of the nn.Module is found in the set.
Raises
CheckerError – If the ONNX checker detects an invalid ONNX graph. Will still export the model to the file f even if this is raised.
torch.onnx.export_to_pretty_string(*args, **kwargs)[source]
Similar to export(), but returns a text representation of the ONNX model. Only differences in args listed below. All other args are the same as export().
Parameters
• add_node_names (bool, default True) – Whether or not to set NodeProto.name. This makes no difference unless google_printer=True.
• google_printer (bool, default False) – If False, will return a custom, compact representation of the model. If True will return the protobuf’s Message::DebugString(), which is more verbose.
Returns
A UTF-8 str containing a human-readable representation of the ONNX model.
torch.onnx.register_custom_op_symbolic(symbolic_name, symbolic_fn, opset_version)[source]
Registers symbolic_fn to handle symbolic_name. See “Custom Operators” in the module documentation for an example usage.
Parameters
• symbolic_name (str) – The name of the custom operator in “<domain>::<op>” format.
• symbolic_fn (Callable) – A function that takes in the ONNX graph and the input arguments to the current operator, and returns new operator nodes to add to the graph.
• opset_version (int) – The ONNX opset version in which to register.
torch.onnx.select_model_mode_for_export(model, mode)[source]
A context manager to temporarily set the training mode of model to mode, resetting it when we exit the with-block. A no-op if mode is None.
Parameters
torch.onnx.is_in_onnx_export()[source]
Returns True iff export() is running in the current thread
torch.onnx.is_onnx_log_enabled()[source]
Returns True iff ONNX logging is turned on.
torch.onnx.enable_log()[source]
Enables ONNX logging.
torch.onnx.disable_log()[source]
Disables ONNX logging.
torch.onnx.set_log_stream(stream_name='stdout')[source]
Set output stream for ONNX logging.
Parameters
stream_name (str, default "stdout") – Only stdout and stderr are supported as stream_name.
torch.onnx.log(*args)[source]
A simple logging facility for ONNX exporter.
Parameters
args – Arguments are converted to string, concatenated together with a newline character appended to the end, and flushed to output stream.
## Classes¶
SymbolicContext` Provides extra context for symbolic functions.
|
2023-02-06 21:37:44
|
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|
https://chemistry.stackexchange.com/questions/131572/what-does-the-intramolecular-aldol-condensation-of-6-oxoheptanal-form/131577
|
# What does the intramolecular aldol condensation of 6-oxoheptanal form?
I know that when 6-oxoheptanal is treated with NaOH, an intramolecular aldol condensation will take place. But will a five-membered ring or seven-membered ring be formed?
I know that the five-membered ring is more stable in comparison to the seven- membered ring, but as per the mechanism of aldol condensation the first step is removal of acidic hydrogen (which is also the slowest step i.e. the rate determining step {if I am not wrong}).
So if I remove the acidic hydrogen on C-7 (the terminal methyl group), then the (primary) carbanion formed will be more stable than if I were to deprotonate on C-5.
So I think that the major product should be the seven-membered ring.
Am I right? Or is my reasoning incorrect?
• You need to remember that every step in this process is reversible. – Waylander Apr 11 '20 at 8:19
B.Anshuman has given a short answer, but I felt more explanation is needed to understand this situation. This reaction can be done in kineticlly controlled conditions (e.g., using $$\ce{LDA/THF}$$ as a base at $$\pu{-78 ^\circ C}$$) or in thermodynamically controlled conditions as in this case. Each case may give totally different major products. For example, alkylation of 2-methylcyclohexanone gives 6-alkyl-2-methylcyclohexanone as the predominanet product under kinetic conditions (1. $$\ce{LDA/THF}/\pu{-78 ^\circ C}$$; 2. $$\ce{R-Cl/THF}/\pu{-78 ^\circ C}$$, alkyl chloride is $$1^\circ$$ or $$2^\circ$$). However the same starting material would gives 2-alkyl-2-methylcyclohexanone as the amjor product under thermodynamic condition such as $$\ce{NaOEt/EtOH}/\gt \pu{25 ^\circ C}$$.
In above example shows that the thermodynamic conditions allow equilibration of the two possible enolates (which are derived from $$2^\circ$$ and $$3^\circ$$ carbanions); the one enolate with greater alkyl substitution on $$\ce{C=C}$$ would form in greater concentration (Zaitsev-like). The question in hand can also be treated like the same because the situation is similar ($$1^\circ$$ vs $$2^\circ$$ carbanions).
The literature has shown that the evidence for carbanion formation might not be the rate determining step (Ref.1). The abstract if the reference is given below for your benefit for reasoning:
Rate and equilibrium constants have been determined for both the aldol addition and the elimination steps in the intramolecular condensation reactions of 2,5-hexanedione, 2,6-heptanedione, 1-phenyl-1,5-hexanedione, and 5-oxohexanal. The overall thermodynamics are similar for cyclization of 2,5-hexanedione and 2,6-heptanedione; conversion of 2,5-hexanedione to the corresponding enone is actually more favorable, but the cyclization of 2,5-hexanedione is 2400 times slower than that of 2,6-heptanedione. As expected on the basis of intermolecular analogs, the addition step is less favorable and slower for 1-phenyl-1,5-hexanedione, and the addition step for 5-oxohexanal is more favorable though similar in rate to that for heptanedione. Detailed analysis of the kinetics and equilibrium for all of these compounds, as well as 2-(2-oxopropyl)benzaldehyde, in terms of Marcus theory, leads to the same intrinsic barriers for the intramolecular reactions as were seen previously for the intermolecular reactions. This means that rate constants for intramolecular aldol reactions should be predictable from the energetics of the reactions and that the effective molarity can be calculated. Methods for estimating thermodynamic quantities for reactants and products of these reactions have been examined.
Thus, my conclusion is formation of 5-membered ring is predominated here to give 1-acetylcyclohexene as the major product among three theoretically possible products.
Reference:
1. J. Peter Guthrie, Junan Guo, "Intramolecular Aldol Condensations: Rate and Equilibrium Constants," J. Am. Chem. Soc. 1996, 118(46), 11472-11487 (https://doi.org/10.1021/ja954247l).
Intramolecular-Aldol reactions involves heat, as you have mentioned, and the base used (NaOH) isn't much affected by steric factors. Therefore, the preferred product should be thermodynamically controlled product. Moreover, as you commented about the formed carbanion being less stable in case of 2° carbon, actually the carbanion is converted to an enolate then the reaction proceeds further. So, the 5-membered cyclic ring should be the major product.
• The carbanion isn't converted to an enolate; it is an enolate. – orthocresol Apr 11 '20 at 11:01
• The base extracts the hydrogen from the carbon, carbon accepting the electron. So though temporarily, the carbon must have an excess negative charge, i.e a carbanion, which becomes an enolate. Please explain, if otherwise. – B.Anshuman Apr 11 '20 at 11:03
• Nope; you deprotonate a carbonyl, you get an enolate. The carbanion that you're thinking of is a resonance form. It doesn't exist as an independent entity. Not even transiently. – orthocresol Apr 11 '20 at 11:06
• Enolate can also be considered as a one of the resonating structure, so actually, none of them should be an individual identity. Sorry for taking your time. – B.Anshuman Apr 11 '20 at 11:10
• Well, okay; let's call it a "resonance hybrid which looks almost like an enolate", or RHWLALAE for short. When you deprotonate a carbonyl compound, you get a RHWLALAE; there's no carbanion intermediate or transition state. So we can rephrase my original comment: the carbanion isn't converted to an enolate; it is a RHWLALAE. If you accept that the RHWLALAE is close enough to an enolate, then we can call it an enolate, but if not we can agree to call it a RHWLALAE. But that doesn't affect my point about using the word "converted" - it's not right to say that. – orthocresol Apr 11 '20 at 11:29
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2021-02-26 16:15:47
|
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http://webdevfunda.blogspot.in/2013/01/installing-zend-framework.html
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Friday, 18 January 2013
Installing Zend Framework
Zend Framework
Installing Zend Framework in WAMP (Windows (7) , Apache , MySQL , PHP) :-
I used the following version of the softwares :-
1) Apache 2.4.2
2) PHP 5.4.3
3) MySQL 5.5.24
4) Zend Framework 1. 12.0
You can use any other databases or packages as you like , the general installation procedure is the same !
My installation directory :- F:\wamp\www\zend , i will use this for future reference .
Step 2 . Extract the zipped archive into any directory (in this case F:/wamp/www)
Step 3 . Open php.ini , normally this is situated inside apache/apache2.x.x/bin/ (F:/wamp/bin/apache/apache2.4.2/bin) . If u don't know where it is situated , create a php file with the following as it's contents :-
<?php
echo phpinfo();
?>
Then open it from your browser , it'll list all your php settings and which php.ini file was loaded , you can get the location of php.ini from there !
Step 4 . Find the following lines :-
; Windows: "\path1;\path2"
include_path = ".;f:\php" <Your drive , or directory may differ>
Now add the following to include_path :- ;Path\to\zend\library
In my case i had something like this :-
; Windows: "\path1;\path2"
include_path = ".;c:\php\includes;f:\wamp\www\zend\library"
Step 5 . Restart Apache
That's it , you are all done now !!
To test your installation of Zend Framework , type this at the command prompt :-
zf show version , it should output the current Zend Framework Version !!
|
2018-02-24 04:17:04
|
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https://icesp.org.br/9f7cwj/relative-atomic-mass-a02012
|
There is no unit as it is a relative value. Atoms with an Ar that is more than this have a larger mass than a carbon atom. It is the ratio of the average mass per atom of an element from a given sample to 1/12 the mass of a carbon-12 atom. A relative atomic mass (also called atomic weight; symbol: Ar) is a measure of how heavy atoms are. It is the ratio of the average mass per atom of an element from a given sample to 1/12 the mass of a carbon-12 atom. It can be best defined as \frac{1}{12}of the mass of a carbon-12 atom in its ground state. relative atomic mass For the natural isotopic composition of each element it shows "name", "atomic number", "symbol", "atomic weight" (or relative atomic mass) and a link to the element's "isotopes". It has no unit. The relative atomic mass scale is now based on an isotope of carbon, namely, carbon-12, nuclide symbol, which is given the value of 12.0000 amu. n. (Chemistry) the ratio of the average mass per atom of the naturally occurring form of an element to one-twelfth the mass of an atom of carbon-12. Thus, the atomic mass of a carbon-12 atom is 12 Da, but the relative isotopic mass of a carbon-12 atom is simply 12. It is a weighed average of the different isotopes of an element. 12.15 g Mg i. Electrons contribute so little mass that they aren't counted. Note: the average atomic weight is dimensionless quantity while atomic mass has the dimension of unified mass unit (u), But both has the same numerical value. The relative atomic mass is represented by the symbol Ar. relative atomic mass. (chlorine-37). This list contains the 118 elements of chemistry. For example, a sample from another planet could have a relative atomic mass very different to the standard Earth-based value. Atoms consist of a nucleus containing protons and neutrons, surrounded by electrons in shells. A single atom has a set number of protons and neutrons, so the mass is unequivocal (won't change) and is the sum of the number of protons and neutrons in the atom. The atomic mass of an isotope and the relative isotopic mass refers to a … Molecular M ass (M r) is the sum of all the relative atomic masses for all the atoms in a given formula. The word relative in relative atomic mass refers to this scaling relative to carbon-12. 3.01 × 1023 atoms Cl i. relative atomic mass (physics) Ratio of the atomic mass of one atom of an isotope to 1/12 (one twelfth) the mass of a Carbon-12 atom. Our tips from experts and exam survivors will help you through. Chlorine naturally exists as two isotopes. .5000 moles b. The number of protons an atom has determines what element it is. click on any element's name for further information on chemical properties, environmental data or health effects.. Each isotope has its own mass, called its isotopic mass. First, determine the fractional percent of each isotope in the substance. In other words, in every 100 chlorine atoms, 75 atoms have a mass number of 35, and 25 atoms have a mass number of 37. The atomic mass or relative isotopic mass refers to the mass of a single particle, and therefore is tied to a certain specific isotope of an element. Read about our approach to external linking. a. Molecular Weight, Atomic Weight, Weight vs. Mass. To calculate the relative atomic mass, Ar, of chlorine: $A_{r} = \frac{total~mass~of~atoms}{total~number~of~atoms} = \frac{(75 \times 35)+(25 \times 37)}{(75+25)}$, $A_{r} = \frac{2625+925}{100} = \frac{3550}{100}$. Mass of an atomic particle is called the atomic mass. Relative atomic mass definition at Dictionary.com, a free online dictionary with pronunciation, synonyms and translation. How to calculate average atomic mass. • However, this does not tell us the mass in grams. The abundance of chlorine-35 is 75% and the abundance of chlorine-37 is 25%. Relative atomic mass or atomic weight is the average atomic mass divided by one unified atomic unit. relative atomic mass For the natural isotopic composition of each element it shows "name", "atomic number", "symbol", "atomic weight" (or relative atomic mass) and a link to the element's "isotopes". The relative isotopic mass of an isotope is roughly the same as its mass number, which is the number of protons and neutrons in the nucleus. It indicates how many times an element's average atom is weightier compared to one-twelfth of a carbon atom-12 from a given sample. The abundance of chlorine-35 is 75% and the abundance of chlorine-37 is 25%. An atomic mass unit is thus defined as 1/12 th of the mass of one atom of carbon-12. Determine the mass in grams of the following: a. Chemical elements listed by atomic mass The elements of the periodic table sorted by atomic mass. Therefore one atom of carbon, isotopic mass 12, equals 12 u, or, Give your answer to 1 decimal place. For covalent compounds it is called the Relative Molecular Mass. The atomic mass unit (amu) is the unit of relative atomic mass. The. From Simple English Wikipedia, the free encyclopedia, International Union of Pure and Applied Chemistry, "Atomic weight: The Name, its History, Definition, and Units", https://simple.wikipedia.org/w/index.php?title=Relative_atomic_mass&oldid=7329713, Creative Commons Attribution/Share-Alike License. .249 moles 3. The mass of one hydrogen atom was assigned 1 unit. The relative atomic mass. The atomic mass unit (u) is defined as a mass equivalent to 1 / 12 of the mass of one atom of carbon-12. The relative atomic mass (A r) of an element is the average mass of the naturally occurring atoms of the element. b) Atomic numbers, mass numbers and isotopes; An atom is named after the number of protons in its nucleus. The table shows the mass numbers and abundances of naturally occurring copper isotopes. Everything else is measured relative to this quantity. Dictionary entry overview: What does relative atomic mass mean? Relative atomic mass values are ratios;[3]:1 relative atomic mass is a dimensionless quantity. Hence the relative atomic mass of the mass m is defined as: $A_r = \dfrac{m}{m_u}$ The quantity is now dimensionless. The relative atomic mass of an element is the average mass of its atoms, compared to 1/12th the mass of a carbon-12 atom. A standard atomic weight is the mean value of relative atomic masses of a number of normal samples of the element. The relative atomic mass of an element is the average mass of its atoms, compared to 1/12th the mass of a carbon-12 atom. At first, chemists use the hydrogen atom as the standard atom because it is the lightest. [3]:17 For example, if a sample of thallium is made up of 30% thallium-203 and 70% thallium-205. Award winning periodic table, by relative atomic mass, with user-friendly element data and facts. • What it tells us is the relative masses of atoms – or relative atomic mass (A r) • The element carbon is the atom against which the mass of all other atoms are compared. An atom consists of a small, positively charged nucleus surrounded by electrons. The nucleus contains protons and neutrons; its diameter is about 100,000 times smaller than that of the atom. Look it up now! 1.50 × 1023 atoms F i. However, most elements in nature consist of atoms with different numbers of neutrons. Notice that the answer is closer to 35 than it is to 37. [1] Individual samples of an element could have a relative atomic mass different to the standard atomic weight for the element. 1 u = 1.66 × 10 -27 kg So, average atomic weight of carbon is 12.011 12 u ÷ 1 u = 12.011 12. Key Takeaways: Atomic Mass Versus Mass Number The mass number is the sum of the number of protons and neutrons in an atom. Carbon is given an A r value of 12. This is not quite correct, because relative atomic mass is a less specific term that refers to individual samples. Atomic mass is the weighted average mass of an atom of an element based on the relative natural abundance of that element's isotopes. This sma… of an element is the average mass of its atoms, compared to 1/12th the mass of a carbon-12 atom. (chemistry) the mass of an atom of a chemical element expressed in atomic mass units Familiarity information: RELATIVE ATOMIC MASS used as a noun is very rare. This quantity takes into account the percentage abundance of all the isotopes of an element which exist. The relative molecular mass (Mr) of an element is the average mass of one molecule of the element/compound when compared with the mass of an atom of carbon-12, which taken as 12 units. In other words, a relative atomic mass tells you the number of times an average atom of an element from a given sample is heavier than one-twelfth of an atom of carbon-12. Sample exam questions - atomic structure and the periodic table - AQA, Home Economics: Food and Nutrition (CCEA). A relative isotopic mass is the mass of an isotope relative to 1/12 the mass of a carbon-12 atom. Answers provided. The standard atomic weight for each element is on the periodic table. A relative atomic mass (also called atomic weight; symbol: A r) is a measure of how heavy atoms are. Both isotopes of thallium have 81 protons, but thallium-205 has 124 neutrons, 2 more than thallium-203, which has 122. Every particle of matter has some amount of mass associated with it whether small or large. Everything is made up of atoms. The mass of an atom can be accounted for by the sum of the mass of protons and neutrons which is almost equal to the atomic mass. Calculate the relative atomic mass of copper. The numbers of subatomic particles in an atom can be calculated from its atomic number and mass number. The normal unit of atomic mass has been one-twelfth of the atomic mass of the carbon-12 isotope since the year 1961. We can find the relative atomic mass of a sample of an element by working out the abundance-weighted mean of the relative isotopic masses. [3]:17 An atom of an element with a certain number of neutrons is called an isotope. The unit 'amu' is now being replaced by a lower case u, where u is the symbol for the unified atomic mass unit. Often, the term relative atomic mass is used to mean standard atomic weight. This is because the chlorine-35 isotope is much more abundant than the chlorine-37 isotope. • RELATIVE ATOMIC MASS (noun) The noun RELATIVE ATOMIC MASS has 1 sense: 1. Two samples of an element that consists of more than one isotope, collected from two widely spaced sources on Earth, are expected to have slightly different relative atomic masses. The formula for relative atomic mass is; The relative molecular mass of a molecule is equal to the sum … To see all my Chemistry videos, check outhttp://socratic.org/chemistryWhat is atomic mass? The relative atomic mass, Ar, of an element is calculated from: Chlorine naturally exists as two isotopes, $$_{17}^{35}\textrm{Cl}$$ (chlorine-35) and $$_{17}^{37}\textrm{Cl}$$ (chlorine-37). As this unit is confusing and against the standards of modern metrology, the use of relative mass is discouraged. Relative Atomic Mass 1 • The deflection in the mass spectrometer varies with the mass of the atom. is the standard atom against which the masses of other atoms are compared. A scaffolded worksheet giving students practise in calculating relative atomic mass from masses of isotopes and percentage abundance. In other words, in every 100 chlorine atoms, 75 atoms have a mass number of 35, and 25 atoms have a mass number of 37. Relative atomic mass Atoms with an Ar of less than this have a smaller mass than a carbon atom. [1][2] In other words, a relative atomic mass tells you the number of times an average atom of an element from a given sample is heavier than one-twelfth of an atom of carbon-12. Former name: atomic weight. The sum of relative isotopic masses of all atoms in a molecule is the relative molecular mass. Determine the amount in moles of the following: Video $$\PageIndex{6}$$: Watch this video for a review of relative atomic mass and isotopes. 17.7 grams 2. Atomic weight, also referred to as relative atomic mass, is the ratio of the mean mass of the atoms of a chemical element to a certain standard. The mass of an atom when compared to a standard atom is known as its relative atomic mass (Ar). The relative atomic mass of Copper is therefore (70 / 100 x 63) + (30 / 100 x 65) = 63.6. For example, chlorine has two major isotopes. The carbon-12 atom, $$_{6}^{12}\textrm{C}$$ is the standard atom against which the masses of other atoms are compared. Atomic mass (m a) is the mass of an atom. This is commonly expressed as per the international agreement in terms of a unified atomic mass unit (amu). This is because the proportions of each isotope are slightly different at different locations. This is a list of chemical elements, sorted by atomic mass (or most stable isotope) and color coded according to type of element.Each element's atomic number, name, element symbol, and group and period numbers on the periodic table are given. [4] For example, the element thallium has two common isotopes: thallium-203 and thallium-205. Relative atomic mass is the same as atomic weight, which is the older term. 1 with 75.77 percent of atoms and 1 with 24.23 percent of atoms. The mass number is a count of the total number of protons and neutrons in an atom's nucleus. Standard atomic weight values are published at regular intervals by the Commission on Isotopic Abundances and Atomic Weights of the International Union of Pure and Applied Chemistry (IUPAC). This page was last changed on 26 January 2021, at 01:14. Symbol: Ar Abbreviation: r.a.m. To two decimal places, what is the relative atomic mass and the molar mass of the element potassium, K? 28.0 grams b. 2.00 mol N i. Like relative atomic mass values, relative isotopic mass values are ratios with no units. $A_{r} = \frac{(69 \times 63)+(31 \times 65)}{(69+31)}$, $A_{r} = \frac{4347+2015}{100} = \frac{6362}{100}$. The relative atomic mass of an element is the weighted average of the masses of the isotopes in the naturally occurring element relative to the mass of an atom of the carbon-12 isotope which is taken to be exactly 12. 'S average atom is named after the number of protons an atom 's nucleus weighed average of the.! 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Word relative in relative atomic mass unit ( amu ) is the unit of relative masses. Given sample, average atomic weight for the element ]:17 an consists... Number the mass of a carbon-12 atom so, average atomic mass unit amu! Of a number of protons and neutrons in an atom can be from!
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2021-05-18 09:54:08
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http://mathoverflow.net/revisions/100445/list
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2 Incorporated image.
I know this thread is already two years old, but, while preparing for a path integration exam, I arrived at an intuitive picture that sheds some light on the origin of the extra term. The picture represents an integral of a smooth function with respect to a concrete realization of Brownian motion. The sum of the areas of the green rectangles represents the difference between Ito (using the left point of each interval) and "anti-Ito" (using the right point of each interval) for sampling of the Brownian motion represented by the red line. Finer sampling leads to smaller rectangles, but they overlap more and more (because Brownian motion is not monotonic), so even if the area occupied by them tends to zero, the sum of their areas does not. This suggests (only suggests -- it is an upper bound on the difference, not a lower bound) that there is a "room" for Ito and "anti-Ito" to differ in their values. Stratonovich can be expected to lie somewhere in between.
Look at the following image:
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2013-05-23 01:58:21
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https://cs.ericyy.me/computational-thinking/lecture-8/index.html
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# Lecture 8
## The Central Limit Theorem (CLT)
• Given a sufficiently large sample:
1. The means of the samples in a set of samples (the sample means) will be approximately normally distributed,
2. This normal distribution will have a mean close to the mean of the population, and
3. The variance of the sample means ($\sigma_{\bar{x}}^2$) will be close to the variance of the population ($\sigma^2$) divided by the sample size (N).
• $\sigma_{\bar{x}}^2=\frac{\sigma^2}{N}$
• $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{N}}$
• Reference:
• More details in lecture 9
• A sample to prove: point 1 and point 2.
• Test with a six-sided die
• roll 100000 times(samples). Every time roll once, and get the mean values(the means of the samples).
• Get mean and STD with the 100000 results
• roll 20000 times(samples). Every time roll 50 dies, and get the mean values(the means of the samples).
• Get mean and STD with the 20000 results
• Base on the CLT, the test results should be normal distributed
• Conclusion:
• It doesn’t matter what the shape of the distribution of values happens to be
• If we are trying to estimate the mean of a population using sufficiently large samples
• The CLT allows us to use the empirical rule when computing confidence intervals
## Monte Carlo Simulation
### Finding π
• Think about inscribing a circle in a square with sides of length 2, so that the radius, r, of the circle is of length 1. (Invented by the French mathematicians Buffon (17071788) and Laplace (1749-1827))
• By the definition of π, area = πr^2 . Since r is 1, π = area.
• If the locations of the needles are truly random, we know that,
• $\frac{\text{needles in circle}}{\text{needles in square}}=\frac{\text{area of circle}}{\text{area of square}}$
• solving for the area of the circle,
• $\text{area of circle} = \frac{\text{area of sqaure}\ *\ \text{needles in circle}}{\text{needles in square}}$
• Recall that the area of a 2 by 2 square is 4, so,
• $\text{area of circle} = \frac{4 * \text{needles in circle}}{\text{needles in square}}$
• in this case $\text{area of circle} = {\pi}r^2$, and r=1, so:
• $\pi = \frac{4 * \text{needles in circle}}{\text{needles in square}}$
• more exercise
• In general, to estimate the area of some region R
1. Pick an enclosing region, E, such that the area of E is easy to calculate and R lies completely within E.
2. Pick a set of random points that lie within E.
3. Let F be the fraction of the points that fall within R.
4. Multiply the area of E by F.
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2020-10-20 06:53:32
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https://tikv.org/docs/4.0/tasks/configure/raftstore/
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# Raftstore Config
## Learn how to configure Raftstore in TiKV.
This document describes the configuration parameters related to Raftstore.
### sync-log
• Enables or disables synchronous write mode. In the synchronous write mode, each commit is forced to be flushed to raft-log synchronously for persistent storage.
• Default value: true
Setting this value to false might lead to data loss. It is strongly recommended that you do not modify this configuration.
### prevote
• Enables or disables prevote. Enabling this feature helps reduce jitter on the system after recovery from network partition.
• Default value: true
### raftdb-path
• The path to the Raft library, which is storage.data-dir/raft by default
• Default value: "”
### raft-base-tick-interval
• The time interval at which the Raft state machine ticks
• Default value: 1s
• Minimum value: greater than 0
### raft-heartbeat-ticks
• The number of passed ticks when the heartbeat is sent. This means that a heartbeat is sent at the time interval of raft-base-tick-interval * raft-heartbeat-ticks.
• Default value: 2
• Minimum value: greater than 0
### raft-election-timeout-ticks
• The number of passed ticks when Raft election is initiated. This means that if Raft group is missing the leader, a leader election is initiated approximately after the time interval of raft-base-tick-interval * raft-election-timeout-ticks.
• Default value: 10
• Minimum value: raft-heartbeat-ticks
### raft-min-election-timeout-ticks
• The minimum number of ticks during which the Raft election is initiated. If the number is 0, the value of raft-election-timeout-ticks is used. The value of this parameter must be greater than or equal to raft-election-timeout-ticks.
• Default value: 0
• Minimum value: 0
### raft-max-election-timeout-ticks
• The maximum number of ticks during which the Raft election is initiated. If the number is 0, the value of raft-election-timeout-ticks * 2 is used.
• Default value: 0
• Minimum value: 0
### raft-max-size-per-message
• The soft limit on the size of a single message packet
• Default value: 1MB
• Minimum value: 0
• Unit: MB
### raft-max-inflight-msgs
• The number of Raft logs to be confirmed. If this number is exceeded, log sending slows down.
• Default value: 256
• Minimum value: greater than 0
### raft-entry-max-size
• The hard limit on the maximum size of a single log
• Default value: 8MB
• Minimum value: 0
• Unit: MB|GB
### raft-log-gc-tick-interval
• The time interval at which the polling task of deleting Raft logs is scheduled. 0 means that this feature is disabled.
• Default value: 10s
• Minimum value: 0
### raft-log-gc-threshold
• The soft limit on the maximum allowable count of residual Raft logs
• Default value: 50
• Minimum value: 1
### raft-log-gc-count-limit
• The hard limit on the allowable number of residual Raft logs
• Default value: the log number that can be accommodated in the 3/4 Region size (calculated as 1MB for each log)
• Minimum value: 0
### raft-log-gc-size-limit
• The hard limit on the allowable size of residual Raft logs
• Default value: 3/4 of the Region size
• Minimum value: greater than 0
### raft-entry-cache-life-time
• The maximum remaining time allowed for the log cache in memory.
• Default value: 30s
• Minimum value: 0
### raft-reject-transfer-leader-duration
• The protection time for new nodes, which is used to control the shortest interval to migrate a leader to the newly added node. Setting this value too small might cause the failure of leader transfer.
• Default value: 3s
• Minimum value: 0
### raftstore.hibernate-regions (Experimental)
• Enables or disables Hibernate Region. When this option is enabled, a Region idle for a long time is automatically set as hibernated. This reduces the extra overhead caused by heartbeat messages between the Raft leader and the followers for idle Regions. You can use raftstore.peer-stale-state-check-interval to modify the heartbeat interval between the leader and the followers of hibernated Regions.
• Default value: false
### raftstore.peer-stale-state-check-interval
• Modifies the state check interval for Regions.
• Default value: 5 min
### split-region-check-tick-interval
• Specifies the interval at which to check whether the Region split is needed. 0 means that this feature is disabled.
• Default value: 10s
• Minimum value: 0
### region-split-check-diff
• The maximum value by which the Region data is allowed to exceed before Region split
• Default value: 1/16 of the Region size.
• Minimum value: 0
### region-compact-check-interval
• The time interval at which to check whether it is necessary to manually trigger RocksDB compaction. 0 means that this feature is disabled.
• Default value: 5m
• Minimum value: 0
### clean-stale-peer-delay
• Delays the time in deleting expired replica data
• Default value: 10m
• Minimum value: 0
### region-compact-check-step
• The number of Regions checked at one time for each round of manual compaction
• Default value: 100
• Minimum value: 0
### region-compact-min-tombstones
• The number of tombstones required to trigger RocksDB compaction
• Default value: 10000
• Minimum value: 0
### region-compact-tombstones-percent
• The proportion of tombstone required to trigger RocksDB compaction
• Default value: 30
• Minimum value: 1
• Maximum value: 100
### pd-heartbeat-tick-interval
• The time interval at which a Region’s heartbeat to PD is triggered. 0 means that this feature is disabled.
• Default value: 1m
• Minimum value: 0
### pd-store-heartbeat-tick-interval
• The time interval at which a store’s heartbeat to PD is triggered. 0 means that this feature is disabled.
• Default value: 10s
• Minimum value: 0
### snap-mgr-gc-tick-interval
• The time interval at which the recycle of expired snapshot files is triggered. 0 means that this feature is disabled.
• Default value: 5s
• Minimum value: 0
### snap-gc-timeout
• The longest time for which a snapshot file is saved
• Default value: 4h
• Minimum value: 0
### lock-cf-compact-interval
• The time interval at which TiKV triggers a manual compaction for the Lock Column Family
• Default value: 256MB
• Default value: 10m
• Minimum value: 0
### lock-cf-compact-bytes-threshold
• The size out of which TiKV triggers a manual compaction for the Lock Column Family
• Default value: 256MB
• Minimum value: 0
• Unit: MB
### notify-capacity
• The longest length of the Region message queue.
