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# South Africa Powerball predictions

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Our best prediction for Tuesday 18th of September 2018 was 2+1 winning numbers (on 45 different tickets).

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We have provided a total of 2,452 winning tickets for this draw. Approximately one ticket in 20 was a winner!
Our best prediction for Friday 14th of September 2018 was 2+1 winning numbers (on 56 different tickets).

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We have provided a total of 2,243 winning tickets for this draw. Approximately one ticket in 17 was a winner!
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We have provided a total of 1,428 winning tickets for this draw. Approximately one ticket in 20 was a winner!
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We have provided a total of 2,303 winning tickets for this draw. Approximately one ticket in 20 was a winner!
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Choice of Hilbert space $L^2$ for particles

While studying quantum mechanics, I have been calculating problems and examples of quantum mechanical models for a particle by assuming that its underlying Hilbert space is $L^2 (\mathbb{R}^3, \textrm{d}x)$, for example.

Why do we choose Hilbert space to be $L^2$? I understand that it is the only Hilbert space if we choose from spaces $L^p$, but I have trouble understanding the specific choice. I also know that every separable Hilbert space is unitarily isomorphic to each other. But why don't we then use different "version" of Hilbert space on $\mathbb{R}^3$? I guess it probably makes calculations simpler and more "intuitive". But are there examples where another Hilbert space is used to describe particle in, say, three spatial dimensions?

I would appreciate both intuitive answers and some proofs/sources of proofs that are related to this subject.

The question is basically what basis for the (unique separable) Hilbert space do you use. The $L^2(\mathbb{R}^3,\mathrm dx)$ corresponds to working in eigenbasis of position operator. You can work in any other basis which is convenient for your problem. Say, if you are doing a problem where Hamiltonian has purely discrete spectrum (like harmonic oscillator), you might want to use the energy eigenbasis, and then the Hilbert space "looks" more like square summable sequences $\ell^2$.
Short answer: Because of the Schrödinger equation and the invariance of $|\psi|^2$.
Long answer: $L^2(\mathbb{R}^3)$ is the natural choice of Hilbert space if you start off with the one-particle Schrödinger equation $$i \partial_t \psi(x,t) = \left( - \tfrac{\hbar^2}{2m}\triangle + V(x) \right) \psi(x,t).$$ with $x \in \mathbb{R}^3$. Now wonder what properties $\psi$ will have. If you know something about the initial values (e.g. they could be smooth), you might be able to infer something about special properties of $\psi$. But one thing that is always true is the continuity equation $$\frac{\partial |\psi|^2(t,x)}{\partial_t} = \mathrm{div} \ j.$$ It implies that, as long as no current flows into the boundary of your system, $\int_{\mathbb{R}^3} |\psi|^2$ is constant in time. And if you want it to describe a probability, you better have $$\int_{\mathbb{R}^3} dx |\psi|^2(x,t) = 1, \quad \forall t \in \mathbb{R}.$$ This directly implies that $\psi$ is an element of $L^2(\mathbb{R}^3)$. In this way of telling the story, which is also the historic one, you see that we were not just looking for some Hilbert space, but the very concrete dynamics of the Schrödinger equation implies that all relevant wave functions are in $L^2$.

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## non trival solution

nontrivial solution是什么意思_百度知道

non-trivial solution是什么意思_百度知道

[深度探索C++对象模型]trival constructor和non-trival ...

trivial and non-trivial solution of homogeneous equation

\$\begingroup\$ Since the zero solution is the "obvious" solution, hence it is called a trivial solution. Any solution which has at least one component non-zero (thereby making it a non-obvious solution) is termed as a "non-trivial" solution. \$\endgroup\$ – Anurag A Aug 13 '15 at 17:26

trival trivial区别 完美作业网 www.wanmeila.com

trivial and nontrivial-玮哥的博客-CSDN博客

A solution or example that is ridiculously simple and of little interest.Often,solutions or examples...


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## Some remarks on first passage of Lévy processes, the American put and pasting principles.(English)Zbl 1083.60034

Let $$X= (X_t)$$ be a Lévy process defined on a filtered probability space $$(\Omega,{\mathcal F},({\mathcal F}_t),P)$$. For given constants $$K> 0$$ and $$r\geq 0$$ consider the following American put optimal stopping problem: $v(x)= \sup_\tau\,E_x[e^{-r\tau}(K- e^{X_\tau})^+]\tag{*}$ (the supremum taken over all $$({\mathcal F}_t)$$-stopping times $$\tau$$). Various authors [e.g. E. Mordecki, Finance Stoch. 6, No. 4, 473–493 (2002; Zbl 1035.60038)] have shown that for many Lévy processes (*) is solved by a strategy of the form $$\tau^*= \text{inf}\{t\geq 0: X_t< x^*\}$$ for a specific value $$x^*<\log K$$ so that $v(x)= KE_x[e^{-r\tau^*}]- E_x[e^{-r\tau^*+ X_{\tau^*}}].$ The aim of the present paper is to comprehensively link a variety of identities for first passage problems of different Lévy processes and their connection to the American put optimal stopping problem and to explain precisely when smooth pasting is absent. In particular, it is thus possible to give alternative proofs of results obtained in Mordecki’s paper.

### MSC:

 60G40 Stopping times; optimal stopping problems; gambling theory 60J75 Jump processes (MSC2010) 91B70 Stochastic models in economics 60G51 Processes with independent increments; Lévy processes

Zbl 1035.60038
Full Text:


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Encyclopedia of electromagnetics

μ_e

μ e

You will be redirected to Effective magnetic permeability in approximately 2 seconds.


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Exploring Human Psychology with Mechanical Turk Data

Posted by Cameron Davidson-Pilon at

This blog post is a little different: it's a whole data collection and data analysis story. I become interested in some theories from behavioural economics, and wanted to verify them. So I used Mechanical Turkers to gather data, and then did some exploratory data analysis in Python and Pandas (bonus: I recorded my data analysis and visualization, see below).

Prospect Theory and Expected Values

It's clear that humans are irrational, but how irrational are they? After some research into behavourial economics, I became very interested in Prospect Theory, a modern theory about human behaviours. Part of Prospect Theory is determining how our brains value different outcomes under uncertainty, that is, how expected values are mentally calculated. A very interesting part of Prospect theory is that it is not probabilities that are used in the calculation of expected value:

$E^{\mathbf{Q}}[&space;Z&space;]&space;=&space;\sum_{z_i}&space;q_i&space;z_i$

Here, the q's are not the probabilities of outcome z, but it is from another probability measure called decision weights that humans actually use to weigh outcomes. Using a change of measure, we can observe the relationship between the actual probabilities and the decision weights:

$E^{\mathbf{Q}}[ Z ] = E^{\mathbf{P}}\left[ Z \frac{dQ}{dP}\right] = \sum_{z} p_i \left(\frac{q_i}{p_i} z\right)$

My interest is in this change of measure.

The Setup

Suppose you have two choices:

1. Lottery A: have a 1% chance to win $10 000, 2. Lottery B: have a 99% chance to win$101

Which would you prefer?

Well, under the real world probabilty measure, these two choices are equal: .99 * 101 = .01 * 10000. Thus a rational human would be indifferent to either option. But an actualhuman would have a preference: they would see one more valuable than the other. Thus:

$E^{\mathbf{P}}[ Z_1 ] = E^{\mathbf{P}}[ Z_2 ], \;\; E^{\mathbf{Q}}[ Z_1 ] > E^{\mathbf{Q}}[ Z_2 ]$

rewritten:

$p_1Z_1 = p_2Z_2, \;\; q_1Z_1 > q_2Z_2$

and dividing:

$\frac{q_1}{p_1} > \frac{q_2}{p_2} \\ \Rightarrow \frac{dQ}{dP}(p_1) > \frac{dQ}{dP}(p_2)$

What's left to do is determine the direction of the first inequality.

Mechanical Turk it.

So I created combinations of probabilities and prizes, all with equal real-world expected value, and asked Mechanical Turkers to pick which one they preferred:

The original HIT data and the python scripts that generate are below, plus the data that I just now recieved back from MTurk. Each pair of lotteries received 10 turkers votes.

Note: The Turking cost me $88.40, if you'd like to give back over Gittip, that would be great =) Note: I called the first choice Lottery A and the second choice Lottery B. Analysis Below is a slightly inappropriate heatmap of the choices people made. If everyone was rational, and hence indifferent to the two choices, the probabilities should hover around 0.5. This is clearly not the case. What else do we see here? 1. As expected, people are loss averse: every point in the lower-diagonal is where lottery A had a high probability of success than B. The matrix shows that most points in here are greater than 50%, thus people chose the safer bet more often. 2. The exception to the above point is the fact that 1% is choosen more favourably over 2%. This is an instance of the possibility effect. People are indifferent between 1% and 2%, as they are both so rare, thus will pick the one with larger payoff. Bonus: video of data munging in Pandas I recorded my steps to take the raw Turker data, and create the visualization above: FAQ 1. Why did I ask the Turkers to deeply imagine winning$50 dollars before answering the question? This was to offset a potential anchoring effect: if a Turkers first choice had prize $10 000, then any other prize would have looked pitiful, as the anchor had been set at$10 000. By having them imagine winning \$50 (lower than any prize), then any prize they latter saw would appear better than this anchor.

