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# Alternative transitional measures to liberalize quantitative trade restrictions implications for developing countries by Wendy E. Takacs

Written in English

## Places:

• Developing countries,
• Developing countries.

## Subjects:

• Tariff -- Developing countries.,
• Balance of payments -- Developing countries.,
• Developing countries -- Commercial policy.

Edition Notes

## Book details

Classifications The Physical Object Statement Wendy E. Takacs. Series UNDP-World Bank Trade Expansion Program occasional paper ;, 3 LC Classifications HF1413 .T35 1989 Pagination 28 p. ; Number of Pages 28 Open Library OL2278960M LC Control Number 89170470

Alternative transitional measures to liberalize quantitative trade restrictions. Washington, DC: Trade Policy Division, World Bank, [] (OCoLC) Material Type: Government publication, International government publication: Document Type: Book: All Authors / Contributors: Wendy E Takacs.

Trade liberalization is the removal or reduction of restrictions or barriers, such as tariffs, on the free exchange of goods between nations.

Measures of Restrictions on Trade in Services Database Research memorandum. In a collaborative project, the Productivity Commission and the Australian National University have been measuring restrictions on trade in services for a number of economies in Europe, Asia.

Quantitative restrictions of the core ideas of international trade. The book updates the classic monograph of Professor Gandolfo and is now the single most comprehensive and up-to-date book. distribution.

Thus, the objective of this book is to guide economists with an interest in the applied analysis of trade and trade policies towards the main sources of data and the most useful tools available to analyse real world trade and trade policies.

The book starts with a discussion of the quantification of trade flows and trade policies. China claimed that its export restrictions are necessary in addition to the other measures discussed above—as it argued that export restrictions complement, and provide benefits in addition to those resulting from, the alternative measures proposed by the complainants.

However, the panel concluded that China, which admitted that the. First, the GATT placed many fewer restrictions on agricultural trade than on any other product sector. 23 Article XI of the GATT prohibits quantitative trade restrictions, but contains two exceptions relevant to agriculture.

The first, Article XI.2(a), permits export prohibitions or restrictions temporarily applied to prevent or relieve. Update May The General Court’s judgment upon appeal was annulled by the CJEU in Case C/16 P at the end of March. In essence, as TaylorWessing point out, although the German State controlled the implementation of the EEG surcharge, it did not control the sums generated, so that the existence of State aid is ruled out.

The rather long judgment in T/15 Germany v Commission is. measure the degree of substitutability. In the language of AGE modeling, we need better estimates of Armington elasticities.

We use the least traded products (LTP) methodology of Kehoe et al. (), which focuses on product-level data on bilateral trade relations by industry, to provide guidance on how improvements can be made.

This study examines the impact of trade liberalization on the Ethiopia's trade balance using the data over the period to from NBE (National Bank of Ethiopia). The country has undertaken serious trade reforms, either as a part of major macroeconomic reforms and commitments with international regulations, or by decisions driven by a process of internal adjustment for the last.

quantitative restrictions (QR): A trade restriction placed on the amount of an item or service that can be imported into a country. These are frequently enacted to protect the price of domestically produced goods or to decrease or eliminate a trade deficit.

Also called trade quota. forms of trade intervention are quantitative restrictions (QRS) of dif-ferent sorts. Tariffs distort prices but still leave the economy's price mechanism to allocate resources.

Under extensive QRs the price mech-anism has a much reduced role, so many opportunities to use re-sources more efficiently are lost. Moves toward liberalization. Q5: What is the alternative to export restrictions. A5: There are no easy solutions to this problem. Doctors, nurses, and patients infected by Covid are not presently concerned with the United States upholding a commitment to free trade, especially given equipment shortages and the protectionist measures in other countries such as China.

The total value of trade between Indonesia and Japan reached US$billion ina Alternative transitional measures to liberalize quantitative trade restrictions book decrease from$ billion inaccording to Trade Ministry data.

Indonesia also posted a. Europe's Free Trade Area Experiment: EFTA and Economic Integration is an chapter book that begins by exploring the role of free trade area. This book then describes the first decade of European Free Trade Association (EFTA) and its effects on member countries.

Trade measures are wasteful too if it is only the poorest consumers who need to be helped, since a trade measure affects all food consumers in the country and in proportion to their expenditure on food. 5 That is, altering restrictions on trade is far from the first-best way for a national government to prevent food insecurity from increasing.