• Default value: 40960
• Minimum value: 0
### messages-per-tick
• The maximum number of messages processed per batch
• Default value: 4096
• Minimum value: 0
### max-peer-down-duration
• The longest inactive duration allowed for a peer. A peer with timeout is marked as down, and PD tries to delete it later.
• Default value: 5m
• Minimum value: 0
### max-leader-missing-duration
• The longest duration allowed for a peer to be in the state where a Raft group is missing the leader. If this value is exceeded, the peer verifies with PD whether the peer has been deleted.
• Default value: 2h
• Minimum value: greater than abnormal-leader-missing-duration
### abnormal-leader-missing-duration
• The longest duration allowed for a peer to be in the state where a Raft group is missing the leader. If this value is exceeded, the peer is seen as abnormal and marked in metrics and logs.
• Default value: 10m
• Minimum value: greater than peer-stale-state-check-interval
### peer-stale-state-check-interval
• The time interval to trigger the check for whether a peer is in the state where a Raft group is missing the leader.
• Default value: 5m
• Minimum value: greater than 2 * election-timeout
### leader-transfer-max-log-lag
• The maximum number of missing logs allowed for the transferee during a Raft leader transfer
• Default value: 10
• Minimum value: 10
### snap-apply-batch-size
• The memory cache size required when the imported snapshot file is written into the disk
• Default value: 10MB
• Minimum value: 0
• Unit: MB
### consistency-check-interval
• The time interval at which the consistency check is triggered. 0 means that this feature is disabled.
• Default value: 0s
• Minimum value: 0
### raft-store-max-leader-lease
• The longest trusted period of a Raft leader
• Default value: 9s
• Minimum value: 0
### allow-remove-leader
• Determines whether to allow deleting the main switch
• Default value: false
### merge-max-log-gap
• The maximum number of missing logs allowed when merge is performed
• Default value: 10
• Minimum value: greater than raft-log-gc-count-limit
### merge-check-tick-interval
• The time interval at which TiKV checks whether a Region needs merge
• Default value: 10s
• Minimum value: greater than 0
### use-delete-range
• Determines whether to delete data from the rocksdb delete_range interface
• Default value: false
### cleanup-import-sst-interval
• The time interval at which the expired SST file is checked. 0 means that this feature is disabled.
• Default value: 10m
• Minimum value: 0
### local-read-batch-size
• The maximum number of read requests processed in one batch
• Default value: 1024
• Minimum value: greater than 0
### apply-max-batch-size
• The maximum number of requests for data flushing in one batch
• Default value: 1024
• Minimum value: greater than 0
### apply-pool-size
• The allowable number of threads in the pool that flushes data to storage
• Default value: 2
• Minimum value: greater than 0
### store-max-batch-size
• The maximum number of requests processed in one batch
• Default value: 1024
• Minimum value: greater than 0
### store-pool-size
• The allowable number of threads that process Raft
• Default value: 2
• Minimum value: greater than 0
### future-poll-size
• The allowable number of threads that drive future
• Default value: 1
• Minimum value: greater than 0
|
2022-01-22 16:52:41
|
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|
https://www.semanticscholar.org/paper/Meridian-Surfaces-with-Constant-Mean-Curvature-in-Bulca-Milousheva/6cb41d7d05db4b9f31e32cad1b46b1cad19d3b5d
|
# Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-Space with Neutral Metric
```@article{Bulca2016MeridianSW,
title={Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-Space with Neutral Metric},
author={Bet{\"u}l Bulca and Velichka Milousheva},
journal={Mediterranean Journal of Mathematics},
year={2016},
volume={14},
pages={1-21}
}```
• Published 31 May 2016
• Mathematics
• Mediterranean Journal of Mathematics
In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike, spacelike, or lightlike axis and call them meridian surfaces. We give the complete classification of minimal and quasi-minimal meridian surfaces. We also classify the meridian surfaces with non-zero constant mean curvature.
3 Citations
Meridian Surfaces with Parallel Normalized Mean Curvature Vector Field in Pseudo-Euclidean 4-space with Neutral Metric
• Mathematics
• 2016
We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or
Meridian Surfaces on Rotational Hypersurfaces with Lightlike Axis in \${\mathbb E}^4_2\$
We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and
Meridian Surfaces on Rotational Hypersurfaces with Lightlike Axis in E 42
We construct a special class of Lorentz surfaces in the pseudoEuclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and
## References
SHOWING 1-10 OF 30 REFERENCES
Special Classes of Meridian Surfaces in the Four-dimensional Euclidean Space
• Mathematics
• 2014
Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of
Meridian Surfaces of Elliptic or Hyperbolic Type in the Four-dimensional Minkowski Space
• Mathematics
• 2014
We consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. We call these
Timelike surfaces with constant mean curvature in Lorentz three-space
Abstract. A cyclic surface in Lorentz-Minkowski 3-space L is one that is foliated by circles. We classify all maximal cyclic timelike surfaces in this space, obtaining different families of
Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space
• Mathematics
• 2014
In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis — rotational surfaces of elliptic, hyperbolic or parabolic
Chen Rotational Surfaces of Hyperbolic or Elliptic Type in the Four-dimensional Minkowski Space
• Mathematics
• 2010
We study the class of spacelike surfaces in the four-dimensional Minkowski space whose mean curvature vector at any point is a non-zero spacelike vector or timelike vector. These surfaces are
Meridian Surfaces in E^4 with Pointwise 1-type Gauss Map
• Mathematics
• 2014
In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are
Meridian Surfaces of Elliptic or Hyperbolic Type with Pointwise 1-type Gauss Map in Minkowski 4-Space
• Mathematics
• 2014
In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike
Meridian Surfaces of Elliptic or Hyperbolic Type with Pointwise \$1\$-type Gauss Map in Minkowski \$4\$-space
• Mathematics
• 2016
In the present paper we consider a special class of spacelike surfaces in the Minkowski \$4\$-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or
AN INVARIANT THEORY OF MARGINALLY TRAPPED SURFACES IN THE FOUR-DIMENSIONAL MINKOWSKI SPACE
• Physics, Mathematics
• 2012
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving
Screw invariant marginally trapped surfaces in Minkowski 4-space
• Mathematics
• 2009
A spacelike surface in a Lorentzian manifold whose mean curvature vector is lightlike everywhere is called marginally trapped. The classification of marginally trapped surfaces in Minkowski 4-space
|
2022-01-17 07:24:53
|
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|
https://mikespivey.wordpress.com/2011/12/20/combinatorial-totient/
|
## A Combinatorial Proof of a Formula Involving Euler’s Totient Function
Several years ago I found a proof of the following identity, which I had not seen before (and still haven’t seen anywhere else):
$\displaystyle \sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2},$
where $\varphi(n)$ is Euler’s totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$.
Over a year ago I asked for additional proofs of this identity on Math Stack Exchange and got some nice ones. My favorite, though, is a combinatorial proof that I managed to come up with after I had asked the question. (It was a modification of one of my original proofs.)
Proof: Both sides count the number of fractions (reducible or irreducible) in the interval $(0,1]$ with denominator $n$ or smaller.
Right side: the number of ways to pick a numerator and a denominator is the number of ways to choose two numbers with replacement from the set $\{1, 2, \ldots, n\}$. This is known to be
$\displaystyle \binom{n+2-1}{2} = \frac{n(n+1)}{2}.$
Left side: The number of irreducible fractions in $(0,1]$ with denominator $k$ is equal to the number of positive integers less than or equal to $k$ and relatively prime to $k$; i.e., $\varphi(k)$. For a given irreducible fraction $\frac{a}{k}$, there are $\left\lfloor \frac{n}{k} \right\rfloor$ total fractions with denominators $n$ or smaller in its equivalence class. (For example, if $n = 20$ and $\frac{a}{k} = \frac{1}{6}$, then the fractions $\frac{1}{6}, \frac{2}{12}$, and $\frac{3}{18}$ are those in its equivalence class.) Thus the sum
$\displaystyle \sum_{k=1}^n \left\lfloor\frac{n}{k} \right\rfloor \varphi (k)$
also gives the desired quantity.
This identity comes from the more general setting of summing a multiplicative function convoluted with the constant function, that is $1*f(n)=\sum_{d|n} f(d)$ where * represents Dirichlet convolution. Looking at the sum $\sum_{n\leq x}1*f(n)=\sum_{n\leq x}\sum_{d|n} f(d)$, we see that by switching the order of summation, the above equals $\sum_{d\leq x} f(d)\sum_{n\leq x:\ d|n}1=\sum_{d\leq x} f(d) \left[\frac{x}{d}\right]$. Since $\sum_{d|n}\phi(d)=n$, and $\sum_{n\leq x} n =\frac{x(x+1)}{2}$, we recover your identity $\frac{x(x+1)}{2}=\sum_{n\leq x }\phi(n) \left[\frac{x}{n}\right]$.
|
2017-06-25 22:36:07
|
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|
https://stats.stackexchange.com/questions/161277/just-take-the-average-they-say-its-not-that-straightforward-right
|
# Just “take the average” they say. It's not that straightforward, right?
I have an acquaintance who does not study statistics and doesn't understand that summing data and dividing by the number of data is a summary statistic, i.e. that information is lost.
For example, say that there are data which are measurements of some sort: $x_1, ..., x_{100}$. The most common measure of centrality is $$\hat{\mu}_1 = \bar{x} = \frac{1}{100}(x_1 + ... + x_{100})$$ However, if the data are skewed then $$\hat{\mu}_2 = \tilde{x} = \frac{1}{2}(x_{(50)} + x_{(51)})$$ is a better estimator. Then, of course there is the midrange $$\hat{\mu}_3 = \frac{1}{2}(x_{(1)} + x_{(100)})$$ that is an option.
My question is: defining some loss function— for simplicity's sake L2-loss— how to judge which $\hat{\mu}$ is best? Obviously the answer is specific to the data, but what is the MSE of the midrange, for example?
• Good question, but I believe once you've said L2-loss is what you care about, $\hat \mu_1$ is right by definition. I'm hard pressed to say that I would ever use the midrange, because min & max are not very stable; I might consider the midhinge (mean of 1st & 3rd quartile), though. You might be interested in reading this: Which “mean” to use and when? – gung - Reinstate Monica Jul 13 '15 at 21:53
• $\hat \mu_2$ is a strange statistic. If you intend it to be the median, then change "$49$" to "$51$". @Gung The midrange leads to low expected losses for many loss functions when the underlying distribution is close to uniform or symmetrically u-shaped. – whuber Jul 13 '15 at 22:28
• Your question in the last paragraph "defining some loss function— for simplicity's sake L2-loss— how to judge which $\hat{μ}$ is best?" was unclear to me. It seems to be missing some words. However, if you define a loss function, then that will define "what's best" (whatever minimizes the loss function). The MSE of the midrange depends on the distribution (but whether it's best depends on the loss function, since that defines what "best" means - why use MSE to compare, rather than the loss function?) – Glen_b Jul 13 '15 at 22:34
• @ScouserInTrousers what condition are using to define the median as a better estimator? The median is less efficient than the mean. The halfpoint of the range even moreso. – AdamO Jul 13 '15 at 22:45
• @whuber of course that's what I mean. Fixed. – call-in-co Jul 14 '15 at 14:39
This is not a direct answer to your question about loss functions, but I am a Statistician, and I use the jargon of my domain, not the jargon of machine learning. I will attempt to answer the question: "which statistic is the best estimator of the population mean?"
It's incorrect to say generally that the arithmetic mean results in a loss of information. In fact in some circumstances, it can be proven that the arithmetic mean or some function of it contains as much information (Fisher information) as the data themselves. This is the concept of a sufficient statistic, i.e. some summary of the data that is sufficient for the data.
For example, if you know that your data follow a Poisson distribution then the sufficient statistic is $T(X) = X_1 + ... + X_n$. Which is simply the sum of the data. For a Normal distribution where you know the variance then the arithmetic sample mean is the sufficient statistic for the population mean. That is, it contains all of the information and no other statistic will do better. Now granted, we are never in the situation where we know are data are normally distributed and happen to know exactly the variance. But that is why we have the central limit theorem. Even for skewed data, if what you really care about is the population mean, then the arithmetic mean is pretty good bet, especially if you have a lot of observations. So to that end, I would say in a lot of circumstances, especially when you have a lot of observations the arithmetic mean is best if what you care about is the population mean.
Now, if you happen to be in the privileged position of knowing your data come from some other distribution, perhaps some pathological negative exponential distribution, then you're correct there may be a better sufficient statistic. In that circumstance the sufficient statistic for $\mu$ is the minimum observation. This is favorite example of Mukhopadhyay in Probability and Statistical Inference and you will find all exercises you can stomach in there to demonstrate.
To answer your question more generally, about how to choose the best statistic: plot your data. Look at it. Think about where it came from and how it was collected. Think about what it is you are actually trying to make inference on, and whether the way these data were collected is actually appropriate for that. Think about the form your data take: Are they strictly integer data? Proportions with a known denominator? Are they skewed, if so would a log-normal make a for a good approximation? Choose a parametric family that seems to satisfy and caveat if you must.
• "I will attempt to answer the question: 'which statistic is the best estimator of the population mean?'"--that restatement really is key here, otherwise the OP's question is ill-posed. – Andrew M Jul 14 '15 at 5:58
Piggy-backing / building off of Dalton's answer:
Your question, as posed, is incomplete. "Information" as a statistical concept is only defined with reference to an unknown parameter as well as some statistic (function of the full data)—including the degenerate case of the full data. Exactly no statistic of the data has more information about any parameter than the full data, but sometimes statistics have just as much information as the full data about a specific parameter.
Your intuition seems to be that summary statistics reduce the informational content of the full data, but again, there is no informational content to even the full data unless it is with reference to some parameter. It is true that, if you are estimating the variance of a Normal population, the statistic $S^2$ (sample variance) has no informational content about $\mu$ (the population mean). But as pointed out above, $\bar{X}$ contains equal information about $\mu$ as does the full data.
Your question is incomplete because the intuitive, casual definition of "information" is very different from its mathematical definition. For any specific real-world scenario, you will of course have to judge what is an appropriate assumption for the distribution (accounting for skewness, support, etc) and, consequently, what specific statistics preserve the information you need.
As an aside, $\bar{X}$ gives you no information about the variance of a Normal($\mu$, $\sigma^2$) distribution, either, but knowing the pair $(\bar{X}, S^2)$ provides the same information as the full data about both parameters.
|
2021-04-13 20:37:37
|
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|
https://cs.stackexchange.com/questions/78081/syntax-directed-translation-definition-and-actions
|
# Syntax directed translation , definition and actions
I have trouble understanding Syntax directed translation/definition/actions. I am reading dragon book but it confuses me even more.
Having production for simple arithmetic
$$\mathit{expr} \rightarrow \mathit{expr} + \mathit{expr} \mathbin{|} \mathit{expr} - \mathit{expr} \mathbin{|} \mathit{num}$$
$$\mathit{num} \rightarrow 0 \mathbin{|} 1 \mathbin{|} 2 \mathbin{|} \dots \mathbin{|} 9$$
If I understood it correctly, SDT just adds attributes to the nodes and define rules.
for this production rules can looks like
$$\mathit{expr}.\mathit{val} = \mathit{expr}.\mathit{val} + \mathit{expr}.\mathit{val} \mathbin{|} \mathit{expr}.\mathit{val} - \mathit{expr}.\mathit{val} \mathbin{|} \mathit{num}.\mathit{val}$$
$$\mathit{num}.\mathit{val} = 0 \mathbin{|} 1 \mathbin{|} \dots \mathbin{|} 9$$
this all together is called Syntax directed definiton.
Now what starts to confuse me:
It is often written that SDT is used to transfer infix to postfix notation e.g
$\mathit{expr} \rightarrow \mathit{expr} + \mathit{expr}$ will have semantic rule such as $\mathit{expr}.\mathit{val} = \mathit{expr}.\mathit{val} \mathbin{\|} \mathit{expr}.\mathit{val} \mathbin{\|} \text{'+'}$, where $\|$ represents concating of strings.
Is STD strictly defined to trasnform notation to postfix?
Also, where in this fall actions?
By definition
A syntax-directed translation scheme is a notation for specifying a translation by attaching program fragments to productions in a grammar. A translation scheme is like a syntax-directed definition, except that the order of evaluation of the semantic rules is explicitly specified. Program fragments embedded within production bodies are called semantic actions
and example
$$\mathit{expr} \rightarrow^{+} \mathit{expr}\ \ \{\, \mathit{print}(\text{'+'}) \,\}\ \ \mathit{expr}$$
I cannot find what is the usage of this. Which leads to sum of questions: What is STD (rules) for and what are uses of actions in it?
Thanks.
Syntax directed translation uses the hierarchical structure of the input to generate the output. In other words, you exploit the grammar structure to help you to translate the program.
Translating using semantic rules
Semantic actions allow to embed code into a cfg which helps you to translate the program/source code. The following Ruby code should help you to figure out how things work. The following simple translator translates a single line infix expression into postfix expression.
First of all note that I use grammar that is suitable for top-down recursive descent parser, namely without left recursion and unambiguous. Transformation of this grammar is given in your book. So, you have to be familiar at least with a top-down recursive descent parsing method which is simple if you understand and have no problems with direct/indirect recursive calls.
Also please pay attention to how I embedded prints into while loops in order to emit postfix notation. As an exercise try to move the print to another line to get infix notation.
I wrote deliberately in Ruby so that you could play with the code and see it in action and modify it easily. If you have Ruby installed just type ruby parser.rb and see output.
@source = "1*2-3*4"
@index = 0
def get_token()
@token = @source[@index]
@index+=1
end
def exp()
term()
while(@token == '+' || @token == '-')
op = @token #store the operator + or -
get_token() #move to the next input symbol
term()
print op # <= semantic action (print operator + or -)
end
end
def term()
factor()
while(@token == '*' || @token == '/')
op = @token #store the operator
get_token() #move to the next input symbol
factor()
print op # <= semantic action (print operator * or /)
end
end
def factor()
if ('0'..'9').include?(@token) #if @token is a digit
tk = @token #store in temp variable
get_token() #move to the next input symbol
print tk # <= semantic action (print a digit)
else
puts 'Syntax Error'
end
end
get_token()
puts exp()
Example outputs:
1*2-3*4 is translated into 12*34*-
1+2-3*7 is translated into 12+37*-
Computing expression value using node attributes
The following piece of code demonstrates node attributes in action to compute the value of an infix expression. Semantic actions this time around are lines which compute attribute of each tree node, that is, actual value of the expression.
@source = "9*3-3*7-2"
@index = 0
def get_token()
@token = @source[@index]
@index+=1
end
def exp()
exp_t = term()
while(@token == '+' || @token == '-')
op = @token
get_token()
term_t = term()
# compute attribute of the tree node
if op == '+'
exp_t += term_t
else
exp_t -= term_t
end
end
return exp_t #return attribute to one level up in the tree
end
def term()
term_t = factor()
while(@token == '*' || @token == '/')
op = @token
get_token()
val = factor()
# compute attribute of the tree node
if op == '*'
term_t *= val
else
term_t /= val
end
end
return term_t #return attribute to one level up in the tree
end
def factor()
if ('0'..'9').include?(@token) #if @token is a digit
tk = @token #store in temp variable
get_token() #move to the next input symbol
return tk.to_i #to_i means convert to integer
else
puts 'Syntax Error'
end
end
get_token()
puts exp()
Examples:
9*3-3*7-2 results in 4
9*3-3*7 results in 6
1-1+1*1 results in 1
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2019-06-19 17:29:00
|
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https://gigaom.com/2006/03/21/yahoo-messenger-tomorrowscreenshots/
|
# New Yahoo Messenger …Screenshots
I had reported earlier that the new and improved (hopefully) Yahoo Messenger Beta (code named Postman) was going to hit the web this week. This is a Windows PC only release and is still a beta version of the IM+VoIP client is going to be released tomorrow. More photos are here and here. This is Yahoo turning the heat on Skype!
Clearly, the Yahoo has made a strong bid to integrate most of its communications properties into the IM client. I like the fact that you can send emails, make calls, or even send text messages from the same client. These and more such enhancements are the best way for IM services to differentiate from the new, more slick web-based IM aggregators such as Meebo, and still retain unique brand presence.
There is this little icon – Yahoo Music, and if I think what it is, that should make things interesting. Getting radio piped through an IM without the need for a separate client would be pretty cool. Can someone do a blow-by-blow comparison of Yahoo IM and Skype? I am just too tired to make any effort on this today.
Update: USA Today has the story. While Yahoo VP Brad Garlinghouse says that “what sets Yahoo’s service apart from competitors’ is “aggressive pricing,” Mike Masnick thinks that Brad is giving competitors a head fake. “Given everything that the company is integrating into messenger, this makes a lot more sense. They’re trying to make Yahoo messenger your “console” for communications (not just online communications). That’s a lot more interesting and powerful than starting up yet another price war over cheap calls,” he writes this morning.
* Phone Out: U.S. audiences (including Yahoo! en Espanol.) Calls within the U.S. and to more than 30 other countries can be made for two U.S. cents a minute or less. More information is here.
* Phone In For $2.99 a month or$29.90 a year, people can select a personal phone number, and receive incoming calls free. In the beta service, country-based phone numbers are initially available in France, the United Kingdom, and the United States with additional country-based numbers available in the coming months.
* Free Voicemail. Additionally, Yahoo! Mail now includes useful links to Yahoo! Messenger with Voice, enabling people to easily check their voicemail directly from Yahoo! Mail.
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2021-07-31 11:53:38
|
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|
https://math.stackexchange.com/questions/2824762/why-must-x-be-not-free-in-psi-in-order-for-psi-to-phi-vdash-psi-to-fo
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# Why must $x$ be not free in $\psi$ in order for $(\psi\to\phi)\vDash[\psi\to(\forall x\phi)]$?
In the PDF textbook "A Friendly Introduction to Mathematical Logic 2nd Edition" by Christopher C. Leary and Lars Kristiansen, page 53, the first quantifier inference rule (QR) is defined by the ordered pair $(\Gamma,\theta)$ such that $\Gamma=\{\psi\to\phi\}$ and $\theta=\psi\to(\forall x\phi)$ where $x$ is not free in the formula $\psi$ (note this is important). The book provides a proof of $\Gamma\vDash\theta$ on page 56, but I decided to give my own, very similar proof.
My proof goes as follows:
$$\Gamma\vDash\theta\iff(\psi\to\phi)\vDash[\psi\to(\forall x\phi)]$$
Let $\mathfrak{U}$ be some fixed, arbitrary interpretation. Thus, continuing forth,
$$(\psi\to\phi)\vDash[\psi\to(\forall x\phi)]\iff\mathfrak{U}\vDash(\psi\to\phi)\Rightarrow\mathfrak{U}\vDash[\psi\to(\forall x\phi)]$$
If $\mathfrak{U}\not\vDash(\psi\to\phi),$ then we're done, so let's assume $\mathfrak{U}\vDash(\psi\to\phi)$ which we'll donote (Assumption 1). This leaves us to prove $\mathfrak{U}\vDash[\psi\to(\forall x\phi)]$ under (Assumption 1).
$$\mathfrak{U}\vDash[\psi\to(\forall x\phi)]\iff\mathfrak{U}\vDash\psi\Rightarrow\mathfrak{U}\vDash(\forall x\phi)$$
Like previously, if $\mathfrak{U}\not\vDash\psi$, then we're done, so let's assume $\mathfrak{U}\vDash\psi$ which we'll denote (Assumption 2). This leaves us to prove $\mathfrak{U}\vDash(\forall x\phi)$ under (Assumption 2).
\begin{align} \mathfrak{U}\vDash(\forall x\phi)&\iff(\forall s)(\mathfrak{U}\vDash(\forall x\phi)[s]) \\ &\iff(\forall s)(\forall a)(\mathfrak{U}\vDash\phi[s[x|a]]) \\ \text{Let $s=t$ be some fixed, arbitrary} \\ \text{variable assignment function into $\mathfrak{U}$.} \\ &\iff(\forall a)(\mathfrak{U}\vDash\phi[t[x|a]]) &&(1) \end{align}
From (Assumption 1),
\begin{align} \mathfrak{U}\vDash(\psi\to\phi)&\iff\mathfrak{U}\vDash\psi\Rightarrow\mathfrak{U}\vDash\phi \\ &\iff(\forall s)(\mathfrak{U}\vDash\psi[s])\Rightarrow(\forall s)(\mathfrak{U}\vDash\phi[s]) \\ \text{Let $s=t[x|a]$ be the same x-modified} \\ \text{assignment function seen in (1).} \\ &\iff\mathfrak{U}\vDash\psi[t[x|a]]\Rightarrow\mathfrak{U}\vDash\phi[t[x|a]]) \\ &\iff(\forall a)(\mathfrak{U}\vDash\psi[t[x|a]])\Rightarrow(\forall a)(\mathfrak{U}\vDash\phi[t[x|a]]) && (2) \end{align}
From (Assumption 2),
\begin{align} \mathfrak{U}\vDash\psi&\iff(\forall s)(\mathfrak{U}\vDash\psi[s]) \\ \text{Let $s=t[x|a]$ be the same x-modified} \\ \text{assignment function seen in (1).} \\ &\iff\mathfrak{U}\vDash\psi[t[x|a]] \\ &\iff(\forall a)(\mathfrak{U}\vDash\psi[t[x|a]])&&(3) \end{align}
Frome (2) and (3), we can deduce (1) and thus we have proven $\mathfrak{U}\vDash(\forall x\phi)$ and we are done. (I have left out some of the final steps, but the remaining steps should be trivial to complete from here hopefully.) $\square$
I'm pretty sure this proof is correct, but the only thing I'm unsure about is the necessity that $x$ be not free in $\psi$. The proof shown in the book uses this fact to help prove $\Gamma\vDash\theta$, so it makes me uneasy about the validity of my proof. From my proof, it seems unnecessary that $x$ be not free in the $\psi$, but I'm might be missing something. Can someone show me where I'm mistaken within my proof and explain why $x$ must not be free in $\psi$.
• The reason that we need $x$ to not be free is that $x$ could be mentioned in $\psi$ and $\phi$, but we are only adding a quantifier to $\phi$. For example $x = y \to x =y$ but $x = y \not \to (\forall x)[x = y]$. – Carl Mummert Jun 19 '18 at 14:01
• The error is in the equivalence following : "This leaves us to prove $\mathfrak U \vDash [ψ→(∀xϕ)]$ under (Assumption 1)." See page 38, Ex.1.9.1.4 : $\vDash (\phi \to \psi)$ and $\phi \vDash \psi$ are not equivalent. – Mauro ALLEGRANZA Jun 20 '18 at 7:15
Why must $$x$$ be not free in $$ψ$$ in order for $$(ψ→ϕ) ⊨ [ψ→(∀xϕ)]$$ ?
Because with $$x$$ free the inference is not valid.
In order to see why the condition is necessary, we can consider the following counter-example : $$\psi,\phi := (x=0)$$ and consider the stucture $$\mathbb N$$.
We want to show that :
$$(x=0) \to (x=0) \vDash (x=0) \to \forall x (x=0).$$
This means [see page 36] :
if $$\mathbb N \vDash ((x=0) \to (x=0))$$, then $$\mathbb N \vDash (x=0) \to \forall x (x=0))$$.
See page 37 :
This definition is a little bit tricky. It says that if $$\Delta$$ is true in $$\mathfrak A$$, then $$\Delta$$ is true in $$\mathfrak A$$. Remember, for $$\Delta$$ to be true in $$\mathfrak A$$, it must be the case that $$\mathfrak A \vDash \Delta[s]$$ for every assignment function $$s$$.
Now we cam follow your proof considering $$\mathbb N$$ as $$\mathfrak U$$.
let's assume (1) : $$\mathbb N \vDash ((x=0) → (x=0))$$;
this assumption is clearly correct.
And we have to show that $$(x=0) \to \forall x (x=0)$$ is true in $$\mathbb N$$, i.e. that :
$$\mathbb N \vDash (x=0) \to \forall x (x=0))[s]$$, for every assignment function $$s$$.
If $$s(x) \ne 0$$, we are done (the antecedent is False).
The issue is when $$s(x)=0$$, because we want that : $$\mathbb N \vDash (\forall x(x=0))[s]$$.
In order to have this, we need that :
$$\mathbb N \vDash (x=0)[t[x|n]]$$, for every $$n \in \mathbb N$$,
where $$t[x|n]$$ is an "$$x$$-variant" of $$s$$, i.e. is as $$s$$ except for $$s(x)=n$$.
And here we are stuck....
Clearly :
$$\mathbb N \nvDash (x=0)[t[x|1]]$$.
• I still need help finding at which step my proof fails and why it failed. I understand your counter-example, but I'm having a hard time using it to backtrack through my proof and locate the exact error I made. – Kainoa B Jun 20 '18 at 0:13
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2021-01-20 19:46:21
|
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https://booleanzoo.weizmann.ac.il/index.php/Majority
|
# Majority
## Definition
A function $f:\{-1,1\}^n \to \{-1,1\}$ is called a majority function if $f(x)$ returns the most common bit in the input:
$f(x) = \begin{cases} 1, & if ~ \sum_i x_i \geq 0 \\ -1 & otherwise \end{cases}$
For even $n$, the above definition breaks ties in favor of 1, although any arbitrary rule may be used instead.
Majority is a special case of the perceptron function.
## Properties
• Among all monotone functions, majority has the largest influence: $\mathrm{Inf}(f) \leq \mathrm{Inf}(\mathrm{Maj})$ for all monotone $f$ [1].
• TODO: a description of Majority's Fourier Transform. See http://www.contrib.andrew.cmu.edu/~ryanod/?p=877 for details.
• Majority is the unique function that is symmetric, monotone and odd function. TODO May's theorem, credit.
• Majority is not in AC0, even if we allow using mod q functions as gates for prime $q$. [2]
• For every $\varepsilon \gt 0$, Majority can be $\varepsilon$-approximated by a DNF of size $2^{O(\sqrt{n})}$. [3]
## References
1. Ryan O'Donnell, Analysis of Boolean functions, Theorem 32 in section 2.3
2. A. Razborov, Lower bounds on the size of bounded-depth networks over a complete basis with logical addition (Russian), in Matematicheskie Zametki, Vol. 41, No 4, 1987, pages 598-607. English translation in Mathematical Notes of the Academy of Sci. of the USSR, 41(4):333-338, 1987.
3. O’Donnell R., Wimmer K. (2007) | Approximation by DNF: Examples and Counterexamples. In: Arge L., Cachin C., Jurdziński T., Tarlecki A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg
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2020-05-26 19:41:01
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https://handwiki.org/wiki/Polyconic_projection_class
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Polyconic projection class
American polyconic projection of the world
Van der Grinten projection of the world.