2. Next steps? I'd like to try this again, with more control over the Turkers (have a more diverse set of Turkers on it).

3. Can I see the code and raw data?Sure, here it is on Github!


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# Correlation of the Dzyaloshinskii–Moriya interaction with Heisenberg exchange and orbital asphericity

## Abstract

Chiral spin textures of a ferromagnetic layer in contact to a heavy non-magnetic metal, such as Néel-type domain walls and skyrmions, have been studied intensively because of their potential for future nanomagnetic devices. The Dyzaloshinskii–Moriya interaction (DMI) is an essential phenomenon for the formation of such chiral spin textures. In spite of recent theoretical progress aiming at understanding the microscopic origin of the DMI, an experimental investigation unravelling the physics at stake is still required. Here we experimentally demonstrate the close correlation of the DMI with the anisotropy of the orbital magnetic moment and with the magnetic dipole moment of the ferromagnetic metal in addition to Heisenberg exchange. The density functional theory and the tight-binding model calculations reveal that inversion symmetry breaking with spin–orbit coupling gives rise to the orbital-related correlation. Our study provides the experimental connection between the orbital physics and the spin–orbit-related phenomena, such as DMI.

## Introduction

Chiral interaction between two atomic spins owing to a strong spin–orbit coupling (SOC), which is known as the Dzyaloshinskii–Moriya interaction (DMI), has attracted intense interest1,2. In particular, it has been demonstrated that the DMI at the interface between ferromagnetic (FM) and non-magnetic heavy metals (HMs) plays a major role for the formation of chiral spin textures, such as skyrmions3,4 and homochiral Néel-type domain walls (DWs)5,6,7, which are attractive for the development of future information storage technology8. Understanding the microscopic origin of the DMI is indispensable for the realization of such chiral spin textures9,10. It has been reported that the proximity-induced magnetic moment in HM layers is critical to promote the DMI11. However, this proximity effect is still controversial because it has been also reported that the induced magnetic moment has no direct correlation with the DMI in the case of the Co/Pt system12,13. The scattering of spin-polarized electrons on spin–orbit coupled impurities is known to give rise to the DMI in spin-glass systems as a microscopic viewpoint14,15. However, the orbital hybridization between the spin–orbit coupled ions and the magnetic matrix was explicitly neglected in the theory, which becomes problematic when considering transition metal interfaces. On the other hand, theories have predicted that SOC combined with inversion symmetry breaking (ISB) naturally introduces a chirality to conduction electron spins in equilibrium and the interfacial DMI at an FM/HM interface is related to this spin chirality16,17,18,19,20. It has also been reported that the spin chirality is a manifestation of the chirality of the orbital magnetism in strongly spin–orbit coupled systems with ISB21,22. These previous studies suggest a possible microscopic origin of the interfacial DMI, which has remained experimentally unaddressed so far.

Here we discuss the microscopic origin of the interfacial DMI with experimental and theoretical studies as follows: First, we show the temperature dependence of the DMI for a Pt/Co/MgO trilayer, which is one of the standard structures used for the studies of the DMI7,12,23, using the extended droplet model24. We find that the DMI increases with decreasing temperature in a range from 300 to 100 K. In general, the electron–phonon interaction promotes thermally induced hopping between nearest neighbours when increasing the temperature25. As a result, it is expected that the difference between in-plane and out-of-plane hopping energies is reduced upon temperature increase. Therefore, changing the temperature of the system allows for charge redistribution between in-plane and out-of-plane orbitals while preserving the integrity of electronic states of the trilayer unlike other interface control methods such as ion irradiation26 or thermal annealing technique27, which may cause a permanent atomic rearrangement and thus induce undesired extrinsic effects. To discuss this temperature dependence of the DMI, that of the spin (ms) and orbital (mo) magnetic moments of Co and Pt is studied by X-ray magnetic circular dichroism (XMCD) spectroscopy. We find that ms values of Co and Pt show temperature dependences due to change in Heisenberg exchange. Furthermore, the intra-atomic magnetic dipole moment (mD), which is due to the asymmetric spin density distribution (fSD)28,29, displays strong temperature dependence, suggesting a sizable modification of the charge distribution between the in-plane and the out-of-plane d-orbitals under temperature variation. We also find that the out-of-plane orbital moment ($${\boldsymbol{m}}_{\mathrm{o}}^ \bot$$) shows large temperature dependence while in-plane orbital moment ($${\boldsymbol{m}}_{\mathrm{o}}^\parallel$$) does not, revealing a close connection between the anisotropy of m0 (orbital anisotropy) and the DMI. The ab initio and the tight-binding model calculations suggest that the ISB-dependent electron hopping, which gives rise to the asymmetric charge distribution at the interface of the FM/HM, is a possible microscopic origin of the correlation between the orbital anisotropy and the DMI.

## Results

### Temperature dependence of the DMI in the Pt/Co/MgO trilayer

The temperature dependence of the DMI-induced effective field (HDMI) of the Pt (2 nm)/Co (0.5  nm)/MgO (2 nm) trilayer is determined by measuring the nucleation field (Hn) applied along the in-plane (H x ) and out-of-plane (H z ) direction as schematically shown in Fig. 1a. The measured Hn values are analysed with the extended droplet model24. Since Hn is proportional to square of DW energy ($${\mathrm{\sigma }}_{{\mathrm{DW}}}^2$$), the σDW is a crucial parameter for the nucleation24,30. In case of the droplet with sizable DMI, DW magnetizations are aligned in the radial direction. When we consider the two DW magnetizations with respect to H x , DW1 and DW2 (see the inset of Fig. 1b), $${\mathrm{\sigma }}_{{\mathrm{DW}}}^2$$ of the DW1 and the DW2, respectively, follows the blue and the red curves in terms of H x due to DMI as illustrated in Fig. 1b31. Assuming that total DW energy mainly depends on the two magnetizations at the DW of a droplet with respect to H x , the total $${\mathrm{\sigma }}_{{\mathrm{DW}}}^2$$ follows the trend of the purple-dotted line. Note that there is a threshold where the $${\mathrm{\sigma }}_{{\mathrm{DW}}}^2$$ starts to become H x dependent. Our previous report showed that the extended droplet model allows us to estimate the DMI-induced field from the threshold field24. In spite of such one-dimensional approximation, we have demonstrated that the fitting gives reliable value of the DMI energy density (D).

The measured Hn as a function of H x shows a threshold arising from the DMI as predicted by the extended droplet model24 (see also Supplementary Note 1, 2 and Method section for details about the analysis and the measurement). Figures 2a–d show Hn/HSW(H x =0) as a function of H x /H K at various temperatures (a, T=300 K; b, 200 K; c, 150 K and d, 100 K). Here Hn and H x are normalized by the switching field at H x =0 [HSW(H x =0)] and by the anisotropy field (H K ), respectively. Those normalized values allow us to clearly confirm the DMI-dependent threshold with ruling out the temperature-dependent characteristics of H K and the HSW(H x =0). The temperature-dependent HDMI can be determined from the best fitting using the extended droplet model; HDMI at 300, 200, 150 and 100 K are 166±50, 245±45, 324±15 and 372±30 mT, respectively (Fig. 2e). The temperature-dependent D is readily calculated from HDMI and Δvia $${\boldsymbol{H}}_{{\mathrm{DMI}}} = {\boldsymbol{D}}/\mu _0M_{\mathrm{s}}\varDelta$$, where μ0 is the permeability, Ms is the saturation magnetization and Δ is the DW width6. Δ and Ms values in terms of the temperature are listed in Table 1. We find that D has a strong temperature dependence as shown in Fig. 2e; D increases by a factor of 2.2 as the temperature decreases from 300 to 100 K.

Heisenberg exchange is one of key parameters to understand the microscopic origin of DMI2,32. From the temperature dependence of Ms, the exchange stiffness constant (A) values were quantitatively obtained as detailed in Supplementary Note 3. As listed in Table 1, we find a clear correlation of the Heisenberg exchange with DMI; the A value increases by 57% when the temperature decreases from 300K to 100 K while DMI shows 100% increase. Based on Moriya’s theory, DMI requires three ingredients: (i) SOC, (ii) magnetic exchange, and (iii) ISB. Therefore, we can consider two parameters in addition to Heisenberg exchange: one is the proximity-induced magnetism of the non-magnetic element, which is related to the source ii, and the other is the hybridization between 3d and 5d orbitals, which is related to the sources i and iii. In order to find how DMI correlated with the induced moment in the Pt layer and the orbital structure of Co, XMCD studies were performed using the soft and hard X-ray as discussed in the following sections.