I have joined the International Food Policy Research Institute (IFPRI, Washington DC) in I am a Senior Research Fellow in the Markets, Trade and Institutions Division and the Theme Leader on. included a reduction in the level and dispersion of tariffs, and a removal of quantitative restrictions on imported inputs and capital goods for export production (Chopra et al., ).

The government's trade policy under the Eighth Five‐Year Plan (‐97) ushered. The Agreement on Trade-Related Investment Measures (TRIMs) are rules that are applicable to the domestic regulations a country applies to foreign investors, often as part of an industrial agreement, concluded inwas negotiated under the WTO's predecessor, the General Agreement on Tariffs and Trade (GATT), and came into force in   The Agreement on Trade-Related Investment Measures (TRIMS) recognizes that certain investment measures can restrict and distort trade.

It states that WTO members may not apply any measure that discriminates against foreign products or that leads to quantitative restrictions, both of which violate basic WTO principles. When to trade a stock is the million dollar question. A stock can transition from a downtrend to an uptrend in several different ways.

It can, for example, fall precipitously, turn on a dime, and begin heading higher. Although turning on a dime from downtrending to upward trending sometimes presents profitable trading opportunities, these transitions [ ].

Alternatives to trade restrictions, including subsidies for industries and labour-market policy instruments are also shown to have their drawbacks, and the book emphasises the need for countries to find and exploit policies which fulfil their own political and social needs but which are least injurious to their trading partners.

Government measures other than tariffs that restrict trade flows (e.g. quantitative restrictions on goods or services, import licensing, variable levies, import barriers and TBTs.

Policy space The freedom, scope and mechanisms that governments have to choose, design and implement public policies to fulfil their aims. Role of Trade Measures in Environmental Policy.

Perceived conflicts between efforts to liberalize trade and to protect the environment are driving discussion of trade/environment issues. Liberalization of trade is not a goal in and of itself but rather a means to promote prosperity through improved economic efficiency and development.

The aim of the measure was to limit nitrogen dioxide pollution for reasons of the protection of human health. The European Court found that the measure was contrary to Articles 28 and 29 of the EC treaty which prohibits quantitative restrictions on imports and exports and measures with equivalent effect.

International trade is the framework upon which American prosperity rests. Free trade policies have created a level of competition in today's open.

International trade and investment agreements can have positive outcomes, but also have negative consequences that affect global health and influence fundamental health determinants: poverty, inequality and the environment. This article proposes principles and strategies for designing future international law to attain health and common good objectives.

Trade has long been an axiomatic characteristic of globalization, although international rules governing trade are of more recent vintage. Notably in the post-World War II period, an ever increasing number of countries began negotiating treaties to reduce, first, tariff barriers and, later, non-tariff barriers (government measures of any sort) that could impede the cross-border flow of goods.

in trade which might be termed managed trade, and governments often use different types of QRs to achieve their trade policy objectives (Table 1). Table 1. Quantitative Restrictions on Trade Quantitative restrictions (QRs) on trade in goods are measures that limit the quantity of a product that may be imported or exported.

Article V:1 permits any WTO member to enter into an agreement to further liberalize trade in services with the other countries that are parties to the agreement, provided the agreement has "substantial sectoral coverage", eliminates measures that discriminate against service suppliers of other countries in the group, and prohibits new or more.

Tariffs are custom taxes that governments levy on imported and some exported goods. The tax is a percentage of the total cost of the product, including freight and insurance.

Tariffs are also called customs, import duties, or import fees. In the United States, the U.S. Congress sets the tariffs. Thanks for contributing an answer to Quantitative Finance Stack Exchange. Please be sure to answer the question. Provide details and share your research. But avoid Asking for help, clarification, or responding to other answers.

Making statements based on opinion; back them up with references or personal experience. Use MathJax to format. Smith will analyze international trade in genetically modified (GMO) crops and policies related to this trade.

This project will contribute to scholarly knowledge and practical application of a critical contemporary issue. In practical terms, the adoption and commercialization of GMO crops is occurring at a rapid pace around the world. The global land planted with such crops is estimated. In any event, in the present case, taking into account, inter alia, the share of trade affected by the type of measures at issue (quantitative restrictions on textiles and clothing), we found that there were WTO compatible alternatives available to Turkey if it wanted to conclude a customs union with the European Communities.