Polyconic can refer either to a class of map projections or to a specific projection known less ambiguously as the American polyconic projection. Polyconic as a class refers to those projections whose parallels are all non-concentric circular arcs, except for a straight equator, and the centers of these circles lie along a central axis. This description applies to projections in equatorial aspect.[1]
Polyconic projections
Some of the projections that fall into the polyconic class are:
A series of polyconic projections, each in a circle, was also presented by Hans Mauer in 1922,[3] who also presented an equal-area polyconic in 1935.[4]:248 Another series by Georgiy Aleksandrovich Ginzburg appeared starting in 1949.[4]:258–262
Most polyconic projections, when used to map the entire sphere, produce an "apple-shaped" map of the world. There are many "apple-shaped" projections, almost all of them obscure.[2]
|
2021-10-26 15:52:42
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http://www.physicsforums.com/showpost.php?p=4166881&postcount=3
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View Single Post
P: 2 I know that $x_m = 2 x_M$, so that the small block moves two times as much as the big block. My problem is decomposing the forces.
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2014-04-25 05:38:02
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https://www.esaral.com/chemical-kinetics-jee-main-previous-year-questions-with-solutions/
|
Chemical Kinetics – JEE Main Previous Year Questions with Solutions
JEE Main Previous Year Papers Questions of Chemistry with Solutions are available at eSaral. Practicing JEE Mains chapter wise questions of Chemistry will help the JEE aspirants in realizing the question pattern as well as help in analyzing weak & strong areas. Simulator Previous Years AIEEE/JEE Mains Questions
Q. The half life period of a first order chemical reaction is 6.93 minutes. The time required for the completion of 99% of the chemical reaction will be (log 2 = 0.301) :- (1) 46.06 minutes (2) 460.6 minutes (3) 230.3 minutes (4) 23.03 minutes [aieee-2009]
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Sol. (1)
Q. The time for half life period of a certain reaction A Products is 1 hour, when the initial concentration of the reactant ‘A’ is 2.0 mol L–1, How much time does it take for its concentration to come from 0.50 to 0.25 mol L–1 if it is a zero order reaction? (1) 1 h (2) 4 h (3) 0.5 h (4) 0.29. [aieee-2010]
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Sol. (4) For zero order reaction
Q. Consider the reaction: $\mathrm{Cl}_{2}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{S}(\mathrm{aq}) \rightarrow \mathrm{S}(\mathrm{s})+2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{Cl}^{-}(\mathrm{aq})$ The rate equation for this reaction is rate = $\mathrm{k}\left[\mathrm{Cl}_{2}\right]\left[\mathrm{H}_{2} \mathrm{S}\right]$ Which of these mechanisms is/are consistent with this rate equation? [aieee-2010]
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Sol. (1)
Q. A reactant (A) forms two products : (1) $\mathrm{k}_{1}=2 \mathrm{k}_{2} \mathrm{e}^{\mathrm{E}_{2} / \mathrm{RT}}$ (2) $\mathrm{k}_{1}=\mathrm{k}_{2} \mathrm{e}^{\mathrm{Ea}_{1} / \mathrm{RT}}$ (3) $\mathrm{k}_{2}=\mathrm{k}_{1} \mathrm{e}^{\mathrm{Ea}_{2} / \mathrm{RT}}$ (4) $\mathrm{k}_{1}=\mathrm{A} \mathrm{k}_{2} \mathrm{e}^{\mathrm{Ea}_{1} / \mathrm{RT}}$ [aieee-2011]
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Sol. (2)
Q. The rate of a chemical reaction doubles for every $10^{\circ} \mathrm{C}$ rise of temperature. If the temperature is raised by $50^{\circ} \mathrm{C}, \mathrm$ the rate of the reaction increases by about :- (1) 32 times (2) 64 times (3) 10 times (4) 24 times [aieee-2011]
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Sol. (1) $\mathrm{k}_{2}=\mathrm{k}_{1}(2)^{5}=32 \mathrm{k}_{1}$
Q. For a first order reaction, (A) products, the concentration of A changes from 0.1M to 0.025M in 40 minutes. The rate of reaction when the concentration of A is 0.01 M is : (1) $1.73 \times 10^{-4} \mathrm{M} / \mathrm{min}$ (2) $1.73 \times 10^{-5} \mathrm{M} / \mathrm{min}$ (3) $3.47 \times 10^{-4} \mathrm{M} / \mathrm{min}$ (4) $3.47 \times 10^{-5} \mathrm{M} / \mathrm{m}$ [aieee-2012]
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Sol. (3)
Q. The rate of a reaction doubles when its temperature changes from 300 K to 310 K. Activation energy of such a reaction will be $\left(\mathrm{R}=8.314 \mathrm{JK}^{-1} \mathrm{mol}^{-1} \text { and } \log 2=0.301\right)$ (1) $53.6 \mathrm{kJ} \mathrm{mol}^{-1}$ (2) $48.6 \mathrm{kJ} \mathrm{mol}^{-1}$ (3) $58.5 \mathrm{kJ} \mathrm{mol}^{-1}$ (4) $60.5 \mathrm{kJ} \mathrm{mol}^{-1}$ [J-main 2013]
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Sol. (1)
Q. For the non-stoichiometre reaction 2A + B C + D, the following kinetic data were obtained in three separate experiments, all at 298 K. (1) $\frac{\mathrm{dc}}{\mathrm{dt}}=\mathrm{k}[\mathrm{A}][\mathrm{B}]^{2}$ (2) $\frac{\mathrm{dc}}{\mathrm{dt}}=\mathrm{k}[\mathrm{A}]$ (3) $\frac{\mathrm{dc}}{\mathrm{dt}}=\mathrm{k}[\mathrm{A}][\mathrm{B}]$ (4) $\frac{\mathrm{dc}}{\mathrm{dt}}=\mathrm{k}[\mathrm{A}]^{2}[\mathrm{B}]$ [J-main 2014]
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Sol. (2)
Q. Higher order (>3) reactions are rare due to :- (1) shifting of equilibrium towards reactants due to elastic collision (2) loss of active species on collision (3) low probability of simultaneous collision of all the reacting species (4) increase in entropy and activation energy as more molecules are involved. [JEE-MAIN-(Offline) 2015]
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Sol. (3) Higher order (>3) reaction are rare due to low probability of simulatneous collision of more than three molecuels.
Q. The reaction $2 \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{g}) \rightarrow 4 \mathrm{NO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g})$ follows first order kinetics. The pressure of a vessel containing only $\mathrm{N}_{2} \mathrm{O}_{5}$ was found to increase from 50 mm Hg to 87.5 mm Hg in 30 min. The pressure exerted by the gases after 60 min. will be (Assume temperature remains constant) (1) 106.25 nm Hg (2) 116.25 nm Hg (3) 125 mm Hg (4) 150 mm Hg [JEE-MAIN (Online)2015]
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Sol. (2)
Q. For the equilibrium, $\mathrm{A}(\mathrm{g}) \square \mathrm{B}(\mathrm{g}), \Delta \mathrm{H}$ is is –40 kJ/mol. If the ratio of the activation energies of the forward $\left(\mathrm{E}_{\mathrm{f}}\right)$ and reverse $\left(\mathrm{E}_{\mathrm{b}}\right)$ reactions is $\frac{2}{3}$ then :- (1) $\mathrm{E}_{\mathrm{f}}=60 \mathrm{kJ} / \mathrm{mol} ; \mathrm{E}_{\mathrm{b}}=100 \mathrm{kJ} / \mathrm{mol}$ $|(2) \mathrm{E}_{\mathrm{f}}=30 \mathrm{kJ} / \mathrm{mol} ; \mathrm{E}_{\mathrm{b}}=70 \mathrm{kJ} / \mathrm{mol}$ (3) $\mathrm{E}_{\mathrm{f}}=80 \mathrm{kJ} / \mathrm{mol} ; \mathrm{E}_{\mathrm{b}}=120 \mathrm{kJ} / \mathrm{mol}$ (4) $\mathrm{E}_{\mathrm{f}}=70 \mathrm{kJ} / \mathrm{mol} ; \mathrm{E}_{\mathrm{b}}=30 \mathrm{kJ} / \mathrm{mol}$ [JEE-MAIN (Online)2015]
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Sol. (3)
Q. Decomposition of $\mathrm{H}_{2} \mathrm{O}_{2}$ follows a first order reaction. In fifty minutes the concentration of $\mathrm{H}_{2} \mathrm{O}_{2}$ decreases from 0.5 to 0.125 M in one such decomposition. When the concentration of $\mathrm{H}_{2} \mathrm{O}_{2}$ reaches 0.05 M, the rate of formation of $\mathrm{O}_{2}$ will be :- [JEE – Main 2016]
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Sol. (3) $\mathrm{H}_{2} \mathrm{O}_{2(\mathrm{aq})} \longrightarrow \mathrm{H}_{2} \mathrm{O}_{(\mathrm{aq})}+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g})$
Q. Two reactions R1 and R2 have identical pre-exponential factors. Activation energy of $\mathrm{R}_{1}$ exceeds that of $\mathrm{R}_{2}$ by 10 kJ $\mathrm{mol}^{-1}$. If k1 and k2 are rate constants for reactions $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ respectively at 300 K, then ln $\left(\mathrm{k}_{2} / \mathrm{k}_{1}\right)$ is equal to :- (1) 8 (2) 12 (3) 6 (4) 4 [JEE – Main 2017]
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Sol. (4) From arrhenius equation,
Q. At $518^{\circ} \mathrm{C}$, the rate of decomposition of a sample of gaseous acetaldehyde, initially at a pressure of 363 Torr, was 1.00 Torr $\mathrm{s}^{-1}$ when 5% had reacted and 0.5 Torr s–1 when 33% had reacted. The order of the reaction is : (1)3 (2) 1 (3) 0 (4) 2 [JEE – Main 2018]
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Sol. (4)
• June 13, 2021 at 5:33 am
latest questions plz
0
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2021-06-14 05:45:04
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http://genuinesingaporemaths.blogspot.com/2015/05/s220150523xfds-numbers-that-can-be.html
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## Saturday, May 23, 2015
### [S2_20150523XFDS] Numbers that can be Difference of Squares
Question
Introduction
This is likely an primary mathematics olympiad-type of question, but lower secondary pupils can also try this. It involves deeper thinking. But where do we begin? Sometimes it is good to begin from the beginning, and then follow your nose.
Reminders
Solution
Suppose N is a whole number such that 1 < N < 1000 and N can be expressed as
N = a2b2 = (ab)(a + b)
a difference of squares. So N can be split as a product of two factors (a + b) and (ab). Observe that (a + b) – (ab) = 2b, which is an even number.
The difference between the two factors is an even number. This can only mean that the two factors are both odd or both even. You cannot have one of them odd and the other even, because when you subtract them, you would get an odd number. We now have three cases:-
Case 1a: N is even but not divisible by 4.
Case 1b: N is divisible by 4 (and, of course, is even)
Case 2: N is odd i.e. both (a + b) and (ab) are odd
Ans: 750
Remarks
In the foregoing, it is possible for b to be zero. 0 happens to be a perfect square, because 02 = 0. However, we need not worry about this, because the above algebra is general enough to cover the case where b is 0.
We have solved the problem using logic, even-vs-odd analysis and the three important algebraic identities under reminders (highlighted in orange). We also used the special cases (highlighted in light blue) and made observations based on them.
H04. Look for pattern(s)
H05. Work backwards
H09. Restate the problem in another way
H11. Solve part of the problem
H12* Think of a related problem
H13* Use Equation / write a Mathematical Sentence
Suitable Levels
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2017-11-25 09:31:07
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https://stats.stackexchange.com/questions/346362/what-happens-if-a-is-not-invertible-in-equation-ax-b
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# What happens if A is not invertible in equation Ax=b?
I know that depending on whether consistency condition is satisfied or not we get infinite or no solution. My understanding is that the case for which we have infinite solution are the least squares solution solving normal equation which has inverse of transpose(A)*A, but then if A is not invertible this also won't be invertible.
So how do we get solution in this case ?
I know by hand method in which we eliminate pivotal variables in terms of non pivotal variables and then we can set arbitrary values for non pivotal variables, hence getting infinite solutions.
I want to understand it from the perspective of arriving at the solution by solving normal equation.
So my main point of concern is that (A′A)^-1 = (A^-1)(A'^-1) = (A^-1)(A^-1)' which would require A to be invertible ?
• Typically, $A$ is noninvertible but $A^\prime A$ is. When the latter is noninvertible, there are many methods, but a good keyword to use in searches is "generalized inverse." – whuber May 15 '18 at 15:39
• Isn't (A′A)^-1 = (A^-1)*(A'^-1) which would require A to be invertible ? This is my main point of concern. I have updated it in question too. – Siddharth Shakya May 15 '18 at 18:21
• No. A'A is a matrix, we can't distribute inverse into it. In fact, when we solve the linear system, we do not explicitly calculate the inverse. see here – Haitao Du May 15 '18 at 20:27
• I don't see any help in that link, and for inverse of A'A to exist inverse of A also has to exist. For computing in that case i don't see how will we get pseudo inverse without computing svd in which case eigen-vectors corresponding to 0 eigen-value will not count in the summation. – Siddharth Shakya May 16 '18 at 5:46
• Your logic is incorrect: when $A$ is invertible, then so is $A^\prime A$, but not conversely. A simplest possible counterexample is $A=\pmatrix{1\\0}$ which, not being square, is not invertible, but where $A^\prime A = (1)$ obviously is invertible. – whuber May 16 '18 at 11:36
The pseudo-inverse a.k.a. Moore–Penrose inverse generalizes the matrix inverse for non invertible matrices and even non square matrices. It can be computed using (SVD) singular value decomposition.
When the matrix is invertible, the pseudo-inversion gives the regular inverse of the matrix.
I think OP is confused by $A$ and $A'A$ as @whuber mentioned.
Let $A$ to be the design matrix (for example, if we have 100 data point / persons, and each 2 features /height and weight, $A$ is a $100 \times 2$ matrix)
1. Solving $Ax=b$ will lead to no solutions. NOTE, it is not an underdetermined system, but an overdetermined system!
2. Instead we will solve $A'Ax=A'b$. Note, if $A$ is $100 \times 2$ matrix, $A'A$ is a $2 \times 2$ matrix! There are many nice properties with $A'A$, and if it comes from real data, it is invertable. and the $x$ is the least square solution.
3. For now, let's forget the least square problem. But only consider the math problem: $Ax=b$, i.e., $A$ is not coming from a design matrix transpose times design matrix, it is possible $A$ is not invertable. If that is the case, we can put additional constrains to the system, so we can have unique solutions. Or get one solution from infinite solutions, if that satisfy the needs.
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2019-10-21 22:59:09
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https://www.math.ku.dk/english/calendar/events/algebratopology-seminar-hoekzema/
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# Algebra/Topology seminar
Speaker: Renee Hoekzema
Title: Manifolds with odd Euler characteristic and higher orientability
Abstract: Orientable manifolds have even Euler characteristic unless the dimension is a multiple of 4. I give a generalisation of this theorem: k-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class), unless their dimension is 2^{k+1}m for some m > 0. Here we call a manifold k-orientable if the i^{th} Stiefel-Whitney class vanishes for all i< 2^k. This theorem is strict for k=0,1,2,3, but whether there exist 4-orientable manifolds with an odd Euler characteristic is a new open question. An argument similar to Adams' work on the Hopf invariant one theorem yields that furthermore from k=4 on, m>1. This means that the lowest dimension in which we might hope to find a 4-orientable odd Euler characteristic manifold is 64. I present the results of calculations on the cohomology of the second Rosenfeld plane, a special 64-dimensional manifold with odd Euler characteristic.
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2018-06-19 10:23:58
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http://nxttime.wordpress.com/category/holonomic-wheels/
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Archives for category: Holonomic wheels
I have built a new holonomic robot. It has a smaller wheelbase than Agilis and therefore turns faster. It has a 1:3 gear ratio, it’s maximum controllable speed is about 80 cm/second. It is also very sturdy thanks to all the triangles in its frame. And above all, I think it is prettier than Agilis. However, that is what every father says about his newborn. Below are some pictures
I got some feed-back on my last post with “building instructions” for Agilis. My friend Kirk wondered why I didn’t make proper instructions. He was not impressed by my lame excuses about odd angles or illegal builds and showed me things could be done. He was right, it was about time for me to master some Lego CAD tools. I choose to use LDraw and MLCad for the job. My fears for a steep learning curve proved wrong. The manual is quite good and I was up and running within an hour. The hardest thing to learn was the discipline to work precise immediately. One cannot build a sketch at first and then improve the model. Well, technically you can, but you shouldn’t as every brick you move triggers a chain of new moves that need to be made.
It was fun to create proper building instructions for Agilis. Along the way I also improved the gearbox as was suggested by another reader of my blog, Thimoty. You can freely download the instructions. I do appreciate constructive feedback. This is my first and I am sure there is room for improvement.
By request I created building instructions for Agilis. Well, sort of. It is a series of photos made while taking one of Agilis legs apart. Presented to you in reverse order, so that it looks like a building instruction.
One warning before you start. The gear box can be assambled very easily, but it cannot be disassambled without some tools and, potentially, some damage.
The gear box displayed is for a 1:3.33 gear ratio. Here you find a picture of a 1:2 gear box. I think the 1:2 gear box is a better option.
The part list in the pictures 1 to 5 is for one leg only. Also I did not have the parts to make a nice color scheme so you might end up with a rainbow warrior in black and white if you follow my color scheme.
Please send me a message or picture when you built Agilis.
Today Dexter Industries launched their latest product, an all-color LED called the dLight. I was involved in the development of the dLights so I got them early. I can also give away one set of dLights to one of my readers. More on that later. Let’s take a look at the dLights firs in this video.
I mounted three dLights underneath Agilis, one under each leg pointing to the wheel. I programmed the dLights to give a color that corresponds to the wheel speed of the particular leg. I think the result looks cool.
If you look closely to the sensor ports in the video, you’ll notice that only one is in use. This is one of the great benefits of the dLights, you can daisy chain them. So one free port is all you need to give a robot multiple lights. One set contains four dLights plus the cables to chain them.
As said, I can give away one set of dLights. If you want this set you just have to reply to this message before the first of April.
In case you might wonder how fast or accurate Agilis is, here are some numbers.
Setup
• The gear ratio of the motors to the wheels is 1:2, making the wheels rotate twice as fast as the motors. (It is possible to change the gear ratio to 1:3).
• Prototype orange Rotacaster wheels. This is the hard compound. There isalso a medium compound (gray) and soft compound (black) available.
• The batteries were rechargeable NiMH, 2500 mAh batteries. These were not fully charged.
• The test surface was a clean linoleum floor.
Speed
• Reliable top speed is about 60 cm/sec, equivalent to 2.16 km/h or 1.35 mph. At this speed the robot is still accurate as there is ample margin for the PID controllers of the motors.
• Unreliable top speed is about 75 cm/sec, equivalent to 2.7 kmh or 1.68 mph. At this speed the robot is not very accurate, especially the heading.
Accuracy
• The test track is a square with sides of one meter each. During each run the test track is traveled 4 times. Making the total distance of the test track 16 meters.
• The robot finishes the test track on average within 10 cm of its starting position. Expressed as a percentage of the total distance the error is about 0.6%.
• The movement error is systematic. The robot always ends up above and to the right of the starting position.
• The robot is more accurate at slower speed and acceleration settings.
The images shows the result of the accuracy testing. For each test the robot was placed exactly on the origin (bottom left in the picture). It then traveled a square with sides of one meter for four times, making the total distance traveled 16 meters. The finish location of the robot was then marked on the floor. This test was repeated three times for a single set of settings of speed and acceleration. Three different dynamic sets were used, speed: 50 cm/sec and acceleration at 100 cm/sec^2, speed 50 cm/sec and acceleration at 750 cm/sec^2 and speed 30 cm/sec and acceleration 60 cm/sec^2.
I want to repeat the tests with a 1:3 gear ratio and also with the black Rotacaster wheels.
This Christmas holiday I started working on a new robot, called Agilis. This robot should be a very agile and elegantly moving robot. The frame is based on a triangle equipped with holonomic wheels. So you might think, “What’s new, it is like your last robot?”. From the outside this is true, but it gets new and better brains on the inside. Let me tell you what I envision.
Most robots I built went from point A to point B, only then to decide what to do next. Others just drove around avoiding obstacles. This one should be able to do both at the same time. Agilis must be able to perform complex manouvres, like driving in a straight line while rotating around its centre, or like driving an arc while keeping pointed at an arbitrary spot. It should constantly use sensory input to stay on course, or to alter its course if needed. And all this must go fluently, just like a cat walking through the room.
Over the next several posts I will discuss the different aspects of Agilis. This first post deals with the drive system.
the chassis
Agilis is a three wheeled holonomic platform. This means it can drive in any direction without turning. It can turn around any point, even around its own center. Each wheel is driven by a NXT motor via a gear train that has a 1:2 ratio, the wheels rotate twice as fast as the motors. This makes Agilis a relatively fast robot. The gear train has to be very sturdy to handle the torque of the motors. It also has to be precise to allow for odometry. I used the same setup that I developed for my last robot, Koios the Guard bot.
From robot speed to motor speed
It is not very easy to control a holonomic robot, it takes some math. I created a kinematic model that does just that. The model takes robot speed as input and gives motor speed as output. Robot speed is expressed as speed in x-direction, y-direction and rotational speed. Motor speed is expressed as encoder ticks per second.
So how does this kinematic model look like? For a single wheel this looks like a function that takes the three robot speeds as input. For the three wheels together it looks like a matrix multiplication that multiplies a robot speed vector {xSpeed,ySpeed,angularSpeed} with a kinematic matrix. The resulting vector containers the speed of each of the three wheels. Let’s take a look at the single wheel function first.
To translate robot speed into motor speed one needs to know some physical aspects of the robot, the wheel and the motor. How big is the wheel, how far is it from the center of the robot, under what angle is it mounted, what is the gear ratio of the gear train and what is the number of encoder ticks per full cycle of the motor? With all this information one can write a formula to calculate motor speed from robot speed. Here is the formula.
motorSpeed =
xSpeed * (cosine(wheelAngle) * nEncoderTicks / ( gearRatio * 2 * PI * wheelRadius) -
ySpeed * (sinus(wheelAngle) * nEncoderTicks / (gearRatio * 2 * PI * wheelRadius) +
angularSpeed * distanceToCenter * nEncoderTicks / (gearRatio * 2 * PI * wheelRadius)
This formula might look daunting at first, but on second glance you might notice that there are a lot of constants in it. If you substitute the constants with their respective values you will end up with a much simpler formula.
motorSpeed = xSpeed * aConstantValue - ySpeed * anotherConstantValue + angularSpeed * yetAnotherConstantValue
This formula is not only much simpler, it is also very efficient to calculate, just three multiplications and two additions. A NXT can do this in no time. But remember, these constants are not the same for all the motors because each of the wheels has a different wheelAngle. But, you could also have wheels of different sizes, or differences in any of the other aspects. This means that you will have a formula for each of the motors, each formula is the same in structure but has its own constants. These constants can be stored in a matrix where each row in the matrix contains the 3 constants belonging to a single wheel. The matrix has a row for each of the wheels. If you then take the speed vector and multiply this with the matrix then all formulas are calculated at once and the result, the motorSpeed, is stored in a new vector. Each row in this vector holds the speed of a single motor. In java this matrix multiplication would look like this:
Matrix motorSpeed = kinematicModel.times(robotSpeed);
Wow, now things look simple at last! This is the beauty of matrix algebra.
The same kinematic model can be used to transform robot acceleration into motor acceleration. I use this to make my robot accelerate very smoothly. (the regulated motor class of Lejos supports regulated acceleration).
From tacho counter to robot position
To drive a robot this kinematic model works perfect. But I also want to be able to do things the other way around. I want to be able to calculate robot position from encoder values. At first I couldn’t figure this out at all. The math was just too complex for my mind. That is, until I realized that I just needed to use the inverse of the kinematic model.
deltaRobotPose = deltaMotorPosition * inverseKinematicModel
Here deltaMotorPosition is a vector containing the change in encoder value of each of the motors since the previous measurement. The inverseKinematicModel is the kinematic model inverted. And deltaRobotPose is the change in pose (x and y position and heading) of the robot. Looks simple, doesn’t it? The problem is how to calculate the inverse matrix of the kinematic model. I can’t tell you, because I don’t know. But hey, somebody else already programmed this in Java. I just used the inverse method of the Matrix class.
From the robots coordinates to the worlds coordinates
There is just one more thing to it. The robot can have any heading, this means that x and y coordinates of the robot are not aligned with the x and y coordinates of the world. To be able to steer the robot to a certain position in a room one must be able to convert this location to a location as the robot sees it. The same goes for keeping track of pose. We have seen the formula to calculate change in pose from the wheel encoders. This change however is a change as the robot sees it, not a change in the robots position it the world. The translation from world coordinates to robot coordinates can also be done with a simple matrix multiplication using a rotation matrix. The rotation matrix itself can be calculated from the heading of the robot.
$\begin{bmatrix} cos(heading) & -sin(heading) & 0\\ sin(heading)) & cos(heading) & 0\\ 0& 0 & 1 \end{bmatrix}$
Suppose you want to drive your robot to the left side of the room. The speed matrix in world frame would look like {0, speed, 0}. this can be multiplied with the rotation matrix to get a speed matrix as the robot sees it.
RobotSpeed =worldSpeed * rotationMatrix
If we want to go the other way around, to get the change in pose in world frame we multiply the change in robot frame with the (you guessed it) inverse of the rotation matrix. For rotation matrices the inverse is the same as the transpose of the matrix, the transpose is far more efficient to calculate.
Wrap up
This really is all there is to driving a robot. To sum it up. You have a kinematic model to translate robot speed into motor speed. You have a rotation matrix to translate things from world coordinates to robot coordinates.
The same goes for odometry. You have the inverse of the kinematic model to translate change in encoder values to change in robot pose expressed in robot coordinates. You have the inverse of the rotation matrix to translate change robot pose in robot coordinates into world coordinates.
The kinematic model is a constant, it has to be calculated only once (unless your robot changes shape). The rotation matrix on the other hand has to be updated every time the heading of he robot changes.
The implementation
the robot uses lejos as its brains. Lejos has some excellent classes to drive the NXT motors. The regulated motor class that I used is able to rotate a motor at any given speed. This speed is maintained no matter what the conditions are. It also supports setting an acceleration rate. This is very good for my robot, as for most movements the wheel speed of the three motors is different. If all wheels would accelerate equally, then the slower moving motors would reach their target speed sooner than the faster spinning motors. This results in a robot that shivers and shakes during acceleration (or breaking). All this can be avoided by setting an acceleration value for each of the motors. The ratio of the acceleration values must be the same as the ratio between the (difference between current speed and) target speed of each of the motors. If done properly the robot accelerates very smoothly without jerky movements.
Lejos also has a Matrix class that helps to perform matrix algebra. I used this class to store the (inverse) kinematic models and the rotation matrix. I subclassed it to make a Pose Class that can be used together with the matrix class.
To create the kinematic model I developed a Builder class. This class has all kinds of methods to describe the robot, the wheels the motors and the gear train. When you are done describing the robot, the builder class delivers you a kinematic model and an inverse kinematic model.
To drive the robot I made a kind of pilot class. I plan to discuss it in a future post. This pilot accepts the kinematic model in its constructor.
For odometry I made another class, the Odometer. This class uses the inverse kinematic model.
Remember my plan to make a ball balancing robot? Last year I set myself the goal to make a ball balancing robot. I even build the robot. Since then I wondered off my original goal and made a guardbot, Koios, from this platform. Now I am having another shot at making a balancing robot.
Programming a balancing robot is easy in theory. You just need a sensor that tells you how much the robot is tilted, most often people use a gyro for this. I use my IMU for this, so that I do not suffer from gyro drift. The tilt angle is then feeded to a PID-controller that transformes tilt to motor speed. The hard part is to tune the PID controller, it has to translate tilt into just the right amount of motor speed, too little and the robot falls of the ball, too much and the robot goes over the top and falls of on the other side of the ball. Falling of the ball damages the robot. So I had a problem, how to tune the PID controller without damaging the robot?
To be able to tune the PID-controller without damaging the robot I made a special platform. It is a large disk with a small pole in the middle pointing down Due to the pole the disk will always be tilted when lying on the ground, only when it balances perfectly on the pole it is level. Therefore this disk can be used to tune the controller. The robot can ride off the disk, but it doesn’t fall then, it just comes on the floor with one or two wheels.
When I tested this setup I discovered that the disk whas too smooth, the wheels didn’t have enough grip and slipped. To increase the friction I coated the surface of the disk with sillicon rubber, It is the light blue surface you see in the picture. Now I have a very “slick” surface.I only hope it lasts under the forces the NXT motors generate.But for the moment this problem is solved.
But there are other problems. One is the fact that these holonomic wheels make the robot vibrate, this affects the IMU filter, there is still some drift although it stays within certain limits. I do have prototype rotacaster wheels. The manufacturer told me that the production wheels are more round and generate less vibrations. If you are ever going to by these wheels, and they are a lot of fun, I advice you to take the black ones. They have the best grip. Anyway, I will have to tune the IMU as well.
Tuning PID controllers is difficult and very, very time consuming. There is some theory around tuning PID controllers but in the end it is mostly trial and error. Everytime I want to try a new set of parameters I’ll have to modify the program, download it to the brick, run the program and evaluate the results by watching the robot. It is hard to understand what goes wrong when you see the robot ride of the disk and make a run for the door to the staircase.
But not anymore. Kirk, one of the developers of Lejos made a very nice program that allows you to tune a running PID controller during over bluetooth. The tool is still under development so you won’t find it in Lejos 0.9.1 yet. This program is an add-on to the charting logger I often use to evaluate internals of the robot on the PC. So basicly, this program shows me what is going on in my robot and allows me to modify PID parameters on the fly. I think this is a great tool. Below is a screen shot of it.
So, now I have the robot, a test platform and a efficient tuning tool. That must mean immediate succes! Well, to be honest I don´t think so. I´m still not sure if I can get this robot to work as there are problem with weight and inertia as well. The robot weigths 998 grams. This is quite heavy, even for three powerful NXT motors. The robot is quite stiff, but there it still bends a bit under weight. This affects the IMU sensor. And I´m working on other projects as well. So in the end I think there is a bigger chance to fail than to succeed.
To be continued.
This time I want to introduce you to Koios the guard bot. Koios guards my house, it detects intruders and scares them away.
To perform this complicated task I have split Koios’ behavior into different subtasks and put these in a subsumption architecture.
The first task of Koios is to map its surrounding using two range sensors. These are an ultrasonic sensor for long range (<200 cm) and a Mindsensors dist sensor for accuracy on the short range (<80 cm). To map the surrounding Koios makes a full turn in place while scanning. The resulting map is what I call a circular range map. This means that the map stores the distance to the nearest obstacle for all directions (I recognize 24 directions, each 15 degrees wide). The map looks like a radar scan when plotted. This map is not permanent, it will be thrown away when Koios moves. As Koios does not try to build and maintain a more permanent map of its surrounding I did not have to deal with uncertainties about its absolute position. Therefore the mapping algorithm could be kept simple.