### Temperature dependence of the proximity-induced magnetic moment of Pt

In this section, we first investigate the role of the induced magnetic moment in the Pt layer as mentioned in previous section. The temperature dependence of the Pt-induced magnetic moment was measured using the XMCD method, which enables element-specific analyses of spin and orbital magnetism33,34. Figure 3a presents the XMCD and integration of XMCD spectra measured at the Pt L2,3 edge. The intensities of XMCD are ~3% of the X-ray absorption spectra (XAS) edge heights. At both the L3 and L2 edges (around 11.57 keV and 13.28 keV, respectively), the integrated XMCD spectra show temperature dependences. In contrast, there is no temperature dependence in XAS spectra (see inset of Fig. 3a). The total magnetic moment (mtotal), which is the sum of an effective spin magnetic moment $$\left( {{\boldsymbol{m}}_{\mathrm{s}}^{{\mathrm{eff}}} = {\boldsymbol{m}}_{\mathrm{s}} + {\boldsymbol{m}}_{\mathrm{D}}} \right)$$ and mo, was estimated by a sum rule calculation35 (see the details about the sum rule calculation in Supplementary Note 4). In this study, we use moment values per single hole rather than those per atom because it is difficult to precisely determine the hole number (nh) of Pt in the Pt/Co/MgO. The moment values normalized by nh are directly obtained from the sum rule formula without considering nh35. The changes in the induced magnetic moments with temperature are small (~15%) and comparable to the error range of the analysis as shown in Fig. 3b. This suggests that there is a weak correlation between the temperature dependences of the proximity-induced magnetic moments of the Pt layer and the DMI in the Pt/Co/MgO system11.

### Correlation between asymmetric orbital structure of the FM layer and the DMI

In this section, we study the correlation of DMI with various physical quantities associated with the magnetism of Co such as ms, $${\boldsymbol{m}}_{\mathrm{o}}^\parallel$$ and $${\boldsymbol{m}}_{\mathrm{o}}^ \bot$$ and the intra-atomic dipole moment mD. As explained in the previous section, ms,eff is the sum of ms and mD. The ms and mD values in terms of temperature were obtained from the relation $${\boldsymbol{m}}_{\mathrm{D}}\left( \theta \right) = {\boldsymbol{m}}_{\mathrm{D}}(0^\circ ) \cdot (1 - 3{\mathrm{cos}}^2\theta )$$, thereby $${\boldsymbol{m}}_{{\mathrm{s}},{\mathrm{eff}}}\left( \theta \right) = {\boldsymbol{m}}_{\mathrm{s}} + {\boldsymbol{m}}_{\mathrm{D}}(0^\circ ) \cdot (1 - 3{\mathrm{cos}}^2\theta )$$28,34.

Details about the sum rule calculation are explained in Supplementary Note 4. The XAS and XMCD of the film are measured at 100, 200 and 300 K. Two incident angles (θ=0° and 70° with respect to the film normal) were used to separately estimate $${\boldsymbol{m}}_{\mathrm{o}}^\parallel$$ and $${\boldsymbol{m}}_{\mathrm{o}}^ \bot$$ values using the relation $${\boldsymbol{m}}_{\mathrm{o}}\left( \theta \right) = {\boldsymbol{m}}_{\mathrm{o}}^ \bot {\mathrm{cos}}^2\theta + {\boldsymbol{m}}_{\mathrm{o}}^\parallel {\mathrm{sin}}^2\theta$$36. Figure 4a, b are typical XAS and XMCD spectra at the Co L2,3 edges obtained at 0° and 300 K. The XMCD spectra show a clear temperature dependence of their intensity at both 0° and 70° as shown in Fig. 4c–f. In case of the ms, it increases by 20% (from 0.87 μB/μh to 1.11 μB/nh) as temperature decreases from 300 to 100 K (see Fig. 4g), which is consistent with the observed temperature dependence of Ms.

In addition to ms, mD shows a strong temperature dependence. The mD reflects the anisotropy of fSD distorted by the SOC or the crystal-field effect.28,29. When the electron occupation becomes asymmetric, for instance, more electrons in the in-plane orbitals than the out-of-plane orbitals, the situation gives rise to both non-zero mD term and asymmetry in the charge distribution. As a result, asymmetry of the charge distribution naturally implies non-vanishing fSD at transition metals' interfaces. As a result, the value of mD determined from the sum rule analysis increases from 0.014 μB/nh to 0.094 μB/nh (Fig. 4g). In addition, the orbital anisotropy also increases as temperature decreases; $${\boldsymbol{m}}_o^ \bot$$ increases from 0.058 μB/nh to 0.080μB/nh, whereas $${\boldsymbol{m}}_{\mathrm{o}}^\parallel$$ slightly decreases from 0.044 μB/nh to 0.039 μB/nh (Fig. 4h). These results imply a correlation of DMI with mD and orbital anisotropy. Since the variation of the perpendicular component of the mo is also related to magnetocrystalline anisotropy (KU) based on the Bruno theory37, the correlation between DMI and KU is also reasonable (see the Supplementary Note 5). Figure 5a shows $${\boldsymbol{m}}_{\mathrm{o}}^ \bot /{\boldsymbol{m}}_{\mathrm{o}}^\parallel$$(=[$${\boldsymbol{m}}_{\mathrm{o}}^ \bot (T)/{\boldsymbol{m}}_{\mathrm{o}}^\parallel (T)$$]/[$${\boldsymbol{m}}_{\mathrm{o}}^ \bot (300{\mathrm{K}})/{\boldsymbol{m}}_{\mathrm{o}}^\parallel (300{\mathrm{K}})$$]), and $${\boldsymbol{m}}_{\mathrm{D}}(T)/{\boldsymbol{m}}_{\mathrm{D}}(300{\mathrm{K}})$$ plots as a function of the normalized D=D=(T)/D(300K) where T inside of parentheses is the measurement temperature. The ratio $${\boldsymbol{m}}_{\mathrm{o}}^ \bot /{\boldsymbol{m}}_{\mathrm{o}}^\parallel$$ increases by 53% as D increases, and mD also shows a clear correlation with the DMI; $${\boldsymbol{m}}_{\mathrm{D}}(100{\mathrm{K}})/{\boldsymbol{m}}_{\mathrm{D}}(300{\mathrm{K}})$$≈ 6.4. Because both orbital anisotropy and mD are closely related to the orbital occupation with ISB28,37,38, these results suggest that the temperature dependence of DMI is governed by the change in asymmetric electron occupation in orbitals in addition to spin magnetic moments as we discuss with the following theoretical studies. We also note that peak intensity of the XAS spectra reflects the hole number in 3d orbitals. We found small change in the absorption intensity when varying the temperature. Quantitatively, difference in the XAS intensity integral (IXAS) between 300 and 100 K is about 4% $$\{ [I_{{\mathrm{XAS}}}\left( {300{\mathrm{K}}} \right) - I_{{\mathrm{XAS}}}\left( {100{\mathrm{K}}} \right)]/ I_{{\mathrm{XAS}}}\left( {300{\mathrm{K}}} \right) = 0.04\}$$. Hence, this indicates that there is a small change in the hole number within the temperature range, reflecting temperature dependence of the charge distribution in orbitals. Nonetheless, the value is so small that it does not affect the trend in the respective quantities normalized by nh from the sum rule calculation.

### Theoretical consideration about the microscopic origin of the DMI

In order to support the insights obtained from our experimental study, we carry out two types of theoretical calculations: a tight-binding model calculation and ab initio calculation based on density function theory (DFT), and the main result is shown in Fig. 5b, c. The details of these two complementary theoretical studies can be found in Supplementary Notes 6 and 7.

For the tight-binding model, we extend the trimer model suggested by Kashid et al.10, which contains two magnetic atoms coupled to a spin–orbit coupled non-magnetic ion. Compared to Kashid’s trimer model, we add one more orbital (d xz ) on the non-magnetic site, enabling the computation of the orbital anisotropy (see details in Supplementary Note 6). In order to describe the temperature dependence of parameters in our trimer model, we assume the level broadening increases with the temperature. This broadening can be phenomenologically explained by magnetization fluctuations and the electron–phonon interaction given by atomic vibrations. As this model calculation is too simple, we do not aim to give a quantitative explanation of the experimental data but provide a qualitative understanding of the experimental result.

Figure 5b shows that the tight-binding model calculation reproduces qualitatively our experimental observations; both the DMI and $${\boldsymbol{m}}_{\mathrm{o}}^ \bot$$ decrease with temperature, while the $${\boldsymbol{m}}_{\mathrm{o}}^\parallel$$ is almost temperature independent. Note that the $${\boldsymbol{m}}_{\mathrm{o}}^ \bot$$ and $${\boldsymbol{m}}_{\mathrm{o}}^\parallel$$ cases are related to electron hopping with and without ISB, respectively (Supplementary Note 6). In addition, the ratio of the spin density distribution ($$f_{{\mathrm{SD}}}^ \bot$$/$$f_{{\mathrm{SD}}}^\parallel$$) between the in-plane (xy) and out-of-plane (yz, xz) orbital states decreases as temperature increases as shown in the inset in Fig. 5b (In our tight-binding model, the spin distribution is defined as SD n =fn,↑fn,↓, where fn,↑(fn,↓) is the occupation of spin up (down) state and n represents the orbital quantum state $$d_{xy}^A,d_{xy}^B,d_{xy}^C,d_{yz}^C,d_{xz}^C$$.). This result implies that the spin distribution variation of the orbital states with ISB is key to change of the orbital moment and the DMI, which is in line with the following DFT result.