Finally we found. Protective trade restrictions still persist, but tend to be in terms of more subtle non-tariff barriers (such as sanitary or phyto-sanitary standards), though anti-dumping measures and temporary quantity restrictions are still used by many countries to shield domestic producers.

removed quantitative restrictions before the arrangements mentioned above. Some other transition countries failed to take part in these efforts for a long time due to wars and domestic civil unrests -e.g. Albania, Croatia, Bosnia Herzegovina, Macedonia, and Uzbekistan.

A number of regional bilateral free trade agreements were also signed by. Get this from a library. Trade and transitions: a comparative analysis of adjustment policies. [M J Trebilcock; Marsha A Chandler; Robert Howse] -- Faced with increased levels of international competition and mounting budget deficits some developed, Western economies have responded by introducing trade restrictions.

This book uses a comparative. Nontraditional measures to impede trade are harder to quantify and assess, but they are becoming more significant as traditional tariff protection and such barriers as import quotas decline.

Antidumping measures are on the rise in both industrial and developing countries, but are faced disproportionately by developing countries. and Phytosanitary Measures 78 The Agreement on Technical Barriers to Trade 79 The Agreement on Subsidies and Countervailing Measures 79 The Anti-dumping Agreement 79 The Agreement on Safeguards 79 Test Your Understanding 80 7 Case Studies 81 8 Further Reading 83 Books and Monographs 83 Documents and Information   In reality, however, governments with generally free-trade policies still impose some measures to control imports and exports.

Like the United States, most industrialized nations negotiate “free trade agreements,” or FTAs with other nations which determine the tariffs, duties, and subsidies the countries can impose on their imports and exports.

4)FTAs should cover other non-tariff measures such as Sanitary and Phytosanitary (SPS) measures, technical barriers to trade, and quantitative restrictions. 5)FTAs should contain a provision to safeguard against the adverse effects on domestic industries, by providing for measures such as anti-dumping measures (AD), countervailing duty measures.

US-China economic issues: Implications for US policy, testimony before the House East Asia Subcommitte, Apby Nicholas Lardy, senior fellow, foreign policy studies, the Brookings.The term 'Tool of Trade' in regards to commercial vehicle insurance means the vehicle is needed to ply the trade (transportation of equipment, etc.) Asked in Car Buying, Auto Loans and Financing.

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11.3.1 Index Number Long Questions (Question 1 & 2)

Question 1:

Table below shows the price indices and percentage of usage of four items, P , Q , R and S , which are the main ingredients in the production of a type of toy.

 Item Price index for the year 2007 based on the year 2004 Percentage of usage (%) P 125 30 Q x 20 R 115 10 S 135 40

(a)  Calculate
(i) the price of S in the year 2004 if its price in the year 2007 is RM43.20 ,
(ii) the price index of P in the year 2007 based on the year 2002 if its price index in
the year 2004 based on the year 2002 is 110.
(b)   The composite index of the cost of toy production for the year 2007 based on the
year 2004 is 125.
Calculate
(i) the value of  x ,
(ii) the price of the toy in the year 2004 if the corresponding price in the year 2007 is
RM75.

Solution:
(a)(i)

(a)(ii)

(b)(i)

(b)(ii)

Question 2:
Diagram below is a bar chart indicating the weekly cost of the items A, B, C, D and E  for the year 2000. Table below shows the prices and the price indices for the items.

 Items Price in 2000 Price in 2006 Price Index in 2006 based on 2000 A x RM 1.75 140 B RM 2.00 RM 2.50 125 C RM 4.00 RM4.80 y D RM 6.00 z 125 E RM 2.50 2.75 110

(a) Find the value of
(i) x
(ii) y
(iii) z

(b) Calculate the composite index for items in the year 2006 based on the year 2000.
(c) The total monthly cost of the items in the year 2000 is RM456. Calculate the
corresponding total monthly cost for the year 2006.
(d) The cost of the items increases by 15 % from the year 2006 to the year 2009.
Find the composite index for the year 2009 based on the year 2000.

Solution:
(a)(i)
$\begin{array}{l}I=\frac{{P}_{1}}{{P}_{0}}×100\\ I=\frac{{P}_{2006}}{{P}_{2000}}×100\\ 140=\frac{1.75}{x}×100\\ x=\text{RM}1.25\end{array}$
(a)(ii)
$\begin{array}{l}I=\frac{{P}_{2006}}{{P}_{2000}}×100\\ y=\frac{4.80}{4.00}×100\\ y=120\end{array}$
(a)(iii)
$\begin{array}{l}I=\frac{{P}_{2006}}{{P}_{2000}}×100\\ 125=\frac{z}{6.00}×100\\ z=\text{RM7}\text{.50}\end{array}$
(b)
 Items Price Index in 2006 based on 2000 Weightage (from the diagram above) A 140 15 B 125 30 C 120 24 D 125 33 E 110 12

(c)

(d)

1 thought on “11.3.1 Index Number Long Questions (Question 1 & 2)”

1. I need the longest table with x, y, z

format: plain


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### Home > AC > Chapter 9 > Lesson 9.3.3 > Problem9-82

9-82.