The second task of Koios is to find a safe spot on the map and then to move to this spot. A safe spot for Koios is a spot where Koios is surrounded by objects. A place next to a wall is good, a place in a corner is better. Koios finds a safe spot by examining the map for a place between obstacles. When the safest spot on the map is found Koios travels to this place in a straight line.
Once arrived at the new location Koios again makes a map of the surrounding. If, at second glance, the current location is safe enough then Koios will stay there. If not, it will try to find an even safer spot. This process is repeated until a location is found that is safe enough for Koios.
The video below shows Koios scanning the area and finding a safe spot. I actually had some trouble shooting the video. At first I had placed my webcam on a chair to shoot the video. But to my surprise Koios hid itself under the chair. This indeed is a very safe spot, but it was outside the frame of the video. In the end I placed the camero on my desk to shoot the clip.
When Koios has found a really safe spot it will start guarding the area. It will slowly turn in place while scanning the area again. If an obstacle is detected its location will be compared to the map. When the obstacle is already on the map it will be regarded as a part of the surrounding and ignored. If on the other hand it isn’t on the map then it must be something new and it will be treated as an intruder.
The task of guarding is rather complicated as there is always some uncertainty in both the map and the range sensor data. Suppose the map tells Koios that there is an obstacle at 150 cm and the range sensor detects an object at 145. Is this an intruder or part of the surrounding? The choice Koios makes is based on statistics. To support this kind of decision making a range map stores more information than just the range to the nearest object. It also stores statistical quantities like the number of measurements taken at the direction and the variance in measured ranges. This makes it possible to calculate the probability of an obstacle being new, an intruder, or old, part of the surrounding. If part of the surrounding the measurement is used to improve the map even further.
But if the object is an intruder Koios will home in on it! Koios will quickly run to the intruder until it is very close. I haven’t written this part of Koios’ behavior yet. So everything you read from now on is just on the drawing board.
For one thing I have not decided yet if Koios just runs blindly to the location where the intruder was detected or that it tries to maintain a lock on the intruder while homing in on it. It would be nice if Koios was able to maintain a lock. But it will also be complicated. Mayby I could use a heat sensor like some air-to-air missiles do to maintain a lock on the intruder.
Anyway, once close to the intruder Koios will scare it away using both light and sound. First it will mimic some kind of scary animal with eyes that glow in the dark while making scary noises. Then it will mimic the police using flash lights and and a siren. Then it will map its surrounding again to find a save spot and make a hasty retreat.
Last week I spent my evenings redesigning my holonomic platform. It had te be wider in order to get the center of gravity (relatively) lower. I also wanted to gear it up to make it more agile. And I wanted the brick to be better accessible, especially the USB port. Two other demands remained from Sidbot, the wheel angle should be adjustable so that the robot can be used both on a flat surface and on top of a large ball. It also had to be sturdy.
As it had to be sturdy I decided that the motors should be fixed to the frame. Sidbot had its motors adjustable to make the wheel angles adjustable. The wheels have to be adjustable on the new robot as well, this meant that the hinge point had to be between motor and wheels somewhere in the gear train. I tried different designs and gears but I always ended up with grinding gears. At last I ended up using a 5×7 liftarm to house part of the gear train. This effectively stopped the grinding but resulted in a very wide wheel housing as well. This is not so pretty so I’m still trying to improve this part. However, I now got a 2:1 gear ratio. With a motor speed of 120 RPM and a wheel diameter of 48 mm this gives the robot a top speed of 30 cm per second.
The frame of the robot consists of two triangles stacked vertically. The triangles are made of three 11 stud liftarms connected at the ends with 3 and 4 stud liftarms. This makes a very rigid frame, the brick lies on top of it making it easy accessible. The motors are mounted horizontally with the flat side down. This gives the robot width and also the ground space that is needed when riding on a ball. To prevent torsion between motor and frame I made a housing with L-shaped for the motors.
I used light bluish gray and dark bluish gray for the color scheme as these are the colors the motors and brick are made from. The result is a bit dull but still rather nice looking. It resembles a Starwars like vehicle. Maybe I should mount some laser guns on top.
The resulting robot does meet all my design specifications. But I have not been able to test it yet as I’m one 40 teeth gear short. I hope to get it this week.
The robot still needs a name. If you have a good suggestion you can post it in the remarks. There is a nice price for the winner a second hand Hitechnic gyro sensor. Submit your entry before November 2011.
As you might know from a previous post I want to put Sidbot on a ball. It must be able to balance on top of it without falling of. If it knows how to do that it must also learn to wander around. There are a number of things I must accomplish to reach this ultimate goal. Here I’ll describe what has been done and what must be done. In later posts I might zoom in on some of the tasks.
The platform
Sidbot has been built. You can see photo’s of it on previous posts. I am able to change the angle of the wheels, this enables me to adapt Sidbot to different ball sizes. I also made a spreadsheet that calculates the exact ball size needed for a given setup.
The ball
I bought a fitness ball with a diameter of 65 cm. My guess is that the bigger the ball the easier it is to balance it. Bigger balls will also be heavier and more difficult to get in motion, having more inertia. The ball also has a flat sticky surface that give grip to the wheels. It is a bit bouncy though.
The sensor
To balance on a ball Sidbot must know what is up and when it is tilting over. A lot of NXT based balancing bots measure the distance to the ground for tilt information. This technique cannot be used for Sidbot, it is a long way from the ground and the ball is in the way when looking at the ground. Therefore I will use gyro sensors to get tilt information.
I need at least two gyro’s, one to detect tilt in the y direction
(rear to front) and one to detect tilt in the x direction (left to
right). I also want to measure the rate of turn around the z axis (down to up) . This is not needed for balancing but for keeping a fixed heading when it is moving. Currently I just have the Hitechnic gyro sensor. This measures rate of turn in just one direction. Instead of buying two more of these I will built my own sensor that takes care of all three axes. This occupies less space and sensor ports, has a higher sampling rate, might be more accurate and also includes a 3 axis accelerometer. It is the IMU digital combo board from Sparkfun.
By chance this sensor fits exactly into the slide on the inside of the lego sensors. I will sacrifice my (never used) sound sensor to house it. Thus far my attempts to use this sensor with the NXT have been unsuccessful. The problem is the different voltage level of the sensor (3.3V) and the odd pull ups required by the NXT. I also lack experience with electronics. But I won’t give up that easy.
The PID controller
To balance on the ball Sidbot needs to adjust its position constantly. To do so sensory information from the gyro’s has to be translated into motor commands. For this I will use a PID controller. PID controllers are
very easy to program (that goes for number two and later) but they are hard to tune. And every time the controller fails Sidbot will hit the deck quite hard. So I want a quick way to find the optimal settings for the PID controller. My idea is to use a genetic algorithm to find the best settings. The algorithm works like this:
1. Generate 10 random PID parameters.
2. Try each of the PID parameters and measure how long
Sidbot stays on the ball.
3. Throw away the 5 least successful parameters.
4. Duplicate the other 5 parameters but change them a little bit when duplicating.
5. Start over at point 2.
This algorithm should run until a successful set of parameters is found. During Christmas holidays I developed as small function library that implement the steps mentioned above. I tested this library to find PID values for another controller that Sidbot uses. This PID is used to keep and adjust the heading of Sidbot. In the end this worked surprisingly well. But I also found out that it is not easy to quantify the success of a PID controller. Does it have to be fast, can it overshoot the desired position or not, etc?
I expect some difficulties with this in the future. However, now I got myself a good set of PID parameters for maintaining/adjusting the heading of Sidbot.
That’s my progress thus far.
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2013-06-20 02:22:06
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http://dyz2102-blog.logdown.com/posts/1065684
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Day02 Week03
Objective
Today we were given clear instructions on how to build the order function (i.e. when a user decides to check out and make a purchase). I spent the morning finishing off the remaining 4 tasks left over from yesterday. Xdite explained that we were actually expected to finish all 6 yesterday, and that the instructions to add order function can be done separately from yesterday's tasks. But I still felt compelled to finish in the order the assignments were given.
Luckily, I figured out the 4 tasks pretty quickly (1. will not have duplicate items in cart 2. can change number of a certain item in cart 3. cannot purchase an item which has 0 quantity, 4. cannot add more quantity than what is in stock), though these functions can be improved (which I will get to when I have more time).
Then I followed the instructions closely to build out the order function, which involved creating many migrations and establishing pretty complex relationships between models.
Reflective
In general I felt pretty good about myself as I completed one task after another, with occasional help from classmates to confirm if I'm on the right track (I've been going to the TA less frequently now).
Interpretive
Some key learning points today:
1. "Member" vs. "Collection" - in routing, "member" sets a particular action routes for individual unit of a resource, while "collection" establishes the action for the whole resource.
2. Redefining a variable - saying a + 1 doesn't change the value of a, but a = a + 1 does. And when a variable's value has been changed, it needs to be saved with .save method.
3. Actions defined in the model can be called in the controller.
Worth noting a lot of these concepts were realized from talking with classmates.
Decisional
One key hurdle that remains is figuring out how to call a particular method or data. For instance, what pre-defined method can be called in which of the MVC type of file? What is the difference between "CartItem", "cart_item", "cart_items", "@cart_item", and "@cart_items"? When to use "current_cart" vs. "cart" vs. "@cart"? I really hope to get a better hang of these over the course of this week.
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2018-07-23 11:48:19
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http://stats.stackexchange.com/questions/35/modelling-a-poisson-distribution-with-overdispersion/72
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# Modelling a Poisson distribution with overdispersion
I have a data set that I'd expect to follow a Poisson distribution, but it is overdispersed by about 3-fold. At the present, I'm modelling this overdispersion using something like the following code in R.
## assuming a median value of 1500
med = 1500
rawdist = rpois(1000000,med)
oDdist = rawDist + ((rawDist-med)*3)
Visually, this seems to fit my empirical data very well. If I'm happy with the fit, is there any reason that I should be doing something more complex, like using a negative binomial distribution, as described here? (If so, any pointers or links on doing so would be much appreciated).
Oh, and I'm aware that this creates a slightly jagged distribution (due to the multiplication by three), but that shouldn't matter for my application.
Update: For the sake of anyone else who searches and finds this question, here's a simple R function to model an overdispersed poisson using a negative binomial distribution. Set d to the desired mean/variance ratio:
rpois.od<-function (n, lambda,d=1) {
if (d==1)
rpois(n, lambda)
else
rnbinom(n, size=(lambda/(d-1)), mu=lambda)
}
(via the R mailing list: https://stat.ethz.ch/pipermail/r-help/2002-June/022425.html)
-
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2013-12-05 04:55:44
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https://www.abs.gov.au/statistics/detailed-methodology-information/concepts-sources-methods/producer-and-international-trade-price-indexes-concepts-sources-and-methods/2022/chapter-3-technical-methodology/imputation-theory-and-methodology
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# Imputation theory and methodology
Latest release
Producer and International Trade Price Indexes: Concepts, Sources and Methods
Reference period
2022
Across all indexes, missing price observations occur on a regular basis and this could be due to factors such as:
• temporary out of stock items,
• discontinued items,
• or seasonal elements.
In any price reference period, these factors can make it impossible to obtain a price measure for a particular product.
The ABS employs a number of imputation methods to address temporarily missing observations within price indexes. These include:
• the imputation a movement for the product based on the price movement for all other products in the sample
• the use the price movement from another price sample, or
• repeat the previous period’s price of the product (also called carry forward method).
These options are known as imputation.
Their purpose is to calculate a price for the temporarily missing product. The aim of imputation is to provide prices such that the resulting movement in the price index is the same as would have been calculated had all prices been observed. In achieving such a result, it is necessary to make an assumption regarding the price behaviour of the temporarily missing product.
### Imputation from price sample
The rationale for imputing a price movement from other products in the sample is that products are bought and sold in a competitive marketplace and in those cases where an individual product has not been observed in the current period, it is assumed that its price behaviour is reflected by similar products in the sample. The design of elementary aggregates to contain products that are homogeneous in terms of price behaviour (as noted above) ensures that the assumption underlying this method of imputation is generally robust.
Imputing from other products in the sample is also mathematically equivalent to excluding the product, for which a price is unavailable in one period, from both periods involved in the index calculation. It strictly maintains the ‘matched sample’ concept.
In order to impute a movement resulting from excluding the product it is necessary to construct a measure of price change from the previous period to the current period for those products common to both periods. This calculation is dependent upon the price index formula used for the elementary aggregate.
When the elementary aggregate is compiled using a Laspeyres formula, it is first necessary to derive the implicit quantity shares underlying the weights of the matched products. This can be achieved by dividing the weight for each product by its reference period price.
The resulting quantity shares for the matched products are then used to calculate the price change from the previous period to the current period.
$$\Large s_{q,i}=\frac{\frac{w_i}{p^0_i}}{\sum_\limits {MATCHED} \frac{w_i}{p^0_i}}$$
$$\large M^t_{t-1} = \frac{\sum_\limits {MATCHED}s_{q,i}p^t_i}{\sum_\limits {MATCHED}s_{q,i}p^{t-1}_i}$$
$$\large \hat{p}_j^t=M^t_{t-1}\times p^{t-1}_j$$
where $$S_{q, i}$$ is the implicit quantity share in the reference period for matched product $$i$$$$w_i$$ is the weight for matched product $$i$$$$p^0_i$$$$p_i^{t-1}$$,$$p^t_i$$ are respectively the reference period price, previous period price, and current period price for matched product $$i$$ (at time t), $$M^t_{t-1}$$ is the price movement between the previous and current period for the matched products, and $$\hat{p}^{t-1}_j$$ is the imputed price for missing product $$j$$ at time $$t$$.
An example of this calculation is shown in Table 3.7 below.
Table 3.7 Example of imputation from other products in the price sample
Reference Period Value ShareReference Period Price (\$)Previous Period PriceCurrent Period Price
Product A305812
Product B60101620
Product C1024n.a.
Implicit quantitiesImplicit quantity shareShare x Previous Period PriceShare x Current Period Price
Product A60.54.06.0
Product B60.58.010.0
Total 12.016.0
Movement 1
Current period Price after imputePrice relative after imputeWeight x relative
Product A122.472
Product B202120
Product C5.3332.66666726.66667
Laspeyres price index 218.6667
### Imputation from another price sample
The second approach to imputation for the Producer and International Trade Price Indexes is to use the price movement from another related sample or comparable product. This approach is used in cases where price changes from a comparable product (or products) from a similar type of provider can be expected to be similar to the missing product.
Carry forward imputation
The rationale for adopting a carry forward imputation is that failure to observe a price for a product reflects no transactions for the product, and hence there can be no price change. However, each product in the price sample represents similar products purchased and sold elsewhere in the marketplace, and such an assumption does not hold in most cases. Application of this method of imputation when transactions are actually occurring in a marketplace (but not observed by the sample) consistently biases the index towards zero (that is, biased downward when prices are rising and biased upward when prices are falling).
It is for these reasons that the price statisticians apply this imputation mechanism only under specific conditions where it is known that failure to observe a transaction means that no transactions are occurring (such as where there is only one sale per year of a type of agricultural crop, for example, or where the price changes only once per year during annual price setting).
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2023-03-23 21:49:31
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https://experts.mcmaster.ca/display/publication1968538
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# Operator algebras with hyperarithmetic theory Academic Article
•
• Overview
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• Research
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• Identity
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• Additional Document Info
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• View All
•
### abstract
• Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma$ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma$ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma$ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).
### publication date
• March 21, 2021
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2021-12-06 08:06:31
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https://cs.stackexchange.com/questions/96742/interchangeable-role-of-public-private-key-pair-and-encoding-decoding-algorithms
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# Interchangeable role of public/private key pair and encoding/decoding algorithms
In designing a public-key cryptosystem, there are often two desirable properties:
1. Public/private key pair of each participant can be exchanged. Meaning that whenever a participant has generated a key pair $(a, b)$, it is free up to him to choose to public either $a$ or $b$ (but not both, of course)
2. Encoding/decoding algorithms $(E, D)$ are interchangeable. It is up to all participants to choose either $E$ or $D$ to be the encoding algorithm, then automatically, $D$ or respectively, $E$ will be the decoding algorithm. But the choice must be made prior to any communication.
Note that if $D$ and $E$ are identical, for e.g. $\mathrm{RSA}$, then the property 2 is vacuously true.
The question is: Assume that $E$ and $D$ are different, is there any known public-key cryptosystem that satisfies property 1 but violates property 2.
Informally, the cryptosystems that we are interested are the ones in which the key generation is in some sense symmetric, but the encoding and decoding algorithms are not.
EDIT: We only consider deterministic encryption scheme. To be meaningful, we assume $\mathrm{M} = \mathrm{C}$ and $\mathrm{PU} = \mathrm{PR}$ where $\mathrm{M}$ is the message space, $\mathrm{C}$ is the ciphertext space, $\mathrm{PU}$ and $\mathrm{PR}$ are public and private key space.
• Obviously it's an interesting theoretical problem to study, but why would these properties be “often desirable”? As an implementer, I've yet to encounter a case where this would be useful. Even RSA doesn't have property 1 in practice because the public key almost always uses a small, guessable exponent. – Gilles Aug 29 '18 at 6:35
• Also what do you call “encoding” and “decoding”? Asymmetric schemes can be used for multiple purposes, the most common ones being encryption and signature. It isn't clear to me that these can be broken up into an “encoding/decoding” step and another step of — what would it be? padding? – Gilles Aug 29 '18 at 6:38
• I've made some edit to better narrow down the intended terms. We only consider theoretical aspects of deterministic encryption scheme. There is NO non-trivial adversarial model here. In fact, the adversary has nothing except the public key, but he does not need to recover the private key, only aims at successful decryption of non-negligible amount of ciphertext. – Thinh D. Nguyen Aug 29 '18 at 7:16
• Also, surely the practical implementation of RSA with small exponent $e$ is not considered here. – Thinh D. Nguyen Aug 29 '18 at 7:17
• For example, OAEP is introduced to solve the following problem: if $c$ is the ciphertext of the plaintext $m$, then $cr^e$ is the ciphertext of $mr$. The plaintext is still not revealed to Eve. But she gains some control to the output ciphertext. – Thinh D. Nguyen Aug 29 '18 at 7:35
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2019-08-20 16:36:24
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https://www.gamedev.net/forums/topic/608772-zoompanning-with-scaletranslate-transforms/
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Public Group
# Zoom+Panning with Scale/Translate Transforms
This topic is 2559 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Wow its been a while since ive come here to ask a question (atleast it feels like it)
So I am trying to create a zoom and panable makeshift viewport using an SFML shape (for now) the panning is fine its just the zooming..
My initial try to solve the problem was to translate to the position of the cursor (i.e. the place I wanted to zoom into) and do a directional scale of the shape so that so what ever is under the cursor will scale relatively uniformly and the whatever is around it will scale out from that position creating the illusion of a zoom-to-point mechanism. unfortunatly this resulted in some strange goings on. I believed this to be my forgetting to take into account the current position of the of the makeshift-shape-viewport which i was correct, but alas my solution was not (Position = Origin) while this went some way to solving my problem my because the zooming began to work as I had originally expected.. however it broke the panning functionality because each time i try to zoom the position of the box is set to "(0, 0)" making it seem as though the viewport is fixed in place when zooming (if you zoom on certain areas of the parent viewport (in this case a WinForms panel) it will snap the makeshift one back into its original place and continue to zoom normally.
Finally I decided that somehow I have to find the position prior to the transform (i.e the position in the previous coordinate system) and find that relative to the new frame of reference after the transformation has taken place.. I drew a little diagram of what a translation looks like and realised that I may be able to get this unknown by performing the inverse opertion on the origin and assigning that to the position.. this did not work to well.. so now I am stumped (or its possible that I did it incorrectly..)..
PS: Im going for something like what Gamemaker 8.1 has in its map editor (downloaded it the otherday because i needed to create some animations, then i found something else )
##### Share on other sites
see my post
And panning:
[font="Courier New"]
ViewCenter.x += MouseDX * CurrentZoom;
ViewCenter.y += MouseDY * CurrentZoom;[/font]
Where the Mouse delta values are in screen coordinates.
The zooming and panning is achieved through the projection matrix:
[font="Courier New"]glOrtho(
ViewCenter.x - width* CurrentZoom,
ViewCenter.x + width* CurrentZoom,
ViewCenter.y - height* CurrentZoom,
ViewCenter.y + height* CurrentZoom,-100,100);[/font]
Where width and height are the half sizes of the viewport.
I hope that helps.
##### Share on other sites
see my post
And panning:
[font="Courier New"]
ViewCenter.x += MouseDX * CurrentZoom;
ViewCenter.y += MouseDY * CurrentZoom;[/font]
Where the Mouse delta values are in screen coordinates.
The zooming and panning is achieved through the projection matrix:
[font="Courier New"]glOrtho(
ViewCenter.x - width* CurrentZoom,
ViewCenter.x + width* CurrentZoom,
ViewCenter.y - height* CurrentZoom,
ViewCenter.y + height* CurrentZoom,-100,100);[/font]
Where width and height are the half sizes of the viewport.
I hope that helps.
If you could forgive me, I am not quite as well endowed with mathematical talent such as yourself, though from the thread you linked I read through and I dont think we are suffering from the same problem exactly.
I should have posted this earlier but here it is:
C#/.Net/Winforms/SFML.Net
void PanelView_MouseWheel(object sender, MouseEventArgs e) { float oldzoom = zoom; float zoomfactor = (float)(e.Delta/120) * 0.1f; zoom = Math.Max( 0.1f, Math.Min( 3.0f, zoom + zoomfactor ) ); SFML.Window.Vector2f origin = Map.Position; Map.Origin = new SFML.Window.Vector2f( e.X, e.Y ); Map.Scale = new SFML.Window.Vector2f( zoom, zoom ); Render( ); } void PanelView_MouseMove(object sender, MouseEventArgs e) { if ( e.Button != System.Windows.Forms.MouseButtons.Left || ModifierKeys != Keys.Control) return ; Map.Position = new SFML.Window.Vector2f (e.X - DragDiff.X, e.Y - DragDiff.Y); Render( ); }
What im trying to do is have the map (i.e. just a rectangle with a few gridlines across it) behave as though it were an object of the view which I am trying to zoom not the view itself, so i can zoom somewhere outside of the map area and it would (if position your trying to zoom to is far enough and the zoom level deep enough) go out of the actual viewing area the map is "a child of"
I will continue to study your post(s) and see if I can figure this thing out..
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JoeJ
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• ### Forum Statistics
• Total Topics
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2018-08-20 07:30:13
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https://indico.fnal.gov/event/22303/contributions/243741/
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# Seattle Snowmass Summer Meeting 2022
Jul 16 – 26, 2022
US/Pacific timezone
### Support
Jul 18, 2022, 7:00 PM
2h 20m
211 South Ballroom (HUB)
### Speaker
Stefan Knirck (Fermi National Accelerator Laboratory)
### Description
We present a novel dish antenna for broadband ~$\mu$eV-eV range axion and wave-dark matter detection, which allows to utilize state-of-the-art high-field solenoidal magnets. At these masses it is difficult to scale up traditional resonator setups to the required volume. However, at metallic surfaces in a high magnetic field dark matter axions can convert to photons regardless of axion mass. These photons can be successively focused onto a detector (dish antenna concept). We present progress on BREAD, a dish antenna proposal with a $\sim 10\,{\rm m}^2$ conversion area and a novel rotationally symmetric parabolic focusing reflector designed to take advantage of high-field solenoidal magnets. We recently demonstrated [PRL 128 (2022) 131801] that this concept has the potential to discover QCD axions of several decades in mass range. Besides the experimental concept this poster shows our progress towards first stage hidden photon and axion pilot experiments for two distinct frequency ranges - GigaBREAD and InfraBREAD - with expected sensitivities to unexplored coupling strengths. We detail R&D on reflector characterization, horn antenna & sensor testing and signal readout. We also outline sensitivity estimates for future large-scale versions.
In-person or Virtual? In-person
### Primary author
Stefan Knirck (Fermi National Accelerator Laboratory)
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2022-11-29 02:09:33
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https://quant.stackexchange.com/questions/45555/geometric-brownian-motion-price-probabilities
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# Geometric Brownian Motion - Price Probabilities
I am modeling a stock price that follows Geometric Brownian Motion and have the following:
$$E(X)$$ = .16 (16%)
$$\sigma$$ = .24 (24%)
$$X_0$$ = 95
$$T$$ = 1 (12 months)
I am trying to find the probability that the price of this stock will be below 93 at the end of this time period. I am calculating this analytically, using the Log Normal Distribution given as the following:
$$P(X,t)$$ = $$1\over X \cdot1\over {\sigma \sqrt{2 \pi t}}\cdote^{-(ln(x)- ln(x_0)-(\mu- \sigma^2 /2)t)^2}\over 2\sigma^2t$$
I can plug in the values as the following:
$$P(X,t)$$ = $$1\over X \cdot1\over {(.24) \sqrt{2 \pi (1)}}\cdote^{-(ln(x)- ln(95)-((.16)- (.24)^2 /2)(1))^2}\over 2(.24)^2(1)$$
But then I am still left with the X. My question, is this just the 93 value that should be plugged in? Would this represent the probability of the price being below 93 after this time period? What if we wanted to find the probability that the price would close above this 93 (just 1 - this probability)?
knowing that the log of the prices in a GBM follows the following normal distribution:
$$\operatorname{ln}(S_t) \sim N\left(\operatorname{ln}S_0 + T*\left( \mu - \frac{\sigma^2}{2} \right), \sigma^2 T \right)$$
You can create a normal distribution with these values and then check the CDF. Here is the python code:
from scipy.stats import norm;
mu=0.16; sigma=0.24;S_0=95;T=1
my_var=sigma**2*T
my_norm=norm(np.log(S_0) + (mu-sigma**2/2)*T,np.sqrt(my_var))
my_norm.cdf(np.log(93))
from this normal distribution you get the CDF value for log(93) since you want to know the probability of values below 93, it is 0.26260905311083976
and this probability is time dependent if instead of 1 year it was for 6 months then $$T$$ would be 0.5.
And yes, the probability of the price being above 93 is the complementary of that, i.e. 1-0.26.
Just like the normal density, this will give the probability density of x=93. So to find the probability of $$P\left[ S\le 93\right]$$, you will need to calculate the cumulative probability. See some discussion here. https://math.stackexchange.com/questions/2445900/probability-from-log-normal-distribution
Also try the Matlab free page here: https://uk.mathworks.com/help/stats/logncdf.html to get an understanding of the log normal probabilities, and then just look up the equivalent in whatever software you are using. Excel has a function as well.
• Thank you for your note. – QFII May 12 at 14:50
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2019-12-14 07:04:08
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http://www.scholarpedia.org/article/Jakobson_theorem
|
# Jakobson theorem
Post-publication activity
Curator: Michael Jakobson
Let $$q_{\lambda}: x \rightarrow \lambda x(1-x)\ ,$$ $$x \in [0,1]\ ,$$ $$0 \le \lambda \le 4$$ be the one-parameter family of quadratic maps. Let $$f_{\lambda} : [0,1] \rightarrow [0,1]\ ,$$ $$f_{\lambda}(0)=f_{\lambda}(1)=0\ ,$$ $$\lambda \in [\lambda_0,\lambda_1]$$ be a family $$C^2$$-close to $$q_{\lambda}\ ,$$ and suppose $$f_{\lambda_1}$$ is a map topologically equivalent to the Chebyshev polynomial $$x \rightarrow 4 x(1-x)$$ (Logistic Map). The following theorem was proved in [J1].
Theorem. There is a set $$\Lambda$$ of positive Lebesgue measure such that for $$\lambda \in \Lambda$$ the map $$f_{\lambda}$$ has an invariant measure $$\mu_{\lambda}$$ absolutely continuous with respect to the Lebesgue measure (acim). Moreover for $$a < \lambda_1$$
$\tag{1} \lim_{a \rightarrow \lambda_1}\frac{\mid \lambda \in [a,\lambda_1]\cap \Lambda \mid} {\mid \lambda_1-a \mid } = 1$
In [J2] this Theorem was generalized to families of piecewise smooth maps and $$\mid \Lambda \mid$$ was estimated through finitely many parameters of the family $$f_{\lambda}\ .$$ That makes possible computer assisted proofs of the existence of positive measure sets $$\Lambda$$ and estimates of their measures, see also [LT].
The proof of the Theorem is based on an inductive construction of an increasing sequence of partitions $$\xi_n(\lambda)$$ in the phase space. For each $$\lambda \in \Lambda$$ there is a limit partition $$\xi_{\lambda} = \lim_{n \rightarrow \infty}\xi_n(\lambda)$$ of an interval $$I \subset [0,1]\ .$$ Elements of $$\xi_{\lambda}$$ are countably many intervals $$\Delta$$ which are domains of a piecewise smooth power map $$F_{\lambda} : \Delta \rightarrow I$$ such that $$F_{\lambda} \mid \Delta = f_{\lambda}^k, \ k = k(\Delta)\ .$$ Inductive construction implies that for $$\lambda \in \Lambda$$ the maps $$\ F_{\lambda}$$ are expanding and have uniformly bounded distortions. According to the Folklore Theorem, see [J3], [JS], $$F_{\lambda}$$ has an acim $$\nu_{\lambda}$$ with continuous density bounded away from $$0\ .$$ Then $$\mu_{\lambda}$$ is obtained from $$\nu_{\lambda}$$ by a tower construction.
At step $$n$$ of induction partitions $$\xi_n(\lambda)$$ are defined for $$\lambda \in \Lambda_n\ .$$ By using parameter exclusion one constructs a decreasing sequence of sets $$\Lambda_n$$ in the parameter space such that $$\Lambda = \bigcap_n \Lambda_n\ .$$
For $$\lambda \in \Lambda$$ the systems $$(f_{\lambda},\mu_{\lambda})$$ have strong mixing properties. The rate of decay of correlations is faster than polynomial. However there are $$\lambda \in \Lambda$$ such that $$f_{\lambda}$$ do not satisfy Collet-Eckmann condition (CE) and have the rate of decay of correlations slower than exponential, see [J2]. Several alternative proofs of the Theorem were obtained in subsequent works, see references in [J2], [JS]. Properties of $$f_{\lambda}$$ can vary depending on the construction. In particular for $$\lambda \in \Lambda$$ obtained by Benedicks-Carleson construction [BC1] $$F_{\lambda}$$ do not satisfy Markov property, and $$f_{\lambda}$$ satisfy CE condition. For $$\lambda \in \Lambda$$ obtained by Yoccoz construction, see [S], [Y], both Markov property and CE condition are satisfied. Property (1) implies that most $$\lambda$$ close to $$\lambda_1$$ belong to the intersection of $$\Lambda$$ obtained by different constructions.
See [J3], [JS] for an overview of related topics in one-dimensional dynamics.