To examine the correlation between the DMI and the orbital anisotropy in realistic structures, we also perform ab initio calculations for Pt(111)/X ultra-thin films, where X is a 3d transition metal (X=V, Cr, Mn, Fe, Co, Ni). Based on the experimental observation, the microscopic origin of the correlation between the DMI and the orbital anisotropy involves the impact of the temperature on 3d-orbital magnetization and their electron filling. In this respect, it is instructive to vary the 3d transition metals on the Pt substrate and examine the general chemical trend. Details about this study are also discussed in Supplementary Note 7. Figure 5c shows the summary of the DFT calculation. Here changing the overlayer results in modification of the relative alignment between 3d and 5d orbitals and thereby in a modification of the charge distribution: more hybridization results in less asphericity of the charge distribution between the in-plane and the out-of-plane, as reflected by the change in mD. Our calculation results provide a physical insight on how the asymmetric charge distribution induces DMI. As we mentioned in the second section, DMI requires three ingredients: (i) SOC, (ii) magnetic exchange, and (iii) symmetry breaking. While the first two aspects are easily quantifiable, the latter is more difficult to apprehend. It has been reported that the larger the asymmetric spin distribution with symmetry breaking and SOC, the larger intra-dipole spin moment29. Therefore, our experimental and theoretical studies propose a method to address the symmetry breaking term. Figure 5c explains this aspect explicitly. The Mn 3d orbitals and the Pt 5d orbitals provide large magnetic exchange and SOC, respectively. In addition, the dipole moment term becomes maximized with the hybridization between Mn 3d and Pt 5d orbitals. As a result, DMI is maximal. Furthermore, both the DMI and the orbital anisotropy follow the same trend in their signs: Dtot and $${\boldsymbol{m}}_{\mathrm{o}}^ \bot /{\boldsymbol{m}}_{\mathrm{o}}^\parallel$$ at the Pt/V interface have negative sign, while those of other interfaces have positive sign. We also find the correlation between ms and DMI as confirmed experimentally in this study and in ref. 32. In contrast, our calculations do not reproduce the experimental correlation reported above between DMI and the induced moment. This may be because of the difference between the actual film stack (~3 monolayers of Co) and the simulated system (1 monolayer of Co). In fact, the total DMI is dominated by the contribution of the first Co monolayer at the Co/Pt interface, while the Heisenberg exchange, which is a source of the induced moment, is affected by the total number of Co overlayers. This is why our calculation, limited to one Co overlayer, well reproduces the correlation of DMI with spin and orbital moments, but not with the induced moment.

Now, the remaining question is how the orbital asphericity can give rise to DMI. At the interface, in-plane orbitals of Co and out-of-plane orbitals of Pt are mixed through inter-atomic hybridization (in our tight-binding model: d xy of Co and d yz of Pt). In the presence of ISB, this hybridization results in a non-vanishing orbital angular momentum: the hopping trajectory of an electron produces an effective orbital angular momentum as illustrated in Fig. 6. Moreover, the large SOC on Pt mixes in-plane and out-of-plane orbitals, which is accompanied by spin canting (in our tight-binding model: d xy and d yz orbitals on Pt site). In other words, the SOC on Pt converts the interfacial orbital angular momentum of hopping electrons into spin canting, resulting in a non-collinear magnetic texture. In spite of its simplicity, this picture intuitively explains how the orbital asphericity generates the spin canting. A similar idea has been reported in ref. 21: the interfacial electric dipole induced by the asymmetric charge distribution results in chiral orbital angular momentum, and thereby spin canting via SOC. Although the latter approach is based on p-orbitals, it agrees with the intuitive picture based on d-orbitals.

## Discussion

Our experimental study on the temperature dependence of the DMI suggests that the interfacial DMI in FM/HM bilayers originates from Heisenberg exchange and the asymmetric charge distribution caused by the ISB, as evidenced by the orbital anisotropy and magnetic dipole moment ($${\boldsymbol{m}}_{\mathrm{o}}^ \bot /{\boldsymbol{m}}_{\mathrm{o}}^\parallel ,$$ mD). Our DFT simulation and tight-binding calculation provide a clear evidence of the close link between the DMI and orbital physics. Based on the theoretical discussions, the temperature-dependent mD and mo indicate that increase in the temperature promotes the phonon-induced electron hopping between the out-of-plane and in-plane orbitals, thereby resulting in a reduced asphericity (or reduced asymmetry) of fSD over the d-orbitals and thus quenching mD.

In our experiment, however, the correlation between $${\boldsymbol{m}}_{\mathrm{o}}^ \bot /{\boldsymbol{m}}_{\mathrm{o}}^\parallel$$ and the DMI is only semi-quantitative, i.e. the temperature-dependent change in the DMI does not perfectly scale with the temperature-dependent change in $${\boldsymbol{m}}_{\mathrm{o}}^ \bot /{\boldsymbol{m}}_{\mathrm{o}}^\parallel$$. This semi-quantitative correlation between the DMI and mo demands a more detailed discussion. In systems with ISB, $${\boldsymbol{m}}_{\mathrm{o}}^ \bot$$ can be decomposed into ISB-independent part and ISB-dependent part. Given that ISB is an essential ingredient for the DMI, there should be a direct correlation between the DMI and ISB-dependent mo, as evidenced by our tight-binding model calculation. However, $${\boldsymbol{m}}_{\mathrm{o}}^ \bot$$ also has an ISB-independent part, which precludes a direct and quantitative correlation between the DMI and $${\boldsymbol{m}}_{\mathrm{o}}^ \bot$$. This statement can be rephrased technically as the DMI involves only off-diagonal elements of the SOC operator10, while mo involves all of them. We note, however, that even with this uncertainty, the experimentally observed correlation between the DMI and mo anisotropy for the Pt/Co/MgO structure is rather clear, implying that the ISB-dependent m o would dominate over the ISB-independent one in this structure. Therefore, our findings based on both experimental and theoretical studies provide a link between orbital physics and spin–orbit-related phenomena such as the DMI, which are essential for spin–orbitronic devices.

## Method

### Film preparation and device fabrication

Si/Ta (1.5)/Pt (2)/Co (0.5)/MgO (2)/HfO (5) (in nm) film with perpendicular magnetic anisotropy was deposited on an undoped Si substrate by direct current magnetron sputtering and the atomic layer deposition technique. A 5-μm-wide Hall cross-structure were fabricated using the photo lithography and the Ar ion milling. For the XMCD measurement, the same stack film was prepared.

### Nucleation field measurement

Angular-dependent coercivity of the Co/Pt Hall device was measured to estimate the Hn of the magnetic droplet at 300, 200, 150 and 100 K. The angle between magnetic field and the sample normal was varied from 0° to 89° rotating the electromagnet. At each angle, magnetic field was swept with in ±0.5 T to observe the coercivity. Details about this measurement are also discussed in ref. 24.

### XMCD measurement

Soft X-ray: Soft X-ray absorption spectra were measured using the total electron yield method with 96% circularly polarized incident X-rays at the BL25SU at SPring-8. XMCDs at the Co L3 and L2 edges (in a range between 770 and 840 keV) were recorded in the helicity-switching mode with an applied magnetic field of 1.9 T. Homogeneity of the magnetic field was better than 99% for ɸ10 mm at the sample position. The incident light direction was inclined by 10° with respect to the magnetic field direction. Temperature was varied from 300 to 100 K using a continuous liquid He flow-type cryostat.

Hard X-ray: XMCD experiments using hard X-rays were carried out at BL39XU of SPring-8. A circularly polarized X-ray beam with a high degree of circular polarization (>95%) was produced with a transmission-type diamond X-ray phase retarder of 1.4-mm thickness. XAS of the film were observed at a 0.6 T magnetic field applied parallel to the X-ray propagation direction of which an incident angle was 0° with respect to the surface normal. The X-ray fluorescence yield mode was used to record the spectra. The X-ray energy was scanned around the Pt L3 and L2 edges in a range between 11.5 and 3.5 keV, reversing the X-ray photon helicity at 0.5 Hz. In this manner, two helicity dependent spectra I+ and I were recorded simultaneously. Here I+ and I denote the intensities when the incident photon momentum and the magnetization vectors are parallel and antiparallel, respectively. The XMCD spectrum, ΔI = I+I, is given by the difference of the two spectra. Detailed experimental set-ups for the soft and hard X-ray MCD measurements are described elsewhere39,40.

### DFT calculation

To understand the behaviour of DMI in 3d/Pt(111) ultrathin films and its correlation with the orbital moment anisotropy (OMA) and magnetic dipole moment (Tz), we have performed DFT calculations in the local density approximation41 to the exchange correlation functional, using the full potential linearized augmented plane wave method in film geometry42 as implemented in the Jülich DFT code FLEUR43. Both collinear and non-collinear magnetic states have been studied employing an asymmetric film consisting of six substrate layers of Pt covered by a pseudomorphic 3d monolayer on one side of the film at the distance optimized for the lowest collinear magnetic states. For the non-collinear calculations, we used p(1x1) unit cell applying the generalized Bloch theorem (http://www.flapw.de). The OMA is obtained at relaxed geometry as total energy difference for two different magnetization directions employing the force theorem (the principle axes point along hard and easy axis). We considered 512 and 1024 k-points in the two-dimensional Brillouin zone (2D-BZ) for the scalar relativistic computation and the calculation with SOC treated within first-order perturbation theory, respectively. Note that high computational accuracy is required since energy differences between different magnetic configurations are tiny (~meV) in the present case.