The town you live in has decided to limit the amount of trash thrown out each month. Your town, which has 3280 homes, has asked each household to keep track of how many pounds of trash they produce during a month. In addition, the town council has found that other sources of trash, such as local businesses, combine to create $1500$ lbs of trash each month. If the town has a goal of creating less than $50,000$ lbs of trash, how much trash should a household be limited to? Write an inequality for this situation and solve it.

Write an inequality for which x is the amount of trash per house.
$50,000 \gt 3280x + 1500$

Solve for $x$.
$48,500 \gt 3280x$
$14.7876 \gt x$

$x = \text{ less than } 14.8 \text{ pounds|}$


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# PCA Visualization in MATLAB®

How to do PCA Visualization in MATLAB® with Plotly.

## Principal Components of a Data Set

load hald

The ingredients data has 13 observations for 4 variables.

Find the principal components for the ingredients data.

load hald

coeff = pca(ingredients)

coeff =

Columns 1 through 3

-0.067799985695474  -0.646018286568728   0.567314540990512
-0.678516235418647  -0.019993340484099  -0.543969276583817
0.029020832106229   0.755309622491133   0.403553469172668
0.730873909451461  -0.108480477171676  -0.468397518388289

Column 4

0.506179559977706
0.493268092159297
0.515567418476836
0.484416225289198

The rows of coeff contain the coefficients for the four ingredient variables, and its columns correspond to four principal components.

## PCA in the Presence of Missing Data

Find the principal component coefficients when there are missing values in a data set.

load imports-85

Data matrix X has 13 continuous variables in columns 3 to 15: wheel-base, length, width, height, curb-weight, engine-size, bore, stroke, compression-ratio, horsepower, peak-rpm, city-mpg, and highway-mpg. The variables bore and stroke are missing four values in rows 56 to 59, and the variables horsepower and peak-rpm are missing two values in rows 131 and 132.

Perform principal component analysis.

load imports-85

coeff = pca(X(:,3:15));

By default, pca performs the action specified by the 'Rows','complete' name-value pair argument. This option removes the observations with NaN values before calculation. Rows of NaNs are reinserted into score and tsquared at the corresponding locations, namely rows 56 to 59, 131, and 132.

Use 'pairwise' to perform the principal component analysis.

load imports-85

coeff = pca(X(:,3:15),'Rows','pairwise');

In this case, pca computes the (i,j) element of the covariance matrix using the rows with no NaN values in the columns i or j of X. Note that the resulting covariance matrix might not be positive definite. This option applies when the algorithm pca uses is eigenvalue decomposition. When you don’t specify the algorithm, as in this example, pca sets it to 'eig'. If you require 'svd' as the algorithm, with the 'pairwise' option, then pca returns a warning message, sets the algorithm to 'eig' and continues.

If you use the 'Rows','all' name-value pair argument, pca terminates because this option assumes there are no missing values in the data set.

load imports-85

coeff = pca(X(:,3:15),'Rows','all');

...ERROR...Raw data contains NaN missing value while 'Rows' option is set to 'all'. Consider using 'complete' or 'pairwise' option, or using 'als' algorithm instead.

## Weighted PCA

Use the inverse variable variances as weights while performing the principal components analysis.

load hald

Perform the principal component analysis using the inverse of variances of the ingredients as variable weights.

load hald

[wcoeff,~,latent,~,explained] = pca(ingredients,'VariableWeights','variance');

Note that the coefficient matrix, wcoeff, is not orthonormal.