In [BC2], [MV] similar sets $$\Lambda$$ were constructed for Henon-like maps, which were small perturbations of one-dimensional maps. Respective $$f_{\lambda}$$ have attractors carrying Sinai-Ruelle-Bowen measures, see [BY] .
See [LV] for a survey of results on Henon-like maps.
An important technical ingredient in the above results are distortion estimates for compositions of hyperbolic and parabolic maps, and maps with unbounded derivatives, see [JN],[PY] for related results.
Unsolved problems in that direction include construction of similar sets $$\Lambda$$ for families of 2-dim conservative maps, in particular Standard Family, for multidimensional quadratic-like families and for multidimensional Henon-like families.
## References
[BC1] M. Benedicks and L. Carleson. On iterations of $$1-ax^2$$ on $$(-1,1)\ .$$ Annals of Math., 122: 1--25, 1985. [BC2] M. Benedicks and L. Carleson. The dynamics of the Henon map. Annals of Math., 133: 73--169, 1991. [BY] M. Benedicks and L.-S. Young. Sinai-Bowen-Ruelle measures for certain Henon maps. Invent. Math., 112: 541--576, 1993. [J1] M.V. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Communications Math. Phys., 81:39--88, 1981. [J2] M. Jakobson. Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions. Proceedings in Symposia in Pure Math., 69: 825--881, 2001. [J3] M.V. Jakobson. Ergodic theory of one-dimensional mappings. Dynamical Systems, Ergodic Theory and Applications, Encyclopaedia of Math. Sciences, Springer,Volume 100, Part II, Chapter 9 : 234--263, 2000. [JN] M.V. Jakobson and S.E. Newhouse. Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle. Asterisque, 261 : 103--159, 2000. [JS] M. Jakobson and G. Swiatek. One-dimensional maps. Handbook of Dynamical Systems, Elsevier Science B.V., Volume 1A, Chapter 8: 599--664, 2002. [LV] S. Luzzatto and M. Viana. Parameter exclusion in Henon-like systems. Russian Math. Surveys 58, no. 6: 1053--1092, 2003. [LT] S. Luzzatto and H. Takahasi. Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps. Nonlinearity 19 , no. 7 : 1657--1695, 2006. [MV] L. Mora and M. Viana. Abundance of strange attractors. Acta Math. 171: 1--71. [PY] J. Palis and J.-C. Yoccoz. Implicit formalism for affine-like maps and parabolic compositions. Global Analysis of Dynamical Systems : Festschrift dedicated to Floris Takens, 67--88, 2001. [S] S. Senti. Dimension of weakly expanding points for quadratic maps. Bull. Soc. Math. France 131 (3): 399--420, 2003. [Y] J.-C. Yoccoz. Jakobson's theorem. Manuscript of the course at College de France, 1997
Internal references
• David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
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2020-07-08 08:01:18
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https://byjus.com/question-answer/the-given-reaction-is-an-example-of-nucleophilic-substitutionelectrophilic-additionnucleophilic-additionelectrophilic-sustitutionfree-radical-substitution/
|
Question
# The given reaction is an example of:
A
B
nucleophilic substitution
C
D
electrophilic sustitution
E
Solution
## The correct option is B electrophilic additionAlkene is nucleophilic in nature and attack on $$Br_2$$ which acts as electrophile.Thus on alkene bond addition of electrophile takes place. Thus, the above reaction is an example of Electrophilic addition.Chemistry
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2022-01-24 17:31:05
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https://tug.org/pipermail/xetex/2015-October/026149.html
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# [XeTeX] Discretionary hyphens don't work in paragraphed footnotes
Philip Taylor P.Taylor at Rhul.Ac.Uk
Thu Oct 8 10:25:44 CEST 2015
Zdenek Wagner wrote:
> I have no time to look deep into it but does anybody know what is the
> definition of \- if these packages is used? Isn't is possible that it is
> a fragile macro? I remember one thing from Phil Taylor's tutorial held
> on EuroTeX'92 in Prague. He said that the most frequent error is wrong
> timing of expansion. Is it possible that \- is redefined in such a way
> that it disappears somewhere during expansion of \footnote before t is
> typeset?
More than possible :-( Why /does/ LaTeX have to meddle so ?!
\-::macro:->\x at protect \-\protect \-
Try :
% !TeX Program=XeLaTeX
\documentclass[a4paper]{article}
\usepackage{fontspec}
\usepackage[para]{footmisc}
\begin{document}
\def \-{\discretionary {}{}{}}
\footnote{XXXXXXXXXXXXXX just a few normal words to fill up the line
up to my x x x hy\-phe\-n\nobreak a\-te\-me }
\end{document}
** Phil.
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2023-03-23 17:19:35
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https://digitalcommons.wayne.edu/jmasm/vol15/iss2/8/
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•
•
#### Abstract
When the sample size n is small, the random variable T= √n(\overline{X} – μ)/S is said to follow a central t distribution with degrees of freedom (n – 1), where \overline{X} is the sample mean and S is the sample standard deviation, provided that the data X ~ N (μ, σ2). The random variable T can be used as a test statistic to hypothesize the population mean μ. Some argue that the t-test statistic is robust against the normality of the distribution and claim that the normality assumption is not necessary. In this article we will use simulation to study whether the t-test is really robust if the population distribution is not normally distributed. In particular, we will study how the skewness of a probability distribution will affect the confidence interval as well as the t-test statistic.
#### DOI
10.22237/jmasm/1478001960
COinS
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2018-12-14 10:10:27
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https://notes.jerrywang.website/computer-graphics/summary/ray_tracing.html
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# Project: Ray Tracing¶
## Axis-Aligned Bounding Boxes¶
For better performance, a good bounding volume structure is necessary. Axis-Aligned Bounding Boxes provides a very nice and quick approach for detecting ray collision. We only need to care about whether the object has been hit, and not the point or the normal for that object. The biggest issue with using AABBs is that they need to be recreated every time the object changes orientation. Fortunately, we do not have to consider that issue in a “offline” ray tracer.
Most people use the “slab” method, which is based on the observation that an n-dimensional AABB is just the intersection of n axis-aligned intervals.
For a ray to hit one interval we first need to figure out whether the ray hits the boundaries. For example, this is the ray parameters t0 and t1 in 2D.
Photos adapted from P. Shirley’s Book
In 3D, those boundaries are planes. The equations for the planes are x = x0, and x = x1. Now the question become where does the ray exactly hit that plane? Well, ray can be thought of as just a function that given a t returns a location p(t):
$p(t) = A + tB$
That equation applies to all three coordinates.
The key observation to turn that equation into a hit test is that the t-intervals need to overlap for each “hit”.
The pseudocode would be something like
compute (tx0, tx1) compute (ty0, ty1) return overlap?( (tx0, tx1), (ty0, ty1))
This is incredibly simple, and the fact that it still works in 3D is exactly why we love it.
## Motion Blur¶
Test scene with motion blur
The secret of a successful Motion Blur implementation is in the ray tracer’s camera implementation. We generate rays at random times while the shutter is open and intersect the model at that one time. Although the object is moving, each ray we generate exist at exactly one time. This way the core of the ray tracer can just make sure the objects are where they need to be for the ray. This method is introduced by Rob Cook in 1984.
## Solid Texture Mapping¶
A texture in graphics usually means a function that makes the colors on a surface procedural. We can create a checker texture by simply multiply trig functions in all three dimensions, and the sign of that product forms a 3D checker pattern. We first make all colors a texture, then we can make textured materials by replacing the vec3 color with a texture pointer:
class texture {
public:
virtual vec3 value(float u, float v, const vec3 &p) const = 0;
};
class constant_texture : public texture {
public:
constant_texture() {}
constant_texture(vec3 c) : color(c) {}
virtual vec3 value(float u, float v, const vec3 &p) const {
return color;
}
vec3 color;
};
## Perlin Noise¶
To get cool solid textures, most people use some form of noise. One of the most used noise is Perlin noise, named after inventor Ken Perlin. I used Andrew Kensler’s explaination of perlin noise to implement the perlin texture (http://eastfarthing.com/blog/2015-04-21-noise/).
A great thing about perlin noise is that its repeatable: it takes a 3D point as an input and always return some random number. In addition, Nearby points return similar numbers, and this is important when we’re making a “pattern”. Perlin noise is simple and very fast, I
We first tile all of space with a 3D array of random numbers and use them in blocks. We then use hashing to scramble the arrangement of the tiles.
After Scramble
To make it smooth, we linearly interpolate, and we get this:
After Linear Interpolation
Looks nice! but there are obvious grid features in there. We call these “Mach bands”, a known perceptual artifact from linear interpolation. To get rid of the Mach Bands, we use hermite cubic to round off the interpolation:
After Hermite Cubic
## Rectangles and Lights¶
From P. Shirley’s book: First, here is a rectangle in an xy plane. Such a plane is defined by its z value. For example, z = k. An axis-aligned rectangle is defined by lines x=x0, x=x1, y=y0, y=y1.
Photos adapted from P. Shirley’s Book
To determine whether a ray hits such a rectangle, we first determine where the ray hits the plane. A ray P(t)=a+t*b has its z component defined by $$z(t)=az+t*bz$$ . We can then solve for what the t is when z = k : $$t = (k-az) / bz$$. Once we have t, we can plug that into the equations for x and y. It is a hit if x0 < x < x1 and y0 < y < y1
## Volumes¶
One thing nice to add to the ray tracer is smoke. This is sometimes called “volume”. A very nice approach is to make a volume a random surface. A bunch of smoke/fog can be replaced with a surface that probabilistically might or might not be there at every point in the volume.
Think about a volume of constant density. A ray going through a volume can either scatter inside or make it all the way through. How far the ray has to travel through the volume plays a role in determining how likely it is for the ray to make it through.
The probability that the ray scatters in any small distance dL can be represented by a differential equation:
$probability = c*dl$
where C is porportional to the density of the volume.
## Schlick’s Approximation¶
In the process of building a ray tracer, we initially used the full Fresnel equations,
From Wikipedia
These looked absolutely horrible, and because polarization doesn’t matter that much for most appearance, most ray tracers use R = (Rs+Rp)/2. It turns out there are a very nice simple approximation proposed by Christophe Schlick. According to Schlick’s model, the specular reflection coefficient R can be approximated by:
$\begin{split} R(\theta) &= R_0 + (1 - R_0)(1 - \cos \theta)^5 \\ R_0 &= \left(\frac{n_1-n_2}{n_1+n_2}\right)^2\end{split}$
## Nice Hack: de_nan¶
Photos adapted from P. Shirley’s Book
A serious practical limitation of ray tracing algorithms is that they are often very sensitive to numerical precision.
If there is bad or white “acnes”, that means a bad might have killed a whole pixel. The sample might be NaN or a huge number. Peter Shirley introduces a nice hack in his third book of the Ray Tracing series. He mentioned that any “if” statement test with a NaN in it is false. This means we can apply a nice trick.
inline vec3 de_nan(const vec3 &c)
{
vec3 t = c;
if(!(t[0]==t[0]))
t[0]=0;
if(!(t[1]==t[1]))
t[1]=0;
if(!(t[2]==t[2]))
t[2]=0;
return t;
}
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2020-02-26 19:29:55
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https://brilliant.org/problems/evening-square/
|
# Collapsing Angle
Geometry Level 2
In the square above, $E$, $F$ and $G$ are midpoints of $\overline{DC}$, $\overline{BC}$ and $\overline{FE}$, respectively.
If $\angle EGD=x$, find the value of $\tan x$.
×
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2020-10-24 18:38:47
|
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http://leancrew.com/all-this/2010/03/shortened-flickr-urls/
|
# Shortened Flickr URLs
Have you seen those “flic.kr” URLs? For some reason, I hadn’t noticed them until the past few weeks, even though they seem very useful. Once I learned how they work, I made up a quick TextExpander snippet so I could insert them into tweets and emails.
Here’s how they work. Each photo on Flickr has a unique ID number, say 5169665786. That ID can be part of several Flickr URLs, all of which point to some form of that photo. For example,
1. http://www.flickr.com/photos/drdrang/5169665786/
2. http://www.flickr.com/photos/drdrang/5169665786/in/set-72157617260543082
3. http://www.flickr.com/photos/drdrang/5169665786/in/set-72157617952751035/
4. http://www.flickr.com/photos/drdrang/5169665786/sizes/m/
5. http://www.flickr.com/photos/drdrang/5169665786/sizes/s/
all refer to this photo
but in different contexts. Number 1 is its address in my photostream, 2 and 3 are its addresses as part of photo sets, and 4 and 5 are the addresses of its medium and small sizes.
The shortened address for this photo is
http://flic.kr/p/8SPTwJ
Following that URL will take you to the photostream version. The string at the end of the URL, 8SPTwJ, is derived mathematically from the photo’s ID number, 5169665786, by converting it from base 10 to base 58.
Since there’s no standard for the “digits” of base 58, any set of 58 alphanumeric characters can be used. Flickr has chosen these
123456789abcdefghijkmnopqrstuvwxyzABCDEFGHJKLMNPQRSTUVWXYZ
which are in ascending order. The characters are basically the numerals, the lower case letters, and the upper case letters with four omissions. The omissions are:
• The numeral 0 and the upper case O.
• The lower case l and the upper case I.
The latter two were obviously chosen to avoid confusion with the numeral 1. I can’t figure out why they decided to omit the numeral 0; once upper case O is gone, there’s nothing else one could mistake for a zero. But that’s what they did, so that’s what we work with.
My conversion script is written in Python and is set up as a shell script snippet in TextExpander.
Update 3/22/10
Forgot to mention that you’ll need to install the nonstandard appscript module, which lets you control AppleScriptable applications from within Python. Follow these simple instructions to download and install it.
1: #!/usr/bin/python
2:
3: import appscript
4: import re
5: import sys
6:
7: def b58encode(n):
8: chars = '123456789abcdefghijkmnopqrstuvwxyzABCDEFGHJKLMNPQRSTUVWXYZ'
9: basecount = len(chars)
10: b58 = []
11: while (n >= basecount):
12: (div, mod) = divmod(n, basecount)
13: b58.insert(0, chars[mod])
14: n = div
15: if (n > 0):
16: b58.insert(0, chars[n])
17: return ''.join(b58)
18:
19: url = appscript.app('Safari').documents[0].URL.get()
20: ids = re.findall(r'flickr\.com/photos/.*/(\d+)/?', url)
21:
22: shortflickr = 'http://flic.kr/p/%s' % b58encode(int(ids[0]))
23: sys.stdout.write(shortflickr)
Most of the work is done by the function b58encode, which does what any base conversion routine would do: continually divide the input number by the base and use the remainder as the next digit working from right to left. Lines 19-20 extract the photo’s ID number from the URL of the frontmost Safari tab; it can use any of the various Flickr URLs. Lines 22-23 construct and print the shortened URL.
(If you’re wondering why I used sys.stdout.write instead of a simple print with a comma at the end, the answer is simple: I kept getting a newline at the end of the URL when I tried
23: print shortflickr,
I don’t know why the comma didn’t suppress the newline, but it didn’t.)
I have the snippet bound to the abbreviation ;flickr, so I can type that whenever I want to insert the shortened URL of the frontmost Flickr photo showing in Safari.
Why not just use a conventional shortener, like bit.ly or xrl.us? They don’t tell the reader that the link is to a Flickr photo. The flic.kr URL does.
Unfortunately, I don’t know how to get the same functionality on the iPhone, where it would be very nice to
1. Take a picture.
2. Upload it to Flickr.
3. Tweet it.
in one fell swoop. Or even two fell swoops. If the Flickr iPhone app had a button for putting the shortened URL on the clipboard, that would be really helpful and would make flic.kr URLs more widely used.
Update 3/21/10
Acting on a tip from Tanja in the comments, I bought Mobile Fotos for my iPhone. It does the three steps I wanted, launching Tweetie with the photo’s title and its flic.kr URL preloaded in a new tweet. Perfect.
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2017-11-23 15:06:04
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https://www.gamedev.net/forums/topic/178391-units-to-pixels/
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#### Archived
This topic is now archived and is closed to further replies.
# Units to pixels?
This topic is 5284 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hi... Im trying to create a very simple 2d game, using a bunch of bitmaps for pictures. Each picture is, for example 32x32 pixels big. The question is, how can i create a square that is 32x32 pixels, creating quad using ogl uints doesnt seem to hard, according to the tutorials, but using pixel seems hard to find an article about. Also, how can i tell where the top left corner of the screen is? Well, maybe this question is completly stupid, but i cant figure the answer Thx for reading it through //iDDqD and yes, iddqd is godmode in d00m. The more you know, the more you know that you dont know.
##### Share on other sites
Setup orthographic projection
The top-left corner in OpenGL is x=0.0 y=1.0
units to pixels (x): units*screenwidth
units to pixels (y): units*screenheight
pixels to units (x): pixels/screenwidth
pixels to units (y): pixels/screenheight
##### Share on other sites
try converting it to a one to one ration using glViewport, and make it 2D with the glOrtho functions.
ie - glViewport(640,480);
My Homepage
Some shoot to kill, others shoot to mame. I say clear the chamber and let the lord decide. - Reno 911
##### Share on other sites
I figure i would do something like that, allto, dont think i understad
#define TOP 0px
#define LEFT 0px
glTranslatef(TOP,LEFT);
glVertex3f(0px, 0px, 0px);
glVertex3f(32px, 0px, 0px);
glVertex3f(0px, 32px, 0px);
glVertex3f(32px, 32px, 0px);
glEnd();
// px just illustrates that i want to move it pixels
could any1 give me some samplecode and/or tell me the fuctions and what parametes they use that would do something similar to this?
The more you know, the more you know that you dont know.
##### Share on other sites
// Set up the viewport to 640x480 for 2d glViewport(0,0,640,480); glMatrixMode(GL_PROJECTION); glLoadIdentity(); glOrtho(0.0f,640.0f,0.0f,480.0f,-10.0f,10.0f); glMatrixMode(GL_MODELVIEW);[/source[
My Homepage
Some shoot to kill, others shoot to mame. I say clear the chamber and let the lord decide. - Reno 911
##### Share on other sites
Thx man, that was exactly what i was trying to achive.
My game is starting to take form now
cheers
-----------------------------------------------------------------
The more you know, the more you know that you dont know.
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2018-02-20 20:00:33
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https://lw2.issarice.com/posts/2Msr65X9Y5NyqygaA/why-rationalists-shouldn-t-be-interested-in-topos-theory
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# Why Rationalists Shouldn't be Interested in Topos Theory
post by jollybard · 2020-05-25T05:35:03.193Z · LW · GW · 13 comments
## Contents
Topoï and toposes
Topoï-logical
Where's my Bayesian topos?
(Geometric) topoï aren't reflective
What now?
None
I spent a lot of the last two years getting really into categorical logic (as in, using category theory [LW · GW] to study logic), because I'm really into logic, and category theory seemed to be able to provide cool alternate foundations of mathematics.
Turns out it doesn't really.
Don't get me wrong, I still think it's interesting and useful, and it did provide me with a very cosmopolitan view of logical systems (more on that later). But category theory is not suitable for foundations or even meant to be foundational. Most category theorists use an extended version of set theory as foundations!
In fact, its purpose is best seen as exactly dual to that of foundations: while set theory allows you to build things from the ground up, category theory allows you to organize things from high above. A category by itself is not so interesting; one often studies a category in terms of how it maps from and into other categories (including itself!), with functors, and, most usefully, adjunctions.
Ahem. This wasn't even on topic.
I want to talk about a particular subject in categorical logic, perhaps the most well-studied one, which is topos theory, and why I believe it be to useless for rationality, so that others may avoid retreading my path. The thesis of this post is that probabilities aren't (intuitionistic) truth values.
# Topoï and toposes
A topos is perhaps best seen not even as category, but as an alternate mathematical universe. They are, essentially, "weird set theories". Case in point: itself is a topos, and other toposes are often constructed as categories of functors , for an arbitrary category.
(Functors assemble into categories if you take natural transformations between them. That basically means that you have maps , such that if you compare the images of a path under and , all the little squares commute.)
Consider that natural numbers, with their usual ordering like , can form a category if you take instead So one simple example is to consider the category of all functors , which are really just sequences of sets, like
where the arrows are regular set theoretic functions. You can do practically any kind of mathematical reasoning using sequences of sets! (as long as it is constructive) For example, you have
• an "empty set", which is just a sequence of empty sets;
• a "point" given by a sequence of points;
• "products" of sequences given by
and so on. Most interestingly, you have truth values given by subobjects of the point; accordingly, in those are the empty set and the point itself, since , corresponding to true and false. Notice that ; in fact the truth values in general will have the structure of a partially ordered set.
What are our truth values here? What is a subobject of a sequence of points? For one, each has to be a subset of . And there are no maps ; so each "truth value" will look like
a bunch of empty sets and, at some position , all points, meaning that we have as many truth values as natural numbers. This is our first glance into the cosmopolitan nature of topos theory: weird truth values! Notice, however, that if , their corresponding subobjects will have this ordering reversed (an exercise left for the already knowledgeable reader); so in the end it might have been better to use functors on , natural numbers with their order reversed.
To sum up, we made the category of sequences of sets, and realized that it was a topos with truth values . Isn't it that interesting...
# Topoï-logical
Turns out there's a big connection between toposes and topological spaces.
The open sets of a topological space have the structure of a partially ordered set, if you set whenever . Moreover, in that poset, you can describe as the greatest lower bound of and , and as their least upper bound.
This is in fact (almost) exactly the structure we want of our topos-theoretic poset of truth values: the greatest lower bound corresponds to conjunction and least upper bound is the disjunction . So we can use topological spaces as spaces of truth values, and this is in fact the approach used in Heyting semantics of intuitionistic logic.
(so each open set, and not point, corresponds to a truth value; you take the AND of two open sets to be their intersection and the OR to be their union)
Alright, so as with , the poset of open sets can define a category if you set whenever . So take a topological space and its category of open sets. We'll define a new category of functors , except that they won't be all of the functors, only those that preserve the topo/logical structure (sheaves).
Guess what? Not only is a topos, turns out that its truth values are isomorphic to ! And since the truth values are the subobjects of the point, that means that the points of the topos are in fact shaped like ...
Now we have a fuller view of the logically cosmopolitan view that topos theory can bring us. You can create, just like that, a whole parallel mathematical universe of bizarro sets where everything is made up of, say, donuts. Or coffee cups. It is as you wish.
# Where's my Bayesian topos?
Since I am at heart a LessWronger, and since I care deeply about problems of logical induction and logical counterfactuals and whatnot, I spent a while trying to design a topos that would behave like manipulating probabilities. Or distributions. Or something. With the objective of making something that would represent beliefs.
Well, I'm sorry, but it doesn't work.
At minimum, we would expect that the truth values of this topos be probabilities, yeah? And with the cosmopolitan principle above, we could then just take the sheaves on this poset of probabilistic truth values.
So these truth-values would be order-isomorphic to [0,1]. But for them to actually represent probabilities, we'd want that , and yet the order on already prescribes that , and we are doomed from the start.
Furthermore, even in an intuitionistic logic, the provable statements all have the maximal truth value (which here would be 1); but we all know that 0 and 1 are not probabilities [LW · GW], and so nothing should be provable... which seems like it wouldn't be very useful.
All in all, I'm truly sorry you had to bear through all of the math above just for this conclusion. It's still pretty cool, though, right?
# (Geometric) topoï aren't reflective
In order to legitimately use topos theory for rationality, we should have a way for the topos to "think about itself". Analogously to the situation in Peano arithmetic [LW · GW], for a topos , we'd want some object (specifically, an internal category) to be isomorphic to in some sense.
We can define an "element" of an object as being an arrow . So the objects of the internal category are given by the set , and in fact the functor respects the structure of the internal category enough that it becomes a category internal to , which is just a small category.
But wait. The toposes generated by sheaves on a topological space are at least as big as , but the collection of all sets is too big to be a set, and thus we run into size issues.
It should in principle be possible to do so in small toposes, such as the free (as in syntactic) topos, but I am not sure and will refrain from claiming so. It is however certainly possible to do so in list-arithmetic pretoposes (yes it's a mouthful), as shown by André Joyal in his as of yet unpublished categorical proof of Gödel's incompleteness theorems, which I have studied with him last year.
# What now?
It now seems to me that linear logic might be the "right" weakening of classical logic into something probabilistic. I still need to figure out some of the details, but let's say that the work has already been done, and one need only piece it together into something relevant to rationality and agent foundations. Particularly promising is that some claim that linear logic is a good setting for "paraconsistent" logic (logic that deals gracefully with contradictions), which could make it work for logical counterfactuals.
All this and more in my next post, pretentiously monikered "Probability Monads".
comment by Vanessa Kosoy (vanessa-kosoy) · 2020-05-25T08:42:12.932Z · LW(p) · GW(p)
I spent a lot of the last two years getting really into categorical logic (as in, using category theory to study logic), because I'm really into logic, and category theory seemed to be able to provide cool alternate foundations of mathematics.
Turns out it doesn't really.
Category thing doesn't. But, (the closely related) homotopy type theory does.
Replies from: jollybard
comment by jollybard · 2020-05-25T08:47:19.083Z · LW(p) · GW(p)
Indeed, and -categories can provide semantics of homotopy type theory. But -categories are ultimately based on sets. At some point though maybe we'll use HoTT to "provide semantics" to set theories, who knows.
In general, there's a close syntax-semantics relationship between category theory and type theory. I was expecting to touch on that in my next post, though!
EDIT: Just to be clear, type theory is a good alternate foundation, and type theory is the internal language of categories.
I've ran into this too, and I think that quasitopoi are also a dead-end for this sort of thing. I'm currently interested in linear logic as well!
comment by MrMind · 2020-05-25T10:50:52.460Z · LW(p) · GW(p)
I've also dabbled into the matter, and I have two observation:
• I'm not sure that probabilities should be understood as truth values. I cannot prove it, but my gut feeling is telling me that they are two different things altogether. Sure, operations on truth values should turn into operations on probabilities, but their underlying logic is different (probabilities after all should be measures, while truth values are algebras)
• While 0 and 1 are not (good) epistemic probabilities, they are of paramount importance in any model of probability. For example, P(X|X) = 1, so 0/1 should be included in any model of probability
Replies from: jollybard
comment by jollybard · 2020-05-25T19:49:01.406Z · LW(p) · GW(p)
I'm not sure that probabilities should be understood as truth values. I cannot prove it, but my gut feeling is telling me that they are two different things altogether.
My feeling is that the arguments I give above are pretty decent reasons to think that they're not truth values! As I wrote: "The thesis of this post is that probabilities aren't (intuitionistic) truth values."
Replies from: MrMind
comment by MrMind · 2020-05-26T09:17:00.099Z · LW(p) · GW(p)
Yeah, my point is that they aren't truth values per se, not intuitionistic or linear or MVs or anything else
comment by Krow · 2020-05-25T12:25:26.050Z · LW(p) · GW(p)
• Here is a list of various notions of probability measures developed in category theory, might be useful (with lots of references). I have no idea if any of these notions somehow fit well with linear logic though.
That being said if you are aiming to create a language for bayesian probabilities, I think it would be interesting to look in particular into continuous valuations on dcpos (see previous link), and try to somehow apply this stuff to the framework developed by Paul Taylor , which gives a constructive account of topology. Plus he's quite opinionated and fun to read too.
• And an aside on reflectivity. In general any sufficiently general logical construct does not have this property: set universes are not reflective; types of types in type theories are not either; NF is the one exception I know of but suffers from some pathologies such as not allowing currying. So overall I do not think this is a reasonable expectation to have.
However toposes are partially reflective: their (impoverished) internal version of themselves is their set of truth values. A way to see this is that you get back the poset of truth values if you collapse all hom-sets of your topos into singletons (which is like seing all objects as propositions instead of sets).
This is related to Lawvere-Tierney topologies , which give a way of "modifying" your topos through an operation on its set of truth values only (and this is equivalent to taking sheaves on a site in a presheaf topos, so pretty much all "nice" toposes arise in this way).
I would use this as a guideline for what to expect of a reflective operation in a good enough probabilistic universe.
comment by Clamwinds · 2020-05-25T06:28:26.491Z · LW(p) · GW(p)
Interesting. Ever hear of the work of Jean-Yves-Girard? (You mentioned mathematical logic) I hear his book "The Blind Spot" is an excellent lecture on the nature of mathematical truth by a seasoned veteran of such difficult affairs.
Replies from: rhollerith_dot_com, jollybard
comment by rhollerith_dot_com · 2020-05-25T13:36:46.750Z · LW(p) · GW(p)
I like Girard. The Rust programming language's borrow checker probably wouldn't've been invented yet if it weren't for Girard's 1987 paper, "Linear logic". (The paper got sustained attention from numerous programming-languages researchers; I read many thousands of papers on programming-language design before the appearance of Girard 1987 and I can recall no exploration of the use of linear types, use-once variables or whatever you want to call them before Girard 1987.)
comment by jollybard · 2020-05-25T06:37:16.854Z · LW(p) · GW(p)
Yes, I have! Girard is very... opinionated, he is fun to read for that reason. That is, Jean-Yves has some spicy takes:
Quantum logic is indeed a sort of punishment inflicted on nature, guilty of not yielding to the prejudices of logicians… just like Xerxes had the Hellespont – which had destroyed a boat bridge – whipped.
I enjoyed his book "Proofs and Types" as an introduction to type theory and the Curry-Howard correspondence. I've looked through "The Blind Spot" a bit and it also seemed like a fun read. Of course, you can't avoid his name if you're interested in linear logic (as I currently am), since the guy invented it.
comment by dsatan · 2020-06-19T23:33:41.076Z · LW(p) · GW(p)
If you're so interested in logical induction, aren't you already assuming that classical mathematics is The One True Logic? Why is that? Why not look at ordinary mathematics internal to a topos and then ask what logical induction looks like for that?
And as for reflection, a topos with a NNO has (higher order) primitive recursion so your claim about not having reflection is confusing.
And lastly, your title doesn't match your thesis. All you show is that you can't directly do probability in toposes. Category theory is extraordinarily useful for many areas of mathematics in general, and is more than just a language. See Beck's monadicity theorem, the adjoint functor theorem, the small object argument, Gabriel Ulmer duality, and so on for nontrivial results in category theory.
Maybe you shouldn't base your entire identity around doing probability theory. At the very least, epistemology spills far beyond the purview of probability theory.
Replies from: jollybard
comment by jollybard · 2020-06-20T04:48:08.438Z · LW(p) · GW(p)
This is the logical induction I was thinking of.
Replies from: dsatan
comment by dsatan · 2020-07-28T17:37:26.508Z · LW(p) · GW(p)
Yes. That is the logical induction I was talking about.
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2021-07-30 04:47:56
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http://keytonsiddoway.blogspot.com/2011_02_01_archive.html
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## Sunday, February 13, 2011
### Carrot pie
I bet you were wondering what we were going to do with all of those canned carrots. The winner so far: pumpkin. . .I mean carrot pie.
January 2011
## Wednesday, February 09, 2011
### Newton's laws
#1: An object that is at rest will stay at rest.