In order to investigate the DMI, first we self-consistently calculate the total energy of homogeneous magnetic spin spirals employing the generalized Bloch theorem within the scalar-relativistic approach44. We have considered the energy dispersion E(q) of planar spin spirals, which are the general solution of the Heisenberg Hamiltonian, i.e. states in which the magnetic moment of an atom site R i is given by M i =M[cos(q·R i ), sin(q·R i ), 0] where q is the wave vector propagation of the spin spiral. By imposing the Néel spin spirals along the high symmetry lines of 2D-BZ, we can scan all possible magnetic configurations that can be described by a single q-vector. So, varying the q-vector with small steps along the paths connecting the high symmetry points, we find the well-defined magnetic phases of the hexagonal lattice: FM state at $${\bar{\mathrm \Gamma }}$$-point (q=0), anti-FM state at the $${\bar{\mathrm M}}$$-point, and periodic 120° Néel state the $${\bar{\mathrm K}}$$-point. When the energy E(q) along the high symmetry lines of 2D-BZ is lower than any of the collinear magnetic phases studied previously, the system most likely adopts an incommensurate spin-spiral magnetic ground state structure.

In a second step, we evaluate the DMI contribution from the energy dispersion of spin spirals by applying the SOC treated within first-order perturbation theory combined with the spin spirals5,42. Phenomenologically, the antisymmetric exchange interaction DMI has the typical form $$E_{{\mathrm{DM}}} = \mathop {\sum }\limits_{i,j} {\boldsymbol{D}}_{i,j} \cdot ({\boldsymbol{S}}_i \times {\boldsymbol{S}}_j)$$, where D i , j is the DM vector that determine the strength and sign of DMI and S i and S j are magnetic spin moments located on neighbouring atomic sites i and j, respectively. Considering the Néel-type out-of-plane configuration, the D i , j vector should be oriented in plane and normal to the q-vector. Note that DMI term must vanish for both configurations Néel-type in-plane and Bloch-type spin spirals due to symmetry arguments44. According to our definition, the vector chirality reads $$\boldsymbol{C}= C\hat{\boldsymbol c} = {\boldsymbol{S}}_i \times {\boldsymbol{S}}_{i + 1}$$, where the direction of the vector spin chirality $$\hat{\boldsymbol c}$$ is considered as spin rotation axis. Thus the left-handed (right-handed) spin spiral correspond to C = +1 (C = −1)45.

### Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We also thank H.-W. Lee for fruitful discussion about the relation between the orbital magnetism and the DMI. This work was partly supported by JSPS KAKENHI Grant Numbers 15H05702, 26870300, 26870304, 26103002 and 25220604; JSPS Postdoctoral Fellowship program (Grant Number 2604316, P16314); Collaborative Research Program of the Institute for Chemical Research, Kyoto University; R & D project for ICT Key Technology of MEXT from the Japan Society for the Promotion of Science (JSPS) and the Cooperative Research Project Program of the Research Institute of Electrical Communication, Tohoku University. This work has also been performed with the approval of the SPring-8 Program Advisory Committee (Proposal Nos. 2015A0117, 2015A0125). A.M. and A.B. acknowledge support from King Abdullah University of Science and Technology (KAUST) and fruitful discussiojns with S. Blügel and G. Bihlmayer. G.G., P.-H.J. and K.-J.L. also acknowledge support from the National Research Foundation of Korea (NRF-2015M3D1A1070465, 2017R1A2B2006119). K.-J.K. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIP) (Nos. 2017R1C1B2009686, NRF-2016R1A5A1008184) and by the DGIST R&D Program of the Ministry of Science, ICT and Future Planning (17-BT-02). S.K. was supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093818).

## Author information

Authors

### Contributions

S.K., K.U., K.-J.K., T.M. and T.O. conceived and designed the study. S.K., K.U., T.K. and D.C. fabricated the device and performed the nucleation field measurement. S.K., D.-H.K., P.-H.J., T.M., K.-J.K. and T.O. contributed to determine the DMI-induced effective field using the droplet model. S.K., M.S., Y.K., T.N. and K.-J.K. designed and performed XMCD experiment and analysed the data. K.U. and K.Y characterizes the magnetic properties of films. A.B., A.M. and K.N. supported theoretical analyses with ab initio calculation. G.G. and K.-J.L. supported the tight-binding model for the temperature dependence of the DMI and orbital magnetism. All authors discussed the results and wrote the manuscript.

### Corresponding authors

Correspondence to Sanghoon Kim or Teruo Ono.

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Kim, S., Ueda, K., Go, G. et al. Correlation of the Dzyaloshinskii–Moriya interaction with Heisenberg exchange and orbital asphericity. Nat Commun 9, 1648 (2018). https://doi.org/10.1038/s41467-018-04017-x

• Accepted:

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text:
# File:Fishman3.jpg

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The intensity of atoms as a function of momentum and time (with the permission of Mark Raizen).

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text:
existence of closed forms of certain Dirichlet series

In the following post we find an interesting calculation of $$\sum_{n\ge 1} \frac{\mu(n)^2}{n\varphi(n)}.$$ I had been trying to do this calculation myself, setting the slightly more ambitious goal of finding a closed form of $$G(s) = \sum_{n\ge 1} \frac{\mu(n)^2/(n\varphi(n))}{n^s}$$ in some half plane. I gave up on that (when I should have pursued the evaluation of the constant) and I think I can show that there is no such closed representation as a finite product/quotient using only terms of the form $\zeta(ks)$ ($k>0$) and $\zeta(s+k)$ ($k$ an integer). This is because the Euler products for these two are $$\prod_p \left(1-\frac{1}{p^{ks}}\right)^{-1} \quad \text{and} \quad \prod_p \left(1-\frac{1}{p^s} \frac{1}{p^k}\right)^{-1}$$ Putting $z = \frac{1}{p^s}$ for $s$ a variable we see that any such product consists of products of terms of the form $$\frac{1}{1-z^k} \quad \text{and} \quad \frac{1}{1-\frac{z}{p^k}},$$ i.e. the roots and singularities relative to $z$ of these are on the unit circle or at integer powers of primes. However, the Euler product for $G(s)$ is $$\prod_p \left( 1 + \frac{1}{p(p-1)} \frac{1}{p^s}\right)$$ which in terms of $z$ is $$1 + \frac{1}{p(p-1)} z$$ with root $z = -p (p-1)$. Ergo, no such representation of $G(s)$ as a product/quotient of the above Zeta function terms can exist. My question is, is this correct? Or does there exist a closed form of $G(s)$?

For example, the Euler product for $$H_1(s) = \sum_{n\ge 1} \frac{2^{\omega(n)}}{n^s}$$ where $\omega(n)$ is the number of prime divisors, is $$\prod_p \left( 1 + 2\frac{1/p^s}{1-1/p^s}\right) = \prod_p \left( 1 + 2\frac{z}{1-z}\right)$$ which gives $z=-1$, on the unit circle, and hence $H_1(s)$ has a closed form representation in terms of products and quotients of terms of the form $\zeta(ks)$. The root at $z=-1$ predicts a term in $\zeta(2s)$ and the singularity at $z=1$ a term in $\zeta(s).$ Indeed $$H_1(s) = \frac{\zeta(s)^2}{\zeta(2s)}.$$

Similarly, the Euler product for $$H_2(s) = \sum_{n\ge 1} \frac{\varphi(n)}{n^s}$$ is $$\prod_p \left( 1 + \frac{p-1}{p^s-p}\right) = \prod_p \left( 1 + \frac{p-1}{1/z-p}\right)$$ which gives $z=1$, on the unit circle, and hence $H_2(s)$ has a closed form representation like $H_1(s).$ The root at $z=1$ predicts an inverse of $\zeta(s)$ and the singularity at $1/p$ corresponds to the term in $\zeta(s-1).$ Indeed $$H_2(s) = \frac{\zeta(s-1)}{\zeta(s)}.$$ Note that we discovered this automatically and without knowing about the Dirichlet convolution $$\varphi\star 1 = n.$$

What we have here is an algorithm for the discovery of closed forms of Dirichlet series from their Euler products. Just examine the roots and singularities according to their multiplicities and include the corresponding terms. Any takers? Maple and Mathematica come to mind.

E.g. to find a closed form of $$\prod_p \left(1 + \frac{z}{(1-z)^2}\right)$$ note that it has the two primitive roots of $1-z^6$ as roots. So include $1/\zeta(6s)$ for these roots and multiply by $\zeta(s),\zeta(2s)$ and $\zeta(3s)$ to cancel those that are not primitive. This is a representation of the fact that $${\frac {1-{z}^{6}}{ \left( 1-z \right) \left( 1-{z}^{2} \right) \left( 1-{z}^{3} \right) }} =1+{\frac {z}{ \left( 1-z \right) ^{2}}}.$$

-
$\mu(n)^2/n/\varphi(n)$ is ambiguous. –  joriki Nov 10 '12 at 5:16


text:
# Data mining - Principal Component Analysis on Weka - Stack Overflow

Data mining - Principal Component Analysis on Weka - Stack Overflow

The j in the above output implies the resulting eigenvectors are represented as complex numbers.