Calculate the orthonormal coefficient matrix.

load hald
[wcoeff,~,latent,~,explained] = pca(ingredients,'VariableWeights','variance');

coefforth = inv(diag(std(ingredients)))* wcoeff

coefforth =

Columns 1 through 3

-0.475955172748971   0.508979384806409  -0.675500187964285
-0.563870242191993  -0.413931487136986   0.314420442819292
0.394066533909305  -0.604969078471438  -0.637691091806566
0.547931191260862   0.451235109330016   0.195420962611708

Column 4

0.241052184051094
0.641756074427213
0.268466110294533
0.676734019481284

Check orthonormality of the new coefficient matrix, coefforth.

load hald
[wcoeff,~,latent,~,explained] = pca(ingredients,'VariableWeights','variance');
coefforth = inv(diag(std(ingredients)))* wcoeff;

coefforth*coefforth'

ans =

Columns 1 through 3

1.000000000000000   0.000000000000000  -0.000000000000000
0.000000000000000   1.000000000000000  -0.000000000000000
-0.000000000000000  -0.000000000000000   1.000000000000000
0.000000000000000  -0.000000000000000   0.000000000000000

Column 4

0.000000000000000
-0.000000000000000
0.000000000000000
1.000000000000000

## PCA Using ALS for Missing Data

Find the principal components using the alternating least squares (ALS) algorithm when there are missing values in the data.

load hald

The ingredients data has 13 observations for 4 variables.

Perform principal component analysis using the ALS algorithm and display the component coefficients.

load hald

[coeff,score,latent,tsquared,explained] = pca(ingredients);
coeff

coeff =

Columns 1 through 3

-0.067799985695474  -0.646018286568728   0.567314540990512
-0.678516235418647  -0.019993340484099  -0.543969276583817
0.029020832106229   0.755309622491133   0.403553469172668
0.730873909451461  -0.108480477171676  -0.468397518388289

Column 4

0.506179559977706
0.493268092159297
0.515567418476836
0.484416225289198

Introduce missing values randomly.

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);

y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN

y =

7    26     6   NaN
1    29    15    52
NaN   NaN     8    20
11    31   NaN    47
7    52     6    33
NaN    55   NaN   NaN
NaN    71   NaN     6
1    31   NaN    44
2   NaN   NaN    22
21    47     4    26
NaN    40    23    34
11    66     9   NaN
10    68     8    12

Approximately 30% of the data has missing values now, indicated by NaN.

Perform principal component analysis using the ALS algorithm and display the component coefficients.

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);
y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN;

[coeff1,score1,latent,tsquared,explained,mu1] = pca(y,'algorithm','als');
coeff1

coeff1 =

Columns 1 through 3

-0.036212338541056   0.821521833463708  -0.525180807034514
-0.683141604556494  -0.099847237932409   0.182836485390676
0.016929757180233   0.557510085824821   0.821489828858886
0.729191057256729  -0.065687977768361   0.126136436504135

Column 4

0.219033475985737
0.699942066753553
-0.118534166411288
0.669369173908958

Display the estimated mean.

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);
y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN;

[coeff1,score1,latent,tsquared,explained,mu1] = pca(y,'algorithm','als');
mu1

mu1 =

Columns 1 through 3

8.995597151700494  47.908800869772193   9.045133815910839

Column 4

28.551458070546644

Reconstruct the observed data.

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);
y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN;

[coeff1,score1,latent,tsquared,explained,mu1] = pca(y,'algorithm','als');
t = score1*coeff1' + repmat(mu1,13,1)

t =

Columns 1 through 3

6.999999999999986  25.999999999999982   6.000000000000000
0.999999999999980  29.000000000000007  14.999999999999996
10.781877383662085  53.023021922864487   7.999999999999996
11.000000000000014  31.000000000000004  13.549987432049443
6.999999999999996  51.999999999999943   6.000000000000016
10.481832029571301  55.000000000000007   7.832799509066272
3.098161701207636  70.999999999999972  11.949103806476099
0.999999999999969  31.000000000000021  -0.516112356202470
1.999999999999988  53.791389384174110   5.770961215451539
21.000000000000021  46.999999999999979   3.999999999999996
21.580891857665435  40.000000000000028  23.000000000000000
11.000000000000009  66.000000000000014   8.999999999999998
10.000000000000005  67.999999999999972   8.000000000000018

Column 4

51.524967145094351
52.000000000000043
19.999999999999996
46.999999999999972
32.999999999999943
17.936171158236185
5.999999999999936
43.999999999999986
21.999999999999996
25.999999999999972
33.999999999999993
5.707816613776032
11.999999999999936

The ALS algorithm estimates the missing values in the data.

Another way to compare the results is to find the angle between the two spaces spanned by the coefficient vectors. Find the angle between the coefficients found for complete data and data with missing values using ALS.