#2: $\mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \frac{\mathrm{d}(m\mathbf v)}{\mathrm{d}t},$
#3: For every action there is an equal and opposite reaction.
January 15, 2011
## Sunday, February 06, 2011
### Calvin H Rasmussen
Nick's Grandpa passed away shortly after Christmas. While it was sad to see him go and we will miss him, it was an enjoyable time to spend with family and celebrate his life.
Speaking of family. . . .
Rasmussen siblings
With spouses
Terry and Becky family
TAS grandkids
TAS kids
Grandma and all her grandkids
Grandkids and greatgrandkids
The whole group
January 3, 2011
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2017-08-21 23:28:12
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https://electronics.stackexchange.com/questions/243254/what-design-decisions-must-i-make-to-select-a-suitable-p-channel-mosfet
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# What design decisions must I make to select a suitable P-Channel MOSFET
I need a PFET with a logic level gate to trigger a power connection to a hobby servo that pulls more current than my MCU pin can handle.
The requirement is simple but I'm trying to understand how to go about designing the circuit so I can then choose from available devices that meet my need and can eliminate those that don't. At present I can see there are (literally) thousands of MOSFETS that are available and I don't know what is required to choose ones that will meet my requirement.
The reason for the PFET is to kill power to the servo during sleep mode so that I can get longer battery life. In idle mode the servo pulls 3.58 milliamps and I am trying to use just a single 18650 battery connected to a 5v boost converter to power the project. In testing, with the servo always pulling current I get about 27 days out of a single battery. When I remove the servo from the constant current plan and only calculate the current draw when it rotates (50-100 mA), I can get more than a year from a single battery. Problem is I have no idea what an appropriate PFET is for this project. I went to this site but the sheer number of choices is staggering.
Can people please advise what I need to do to establish a tighter requirement for the MOSFET I require?
Project components:
1. 5v 16MHz ATmega168 Arduino Pro Mini
2. 5v boost converter (quiescent current 130 microamps) (used to boost the 3.7v LiPo battery to 5v; actually boosts to about 5.27v)
3. Knock-off Clone Futaba S3003 servo (4.8v - 6v operating range) (idle current 3.58 milliamps; with a normal load (50-100 milliamps)
4. LDR sensor with 10k resistor
5. LED with 220K resistor for low battery warning
• This site is not a forum. – efox29 Jun 28 '16 at 11:06
• Please read the help before posting. Shopping/recommendation questions are off-topic. – Roger Rowland Jun 28 '16 at 11:10
• This could potentially be a good question, but not as it stands. A good question would perhaps be how to figure out what parameters are important, given a certain use case. – pipe Jun 28 '16 at 11:45
• This IS a design question. (tears more hair out). Spec is bad but it's a design question. Hey - this is a design question. PS - this is a design question. Guess what - this .... – Russell McMahon Jun 28 '16 at 13:21
• ... FET power dissipation = I^2R = Idsmax x Rdson^2 = 1 x 0.1^2 = 10 mW = forget it. | FET is switched by about 5V and maybe 3V min from LiIon so gate switching voltage of <<3V is nice = Bgsth. Vgsth is the "just turned on" Vgs so lower is needed so 1 to 2V range. | That should do for a start. SO set Digikey filters (one at a time is recommended) to PChannel >= 20V >= 1A Rdson <= 100 mOhm Vgsth<= 2V, in stock, quantity 1 or more. -> 35 devices. ... – Russell McMahon Jun 28 '16 at 13:38
I need a PFET with a logic level gate to trigger a power connection to a hobby servo that pulls more current than my MCU pin can handle. ... I'm trying to understand how to go about designing the circuit so I can then choose from available devices that meet my need and can eliminate those that don't.
This is a purposefully excessively detailed response to the query.
It's intended as a tutorial of sorts in component selection generally and in small MOSFET selection specifically. This is not an especially demanding application and in some other cases much more detail may be required.
The aims include showing what may be involved in a typical relatively simple component selection process, what sort of not so obvious factors may hide in the background and how by selective refinement a bewilderingly large number of available devices may be reduced to a more manageable and more appropriate small subset.
_____________________________
Set the target:
From the available thousands of MOSFETs available, a subset that meet the design specification can be chosen by working out what the design specification is. This need not be done in infinitely find detail. Just "roughing out" the acceptable values for a number of parameters rapidly reduces the options to (more) manageable levels. Below I'll use relatively standard abbreviations with either approximate meanings given or none at all as they can be rapidly understood on inspection of the data sheets or selection tables or literature.
Vdsmax - constrained by operating voltage.
The MOSFET needs to operate on 5V, so a 10V or greater Vds rating is potentially OK. But a Vds of 20V or more is common, and more 'headroom' gives more robustness and resistance to 'the little things that go wrong' such as inductive spikes, so start by setting Vds >= 20V. This and other parameters can be loosened if the available range proves too low.
Idsmax - maximum continuous operating current.
The MOSFET must handle maximum servo current easily. You do not say what that is so it is assumed that 1A will be very adequate. If this is not the case a different value can be selected. The MOSFET needs an IDsmax rating of 1A but higher is common and usually useful. Higher Idsmax than required in operation usually results in devices with lower on resistance so lower losses at the desired operating current.
Rdson - Resistance when "fully enhanced".
The voltage drop when turned on needs to be small enough to keep thermal losses in the MOSFET to an acceptable level. Acceptable may be based on temperature control of MOSFET, not too much loss in available voltage for the load and energy budget aspects). In this case a "fully turned on" resistance of 0.1 Ohm or less is probably OK and higher may well be acceptable. At 1A a Rdson of 0.1 Ohm drops 0.1V, dissipates 100 mW or 2% of a 5V supply. At 2A and Rdson of 0.2 Ohm drops 0.4V and dissipates 800 milliwatts or 4% of a 5V supply. The former (0.1 Ohm, 1A) is almost certainly acceptable in most cases and certainly for a servo and can be handled by a SOT23 SMD transistor with sensible PCB copper. The latter (0.2 Ohm, 2A) is thermally annoying for most SOT23 uses but would usually be OK with an eg DPak pkg, the voltage drop may be unacceptable in some cases and energy loss% is probably usually acceptable. so Rdson <= 100 milliOhms will be acceptable at 1A and in many cases a modern PFET will give lower Rdson levels.
Rdson - typical and real world values. It is important to note that MOSFET Rdson values are usually (but not always) specified for stupidly short pulse values at unusually low duty cycles. This means the junction temperature is dominated mostly by thermal of the die and will remain near ambient temperature during the short pulse and have ample time to cool before the next pulse. In most real world situations much longer on times and higher duty cycles are used and junction temperatures are dominated more by the thermal_resistive divider from junction to ambient and will be higher or much higher. As a rule of thumb it is almost always safe to assume that worst case Rdson will be no more than twice the 25C value. In some cases it is close to double and in other cases maybe only 20% more so datasheets need to be checked. Note that data sheet tabular values are "typical" and NOT "worst case" unless otherwise specified. Top manufacturers may provide typical and max values. Design MUST always use worst case values.
Data sheet graphs almost always give a single value for the parameter set involved and this must be assumed to be typical unless otherwise stated. eg a set of curves may be given showing Ids against Vds for a family of curves at various Vgs values. To show worst case and typical values would need addition of error bars or twice as many curves and I've never seen either done.
Junction temperature and acceptable max I values under various pulse conditions can be found in many data sheets from a table designed for this purpose. This can be useful for push it to the limit designs but usually for on/off switching designs assuming dissipation of Rdson x Ix^2 perhaps a duty cycle if <<1 is usually safe and advisable. Here Ix is some magical figure between Imax and Iavg. Using Iavg may be safe for a Ids range that is usually about the same most of the time but as dissipation is proportional to I^2 using I average may be risky for cases where peak I and average I differ substantially.
Example: Given an Rdson of 0.1 Ohm and 0A or 10 A current at 50% duty cycle.
Iavg = 10A x 50% = 5A.
Dissipation using Iavg is
Iavg^2 x Rdson = 25 x 0.1 = 2.5W
Actual = 10A^2 x 0.1V x 50% = 5 Watt
= double value using Iavg.
Steady state. junction temperature = Pd/Rth_total
Rth_total = Rja = sum od thermal resistances from junction to ambient.
Main components of Rth are
Rjc = junction to case thermal resistance +
Rcs = case to sink +
Rsa = sink to ambient thermal resistance.
So Rthtotal = Rja = Rthjc + Rthcs + Rthsa
Rjc is usually designed by the manufacturer to be "sensibly low" such that junction temperature is not too high above case temperature as full power, giving the designer a fighting chance of getting rod of the thermal energy at full dissipation.
At more than trivial power levels it is usual for junction temperature to be in the say 60C - 100C range in continuous use and values of 120C or higher may be acceptable. While notionally a device may be operated at up to the Tjmax value of, typically, around 140 C, this leaves no headroom for thermal transients and begins to have affects on device lifetime. [Cree specify their lighting class LEDs for operation at Tj of 85C and 105C and often no longer give 25C Tj figures in datasheets].
Vgsth - minimum gate voltage for turn on.
The MOSFET has 5V available for gate switching when your boost converter is operating but it is safer to start with a device that is at least half happy at the minimum of 3V available from a minimum voltage LiIon cell. So start with a Vgsth of well under 3V if possible.
Vgsth is the "just turned on" Vgs so lower is needed so start with a 1V to 2V Vgsth figure. Values under 1V are fine if available but are very unusual and almost always are also associated with a lower than usual Vgsmax value indicating an extremelt thing gate oxide thickness and potentially added sensitivity to electrostatic damage.
Selection:
That should do for a start. We could have looked at case type (do you want to solder QFN pkgs?) or power dissipation (unlikely to matter here), or ROHS (RO what :-) ) , or ... . In cases of high switching rates losses from gate charge/discharge may become significant and we may start looking at junction charge values and ..., but in this case and for most 1st passes such details can follow later if needed.
I personally use Digikey as my first choice component search engine due to their utterly vast listing base, good (but not perfect) parameter driven selection, knowing that if they stock it the brand is almost certainly reputable (and if you buy from them its probably genuine). I also sometimes actually buy product from them :-)/ (I'm in NZ so if buying for use here shipping costs can matter). There are many other supplier systems that can be used in this manner.
Set Digikey (or other supplier) selection system filters (one at a time is recommended) to:
MOSFET,
PChannel,
Vds >= 20V,
Ids >= 1A,
Rdson <= 100 mOhm
Vgsth<= 2V,
in stock <- usually desirable
quantity 1 or more <- higher minimum usually gives wider range and lower prices
Yielding 35 devices for Digikey in this case.
Potential candidates
Sort by $ascending to start. All else being equal, lowest cost is nice. Later, reordering by eg Rdson and Vgsth can be informative. Near the bottom for low cost is Panasonic MTM761230LBF at$US0.35 /1.
Despite having an annoyingly small pkg I'll comment on specs in some detail as they are 'more special than some' in good and bad ways.
Product pricing page here
Datasheet here
This is in a mini 6 SMD pkg that is harder than some to solder but doable.
4 of the 6 pins are drain. making soldering much easier than if they eg placed the gate connection between two other terminals (as some do). This is a 20V 3A device - acceptable in this application. Rdson = 36 milliOhms - nicely low. They say "2.5V drive". As Vdrive useful is >> Vgsth the Vgsth must be low. It is.
Vgsth = 0.4/0.85/1.3 V min/typ/max = VERY low and
Vgsmax is only 10V as a consequence (see discussion above). This is acceptable as long as it is realised and designed for. As Panasonice 'know their stuff' as well as most you'll see from the datasheet diagram that there is a bidirectional voltage clamp on the gate. This is highly desirable in devices with such a low Vgsmax. Without such the merest whiff of electrostatically induced voltage can destroy the device. With the protection you still should treat them with due reverence.
At 1A with 2V on the gate it drops 0.05V Vds - this will be higher when/if hot but not more than double = 0.1V.
as you have now said that Iload max is 100 mA this will be very acceptable. (Surge current may well be rather higher but easily handled)
If desired, a much to solder part costing 56c/1 and rated at 30V 24A and 25 milliohm is the St STD30PF03LT4. While Vgsth is specified as 1V min this is misleading - it really, really wants 3V to get going half well. It's in a large enough to see and hold and solder DPAk.
At 24A Ids it's in a different class than the lovely but more fragile Panasonic part. Both would work here - especially if close to 5V drive is used for the St part but even at 3V.
Pricing page
Data sheet
For a small and adequate solution see the 49cents/1 SI2301.
Only 20V 2.8A (adequate), SOT23 pkg, and slightly over the 100 milliOhm on resistance spec but probably very happy at 2V on gate and probably rather less (the graph gets too hard to read).
Datasheet
THE POINT HERE is not so much the specific parts we have arrived at (and quite a few more) but the method of choosing important parameters, whittling the options down by setting the desired parameters to >= the minimum acceptable value (eg Vds) or <= max acceptable value (for eg Rdson) and progressively homing in on parts that meet the spec. If potential components ~= 0 then specifications can be inspected to see which ones can be loosened.
• [SOLVED]Started off on the wrong track with a somewhat vague request but in the end @RussellMcMahon got me back on track. I have a much better understanding of MOSFETs now and I think this post will help other electronic hobbyists make an educated decision on how to employ a PFET in their own low power arduino projects. – darksidekilo6 Jun 28 '16 at 18:33
• June 29th NZT - the phantom downvoter strikes again. I wonder what's wrong with them. Maybe they have a scatter gun, or a pathological hatred or ... . Weird. [Hover over the downvote button and it says "This answer is not useful".] – Russell McMahon Jun 29 '16 at 8:14
• @RussellMcMahon That doesn't mean it was downvoted, does it? I thought it was just a description of why one should click that link. Can you actually see downvotes? Surely the upvotes are winning. I think you did a great job cultivating the question and providing an answer. I think this is excellent modeling for Stack Exchange! – Jeff Jun 30 '16 at 14:50
• @Jeff Yes - I can see the up and down votes. One of the actually useful aspects of enough "rep" :-). 9 up 1 down = 8 up overall. I suspect that the downvoter has problems. The "rep" is not the issue - it's how it can affect other people when there are not many votes or no upvotes early on, and it makes one wonder what drives the downvoter. – Russell McMahon Jun 30 '16 at 17:02
• @Jeff FWIW The Up/Down votes on the actual question are +7/-4, it was originally closed ("on hold") and then reopened (maybe because someone said something :-) ) and still has two close votes. It was always a design related question but the OP didn't have the right flavour of wording for the normal mindset and it was described as a shopping question and a forum question and ... . Thus we too often treat newcomers. Some people do ask badly structured questions and never improve them but the OP was happy to learn from the input. While the question is closed it is very hard to provide good input. – Russell McMahon Jun 30 '16 at 23:31
1. 5v 16MHz ATmega168 Arduino Pro Mini
A 5V Arduino's digital output level is 0~5V, so you want a FET that turns on fully with this voltage. It should be specified as 'Logic Level' or '4.5V Gate drive'.
1. 5v boost converter(...actually boosts to about 5.27v)
VGS and VDS should be rated at least 20% higher than that highest voltages they will ever encounter. For safety I would go for at least 12V.
1. Knock-off Clone Futaba S3003 servo
Assuming the 'clone' has similar specs to a real S3003, it should draw about 1A peak. You want minimal voltage drop at this current, let's say 0.1V. So the FET needs to have Rdson <= 0.1Ω with 4.5V Gate drive.
At this low voltage drop the power dissipation is minimal (1A2 * 0.1Ω = 0.1W), which pretty much guarantees that the FET's current rating will be higher than you need. So just ignore the current rating.
I went to [Digikey] but the sheer number of choices is staggering.
Now that you know what specs to look for, select FET Type: P-Channel, FET Feature: Logic Level, Drain-Source Voltage: 12V/20V/30V and Package / Case whatever you prefer (eg. 8-SOIC). This should narrow the choices down to a few. Finally select an Rds On (Max) @ Id, Vgs which is equal to or less than 100mΩ.
When you have chosen a candidate, read its datasheet to verify that all the parameters are suited to your application.
• Bruce, thanks for the detailed step by step instructions on selecting the appropriate PFET. Your comments as well as @RussellMcMahon took from a zero to hero in my overall understanding of FET parameters. Bravo. – darksidekilo6 Jun 28 '16 at 18:47
Both P-MOSFET and N-MOSFET can be used as a switch. To achieve switching you have to add appropriate voltage between Gate and Source which is usually defined as Vgs. When voltage is higher than threshold the resistance of transistor drops dramatically so current can flow through it. The difference between these two is that Vgs has to be positive for N-MOSFET and negative for P-MOSFET for switch to be opened. So, firstly, choose transistor with appropriate treshold voltage. Secondly, make sure that chosen transistor can handle enough current. You can look it up on datasheets too. Thirdly, as a switch is never an ideal short circuit, take resistance Rds into account because there can be high voltage drop on transistor. Usually, when current is several hundred miliamps, this resistance is not a problem.
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2019-11-22 17:25:48
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https://17calculus.com/integrals/improper/limit-comparison-test/
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## 17Calculus Improper Integrals - Limit Comparison Test
##### 17Calculus
On this page we discuss how to determine if an improper integral converges or diverges using the Limit Comparison Test.
[domain] - [basic integration] - [limits at infinity] - [improper integrals]
Limit Comparison Test For Improper Integrals - Theorem
For positive, continuous and real functions, $$f(x)$$ and $$g(x)$$ on the interval $$[a,\infty]$$ and $\lim_{x \to \infty}{ \frac{f(x)}{g(x)} } = L, ~ \text{ with } ~ 0 \lt L \lt \infty$ then the integrals
$\int_a^{\infty}{ f(x) ~ dx }, ~~~ \int_a^{\infty}{ g(x) ~ dx }$ either both converge or both diverge.
Limit Comparison Test Development
Here is a video explaining the logic behind the Limit Comparison Test for improper integrals.
### JCCCmath - Limit Comparison Test Integrals - Development [7min-10secs]
video by JCCCmath
Practice
Unless otherwise instructed, determine if these improper integrals converge or diverge using the Limit Comparison Test.
$$\displaystyle{ \int_3^{\infty}{ \frac{x^3-x+2}{x^5+x^4-x^3} ~dx } }$$
Problem Statement
Determine if $$\displaystyle{ \int_3^{\infty}{ \frac{x^3-x+2}{x^5+x^4-x^3} ~dx } }$$ converges or diverges using the Limit Comparison Test.
Solution
### JCCCmath - 4300 video solution
video by JCCCmath
Log in to rate this practice problem and to see it's current rating.
$$\displaystyle{ \int_1^{\infty}{ \frac{1}{(x-\ln x)^2} ~dx } }$$
Problem Statement
Determine if $$\displaystyle{ \int_1^{\infty}{ \frac{1}{(x-\ln x)^2} ~dx } }$$ converges or diverges using the Limit Comparison Test.
Solution
### Michael Penn - 4301 video solution
video by Michael Penn
Log in to rate this practice problem and to see it's current rating.
$$\displaystyle{ \int_3^{\infty}{ \frac{1}{2+\cos x + \ln x} ~dx } }$$
Problem Statement
Determine if $$\displaystyle{ \int_3^{\infty}{ \frac{1}{2+\cos x + \ln x} ~dx } }$$ converges or diverges using the Limit Comparison Test.
Hint
Use $$g(x) = 1/\ln x$$ as the comparison function.
Problem Statement
Determine if $$\displaystyle{ \int_3^{\infty}{ \frac{1}{2+\cos x + \ln x} ~dx } }$$ converges or diverges using the Limit Comparison Test.
The improper integral $$\displaystyle{ \int_3^{\infty}{ \frac{1}{2+\cos x + \ln x} ~dx } }$$ diverges by the Limit Comparison Test.
Problem Statement
Determine if $$\displaystyle{ \int_3^{\infty}{ \frac{1}{2+\cos x + \ln x} ~dx } }$$ converges or diverges using the Limit Comparison Test.
Hint
Use $$g(x) = 1/\ln x$$ as the comparison function.
Solution
Set up the limit $$\displaystyle{ \lim_{x \to \infty} { \frac{f(x)}{g(x)} } }$$. Use the hint for $$g(x) = 1/\ln x$$. $$\displaystyle{ L = \lim_{x \to \infty}{ \frac{1/(2+\cos x + \ln x)}{1/\ln x} } }$$ $$\displaystyle{ L = \lim_{x \to \infty}{ \frac{1}{(2+\cos x + \ln x)/(\ln x)} } }$$ $$\displaystyle{ L = \lim_{x \to \infty}{ \frac{1}{(2/\ln x)+(\cos x/\ln x) + 1} } }$$ As $$x$$ gets very large, $$\displaystyle{ \lim_{x \to \infty}{ 2/\ln x } = 0 }$$ And, as $$x$$ gets very large, $$\displaystyle{ \lim_{x \to \infty}{ \cos x/\ln x } = 0 }$$ This leaves $$L = 1$$. Since $$L$$ is finite and non-zero, then the original integral will converge or diverge the same as $$\displaystyle{ \int_3^{\infty}{g(x)~dx} }$$. So now we need to determine if this integral converges or diverges. We can directly compare $$y=\ln x$$ with $$y=x$$. Since $$\ln x \lt x$$, $$1/x \lt 1/\ln x$$. Also, $$\displaystyle{ \int_3^{\infty}{ 1/x ~dx } \lt \int_3^{\infty}{ 1/\ln x ~ dx } }$$ Since $$\displaystyle{ \int_3^{\infty}{ 1/x ~dx } }$$ diverges, then $$\displaystyle{\int_3^{\infty}{ 1/\ln x ~ dx } }$$ also diverges and so does the original integral.
One thing we left off at the first was to check that $$f(x) \gt 0$$. However, we think you can see that since the numerator is always positive, $$2+\cos x$$ is always positive and $$\ln x$$ is always positive, the function $$f(x) \gt 0$$ on the interval of integration. Although not required, we decided to plot the function just to see what it looks like.
The improper integral $$\displaystyle{ \int_3^{\infty}{ \frac{1}{2+\cos x + \ln x} ~dx } }$$ diverges by the Limit Comparison Test.
Log in to rate this practice problem and to see it's current rating.
$$\displaystyle{ \int_3^{\infty}{ \frac{x~dx}{\sqrt{x^5+x^3}} } }$$
Problem Statement
Determine if $$\displaystyle{ \int_3^{\infty}{ \frac{x~dx}{\sqrt{x^5+x^3}} } }$$ converges or diverges using the Limit Comparison Test.
The improper integral $$\displaystyle{ \int_3^{\infty}{ \frac{x~dx}{\sqrt{x^5+x^3}} } }$$ converges by the Limit Comparison Test.
Problem Statement
Determine if $$\displaystyle{ \int_3^{\infty}{ \frac{x~dx}{\sqrt{x^5+x^3}} } }$$ converges or diverges using the Limit Comparison Test.
Solution
We can see that the integrand is positive on the integration interval. So now we need to find a comparison function $$g(x)$$. As we look at the integrand, the $$x^5$$ under the radical dominates the $$x^3$$ term as $$x$$ gets very large. So we can ignore the $$x^3$$ term. This leaves $$x/x^{5/2} = 1/x^{3/2}$$. So we will use $$g(x) = 1/x^{3/2}$$ as our comparison function. Now let's set up the limit $$\displaystyle{ L = \lim_{x\to\infty}{f(x)/g(x)} }$$ $$\displaystyle{ L = \lim_{x\to\infty}{ (x/\sqrt{x^5+x^3})/(1/x^{3/2}) } }$$ $$\displaystyle{ L = \lim_{x\to\infty}{ \frac{x^{5/2}}{\sqrt{x^5+x^3}} } }$$ $$\displaystyle{ L = \lim_{x\to\infty}{ \frac{1}{\sqrt{1+1/x}} } = 1 }$$ The limit is real, finite and positive. So now we need to determine if the integral of $$g(x)$$ converges or diverges. Fortunately, we can evaluate $$\displaystyle{ \int_3^{\infty}{ 1/x^{3/2} ~dx } }$$ $$\displaystyle{ \int_3^{\infty}{ x^{-3/2} ~dx } }$$ $$\displaystyle{ \lim_{b\to\infty}{\int_3^{b}{ x^{-3/2} ~dx }} }$$ $$\displaystyle{ \lim_{b\to\infty}\left[ \frac{x^{-1/2}}{-1/2} \right]_3^{b} }$$ $$\displaystyle{ \lim_{b\to\infty}\left[ \frac{-2}{x^{1/2}} \right]_3^{b} }$$ $$\displaystyle{ \lim_{b\to\infty}\frac{-2}{(b)^{1/2}} - \frac{-2}{3^{1/2}} }$$ $$0 + 2/\sqrt{3} = 2/\sqrt{3}$$ Since the comparison integral converged, so does our original integral.
Note that the improper integral of $$g(x)$$ evaluated to $$2/\sqrt{3}$$ but that does not mean that the original integral also would evaluate to that same number. For this problem, it's just the fact that the comparison integral converged that was significant.
The improper integral $$\displaystyle{ \int_3^{\infty}{ \frac{x~dx}{\sqrt{x^5+x^3}} } }$$ converges by the Limit Comparison Test.
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2022-05-28 03:33:01
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https://en.wikibooks.org/wiki/Mathematical_Proof_and_the_Principles_of_Mathematics/Logic/Logical_connectives
|
# Mathematical Proof and the Principles of Mathematics/Logic/Logical connectives
In the previous section we made clear what mathematical statement is. In this section we talk about how mathematical statements can be combined to make more complex statements. This is done using what are called 'logical connectives' or 'logical operators'. You can think of these as functions of one or more variables, where the variables can be either True or False and the value of the function can be either True or False. The logical connectives commonly used in mathematics are negation, conjunction, disjunction, implication, and equivalence, which are fancy words for things you encounter in everyday English.
In this section the symbols ${\displaystyle P}$ and ${\displaystyle Q}$ denote mathematical statements.
## Negation
The negation of a statement ${\displaystyle P}$ is the statement that ${\displaystyle P}$ is not true. Some ways to phrase this are
Not ${\displaystyle P}$.
It is false that ${\displaystyle P}$.
Examples:
Statement Negation
It rained on 1 September 2005. It did not rain on 1 September 2005.
All teachers are female. Not all teachers are female.
Mike's dog has a black tail. Mike's dog does not have a black tail.
2 + 2 = 4 2 + 2 ≠ 4.
Triangle ABC is equilateral. Triangle ABC is not equilateral.
Negation inverts the truth or falsehood of logical statements. In other words, not ${\displaystyle P}$ is False when ${\displaystyle P}$ is True, and Not ${\displaystyle P}$ is True when ${\displaystyle P}$ is False. In tabular form:
${\displaystyle P}$ Not ${\displaystyle P}$
True False
False True
The logical symbol for negation is "${\displaystyle \lnot }$", so you can write ${\displaystyle \lnot P}$ for Not ${\displaystyle P}$.
Even though "Not" is the simplest logical operator, the negation of statements is important when trying to prove that certain objects have or do not have certain properties. It makes the skill of being able to correctly negate statements an important one.
## Conjunction
The conjunction of two statements ${\displaystyle P}$ and ${\displaystyle Q}$ is the statement that ${\displaystyle P}$ and ${\displaystyle Q}$ are both True. Some ways to phrase this are
${\displaystyle P}$ and ${\displaystyle Q}$.
${\displaystyle P}$ but ${\displaystyle Q}$.
${\displaystyle P}$ however ${\displaystyle Q}$.
Note that phrasing in English can sometimes include meaning that is not captured by the word 'and'. For example the statement
We had a good time even though it rained.
captures the idea that the fact that it rained would lead you to expect that it would be difficult to have a good time. Logically though, the statement is equivalent to
We had a good time and it rained.
since both combine the statements
We had a good time.
and
It rained.
Examples:
First statement Second statement Conjunction
The hall was long. The hall was dark. The hall was long and dark.
All teachers are female. All teachers are humans. All teachers are female humans.
Mike's dog has a black tail. Mike's dog has a wet nose. Mike's dog does has a black tail and a wet nose.
4 is even. 6 is odd. 4 is even and 6 is odd.
Triangle ABC is equilateral. Triangle ABC is equiangular. Triangle ABC is equilateral and equiangular.
Conjunction combines the assertions of two statements into a single statement. It's difficult to be more specific without being circular, but you might say ${\displaystyle P}$ and ${\displaystyle Q}$ is True when both ${\displaystyle P}$ and ${\displaystyle Q}$ are True, and False when either ${\displaystyle P}$ or ${\displaystyle Q}$ are False. In tabular form:
${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P}$ and ${\displaystyle Q}$
True True True
True False False
False True False
False False False
The logical symbol for conjunction is "${\displaystyle \land }$", so you can write ${\displaystyle P\land Q}$ for ${\displaystyle P}$ and ${\displaystyle Q}$.
## Disjunction
The disjunction of two statements ${\displaystyle P}$ and ${\displaystyle Q}$ is the statement that at least one of ${\displaystyle P}$ and ${\displaystyle Q}$ are True. Some ways to phrase this are
${\displaystyle P}$ or ${\displaystyle Q}$.
${\displaystyle P}$ unless ${\displaystyle Q}$.
In mathematics the exclusive or is never used, so
${\displaystyle P}$ or ${\displaystyle Q}$.
always means
${\displaystyle P}$ or ${\displaystyle Q}$ or both.
This contrasts with English where the exclusive or is often implied by context, as in
You can choose either the Big Box or whatever is behind Curtain #2.
In the rare cases where exclusive or is needed in mathematics, the phrase "but not both" can be added to make it clear.
Examples:
First statement Second statement Disjunction
The hall was long. The hall was dark. The hall was either long or dark.
Mike's dog has a black tail. Dave's dog has a black tail. Either Mike's dog or Dave's dog has a black tail.
4 is even. 6 is odd. 4 is even or 6 is odd.
Triangle ABC is isosceles. Triangle ABC is scalene. Triangle ABC is either isosceles or scalene.
Disjunction offers two possibilities which are given by the two statements. Again, it's difficult to be more specific without being circular, but you might say ${\displaystyle P}$ or ${\displaystyle Q}$ is True when either ${\displaystyle P}$ or ${\displaystyle Q}$ (or both) are True, and False when both ${\displaystyle P}$ and ${\displaystyle Q}$ are False. In tabular form:
${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P}$ or ${\displaystyle Q}$
True True True
True False True
False True True
False False False
The logical symbol for disjunction is "${\displaystyle \lor }$", so you can write ${\displaystyle P\lor Q}$ for ${\displaystyle P}$ and ${\displaystyle Q}$.
## Implication
Implication is perhaps the most important, but also the most confusing of the logical connectives. In fact it even has a paradox named after it.
The implication of two statements ${\displaystyle P}$ and ${\displaystyle Q}$ is the statement is that ${\displaystyle Q}$ is True whenever ${\displaystyle P}$ is True. Some ways to phrase this are
${\displaystyle P}$ implies ${\displaystyle Q}$.
If ${\displaystyle P}$ then ${\displaystyle Q}$.