## 618 questions with answers in PRINCIPAL COMPONENT ANALYSIS

I've then gone through the text and made the appropriate changes diff -- I hope I got them all!

### (PDF) New Interpretation of Principal Components Analysis

Таким образом, смысл метода заключается в том, что с каждой главной компонентой связана определ x956 нная доля общей дисперсии исходного набора данных (е x956 называют нагрузкой). В свою очередь, дисперсия, являющаяся мерой изменчивости данных, может отражать уровень их информативности.

#### AStep by Step Explanation of Principal Component Analysis

However, the solution is obvious. We just have to pick the $q$ largest eigenvalues and their corresponding eigenvectors for matrix $X^{\top} X$.

##### Principal Component Analysis | Dr. Sebastian Raschka

To change easily the graphical of any ggplots, you can use the function ggpar() [ggpubr package]

So by definition, $\lambda$ and $w$ are the eigenvalue and eigenvector of matrix $X^{\top} X$.

The average person understands a 8D world. I'm hoping we can write a gentle introduction to PCA accessible to a smart person who is not in a math related career.

I remove the output variable and apply prcomp to the remaining dataset(new_dataset)

The optimality of PCA is also preserved if the noise n {\displaystyle \mathbf {n} } is iid and at least more Gaussian (in terms of the Kullback–Leibler 8697 divergence ) than the information-bearing signal s {\displaystyle \mathbf {s} } . [77] In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise n {\displaystyle \mathbf {n} } becomes dependent.

Maybe something like this: " Given a set of points in two, three, or higher dimensional space, a "best fitting" line can be defined as one that minimizes the average squared distance from a point to the line. The next best-fitting line can be similarly chosen from directions perpendicular to the first. Repeating this process yields an orthogonal basis in which individual dimensions of the data are uncorrelated. " AP795 ( talk ) 69:89, 8 April 7575 (UTC)

Hence ( x7767 ) {\displaystyle (\ast )\,} holds if and only if cov x7566 ( X ) {\displaystyle \operatorname {cov} (X)} were diagonalisable by P {\displaystyle P} .

(поскольку данные центрированы, выборочная дисперсия здесь совпадает со средним квадратом уклонения от нуля).

Step 7: Compute the mean centered data.

Re errors and residuals , residual is clearly more correct within those definitions, and is in common use, . at Matlab's "PCA Residual" function or this book that talks about the amount that is unexplained by the pc model—the residual on , and on the next page they examine the sum of squares of the residuals. Dicklyon ( talk ) 75:76, 77 August 7575 (UTC)

Note that, the supplementary qualitative variables can be also used for coloring individuals by groups. This can help to interpret the data. The data sets decathlon7 contain a supplementary qualitative variable at columns 68 corresponding to the type of competitions.

This is exactly what PCA allows us to do. In a multi-dimensional data, it will help us find the dimensions that are most useful and contain the most information. It will help us extract essential information from data by reducing the dimensions.

http:///7559/56/57/antarctic-warming-the-final-straw/#comment-6777

### Еще интересное

Principal Components Analysis Example. You are here. A bit of reading and searching led me to the conclusion that Principal Component Analysis(PCA) is the best alternative. Can anyone help me with orienting the image with respect to its principal axis?


text:
# mathlibdocumentation

measure_theory.group

# Measures on Groups #

We develop some properties of measures on (topological) groups

• We define properties on measures: left and right invariant measures.
• We define the measure μ.inv : A ↦ μ(A⁻¹) and show that it is right invariant iff μ is left invariant.
def measure_theory.is_mul_left_invariant {G : Type u_1} [has_mul G] (μ : set Gℝ≥0∞) :
Prop

A measure μ on a topological group is left invariant if the measure of left translations of a set are equal to the measure of the set itself. To left translate sets we use preimage under left multiplication, since preimages are nicer to work with than images.

Equations
• = ∀ (g : G) {A : set G}, μ ((λ (h : G), g * h) ⁻¹' A) = μ A
def measure_theory.is_add_left_invariant {G : Type u_1} [has_add G] (μ : set Gℝ≥0∞) :
Prop

A measure on a topological group is left invariant if the measure of left translations of a set are equal to the measure of the set itself. To left translate sets we use preimage under left addition, since preimages are nicer to work with than images.

def measure_theory.is_add_right_invariant {G : Type u_1} [has_add G] (μ : set Gℝ≥0∞) :
Prop

A measure on a topological group is right invariant if the measure of right translations of a set are equal to the measure of the set itself. To right translate sets we use preimage under right addition, since preimages are nicer to work with than images.

def measure_theory.is_mul_right_invariant {G : Type u_1} [has_mul G] (μ : set Gℝ≥0∞) :
Prop

A measure μ on a topological group is right invariant if the measure of right translations of a set are equal to the measure of the set itself. To right translate sets we use preimage under right multiplication, since preimages are nicer to work with than images.

Equations
• = ∀ (g : G) {A : set G}, μ ((λ (h : G), h * g) ⁻¹' A) = μ A
theorem measure_theory.measure.map_add_left_eq_self {G : Type u_1} [has_add G] [borel_space G] {μ : measure_theory.measure G} :
(∀ (g : G), = μ)
theorem measure_theory.measure.map_mul_left_eq_self {G : Type u_1} [has_mul G] [borel_space G] {μ : measure_theory.measure G} :
(∀ (g : G), = μ)
theorem measure_theory.measure.map_add_right_eq_self {G : Type u_1} [has_add G] [borel_space G] {μ : measure_theory.measure G} :
(∀ (g : G), (measure_theory.measure.map (λ (h : G), h + g)) μ = μ)
theorem measure_theory.measure.map_mul_right_eq_self {G : Type u_1} [has_mul G] [borel_space G] {μ : measure_theory.measure G} :
(∀ (g : G), (measure_theory.measure.map (λ (h : G), h * g)) μ = μ)
def measure_theory.measure.inv {G : Type u_1} [has_inv G] (μ : measure_theory.measure G) :

The measure A ↦ μ (A⁻¹), where A⁻¹ is the pointwise inverse of A.

Equations
def measure_theory.measure.neg {G : Type u_1} [has_neg G] (μ : measure_theory.measure G) :

The measure A ↦ μ (- A), where - A is the pointwise negation of A.

theorem measure_theory.measure.neg_apply {G : Type u_1} [add_group G] [borel_space G] (μ : measure_theory.measure G) {s : set G} (hs : measurable_set s) :
(μ.neg) s = μ (-s)
theorem measure_theory.measure.inv_apply {G : Type u_1} [group G] [borel_space G] (μ : measure_theory.measure G) {s : set G} (hs : measurable_set s) :
(μ.inv) s = μ s⁻¹
@[simp]
theorem measure_theory.measure.inv_inv {G : Type u_1} [group G] [borel_space G] (μ : measure_theory.measure G) :
μ.inv.inv = μ
@[simp]
theorem measure_theory.measure.neg_neg {G : Type u_1} [add_group G] [borel_space G] (μ : measure_theory.measure G) :
μ.neg.neg = μ
theorem measure_theory.measure.regular.inv {G : Type u_1} [group G] [borel_space G] {μ : measure_theory.measure G} [t2_space G] (hμ : μ.regular) :
@[simp]
@[simp]
@[simp]
@[simp]

Properties of regular left invariant measures

theorem measure_theory.is_mul_left_invariant.null_iff_empty {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [group G] (hμ : μ.regular) (h3μ : μ 0) {s : set G} (hs : is_open s) :
μ s = 0 s =
theorem measure_theory.measure.is_add_left_invariant.null_iff_empty {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [add_group G] (hμ : μ.regular) (h3μ : μ 0) {s : set G} (hs : is_open s) :
μ s = 0 s =
theorem measure_theory.is_mul_left_invariant.null_iff {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [group G] (hμ : μ.regular) {s : set G} (hs : is_open s) :
μ s = 0 s = μ = 0
theorem measure_theory.measure.is_add_left_invariant.null_iff {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [add_group G] (hμ : μ.regular) {s : set G} (hs : is_open s) :
μ s = 0 s = μ = 0
theorem measure_theory.measure.is_add_left_invariant.measure_ne_zero_iff_nonempty {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [add_group G] (hμ : μ.regular) (h3μ : μ 0) {s : set G} (hs : is_open s) :
theorem measure_theory.is_mul_left_invariant.measure_ne_zero_iff_nonempty {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [group G] (hμ : μ.regular) (h3μ : μ 0) {s : set G} (hs : is_open s) :
theorem measure_theory.lintegral_eq_zero_of_is_mul_left_invariant {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [group G] (hμ : μ.regular) (h3μ : μ 0) {f : G → ℝ≥0∞} (hf : continuous f) :
∫⁻ (x : G), f x μ = 0 f = 0

For nonzero regular left invariant measures, the integral of a continuous nonnegative function f is 0 iff f is 0.

theorem measure_theory.lintegral_mul_left_eq_self {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [group G] (f : G → ℝ≥0∞) (g : G) :
∫⁻ (x : G), f (g * x) μ = ∫⁻ (x : G), f x μ

Translating a function by left-multiplication does not change its lintegral with respect to a left-invariant measure.

theorem measure_theory.lintegral_add_left_eq_self {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [add_group G] (f : G → ℝ≥0∞) (g : G) :
∫⁻ (x : G), f (g + x) μ = ∫⁻ (x : G), f x μ
theorem measure_theory.lintegral_add_right_eq_self {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [add_group G] (f : G → ℝ≥0∞) (g : G) :
∫⁻ (x : G), f (x + g) μ = ∫⁻ (x : G), f x μ
theorem measure_theory.lintegral_mul_right_eq_self {G : Type u_1} [borel_space G] {μ : measure_theory.measure G} [group G] (f : G → ℝ≥0∞) (g : G) :
∫⁻ (x : G), f (x * g) μ = ∫⁻ (x : G), f x μ

Translating a function by right-multiplication does not change its lintegral with respect to a right-invariant measure.


text:
Categories

# Measuring the length in Triangle | AMC-10B, 2011 | Problem 9

Try this beautiful problem from Geometry: Triangle from AMC-10B, 2011, Problem-9. You may use sequential hints to solve the problem.

Try this beautiful problem from Geometry and solve it by measuring the length in triangle.

## Measuring the length in Triangle- AMC-10B, 2011- Problem 9

The area of $\triangle$$EBD is one third of the area of \triangle$$ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$?

• $8\sqrt 3$
• $\frac {4\sqrt3}{3}$
• $6\sqrt 3$

### Key Concepts

Geometry

Triangle

similarity

Answer: $\frac{ 4\sqrt 3}{3}$

AMC-10B (2011) Problem 9

Pre College Mathematics

## Try with Hints

We have to find out the length of $BD$. The given informations are “The area of $\triangle$$EBD is one third of the area of \triangle$$ABC$. Segment $DE$ is perpendicular to segment $AB$”

If you notice very carefully about the side lengths of the $\triangle ABC$ then $AC=3,BC=4,AB=5$ i.e $(AC)^2+(AB)^2=(3)^2+(4)^2=25=(AB)^2$……..So from the pythagorean theorm we can say that $\angle ACB=90^{\circ}$

Therefore area of $\triangle ACB=\frac{1}{2} \times 3 \times 4=6$

so area of the $\triangle BDE=\frac{1}{3} \times 6=2$

Now the $\triangle BDE$ and $\triangle ABC$ If we can show that two triangles are similar then we will get the value of $BD$.Can you prove $\triangle BDE \sim \triangle ABC$ ?

Can you now finish the problem ……….

In $\triangle BDE$ & $\triangle ACB$ we have…..

$\angle B=X$ $\Rightarrow \angle BED=(90-x)$ and $\angle CAB=(90-X)$ (AS $\angle ACB=90$ & sum of the angles of a triangle is 180)

Therefore $\triangle BDE \sim \triangle BCD$

can you finish the problem……..

The value of BD:

Now $\triangle BDE \sim \triangle BCD$ $\Rightarrow \frac{(BD)^2}{(BC)^2}=\frac{Area of \triangle BDE}{Area of triangle ACB}$ =$\frac{(BD)^2}{16}=\frac{2}{6}$

So $BD=\frac{ 4\sqrt 3}{3}$


text:
# How to handle an independent variable that is partially ordered?

We are currently planning an experiment. We plan with the following variables:

• An independent variable "treatment" with four values (a, b, c, d)
• A number of further independent variables for confounding factors (mostly metric or dichotomous).
• Two dependent variables "time" and "correctness", both ratio scale

I'm currently looking into how to analyze the data once we have it. What complicates things (at least for me) is that there is a partial order relation (https://en.wikipedia.org/wiki/Partially_ordered_set) defined among the values of "treatment". In other words, "treatment" is neither fully ordinal, nor fully nominal: a < b < c and a < b < d, but c and d are incomparable (i.e. the theory does not say anything about their effect relative to each other).

Our hypothesis is that for treatments x and y, $x < y \Rightarrow time(x) < time(y)$ and $x \leq y \Rightarrow correctness(x) \leq correctness(y)$

So far, I came up with two ideas to handle the data:

1. Treat "treatment" as nominal and do a MANCOVA. Then manually look at the effect sizes and directions.
2. Forget about all the other independent variables and just calculate/test Spearman's rank-order correlation for three cases: c < d, c > d and c = d

Question: What are the pros and cons of the approaches? Are there other/better possibilities?

P.S.: To (perhaps) complicate things further: We plan to measure two (randomly chosen) treatments per subject, to gather more data and lower the effect of inter-subject variance.


text:
# Regression Inference in R

Here we will discuss how to perform standard inference in regression models.

# Setup

We continue the example we started in R_Regression#A first example and which is replicated here:

    # This is my first R regression!
setwd("T:/ECLR/R/FirstSteps")              # This sets the working directory
mydata <- read.csv("mroz.csv",na.strings = ".")  # Opens mroz.csv from working directory

Before we run our initial regression model we shall restrict the dataframe mydata to those data that do not have missing wage information, using the following subset command:

   mydata <- subset(mydata, wage!="NA")  # select non NA data

Now we can run our initial regression:

    # Run a regression
reg_ex1 <- lm(lwage~exper+log(huswage),data=mydata)
reg_ex1_sm <- summary(reg_ex1)

We will introduce inference in this model.

# t-tests

We use t-tests to test simple coefficient restrictions on regression coefficients. Let's initially have a look at our regression output

   print(reg_ex1_sm)

which delivers the following regression output

   Call:
lm(formula = lwage ~ exper + log(huswage), data = mydata)
Residuals:
Min       1Q   Median       3Q      Max
-3.10089 -0.31219  0.02919  0.37466  2.11402

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.534866   0.139082   3.846 0.000139 ***
exper        0.016684   0.004243   3.933 9.81e-05 ***
log(huswage) 0.236466   0.063684   3.713 0.000232 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7031 on 425 degrees of freedom
Multiple R-squared: 0.05919,	Adjusted R-squared: 0.05477
F-statistic: 13.37 on 2 and 425 DF,  p-value: 2.338e-06

As you can see, this output contains t-statistics and their associated p-values. These test statistics and their p-values are all associate to the following hypothesis test: $H_0: \beta_{i} = 0; H_A: \beta_{i} \neq 0$. Here $\beta_{i}$ represents the ith unknown population parameter. If you want to test any other hypothesis (rather than the two-sided, equal to 0 hypothesis) you will need to access the regression output in order to calculate

   $t-stat = \frac{\widehat{\beta}_{i} - \beta_{i}}{se_{\widehat{\beta}_{i}}}$

As was discussed in R_Regression#Accessing Regression Output the easiest way to get to these is to recognise that the coefficients element of reg_ex1_sm contains the parameter estimates in the first column and the standard errors in the second column. So let's say that we wanted to test the following hypothesis:

   $H_0: \beta_{exper} = 0.01; H_A: \beta_{exper} \gt 0.01$

Noting that the results for the exper variables are in the 2nd row we can calculate the relevant test statistic according to:

   t_test = (reg_ex1_sm$coefficients[2,1]-0.01)/reg_ex1_sm$coefficients[2,2]

where we recognise that that the experience coefficient is saved in the 2nd row of coefficients. As it turns out the value of this t-test is 1.5755. Sometime, especially if you have many explanatory variables, it can be awkward to have to count in which row the relevant coefficients are. But you can also use the name of the relevant variable as follows:

   t_test = (reg_ex1_sm$coefficients["exper",1]-0.01)/reg_ex1_sm$coefficients["exper",2]

This delivers the identical result.

# F-tests

F-tests are used to test multiple coefficient restrictions on regression coefficients.

Let's say we are interested whether two additional variables age and educ should be included into the model. As a good econometrics student, or even master, you know that to calculate a F-test you need residual sum of squares from a restricted model (that is model reg_ex1) and an unrestricted model. The latter we estimate here:

   reg_ex2 <- lm(lwage~exper+log(huswage)+age+educ,data=mydata)
reg_ex2_sm <- summary(reg_ex2)

## F-test 3 ways

As for many things in R you can achieve the same thing in many different ways. Here I will introduce three different ways to get the F-test result. Choose whichever way seems the most intuitive or straightforward way for you.

Calculating the F-test is now very easy. What we need is a restricted and unrestricted model. We use the function anova:

   print(anova(reg_ex1,reg_ex2))

which delivers the following output:

   Analysis of Variance Table
Model 1: lwage ~ exper + log(huswage)
Model 2: lwage ~ exper + log(huswage) + age + educ
Res.Df    RSS Df Sum of Sq      F    Pr(>F)
1    425 210.11
2    423 188.10  2    22.004 24.741 6.895e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The table at the heart of this output delivers the individual residual sum of squares, the F-test statstic and its p-value. The p-value is extremely small which would lead us to reject the null hypothesis, concluding that at least one of age or educ was significant. If you look at the regression output of reg_ex2 you will see that it is the education variable.