${\displaystyle P}$ only if ${\displaystyle Q}$.
${\displaystyle Q}$ if ${\displaystyle P}$.
${\displaystyle Q}$ is a necessary condition for ${\displaystyle P}$.
${\displaystyle P}$ is a sufficient condition for ${\displaystyle Q}$.
When we use the phrase "If ... then ..." in English it usually means there is some sort of causality going on. For example the statement
If it rains the traffic will be terrible.
somehow contains the idea that the rain will cause the traffic to be terrible. But in terms of logic there doesn't have to be any such connection between the two statement. This is where the paradox, one of the 'paradoxes of material implication', comes in. Namely, if ${\displaystyle P}$ is a false statement, then the implication ${\displaystyle P}$ implies ${\displaystyle Q}$ is true, even if there is no connection between ${\displaystyle P}$ and ${\displaystyle Q}$. For example
If 0=1 then the moon is made of cheese.
is logically true even though whether the moon is made of cheese has nothing to do with whether 0 is equal to 1.
This state of affairs may seem rather strange, which is why it's called a paradox. So perhaps it would help to ask when you can say that the statement ${\displaystyle P}$ implies ${\displaystyle Q}$ is False rather than when you can say it's True. Imagine your dentist says to you
If you eat a lot of sugar then you'll get more cavities.
This is an implication between the two statements
You eat a lot of sugar.
and
You'll get more cavities.
Now suppose you want to prove your dentist wrong and say "Ha! You don't know what you're talking about. I shall seek dental care elsewhere." If you stay away from sugar and don't get cavities then your dentist will be right. If you stay away from sugar but get cavities anyway then your dentist can ask "Did you brush after eating?" and you'll say "No," and your dentist will say "There you go!" and will still be right. The only way you can prove your dentist wrong is to eat a lot of sugar but not get cavities.
This fact is actually useful in some situations and since it's logically valid there's nothing wrong with using it in a proof.
Examples:
First statement Second statement Implication
The hall was long. The hall had many doors. If the hall was long then it had many doors.
Mike's dog has a wet nose. Mike's dog is healthy. If Mike's dog has a wet nose then he/she is healthy.
4 is even. 6 is odd. If 4 is even then 6 is odd.
Triangle ABC is equilateral. Triangle ABC is isosceles. If Triangle ABC is equilateral then it is isosceles.
As we've seen, the implication ${\displaystyle P}$ implies ${\displaystyle Q}$ is True when ${\displaystyle P}$ is false. It's also True when ${\displaystyle Q}$ is True and only false when ${\displaystyle P}$ is True and ${\displaystyle Q}$ is False. In tabular form:
${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P}$ or ${\displaystyle Q}$
True True True
True False False
False True True
False False True
The logical symbol for implication is "${\displaystyle \implies }$", though "${\displaystyle \supset }$" is sometimes seen instead. so you can write ${\displaystyle P\implies Q}$ for ${\displaystyle P}$ implies ${\displaystyle Q}$.
Unlike ${\displaystyle P}$ and ${\displaystyle Q}$ and ${\displaystyle P}$ or ${\displaystyle Q}$, the value of ${\displaystyle P}$ implies ${\displaystyle Q}$ may change if you switch ${\displaystyle P}$ with ${\displaystyle Q}$. In other words
${\displaystyle P}$ implies ${\displaystyle Q}$
is not always the same as
${\displaystyle Q}$ implies ${\displaystyle P}$.
The two statements are related though and we call the statement
${\displaystyle Q}$ implies ${\displaystyle P}$
the 'converse' of
${\displaystyle P}$ implies ${\displaystyle Q}$
Implication plays an important role since most theorems take on the form of an implication.
## Equivalence
The last connective we'll be talking about is equivalence. This one does not occur in English very often, so some of the ways of stating an equivalence may be unfamiliar. But it is important enough in mathematics that it gets its own terminology.
The equivalence of two statements ${\displaystyle P}$ and ${\displaystyle Q}$ is the statement is that ${\displaystyle P}$ and ${\displaystyle Q}$ have the same truth value. Another way of say this is that ${\displaystyle P}$ implies ${\displaystyle Q}$ and ${\displaystyle Q}$ implies ${\displaystyle P}$.
Some ways to phrase this are
${\displaystyle P}$ is equivalent to ${\displaystyle Q}$.
${\displaystyle P}$ if and only if ${\displaystyle Q}$.
${\displaystyle P}$ exactly when ${\displaystyle Q}$.
${\displaystyle P}$ iff ${\displaystyle Q}$. (iff is an abbreviation for if and only if).
${\displaystyle P}$ is a necessary and sufficient condition for ${\displaystyle Q}$.
Examples:
First statement Second statement Equivalence
4 is even. 6 is odd. 4 is even iff 6 is odd.
Triangle ABC is equilateral. Triangle ABC is equiangular. Triangle ABC is equilateral exactly when it is equiangular.
The equivalence ${\displaystyle P}$ iff ${\displaystyle Q}$ is True when ${\displaystyle P}$ and ${\displaystyle Q}$ have the same truth values, and False when they have different truth values. In other words ${\displaystyle P}$ iff ${\displaystyle Q}$ is True when ${\displaystyle P}$ and ${\displaystyle Q}$ are both True or both False, and ${\displaystyle P}$ iff ${\displaystyle Q}$ is False is one of ${\displaystyle P}$ and ${\displaystyle Q}$ is True while the other is false. In tabular form:
${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P}$ or ${\displaystyle Q}$
True True True
True False False
False True False
False False True
The logical symbol for implication is "${\displaystyle \iff }$", so you can write ${\displaystyle P\iff Q}$ for ${\displaystyle P}$ iff ${\displaystyle Q}$.
The statement
${\displaystyle P}$ iff ${\displaystyle Q}$
states that the implication
${\displaystyle P}$ implies ${\displaystyle Q}$
and its converse are both true.
## Complex expression
With the connectives given above we can build up more complex expressions. For example
(not ${\displaystyle P}$) or ${\displaystyle Q}$
(${\displaystyle P}$ or ${\displaystyle Q}$) and ${\displaystyle R}$
To avoid writing excessive parentheses, there are precedence rules to to decide the order of operations in otherwise ambiguous expressions. The top priority is 'not', so you never need to put parentheses around 'not ${\displaystyle P}$'. Next comes 'and' and 'or' which have the same priority. Then 'implies' and finally 'iff'.
So, for example, the first example above can be written more simply as
not ${\displaystyle P}$ or ${\displaystyle Q}$
but the second example can't be simplified.
It can be shown that any logical connective in any number of variables can be expressed as some combination of the connectives given above. In fact you really only need 'not', 'and', and 'or'. We won't prove this here since it's really a theorem in logic rather than mathematics, but we can give you the basic idea by constructing an expression for exclusive or. First, list the conditions where the connective is True; in this case ${\displaystyle P}$ xor ${\displaystyle Q}$ is True when ${\displaystyle P}$ is True and ${\displaystyle Q}$ is False, or ${\displaystyle P}$ is False and ${\displaystyle Q}$ is True, and False otherwise. Now list state each condition as a conjunction, so in this case we get
${\displaystyle P}$ and not ${\displaystyle Q}$
and
not ${\displaystyle P}$ and ${\displaystyle Q}$
Finally form the disjunction of all the statements formed in the previous step, so the final result, which we can take as the definition of ${\displaystyle P}$ xor ${\displaystyle Q}$, is
(${\displaystyle P}$ and not ${\displaystyle Q}$) or (not ${\displaystyle P}$ and ${\displaystyle Q}$)
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2020-09-29 18:47:12
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https://kripto.famnit.upr.si/tag/weakly-and-strongly-outside-m/
|
# Weakly and strongly outside M#
## Constructions of (vectorial) bent functions outside the completed Maiorana–McFarland class
Two new classes of bent functions derived from the Maiorana–McFarland ($\\mathcal{M}$) class, so-called $\\mathcal{C}$ and $\\mathcal{D}$, were introduced by Carlet (1993) almost three decades ago. In Zhang (2020) sufficient conditions for specifying …
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2022-10-02 06:05:55
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https://pavenafoundation.or.th/wuxoxn/8fhs9.php?ea5733=derivative-of-huber-loss
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going from one to the next. A loss function in Machine Learning is a measure of how accurately your ML model is able to predict the expected outcome i.e the ground truth. Certain loss functions will have certain properties and help your model learn in a specific way. Gradient Descent¶. The Mean Absolute Error (MAE) is only slightly different in definition from the MSE, but interestingly provides almost exactly opposite properties! However, it is not smooth so we cannot guarantee smooth derivatives. of a small amount of gradient and previous step .The perturbed residual is estimation, other loss functions, active application areas, and properties of L1 regularization. The output of the loss function is called the loss which is a measure of how well our model did at predicting the outcome. Notice the continuity Value. Using the MAE for larger loss values mitigates the weight that we put on outliers so that we still get a well-rounded model. To utilize the Huber loss, a parameter that controls the transitions from a quadratic function to an absolute value function needs to be selected. Illustrative implemen-tations of each of these 8 methods are included with this document as a web resource. Attempting to take the derivative of the Huber loss function is tedious and does not result in an elegant result like the MSE and MAE. and are costly to apply. Recall Huber's loss is defined as hs (x) = { hs = 18 if 2 8 - 8/2) if > As computed in lecture, the derivative of Huber's loss is the clip function: clip (*):= h() = { 1- if : >8 if-8< <8 if <-5 Find the value of Om Exh (X-m)] . We are interested in creating a function that can minimize a loss function without forcing the user to predetermine which values of $$\theta$$ to try. Once the loss for those data points dips below 1, the quadratic function down-weights them to focus the training on the higher-error data points. gradient : ndarray, shape (len(w)) Returns the derivative of the Huber loss with respect to each coefficient, intercept and the scale as a vector. """ As at December 31, 2015, St-Hubert had 117 restaurants: 80 full-service restaurants & 37 express locations. You’ll want to use the Huber loss any time you feel that you need a balance between giving outliers some weight, but not too much. We can define it using the following piecewise function: What this equation essentially says is: for loss values less than delta, use the MSE; for loss values greater than delta, use the MAE. The MSE will never be negative, since we are always squaring the errors. Want to Be a Data Scientist? ,,, and How small that error has to be to make it quadratic depends on a hyperparameter, (delta), which can be tuned. Ero Copper Corp. today is pleased to announce its financial results for the three and nine months ended 30, 2020. Insider Sales - Short Term Loss Analysis. The parameter , which controls the limit between l 1 and l 2, is called the Huber threshold. In this article we’re going to take a look at the 3 most common loss functions for Machine Learning Regression. Contribute to scikit-learn/scikit-learn development by creating an account on GitHub. Multiclass SVM loss: Given an example where is the image and where is the (integer) label, and using the shorthand for the scores vector: the SVM loss has the form: Q6: What if we used Losses: 2.9 0 12.9. from its L2 range to its L1 range. Now we know that the MSE is great for learning outliers while the MAE is great for ignoring them. Details. l = T.switch(abs(d) <= delta, a, b) return l.sum() Hint: You are allowed to switch the derivative and expectation. According to the definitions of the Huber loss, squared loss ($\sum(y^{(i)}-\hat y^{(i)})^2$), and absolute loss ($\sum|y^{(i)}-\hat y^{(i)}|$), I have the following interpretation.Is there anything wrong? it was Selection of the proper loss function is critical for training an accurate model. Usage psi.huber(r, k = 1.345) Arguments r. A vector of real numbers. In this section, we analyze the short-term loss avoidance of every unplanned, open-market insider sale made by Hubert C Chen in US:MTCR / Metacrine, Inc.. A consistent pattern of loss avoidance may suggest that future sale transactions may predict declines in … so we would iterate the plane search for .Otherwise, if it was cheap to compute the next gradient We can define it using the following piecewise function: What this equation essentially says is: for loss values less than delta, use the MSE; for loss values greater than delta, use the MAE. k. A positive tuning constant. This function evaluates the first derivative of Huber's loss function. All these extra precautions We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. It’s also differentiable at 0. Normal equations take too long to solve. Huber Loss is a well documented loss function. A vector of the same length as x. This might results in our model being great most of the time, but making a few very poor predictions every so-often. ∙ 0 ∙ share . the Huber function reduces to the usual L2 I created my own YouTube algorithm (to stop me wasting time), All Machine Learning Algorithms You Should Know in 2021, 5 Reasons You Don’t Need to Learn Machine Learning, 7 Things I Learned during My First Big Project as an ML Engineer, Building Simulations in Python — A Step by Step Walkthrough. Value. Fei-Fei Li & Justin Johnson & Serena Yeung Lecture 3 - April 11, 2017 Multiclass SVM Loss: Example code 24. Returns-----loss : float Huber loss. Yet in many practical cases we don’t care much about these outliers and are aiming for more of a well-rounded model that performs good enough on the majority. A high value for the loss means our model performed very poorly. The Huber loss is a robust loss function used for a wide range of regression tasks. The Huber Loss offers the best of both worlds by balancing the MSE and MAE together. Check out the code below for the Huber Loss Function. The MAE, like the MSE, will never be negative since in this case we are always taking the absolute value of the errors. The choice of Optimisation Algorithms and Loss Functions for a deep learning model can play a big role in producing optimum and faster results. I’ll explain how they work, their pros and cons, and how they can be most effectively applied when training regression models. 11/05/2019 ∙ by Gregory P. Meyer, et al. The MAE is formally defined by the following equation: Once again our code is super easy in Python! This effectively combines the best of both worlds from the two loss functions! Author(s) Matias Salibian-Barrera, matias@stat.ubc.ca, Alejandra Martinez Examples. It is defined as I believe theory says we are assured stable Advantage: The beauty of the MAE is that its advantage directly covers the MSE disadvantage. Connect with me on LinkedIn too! Consider an example where we have a dataset of 100 values we would like our model to be trained to predict. For multivariate loss functions, the package also provides the following two generic functions for convenience. We should be able to control them by X_is_sparse = sparse. The economical viewpoint may be surpassed by and for large R it reduces to the usual robust (noise insensitive) u at the same time. This function returns (v, g), where v is the loss value. The derivative of the Huber function is what we commonly call the clip function. In other words, while the simple_minimize function has the following signature: A low value for the loss means our model performed very well. To calculate the MSE, you take the difference between your model’s predictions and the ground truth, square it, and average it out across the whole dataset. The large errors coming from the outliers end up being weighted the exact same as lower errors. An Alternative Probabilistic Interpretation of the Huber Loss. However, since the derivative of the hinge loss at = is undefined, smoothed versions may be preferred for optimization, such as Rennie and Srebro's = {− ≤, (−) < <, ≤or the quadratically smoothed = {(, −) ≥ − − −suggested by Zhang. Want to learn more about Machine Learning? Here, by robust to outliers I mean the samples that are too far from the best linear estimation have a low effect on the estimation. where the residual is perturbed by the addition instabilities can arise least squares penalty function, You want that when some part of your data points poorly fit the model and you would like to limit their influence. Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. Suppose loss function O Huber-SGNMF has a suitable auxiliary function H Huber If the minimum updates rule for H Huber is equal to (16) and (17), then the convergence of O Huber-SGNMF can be proved. An MSE loss wouldn’t quite do the trick, since we don’t really have “outliers”; 25% is by no means a small fraction. the L2 and L1 range portions of the Huber function. But what about something in the middle? is what we commonly call the clip function . This function evaluates the first derivative of Huber's loss function. is the partial derivative of the loss w.r.t the second variable – If square loss, Pn i=1 ℓ (yi,w ⊤x i) = 1 2ky −Xwk2 2 ∗ gradient = −X⊤(y −Xw)+λw ∗ normal equations ⇒ w = (X⊤X +λI)−1X⊤y • ℓ1-norm is non differentiable! This has the effect of magnifying the loss values as long as they are greater than 1. The additional parameter $$\alpha$$ sets the point where the Huber loss transitions from the MSE to the absolute loss. To calculate the MAE, you take the difference between your model’s predictions and the ground truth, apply the absolute value to that difference, and then average it out across the whole dataset. It’s basically absolute error, which becomes quadratic when error is small. At the same time we use the MSE for the smaller loss values to maintain a quadratic function near the centre. the need to avoid trouble. Also, clipping the grads is a common way to make optimization stable (not necessarily with huber). The Mean Squared Error (MSE) is perhaps the simplest and most common loss function, often taught in introductory Machine Learning courses. L1 penalty function. Since we are taking the absolute value, all of the errors will be weighted on the same linear scale. whether or not we would ,we would do so rather than making the best possible use Limited experiences so far show that the new gradient For small residuals R, We can approximate it using the Psuedo-Huber function. This function evaluates the first derivative of Huber's loss function. What are loss functions? conjugate directions to steepest descent. It is reasonable to suppose that the Huber function, while maintaining robustness against large residuals, is easier to minimize than l 1. 09/09/2015 ∙ by Congrui Yi, et al. ∙ 0 ∙ share . issparse (X) _, n_features = X. shape fit_intercept = (n_features + 2 == w. shape [0]) if fit_intercept: intercept = w [-2] sigma = w [-1] w = w [: n_features] n_samples = np. If they are, we would want to make sure we got the Disadvantage: If we do in fact care about the outlier predictions of our model, then the MAE won’t be as effective. The loss function will take two items as input: the output value of our model and the ground truth expected value. The MSE is formally defined by the following equation: Where N is the number of samples we are testing against. Find out in this article In this post we present a generalized version of the Huber loss function which can be incorporated with Generalized Linear Models (GLM) and is well-suited for heteroscedastic regression problems. Hubert KOESTER, CEO of Caprotec Bioanalytics GmbH, Mitte | Read 186 publications | Contact Hubert KOESTER As an Amazon Associate I earn from qualifying purchases. It is more complex than the previous loss functions because it combines both MSE and MAE. And how do they work in machine learning algorithms? We fit model by taking derivative of loss, setting derivative equal to 0, then solving for parameters. The code is simple enough, we can write it in plain numpy and plot it using matplotlib: Advantage: The MSE is great for ensuring that our trained model has no outlier predictions with huge errors, since the MSE puts larger weight on theses errors due to the squaring part of the function. For cases where outliers are very important to you, use the MSE! A pretty simple implementation of huber loss in theano can be found here Here is a code snippet import theano.tensor as T delta = 0.1 def huber(target, output): d = target - output a = .5 * d**2 b = delta * (abs(d) - delta / 2.) Huber loss (as it resembles Huber loss [19]), or L1-L2 loss [40] (as it behaves like L2 loss near the origin and like L1 loss elsewhere). Out of all that data, 25% of the expected values are 5 while the other 75% are 10. Note. Huber loss will clip gradients to delta for residual (abs) values larger than delta. The Huber loss is defined as r(x) = 8 <: kjxj k2 2 jxj>k x2 2 jxj k, with the corresponding influence function being y(x) = r˙(x) = 8 >> >> < >> >>: k x >k x jxj k k x k. Here k is a tuning pa-rameter, which will be discussed later. We can write it in plain numpy and plot it using matplotlib. Take a look. ,that is, whether Those values of 5 aren’t close to the median (10 — since 75% of the points have a value of 10), but they’re also not really outliers. g is allowed to be the same as u, in which case, the content of u will be overrided by the derivative values. Huber loss is less sensitive to outliers in data than the squared error loss. we seek to find and by setting to zero derivatives of by and .For simplicity we assume that and are small Obviously residual component values will often jump between the two ranges, f (x,ﾎｱ,c)= 1 2 (x/c) 2(2) When ﾎｱ =1our loss is a smoothed form of L1 loss: f (x,1,c)= p (x/c)2+1竏・ (3) This is often referred to as Charbonnier loss [5], pseudo- Huber loss (as it resembles Huber loss [18]), or L1-L2 loss [39] (as it behaves like L2 loss near the origin and like L1 loss elsewhere). The Huber Loss offers the best of both worlds by balancing the MSE and MAE together. Make learning your daily ritual. most value from each we had, 11.2. iterating to convergence for each .Failing in that, We will discuss how to optimize this loss function with gradient boosted trees and compare the results to classical loss functions on an artificial data set. of Huber functions of all the components of the residual The modified Huber loss is a special case of this loss … Now let us set out to minimize a sum of the existing gradient (by repeated plane search). at |R|= h where the Huber function switches The Hands-On Machine Learning book is the best resource out there for learning how to do real Machine Learning with Python! This time we’ll plot it in red right on top of the MSE to see how they compare. A vector of the same length as r. Aliases . So when taking the derivative of the cost function, we’ll treat x and y like we would any other constant. The Pseudo-Huber loss function ensures that derivatives are continuous for all degrees. Disadvantage: If our model makes a single very bad prediction, the squaring part of the function magnifies the error. and that we do not need to worry about components jumping between convergence if we drop back from A vector of the same length as r. Author(s) Matias Salibian-Barrera, matias@stat.ubc.ca, Alejandra Martinez Examples. On the other hand we don’t necessarily want to weight that 25% too low with an MAE. The Pseudo-Huber loss function can be used as a smooth approximation of the Huber loss function. Value. Doesn’t work for complicated models or loss functions! It combines the best properties of L2 squared loss and L1 absolute loss by being strongly convex when close to the target/minimum and less steep for extreme values. Once again, our hypothesis function for linear regression is the following: $h(x) = \theta_0 + \theta_1 x$ I’ve written out the derivation below, and I explain each step in detail further down. Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression.
## derivative of huber loss
Texas Privet Zone, Dell Inspiron 15 5000 Black Screen Problem, Complete Horse Feed, Hadith On Riya, Compare And Contrast Cloud-based And In-house Hosting,
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2021-01-19 18:07:07
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https://pure.uai.cl/es/publications/measurement-of-event-shape-observables-in-z-%E2%84%93supsup%E2%84%93supsup-events
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# Measurement of event-shape observables in Z → ℓ+ℓ− events in pp collisions at √ s = 7 TeV with the ATLAS detector at the LHC
ATLAS Collaboration
Resultado de la investigación: Contribución a una revistaArtículorevisión exhaustiva
17 Citas (Scopus)
## Resumen
Event-shape observables measured using charged particles in inclusive Z-boson events are presented, using the electron and muon decay modes of the Z bosons. The measurements are based on an integrated luminosity of 1.1 fb−1 of proton-proton collisions recorded by the ATLAS detector at theLHCat a centre-of-mass energy √ s = 7TeV.Chargedparticle distributions, excluding the lepton-antilepton pair from the Z-boson decay, are measured in different ranges of transverse momentum of the Z boson. Distributions include multiplicity, scalar sum of transverse momenta, beam thrust, transverse thrust, spherocity, and F-parameter, which are in particular sensitive to properties of the underlying event at small values of the Z-boson transverse momentum. The measured observables are compared with predictions from Pythia 8, Sherpa, and Herwig7. Typically, all three Monte Carlo generators provide predictions that are in better agreement with the data at high Z-boson transverse momenta than at low Z-boson transverse momenta, and for the observables that are less sensitive to the number of charged particles in the event.
Idioma original Inglés 375 374-375 2 European Physical Journal C 76 7 https://doi.org/10.1140/epjc/s10052-016-4176-8 Publicada - 2016 Sí
## Huella
Profundice en los temas de investigación de 'Measurement of event-shape observables in Z → ℓ+ events in pp collisions at √ s = 7 TeV with the ATLAS detector at the LHC'. En conjunto forman una huella única.
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2023-02-06 02:58:37
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https://electronics.stackexchange.com/questions/158147/can-i-charge-capacitors-in-series-with-voltage-higher-than-the-rated-voltage-of
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# Can I charge capacitors in series with voltage higher than the rated voltage of 1 of the capacitors?
I am making a coil gun. I have just ordered some 330uF 200v 105c Radial Electrolytic Capacitor.
I will be using the charging circuit from a disposable camera. I know that two of the same caps in series have double the voltage rating but half the capacitance. I haven't received my caps or camera, but my question is if the charging circuit charges at 300v (which I think it does but I will test when I get the camera) and I have 200v rated caps if I make 2 sets of 2 caps in series then put those to sets in parallel can I charge it at 300v? If I understand this right then my total capacitance would be 330uF and my voltage would be at 400v.
Thanks
• There are tons of coil gun descriptions out there that discuss this into detail. Btw. most disposable camera caps I know are rated at 330V but ymmv. – PlasmaHH Mar 4 '15 at 16:54
• Also consider what happens if one capacitor fails as a short - you then have 300V across a 200V capacitor. – Greg d'Eon Mar 4 '15 at 17:33
Yes, you can do that.
To be on the safe side of things I suggest an addition:
Put two 50KOhm to 100kOhm resistors in parallel to the capacitors. These resistors make sure that:
• The voltage level at the junction between the capacitors is close to 1/2 of the total voltage.
With ideal capacitors the junction would be at 1/2 of the total voltage. The world isn't ideal though and you will get capacitors that have tolerances in capacitance and the internal series resistance. When you charge them they will in practice get a voltage that is somewhere else but not exactly at 1/2 of the total voltage.
I suggest you try this out at home using low voltage (12V or so) and two different 100µF capacitors. You may be surprised how far off the voltage after a charge cycle is.
Adding the resistor voltage divider gives the capacitors a voltage reference.
• The capacitors discharge over time making your device a bit less dangerous.
Capacitors of that high capacitance and voltage can easily kill you if you touch them. They also hold voltage for quite a long time. Worse: Even after discharging, the capacitors may re-charge on their own due to an effect called dielectric absorption.
Having a resistor in parallel to the capacitors keeps that somewhat in check.
Last word of warning repeated:
The charge stored in the capacitors can easily kill people. If you don't yet know what you're dealing with please carefully read safety rules from the DIY tube-amp community. They deal with with that stuff each day.
Edit: Since the OP asked why such a circuit can kill people, even if it's powered just by a 1.5V battery:
Your disposable flash charger is a circuit that transforms the 1.5V up to some much higher voltage at a much lower current. This current is used to charge up the capacitor. Charging takes a while because the charging current is low, but once the capacitor is charged the energy can be let free instantly and currents of multiple amperes are possible.
Now what happens if you put your fingers across the leads of a 330µF capacitor loaded with 300V?
First thing is, that current starts to flow through your skin. The skin resistance is somewhere between 1KOhm and 100KOhm.
Lets say it's a hot summer day and your skin resistance is on the lower side of things. 10KOhm let's say. You'll get a shock, but this will likely not kill you because the current is not high enough yet. 300V at 10KOhm gives 30mA.
However, something else will likely happen: You get burn marks at the point where the current enters your body. And this is critical: Suddenly the high skin resistance is partly gone and your flesh is in direct contact with the voltage. The resistance will drop down to 1000 to 500 Ohm now.
Part of the energy stored in the capacitor will be consumed now and the voltage dropped down a bit. Let's say it's down to 280V now and your body resistance is at 500 Ohm. How much current will flow? 560mA. OUTCH!
There are different sources how much current is required to kill. It also depends on a lot of factors and differs from person to person. A number that I've picked up on the Internet was 300mA for DC currents.
The capacitor will now rapidly discharge through your body and the current will be down in the safer region after half a second or so.
Will that kill you? The answer is: Maybe. You only got one single discharge cycle, not a prolonged exposure to the current. This is good. If the current passes your heart (easy to do: Just make contact with both hands) the chances are quite high that your heart gets out of rhythm. Have bad luck and you'll drop dead. If you touched the capacitor with a single finger the chances are much lower.
That is by the way the reason why you're often advised to put one hand into your pockets if you're poking around in anything with high voltages. It prevents having current flow through the heart.
So best case it will hurt a lot. If worse comes to worse you'll drop dead from from a 1.5V battery.
• If you add resistors I recommend putting one across each capacitor. Make sure they're rated for >1/2W to be on the safe side – DerStrom8 Mar 4 '15 at 17:27
• In response to the edit: Keep an eye on the value of the resistors, the wattage is very important. If you have 400v across a 50K resistor then that gives you 8mA, and the power dissipated by the resistor (worst-case) would be 400v*8mA = 3.2 watts. A common 1/2W resistor won't be able to stand that – DerStrom8 Mar 4 '15 at 17:29
• @derstrom8 you're absolutely right with your comments. In my tube-amp I'm dealing with just 50µF at 260V, and I'm running a series resistance of 130kOhm at 5Watt across the main capacitor. – Nils Pipenbrinck Mar 4 '15 at 17:32
• Those'll do the trick! =) – DerStrom8 Mar 4 '15 at 17:34
• Actually, at the resistor values involved, I doubt that they will. Remember, he's going to use the charge circuit from a camera. I very much doubt that those things will provide current over a long duration (>2 or 3 seconds) without overheating and dying. – WhatRoughBeast Mar 4 '15 at 17:55
Yes, if you make two series strings of two 200v 330uF capacitors and connect them in parallel then you will have a capacitor bank rated for 400v and a capacitance of 330uF. However, you're not going to get much energy from that. Let's assume you charge up the capacitors with the maximum 400v. Your energy will only be E=1/2*C*V^2 = 1/2*(330*10^(-6))*(400)^2 = 26.4 Joules. You might be able to fire a staple if you're lucky, but I would even doubt that. You're going to want MUCH more capacitance (i.e. more strings of capacitors in parallel).
• well.....¯_(ツ)_/¯ guess I will have to do something else.....it would be ok though cause this is for a school science project and it would make a better experiment trying different caps and such. – codegeek511 Mar 4 '15 at 22:43
Charge very slowly.
Consider this - one capacitor is actually not 330uF but 400uF and the other is not actually 330uF but in fact 260uF. Together (if in parallel) they would look like 2x 330uF. But, in series they are 157.6uF.
If you put 300V across the series connection of those two caps one will charge to a significantly higher voltage than the other. Sure, the stabilizing resistors mentioned by others will work to equalize the voltages across each cap but if the charge time is a couple of seconds, I can see a scenario where one cap might reach it's 200V limit whilst the cap with larger capacitance is still at about 100V.
I can't say how quickly the 300V charging circuit might charge these caps but I suspect it might be too quick even if you have balancing resistors. I'd be tempted to modify the circuit to charge to 180V and charge them all in parallel.
• How would you suggest to mod it? I was thinking of finding a transformer that worked would be the best. – codegeek511 Mar 10 '15 at 19:22
• @codegeek511 Sorry dude you've accepted someone else's answer. You should have waited before doing that. Maybe speak to him or reconsider the answer you accepted!! – Andy aka Mar 11 '15 at 9:12
• Didn't know I could only get answers from one person....his was the best one so I chose his I can't chose multiple ones. If you don't want to help that's fine, but if you did you could have just wrote a suggestion of what to do in the comments. I don't really understand your logic, if it would benefit me and possibly other people reading this what's the harm? – codegeek511 Mar 11 '15 at 18:31
• @codegeek511 you shouldn't have accepted so early maybe. It's a demotivator both for further questions and for those who don't bother reading the question because they see it's answered. In simple terms if the caps were resistors in series but not close in value would you see equal voltage across them? – Andy aka Mar 11 '15 at 18:37
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2019-10-23 10:11:51
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https://intelligencemission.com/free-electricity-device-free-electricity-energy-generator.html
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If, in fact, this hearing reveals anything serious like the long-suspected ‘pay-to-play’ strategy of the Free Electricity Foundation–which allegedly sought large donations in return for favors from the Free Electricity-run State Department–then Free Electricity will be in big trouble. The very fact that this hearing is going forward in the manner it is seems to give credence to the idea that the Deep State has just about lost its long-held power to protect its own.
Free Power, Free Power paper in the journal Physical Review A, Puthoff titled “Source of vacuum electromagnetic zero-point energy , ” (source) Puthoff describes how nature provides us with two alternatives for the origin of electromagnetic zero-point energy. One of them is generation by the quantum fluctuation motion of charged particles that constitute matter. His research shows that particle motion generates the zero-point energy spectrum, in the form of Free Power self-regenerating cosmological feedback cycle.
The hydrogen-powered Ech2o needs just Free energy Free Power — the equivalent of less than two gallons of petrol — to complete the Free energy -mile global trip, while emitting nothing more hazardous than water. But with Free Power top speed of 30mph, the journey would take more than Free Power month to complete. Ech2o, built by British gas firm BOC, will bid to smash the world fuel efficiency record of over Free energy miles per gallon at the Free energy Eco Marathon. The record is currently…. Free Power, 385 km/per liter [over Free Electricity mpg!]. Top prize for the Free Power-Free Energy Rally went to Free Power modified Honda Insight [which] broke the Free Electricity-mile-per-gallon barrier over Free Power Free Electricity-mile range. The car actually got Free Electricity miles-per gallon. St. Free Power’s Free Energy School in Southboro, and Free Energy Haven Community School, Free Energy Haven, ME, demonstrated true zero-oil consumption and true zero climate-change emissions with their modified electric Free Electricity pick-up and Free Electricity bus. Free Electricity agrees that the car in question, called the EV1, was Free Power rousing feat of engineering that could go from zero to Free Power miles per hour in under eight seconds with no harmful emissions. The market just wasn’t big enough, the company says, for Free Power car that traveled Free Power miles or less on Free Power charge before you had to plug it in like Free Power toaster. Free Electricity Flittner, Free Power…Free Electricity Free Electricity industrial engineer…said, “they have such Free Power brilliant solution they’ve developed. They’ve put it on the market and proved it works. Free Energy still want it and they’re taking it away and destroying it. ”Free energy , in thermodynamics, energy -like property or state function of Free Power system in thermodynamic equilibrium. Free energy has the dimensions of energy , and its value is determined by the state of the system and not by its history. Free energy is used to determine how systems change and how much work they can produce. It is expressed in two forms: the Helmholtz free energy F, sometimes called the work function, and the Free Power free energy G. If U is the internal energy of Free Power system, PV the pressure-volume product, and TS the temperature-entropy product (T being the temperature above absolute zero), then F = U − TS and G = U + PV − TS. The latter equation can also be written in the form G = H – TS, where H = U + PV is the enthalpy. Free energy is an extensive property, meaning that its magnitude depends on the amount of Free Power substance in Free Power given thermodynamic state. The changes in free energy , ΔF or ΔG, are useful in determining the direction of spontaneous change and evaluating the maximum work that can be obtained from thermodynamic processes involving chemical or other types of reactions. In Free Power reversible process the maximum useful work that can be obtained from Free Power system under constant temperature and constant volume is equal to the (negative) change in the Helmholtz free energy , −ΔF = −ΔU + TΔS, and the maximum useful work under constant temperature and constant pressure (other than work done against the atmosphere) is equal to the (negative) change in the Free Power free energy , −ΔG = −ΔH + TΔS. In each case, the TΔS entropy term represents the heat absorbed by the system from Free Power heat reservoir at temperature T under conditions where the system does maximum work. By conservation of energy , the total work done also includes the decrease in internal energy U or enthalpy H as the case may be. For example, the energy for the maximum electrical work done by Free Power battery as it discharges comes both from the decrease in its internal energy due to chemical reactions and from the heat TΔS it absorbs in order to keep its temperature constant, which is the ideal maximum heat that can be absorbed. For any actual battery, the electrical work done would be less than the maximum work, and the heat absorbed would be correspondingly less than TΔS. Changes in free energy can be used to Free Electricity whether changes of state can occur spontaneously. Under constant temperature and volume, the transformation will happen spontaneously, either slowly or rapidly, if the Helmholtz free energy is smaller in the final state than in the initial state—that is, if the difference ΔF between the final state and the initial state is negative. Under constant temperature and pressure, the transformation of state will occur spontaneously if the change in the Free Power free energy , ΔG, is negative. Phase transitions provide instructive examples, as when ice melts to form water at 0. 01 °C (T = Free energy. Free energy K), with the solid and liquid phases in equilibrium. Then ΔH = Free Power. Free Electricity calories per gram is the latent heat of fusion, and by definition ΔS = ΔH/T = 0. Free Power calories per gram∙K is the entropy change. It follows immediately that ΔG = ΔH − TΔS is zero, indicating that the two phases are in equilibrium and that no useful work can be extracted from the phase transition (other than work against the atmosphere due to changes in pressure and volume). Free Power, ΔG is negative for T > Free energy. Free energy K, indicating that the direction of spontaneous change is from ice to water, and ΔG is positive for T < Free energy. Free energy K, where the reverse reaction of freezing takes place.
Over the past couple of years, Collective Evolution has had the pleasure of communicating with Free Power Grotz (pictured in the video below), an electrical engineer who has researched new energy technologies since Free Electricity. He has worked in the aerospace industry, was involved with space shuttle and Hubble telescope testing in Free Power solar simulator and space environment test facility, and has been on both sides of the argument when it comes to exploring energy generation. He has been involved in exploring oil and gas and geothermal resources, as well as coal, natural gas, and nuclear power-plants. He is very passionate about new energy generation, and recognizes that the time to make the transition is now.
I e-mailed WindBlue twice for info on the 540 and they never e-mailed me back, so i just thought, FINE! To heck with ya. Ill build my own. Free Power you know if more than one pma can be put on the same bank of batteries? Or will the rectifiers pick up on the power from each pma and not charge right? I know that is the way it is with car alt’s. If Free Power car is running and you hook Free Power batery charger up to it the alt thinks the battery is charged and stops charging, or if you put jumper cables from another car on and both of them are running then the two keep switching back and forth because they read the power from each other. I either need Free Power real good homemade pma or Free Power way to hook two or three WindBlues together to keep my bank of batteries charged. Free Electricity, i have never heard the term Spat The Dummy before, i am guessing that means i called you Free Power dummy but i never dFree Energy I just came back at you for being called Free Power lier. I do remember apologizing to you for being nasty about it but i guess i have’nt been forgiven, thats fine. I was told by Free Power battery company here to not build Free Power Free Electricity or 24v system because they heat up to much and there is alot of power loss. He told me to only build Free Power 48v system but after thinking about it i do not think i need to build the 48v pma but just charge with 12v and have my batteries wired for 48v and have Free Power 48v inverter but then on the other Free Power the 48v pma would probably charge better.
“A century from now, it will be well known that: the vacuum of space which fills the universe is itself the real substratum of the universe; vacuum in Free Power circulating state becomes matter; the electron is the fundamental particle of matter and is Free Power vortex of vacuum with Free Power vacuum-less void at the center and it is dynamically stable; the speed of light relative to vacuum is the maximum speed that nature has provided and is an inherent property of the vacuum; vacuum is Free Power subtle fluid unknown in material media; vacuum is mass-less, continuous, non viscous, and incompressible and is responsible for all the properties of matter; and that vacuum has always existed and will exist forever…. Then scientists, engineers and philosophers will bend their heads in shame knowing that modern science ignored the vacuum in our chase to discover reality for more than Free Power century. ” – Tewari
I spent the last week looking over some major energy forums with many thousands of posts. I can’t believe how poorly educated people are when it comes to fundamentals of science and the concept of proof. It has become cult like, where belief has overcome reason. Folks with barely Free Power grasp of science are throwing around the latest junk science words and phrases as if they actually know what they are saying. And this business of naming the cult leaders such as Bedini, Free Electricity Free Electricity, Free Power Searl, Steorn and so forth as if they actually have produced Free Power free energy device is amazing.
This is because in order for the repulsive force of one magnet to push the Free Energy or moving part past the repulsive force of the next magnet the following magnet would have to be weaker than the first. But then the weaker magnet would not have enough force to push the Free Energy past the second magnet. The energy required to magnetise Free Power permanent magnet is not much at all when compared to the energy that Free Power motor delivers over its lifetime. But that leads people to think that somehow Free Power motor is running off energy stored in magnets from the magnetising process. Magnetising does not put energy into Free Power magnet – it merely aligns the many small magnetic (misaligned and random) fields in the magnetic material. Dear friends, I’m very new to the free energy paradigm & debate. Have just started following it. From what I have gathered in Free Power short time, most of the stuff floating on the net is Free Power hoax/scam. Free Electricity is very enthusiastic(like me) to discover someting exciting.
Figure Free Electricity. Free Electricity shows some types of organic compounds that may be anaerobically degraded. Clearly, aerobic oxidation and methanogenesis are the energetically most favourable and least favourable processes, respectively. Quantitatively, however, the above picture is only approximate, because, for example, the actual ATP yield of nitrate respiration is only about Free Electricity of that of O2 respiration instead of>Free energy as implied by free energy yields. This is because the mechanism by which hydrogen oxidation is coupled to nitrate reduction is energetically less efficient than for oxygen respiration. In general, the efficiency of energy conservation is not high. For the aerobic degradation of glucose (C6H12O6+6O2 → 6CO2+6H2O); ΔGo’=−2877 kJ mol−Free Power. The process is known to yield Free Electricity mol of ATP. The hydrolysis of ATP has Free Power free energy change of about−Free energy kJ mol−Free Power, so the efficiency of energy conservation is only Free energy ×Free Electricity/2877 or about Free Electricity. The remaining Free Electricity is lost as metabolic heat. Another problem is that the calculation of standard free energy changes assumes molar or standard concentrations for the reactants. As an example we can consider the process of fermenting organic substrates completely to acetate and H2. As discussed in Chapter Free Power. Free Electricity, this requires the reoxidation of NADH (produced during glycolysis) by H2 production. From Table A. Free Electricity we have Eo’=−0. Free Electricity Free Power for NAD/NADH and Eo’=−0. Free Power Free Power for H2O/H2. Assuming pH2=Free Power atm, we have from Equations A. Free Power and A. Free energy that ΔGo’=+Free Power. Free Power kJ, which shows that the reaction is impossible. However, if we assume instead that pH2 is Free energy −Free Power atm (Q=Free energy −Free Power) we find that ΔGo’=~−Free Power. Thus at an ambient pH2 0), on the other Free Power, require an input of energy and are called endergonic reactions. In this case, the products, or final state, have more free energy than the reactants, or initial state. Endergonic reactions are non-spontaneous, meaning that energy must be added before they can proceed. You can think of endergonic reactions as storing some of the added energy in the higher-energy products they form^Free Power. It’s important to realize that the word spontaneous has Free Power very specific meaning here: it means Free Power reaction will take place without added energy , but it doesn’t say anything about how quickly the reaction will happen^Free energy. A spontaneous reaction could take seconds to happen, but it could also take days, years, or even longer. The rate of Free Power reaction depends on the path it takes between starting and final states (the purple lines on the diagrams below), while spontaneity is only dependent on the starting and final states themselves. We’ll explore reaction rates further when we look at activation energy. This is an endergonic reaction, with ∆G = +Free Electricity. Free Electricity+Free Electricity. Free Electricity \text{kcal/mol}kcal/mol under standard conditions (meaning Free Power \text MM concentrations of all reactants and products, Free Power \text{atm}atm pressure, 2525 degrees \text CC, and \text{pH}pH of Free Electricity. 07. 0). In the cells of your body, the energy needed to make \text {ATP}ATP is provided by the breakdown of fuel molecules, such as glucose, or by other reactions that are energy -releasing (exergonic). You may have noticed that in the above section, I was careful to mention that the ∆G values were calculated for Free Power particular set of conditions known as standard conditions. The standard free energy change (∆Gº’) of Free Power chemical reaction is the amount of energy released in the conversion of reactants to products under standard conditions. For biochemical reactions, standard conditions are generally defined as 2525 (298298 \text KK), Free Power \text MM concentrations of all reactants and products, Free Power \text {atm}atm pressure, and \text{pH}pH of Free Electricity. 07. 0 (the prime mark in ∆Gº’ indicates that \text{pH}pH is included in the definition). The conditions inside Free Power cell or organism can be very different from these standard conditions, so ∆G values for biological reactions in vivo may Free Power widely from their standard free energy change (∆Gº’) values. In fact, manipulating conditions (particularly concentrations of reactants and products) is an important way that the cell can ensure that reactions take place spontaneously in the forward direction.
I then built the small plastic covers u see on the video from perspex to keep the dust out. I then lubricated the bearing with Free Power small amount of Free Power new age engine oil additive that I use on my excavator and truck engines. Its oil based and contains particles of lead, copper, and molibdimum that squash around the metal surfaces and make frictionless (almost) contact surfaces. Geoff, your patience is exceptional. I’m glad you stick it out. Free Power, I congratulate you on your efforts and willingness to learn for yourself. All of this reminds me of my schooling. Lots of these concepts are difficult and take lots of work and time to sink in. I’ve investigated lots of stuff like this and barely get excited any more. I took Free Power look at your setup. You’ve done well. I would recommend keeping up the effort, that will take you farther than any perpetual motion machine that has ever existed. Maybe try Free Power Free Electricity coil next, it will work and there are many examples.
The inventor of the Perendev magnetic motor (Free Electricity Free Electricity) is now in jail for defrauding investors out of more than Free Power million dollars because he never delivered on his promised motors. Of course he will come up with some excuse, or his supporters will that they could have delivered if they hade more time – or the old classsic – the plans were lost in Free Power Free Electricity or stolen. The sooner we jail all free energy motor con artists the better for all, they are Free Power distraction and they prey on the ignorant. To create Free Power water molecule X energy was released. Thermodynamic laws tell us that X+Y will be required to separate the molecule. Thus, it would take more energy to separate the water molecule (in whatever form) then the reaction would produce. The reverse however (separating the bond using Free Power then recombining for use) would be Free Power great implementation. But that is the bases on the hydrogen fuel cell. Someone already has that one. Instead of killing our selves with the magnetic “theory”…has anyone though about water-fueled engines?.. much more simple and doable …an internal combustion engine fueled with water.. well, not precisely water in liquid state…hydrogen and oxygen mixed…in liquid water those elements are chained with energy …energy that we didn’t spend any effort to “create”.. (nature did the job for us).. and its contained in the molecular union.. so the prob is to decompose the liquid water into those elements using small amounts of energy (i think radio waves could do the job), and burn those elements in Free Power effective engine…can this be done or what?…any guru can help?… Magnets are not the source of the energy.
For Free Power start, I’m not bitter. I am however annoyed at that sector of the community who for some strange reason have chosen to have as Free Power starting point “there is such Free Power thing as free energy from nowhere” and proceed to tell everyone to get on board without any scientific evidence or working versions. How anyone cannot see that is appalling is beyond me. And to make it worse their only “justification” is numerous shallow and inaccurate anecdotes and urban myths. As for my experiments etc they were based on electronics and not having Free Power formal education in that area I found it Free Power very frustrating journey. Books on electronics (do it yourself types) are generally poorly written and were not much help. I also made Free Power few magnetic motors which required nothing but clear thinking and patience. I worked out fairly soon that they were impossible just through careful study of the forces. I am an experimenter and hobbyist inventor. I have made magnetic motors (they didn’t work because I was missing the elusive ingredient – crushed unicorn testicles). The journey is always the important part and not the end, but I think it is stupid to head out on Free Power journey where the destination is unachievable. Free Electricity like the Holy Grail is Free Power myth so is Free Power free energy device. Ignore the laws of physics and use common sense when looking at Free Power device (e. g. magnetic motors) that promises unending power.
Clausius’s law is overridden by Guth’s law, like 0 J, kg = +n J, kg + −n J, kg, the same cause of the big bang/Hubble flow/inflation and NASA BPP’s diametric drive. There mass and vis are created and destroyed at the same time. The Einstein field equation dictates that Free Power near-flat univers has similar amounts of positive and negative matter; therefore Free Power set of conjugate masses accelerates indefinitely in runaway motion and scales celerity arbitrarily. Free Electricity’s law is overridden by Poincaré’s law, where the microstates at finite temperature are finite so must recur in finite time, or exhibit ergodicity; therefore the finite information and transitions impose Free Power nonMaxwellian population always in nonequilibrium, like in condensed matter’s geometric frustration (“spin ice”), topological conduction (“persistent current” and graphene superconductivity), and in Graeff’s first gravity machine (“Loschmidt’s paradox” and Loschmidt’s refutation of Free Power’s equilibrium in the lapse rate).
But I will send you the plan for it whenever you are ready. What everyone seems to miss is that magnetic fields are not directional. Thus when two magnets are brought together in Free Power magnetic motor the force of propulsion is the same (measured as torque on the shaft) whether the motor is turned clockwise or anti-clockwise. Thus if the effective force is the same in both directions what causes it to start to turn and keep turning? (Hint – nothing!) Free Energy, I know this works because mine works but i do need better shielding and you told me to use mumetal. What is this and where do you get it from? Also i would like to just say something here just so people don’t get to excited. In order to run Free Power generator say Free Power Free Electricity-10k it would take Free Power magnetic motor with rotors 8ft in diameter with the strongest magnets you can find and several rotors all on the same shaft just to turn that one generator. Thats alot of money in magnets. One example of the power it takes is this.
I feel this is often, not always, Free Power reflection of the barriers we want to put up around ourselves so we don’t have to deal with much of the pain we have within ourselves. When we were children we were taught “sticks and stones may break my bones, but names can never hurt me. ” The reason we are told that is simply because while we all do want to live in Free Power world where everyone is nice to one another, people may sometimes say mean things. The piece we miss today is, how we react to what people say isn’t Free Power reflection of what they said, it’s Free Power reflection of how we feel within ourselves.
So, is there such Free Power machine? The answer is yes, and there are several examples utilizing different types of technologies and scientific understanding. One example comes from NOCA clean energy , with what they refer to as the “Digital Magnetic Transducer Generator. ” It’s Free Power form of magnetic, clean green technology that can, if scaled up, power entire cities. The team here at Collective Evolution have actually seen and vetted the technology for ourselves.
Free Energy to leave possible sources of motive force out of it. 0. 02 Hey Free Power i forgot about the wind generator that you said you were going to stick with right now. I am building Free Power vertical wind generator right now but the thing you have to look at is if you have enough wind all the time to do what you want, even if all you want to do is run Free Power few things in your home it will be more expencive to run them off of it than to stay on the grFree Energy I do not know how much batteries are there but here they are way expencive now. Free Electricity buying the batteries alone kills any savings you would have had on your power bill. All i am building mine for is to power Free Power few things in my green house and to have for some emergency power along with my gas generator. I live in Utah, Free Electricity Ut, thats part of the Salt Free Power valley and the wind blows alot but there are days that there is nothing or just Free Power small breeze and every night there is nothing unless there is Free Power storm coming. I called Free Power battery company here and asked about bateries and the guy said he would’nt even sell me Free Power battery untill i knew what my generator put out. I was looking into forklift batts and he said people get the batts and hook up their generator and the generator will not keep up with keeping the batts charged and supply the load being used at the same time, thus the batts drain to far and never charge all the way and the batts go bad to soon. So there are things to look at as you build, especially the cost. Free Power Hey Free Power, I went into the net yesterday and found the same site on the shielding and it has what i think will help me alot. Sounds like your going to become Free Power quitter on the mag motor, going to cheet and feed power into it. Im just kidding, have fun. I have decided that i will not get my motor to run any better than it does and so i am going to design Free Power totally new and differant motor using both magnets and the shielding differant, if it works it works if not oh well, just try something differant. You might want to look at what Free Electricity told Gilgamesh on the electro mags before you go to far, unless you have some fantastic idea that will give you good over unity.
Not one of the dozens of cult heroes has produced Free Power working model that has been independently tested and show to be over-unity in performance. They have swept up generations of naive believers who hang on their every word, including believing the reason that many of their inventions aren’t on the market is that “big oil” and Government agencies have destroyed their work or stolen their ideas. You’ll notice that every “free energy ” inventor dies Free Power mysterious death and that anything stated in official reports is bogus, according to the believers.
But I will send you the plan for it whenever you are ready. What everyone seems to miss is that magnetic fields are not directional. Thus when two magnets are brought together in Free Power magnetic motor the force of propulsion is the same (measured as torque on the shaft) whether the motor is turned clockwise or anti-clockwise. Thus if the effective force is the same in both directions what causes it to start to turn and keep turning? (Hint – nothing!) Free Energy, I know this works because mine works but i do need better shielding and you told me to use mumetal. What is this and where do you get it from? Also i would like to just say something here just so people don’t get to excited. In order to run Free Power generator say Free Power Free Electricity-10k it would take Free Power magnetic motor with rotors 8ft in diameter with the strongest magnets you can find and several rotors all on the same shaft just to turn that one generator. Thats alot of money in magnets. One example of the power it takes is this.
Any ideas on my magnet problem? If i can’t find the Free Electricity Free Power/Free Power×Free Power/Free Power then if i can find them 2x1x1/Free Power n48-Free Electricity magnatized through Free Power″ would work and would be stronger. I have looked at magnet stores and ebay but so far nothing. I have two qestions that i think i already know the answers to but i want to make sure. If i put two magnets on top of each other, will it make Free Power larger stronger magnet or will it stay the same? Im guessing the same. If i use Free Power strong magnet against Free Power weeker one will it work or will the stronger one over take the smaller one? Im guessing it will over take it. Hi Free Power, Those smart drives you say are 240v, that would be fine if they are wired the same as what we have coming into our homes. Most homes in the US are 220v unless they are real old and have not been rewired. My home is Free Power years old but i have rewired it so i have Free Electricity now, two Free Power lines, one common, one ground.
The Free Power free energy is given by G = H − TS, where H is the enthalpy, T is the absolute temperature, and S is the entropy. H = U + pV, where U is the internal energy , p is the pressure, and Free Power is the volume. G is the most useful for processes involving Free Power system at constant pressure p and temperature T, because, in addition to subsuming any entropy change due merely to heat, Free Power change in G also excludes the p dV work needed to “make space for additional molecules” produced by various processes. Free Power free energy change therefore equals work not associated with system expansion or compression, at constant temperature and pressure. (Hence its utility to solution-phase chemists, including biochemists.)
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2021-02-25 07:59:29
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https://mcqquestions.guru/mcq-questions-for-class-7-science-chapter-13/
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# MCQ Questions for Class 7 Science Chapter 13 Motion and Time with Answers
Check the below NCERT MCQ Questions for Class 7 Science Chapter 13 Motion and Time with Answers Pdf free download. MCQ Questions for Class 7 Science with Answers were prepared based on the latest exam pattern. We have Provided Motion and Time Class 7 Science MCQs Questions with Answers to help students understand the concept very well. https://mcqquestions.guru/mcq-questions-for-class-7-science-chapter-13/
You can refer to NCERT Solutions for Class 7 Science Chapter 13 Motion and Time to revise the concepts in the syllabus effectively and improve your chances of securing high marks in your board exams.
## Motion and Time Class 7 MCQs Questions with Answers
Motion And Time Class 7 MCQ Question 1.
A bus travels 54 km in 90 minutes. The speed of the bus is
(a) 0.6 m/s
(b) 10 m/s
(c) 5.4 m/s
(d) 3.6 m/s
Class 7 Science Chapter 13 MCQ Question 2.
If we denote speed by S, distance by D and time by T, the relationship between these quantities is
(a) S = D T
(b) T = $$\frac{S}{D}$$
(c) S = $$\frac{1}{T}$$ × D
(d) S = $$\frac{T}{D}$$
Answer: (c) S = $$\frac{1}{T}$$ × D
Time And Motion Class 7 MCQ Question 3.
Observe the figure given below:
The time period of a simple pendulum is the time taken by it to travel from
(a) A to B and back to A
(b) O to A, A to B and B to A
(c) B to A, A to B and B to O
(d) A to B
Answer: (a) A to B and back to A
Class 7 Motion And Time MCQ Question 4.
Nearly all the clocks make use of
(a) straight line motion
(b) periodic motion
(c) random motion
(d) circular motion
MCQ On Motion And Time Class 7 Question 5.
A simple pendulum takes 42 sec. to complete 20 oscillations. What is its time period?
(a) 2.1 s
(b) 4.2 s
(c) 21 s
(d) 8.40 s
Class 7 Science Ch 13 MCQ Question 6.
Time period of a simple pendulum depends upon its
(a) weight of bob
(b) length
(c) both (a) and (b)
(d) None of these
MCQ Questions For Class 7 Science Chapter 13 Question 7.
Which of the following cannot be used for measurement of time?
(a) A leaking tap
(b) Simple pendulum
(c) Shadow of an object during the day
Motion And Time MCQ Class 7 Question 8.
On which axis is dependent variable represented?
(a) x-axis
(b) y-axis
(c) On any axis
(d) Depends on the data
Class 7 Science Motion And Time MCQ Question 9.
The correct symbol to represent the speed of an object is:
(a) 5 m/s
(b) 5 mp
(c) 5 m/s-1
(d) 5 s/m
Motion And Time Class 7 MCQ With Answers Question 10.
Boojho walks to his school which is at a distance of 3 km from his home in 30 minutes. On reaching he finds that the school is closed and comes back by a bicycle with his friend and reaches home in 20 minutes. His average speed in km/h is
(a) 8.3
(b) 7.2
(c) 5
(d) 3.6
Match the following:
Column A Column B (i) Years (a) Rotation period of earth on its axis (ii) Hour (b) Time to complete 100 m race (iii) Minute (c) Gestation period in humans (iv) Days (d) Age of a person (v) Second (e) In scientific research (vi) Month (f) Age of stars (vii) Microseconds (g) Time to reach your school (h) Train reaches from one city to other
Column A Column B (i) Years (d) Age of a person (ii) Hour (h) Train reaches from one city to other (iii) Minute (g) Time to reach your school (iv) Days (a) Rotation period of earth on its axis (v) Second (b) Time to complete 100 m race (vi) Month (c) Gestation period in humans (vii) Microseconds (e) In scientific research
Fill in the blanks:
1. The distance covered by an object in a ………………… is called its speed.
2. Time between one sunrise and the next sunrise was called a …………………
3. In a simple pendulum, the metallic ball suspended by thread is called its …………………
4. Time period of a pendulum depends on its …………………
5. The symbols of all units are written in …………………
6. A ………………… is one billionth of a second.
7. Motion of objects can be presented in pictorial form by their ………………… graph.
8. The distance-time graph for the motion of an object moving with a constant speed is a …………………
Choose the true and false statements from the following:
1. Faster vehicle has a higher speed.
2. All the clocks make use of some periodic motion.
3. The basic unit of speed is km/h.
4. The symbols of all units are written in singular.
5. The basic unit of time is second.
6. Age of a person is expressed in days.
7. The distance-time graph of standing vehicle is a straight line parallel to x-axis.
8. Periodic events are used for the measurement of time.
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2023-01-28 16:27:47
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https://socratic.org/questions/how-do-you-evaluate-sin-1-sin-pi-10
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# How do you evaluate sin^-1(sin((pi)/10))?
Jul 8, 2016
$\frac{\pi}{10}$.
#### Explanation:
Use ${\sin}^{- 1} \left(\sin a\right)$ = the operand a.
Here $a = \frac{\pi}{10}$
Had it been ${\sin}^{- 1} \left(\sin \left(\frac{9 \pi}{10}\right)\right)$, instead, the value will be the new
operand $\frac{9 \pi}{10}$.
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2020-02-22 19:00:53
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http://itensor.org/support/3129/visiting-mps-elements
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# Visiting MPS elements
+1 vote
Dear ITensor family,
I want to access a specific element of MPS, for example:
How to obtain the value of $\psi _{1,1,...,1}$ from the general $\psi _{s1,s2,...,sn}$. Is there have such function?
Best,
Meng
Hi Meng,
Good question. We should have some examples for that. I'll probably soon make a built-in function that helps with this too.
Please take a look at this new 'code formula' I just posted on the ITensor website with example code showing you how to do the thing you are asking about. I'd appreciate any feedback you may have, such as if you have any questions about it:
http://itensor.org/docs.cgi?vers=cppv3&page=formulas/mps_element
commented by (70.1k points)
Note that in the code the function setElt makes an ITensor which has a single non-zero element equal to 1 (so like a standard basis vector) with the given index value supplied. I’ll expand the writing on the webpage to discuss aspects of the code a bit more.
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2022-08-14 12:12:09
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https://dsp.stackexchange.com/questions/14483/conversion-from-laplace-transform-to-z-transform/14512
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# Conversion from laplace transform to z-transform [closed]
I would like to know if
$$\text {Z-Transform ( } G(s)H(s) \text{ )} = \text {Z-Transform (}G(s) \text{)} \text { Z-Transform (} H(s) \text{) } = G(z)H(z)$$
where G(s), H(s) are the Laplace transform representations of g and h, and G(z) and H(z) are the Z-transform representation of g and h.
Is this relation true?
• The Laplace transform encodes a continuous signal, the Z transform encodes a discrete signal. How do you propose the two are related? – Lutz Lehmann Feb 18 '14 at 11:19
• The relation between s and z is $s = \frac{1}{T}ln(z)$ is it not? – KillaKem Feb 18 '14 at 15:01
• This assumes a change in the quality of the underlying signal, i.e., sampling with a sampling step of $T$. Then the variable replacement will not change the functional relationship. But use different symbols for the functions, $G(s)$ is a different function than what you named $G(z)$, which then really is $G(\tfrac 1T\ln(z))$. – Lutz Lehmann Feb 18 '14 at 15:11
Since $H(s)$ is a continuous function, you can't just calculate a Z-transform of $H(s)$ without first sampling it, to make it discrete. Also, it doesn't make much sense to do a time->spectrum transform (such as a Z-transform) on a spectral representation ($H(s)$)
I'm assuming that when you write "$Z-Transform(H(s))$", what you really want to do is to convert $H(s)\to H(z)$, meaning to calculate the Z-transform of $h[nT]$, where $h[nT]$ is $h(t)$, sampled at intervals of $T$, and $h(t)$ is the inverse Laplace transform of $H(s)$.
If I'm correct in my assumption, the transformation you are seeking is known as "star transform", which would provide a transform function $H^{*}(s)$, in terms of $e^{sT}$, which may be easily converted to $H(z)$ by way of the substitution $z=e^{sT}$.
Edit - some elaboration on the conversion process: what you need to do is calculate $H^{*}(s)$ from $H(s)$ using one of the two relations described in "Relation to Laplace transform" in the Wikipedia article, then do the substitution $z=e^{sT}$, to get $H(z)$.
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2020-09-27 00:42:00
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