Slighly easier, use the linear hypothesis testing function (lht) which is also part of the car package:

     lht1 <- lht(reg_ex2, c("age = 0"," educ = 0"))

which requires as input the unrestricted model, (here reg_ex2) and the restrictions. Here we have two restrictions, i.e. that the coefficients to the two variables age and educ are 0 and hence both variables irrelevant. You could add more restrictions by adding them into the vector list c( ). Also note that instead of the short version (lht) you could use the slightly longer function name (linearHypothesis). But everything else remains unchanged.

Look at the output and you will find that it is essentially exactly the same as for the anova function. The lht function is more convenient to use for testing purposes, as you will only need the unrestricted model and then add the restriction. Internally the function will then estimate the restricted model, but you as the user will not see it.

The last way to calculate an F-test is by using the waldtest function which is a part of the AER package. In some ways this is the most powerful of the three options, in particular if you also want to allow for heteroskedastic error terms (see R_robust_se). All you need to do to calculate the above F-test is to call the following line

    > waldtest(reg_ex2, .~. - age - educ)
Wald test

Model 1: lwage ~ exper + log(huswage) + age + educ
Model 2: lwage ~ exper + log(huswage)
Res.Df Df      F    Pr(>F)
1    423
2    425 -2 24.741 6.895e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Don't be confused by the name "waldtest" in the end you get the same F-test. As for the lht function you will only need the unrestricted model (reg_ex2) as the first input. The second input into the function specifies the restriction to be tested, or more literally specifies how the restricted model should look like relative to the unrestricted one. We start this specification by the term .~. which basically says "use the same terms to the left and right of the "~" sign, but then remove the age and education variables, -age-educ.

Of course, the results are absolutely identical.

# p-value

Often you will want to calculate p-values for test statistics. You need a number of ingredients to do that:

• You need to know the value of a calculated test statistic
• You need to know what the distribution of the test statistic is (assuming that the null hypothesis is true) - this includes knowledge of degrees of freedom parameters if these are required for your distribution.
• You need to know whether you are working with a two-tailed, left-tailed or right-tailed test.

Once you know all these ingredients you can use some R internal functions to get p-values. Let's go to the t-test we calculated earlier (and saved in t_test) to test $H_0: \beta_{exper} = 0.01; H_A: \beta_{exper} \gt 0.01$. To get the p-value here (right-tailed area) we can call on the function pt, which calculates probabilities from a t distribution (probability under the t-distribution with 425 degrees of freedom to the left of t_test:

   p_t_test = (1-pt(t_test,425))  # right tailed p-value

The result is 0.05794. The (1 - ... ) part ensures that we calculate the right tail. Alternatively (as usual there are several ways to achieve the same result - all equally valied) we could have used an option in the pt function to ask for the right tail:

   p_t_test = pt(t_test,425,lower.tail = FALSE)

Of course different test statistics require different distributions. But the principle is the same. The relevant function to get p-values for a F-test is pf. Type ?pf into R to get to the help function where you can see how to use it exactly.

These probability functions exist for a range of common distributions. For a list see [1].


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# Binomial Random number Generation in R

We will learn here how to generate Bernoulli or Binomial distribution in R with example of flip of a coin. This tutorial is based on how to generate random numbers according to different statistical distributions in R. Our focus is in binomial random number generation in R.

We know that in Bernoulli distribution, either something will happen or not such as coin flip has to outcomes head or tail (either head will occur or head will not occur i.e. tail will occur). For unbiased coin there will be 50%  chances that head or tail will occur in the long run. To generate a random number that are binomial in R, use rbinom(n, size,prob) command.

rbinom(n, size, prob) command has three parameters, namely

where
n is number of observations
size is number of trials (it may be zero or more)
prob is probability of success on each trial for example 1/2

Some Examples

• One coin is tossed 10 times with probability of success=0.5
coin will be fair (unbiased coin as p=1/2)
>rbinom(n=10, size=1, prob=1/2)
OUTPUT: 1 1 0 0 1 1 1 1 0 1
• Two coins are tossed 10 times with probability of success=0.5
• > rbinom(n=10, size=2, prob=1/2)
OUTPUT: 2 1 2 1 2 0 1 0 0 1
• One coin is tossed one hundred thousand times with probability of success=0.5
> rbinom(n=100,000, size=1, prob=1/2)
• store simulation results in $x$ vector
> x<- rbinom(n=100,000, size=5, prob=1/2)
count 1’s in x vector
> sum(x)
find the frequency distribution
> table(x)
creates a frequency distribution table with frequency
> t=(table(x)/n *100)}
plot frequency distribution table
>plot(table(x),ylab=”Probability”,main=”size=5,prob=0.5″)

View Video tutorial on rbinom command

Updated: Apr 1, 2014 — 11:26 am


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# Central Limit Theorem for independent but non identically distributed random variables

My question is the following:

Given the sum of R.V.s, $Z_N = X_1 + X_2 + ... +X_N$, where $X_i$ are independent, Rice distributed ($X_i\sim Rice(\mu_i,\sigma)$), is there any way to approximate $Z_N$ by a Gaussian distribution making use of the Central Limit Theorem?

$X_i$ have the same variance, but different mean $\mu_i$.

• Try Lyapunov's central limit theorem. Does your set up satisfy Lyapunov's condition? If so, the answer is yes. – David Simmons Sep 17 '14 at 10:01
• I suggest you work out whether the Liapounov condition is met. Recall, that the requirement is the r.v.'s are i.i.d. where the second "i" refers to identical be it whatever, in your case Rice. – Hirek Dec 19 '15 at 18:41


text: A person starts from rest at the top of a large, frictionless, spherical surface, and slides into the water below (see the drawing). At what angle θ does the person leave the surface? (Hint: When the person leaves the surface, the normal force is zero.)


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# What can we conclude about the natural projection maps?

In an arbitrary category, we have that even if $X$ and $Y$ have a product $X \times Y$, the natural projections needn't be epimorphisms.

Two questions:

1. Are there (preferably simple!) conditions we can place on the category such that all the natural projections are, in fact, epimorphisms?

2. Without assuming anything about the category, is there a (preferably interesting!) weaker property that the natural projections always possess?

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To answer 1 : Note sure about any condition more general than requiring that the category contain a zero object. Not sure about 2 either. –  Prahlad Vaidyanathan Sep 1 '13 at 16:11
(1) is not true for $\mathbf{Set}$, so I would say there are no reasonable conditions. –  Zhen Lin Sep 1 '13 at 16:37
@ZhenLin, ah of course. Take $X=0$ and $Y$ non-empty, then $X \times Y \rightarrow Y$ isn't epic. I always forget the $X=0$ case.... –  goblin Sep 1 '13 at 16:43
@ZhenLin does $\mathrm{Set}^{\mathrm{Op}}$ have the property of interest? I think so but I may be getting confused about the zero case. –  goblin Sep 1 '13 at 16:44
Yes, it is true for $\mathbf{Set}^\mathrm{op}$, or more generally, the opposite of any elementary topos. –  Zhen Lin Sep 1 '13 at 17:05

You might like this theorem:

If $\pi _i:P {\to} A_i$ for $i\in I$ is a product and if $i_0 \in I$ is such that, for each $i \in I$, $hom(A_{i0} ,A_i)$ is not empty then $\pi _{i0}$ is a retraction.

In general the $\pi _i$ form an extremal mono-source

See:

Abstract and concrete categories: the joy of cats. Propositions 10.21 and 10.28

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I do like that theorem! Is there a special name for objects $Y$ such that for all objects $X$ we can find a morphism $f : X \rightarrow Y$? The projections associated with any product of such objects are necessarily retractions, by the theorem you give. –  goblin Sep 1 '13 at 17:41
I would call $Y$ weakly terminal. But note that this assumption is too strong, we really only need that the factors map to each other. –  Martin Brandenburg Sep 1 '13 at 20:38


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### Proving that the sum of the interior angles of a triangle is 180°

The statement of the result is called a theorem. This comes from a Greek word meaning 'a thing to be gazed upon' or 'a thing contemplated by the mind'. The word 'theatre' comes from the same Greek root.

#### Theorem

The sum of the angles of a triangle is 180°.

#### Proof

Let ∆ABC be a triangle.

Let $$\angle$$BAC = α, $$\angle$$CBA = β and $$\angle$$BCA = γ.

We must prove that α + β + γ = 180°.

The line XY is drawn parallel to the line AB through the vertex C.

\begin{align}\text{Then}\ \angle XCA &= α\ (\text{alternate angles}\ XY || AB).\\\\ \angle YCB&= β\ (\text{alternate angles}\ XY||AB)\end{align}

Hence α + β + γ = 180° (straight angle at C).

#### Example 1

Find the value of θ in each of the following.

#### Solution

1. θ + 83° + 63° = 180° (angle sum of ∆ABC)
Hence θ = 34°.
2. θ + 90° + 62° = 180° (angle sum of ∆ABC)
Hence θ = 28°.