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);
y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN;
[coeff1,score1,latent,tsquared,explained,mu1] = pca(y,'algorithm','als');
t = score1*coeff1' + repmat(mu1,13,1);

subspace(coeff,coeff1)

ans =

9.133575854440947e-16

This is a small value. It indicates that the results if you use pca with 'Rows','complete' name-value pair argument when there is no missing data and if you use pca with 'algorithm','als' name-value pair argument when there is missing data are close to each other.

Perform the principal component analysis using 'Rows','complete' name-value pair argument and display the component coefficients.

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);
y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN;

[coeff2,score2,latent,tsquared,explained,mu2] = pca(y,'Rows','complete');
coeff2

coeff2 =

-0.205420225953307   0.858688026698708   0.049181247828161
-0.669364063599413  -0.371998098041251   0.551021387596589
0.147356319504436  -0.351257041216125  -0.518714827525170
0.698598880784301  -0.029845918549515   0.651837067815822

In this case, pca removes the rows with missing values, and y has only four rows with no missing values. pca returns only three principal components. You cannot use the 'Rows','pairwise' option because the covariance matrix is not positive semidefinite and pca returns an error message.

Find the angle between the coefficients found for complete data and data with missing values using listwise deletion (when 'Rows','complete').

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);
y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN;
[coeff2,score2,latent,tsquared,explained,mu2] = pca(y,'Rows','complete');

subspace(coeff(:,1:3),coeff2)

ans =

0.357607357590711

The angle between the two spaces is substantially larger. This indicates that these two results are different.

Display the estimated mean.

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);
y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN;
[coeff2,score2,latent,tsquared,explained,mu2] = pca(y,'Rows','complete');

mu2

mu2 =

Columns 1 through 3

7.888888888888889  46.909090909090907   9.875000000000000

Column 4

29.600000000000001

In this case, the mean is just the sample mean of y.

Reconstruct the observed data.

load hald
[coeff,score,latent,tsquared,explained] = pca(ingredients);
y = ingredients;
rng('default'); % for reproducibility
ix = random('unif',0,1,size(y))<0.30;
y(ix) = NaN;
[coeff2,score2,latent,tsquared,explained,mu2] = pca(y,'Rows','complete');

score2*coeff2'

ans =

Columns 1 through 3

NaN                 NaN                 NaN
-7.516185143683813 -18.354534728448066   4.096758018165185
NaN                 NaN                 NaN
NaN                 NaN                 NaN
-0.564429969715416   5.321307756733086  -3.343158339022808
NaN                 NaN                 NaN
NaN                 NaN                 NaN
NaN                 NaN                 NaN
NaN                 NaN                 NaN
12.831514933168062  -0.107632488812892  -6.333304231731043
NaN                 NaN                 NaN
NaN                 NaN                 NaN
1.467986895710465  20.634225817831361  -2.929186618770285

Column 4

NaN
22.005631390194360
NaN
NaN
3.603980833482452
NaN
NaN
NaN
NaN
-3.775776525301393
NaN
NaN
-18.004319332087810

This shows that deleting rows containing NaN values does not work as well as the ALS algorithm. Using ALS is better when the data has too many missing values.

## Principal Component Coefficients, Scores, and Variances

Find the coefficients, scores, and variances of the principal components.

load hald

The ingredients data has 13 observations for 4 variables.

Find the principal component coefficients, scores, and variances of the components for the ingredients data.

load hald

[coeff,score,latent] = pca(ingredients);

Each column of score corresponds to one principal component. The vector, latent, stores the variances of the four principal components.

Reconstruct the centered ingredients data.

load hald
[coeff,score,latent] = pca(ingredients);

Xcentered = score*coeff'

Xcentered =

Columns 1 through 3

-0.461538461538460 -22.153846153846171  -5.769230769230778
-6.461538461538460 -19.153846153846168   3.230769230769222
3.538461538461538   7.846153846153854  -3.769230769230770
3.538461538461537 -17.153846153846164  -3.769230769230778
-0.461538461538463   3.846153846153853  -5.769230769230771
3.538461538461539   6.846153846153851  -2.769230769230769
-4.461538461538460  22.846153846153854   5.230769230769241
-6.461538461538457 -17.153846153846171  10.230769230769225
-5.461538461538460   5.846153846153847   6.230769230769234
13.538461538461533  -1.153846153846157  -7.769230769230772
-6.461538461538459  -8.153846153846164  11.230769230769230
3.538461538461539  17.846153846153861  -2.769230769230764
2.538461538461538  19.846153846153868  -3.769230769230763

Column 4

30.000000000000028
22.000000000000014
-10.000000000000004
17.000000000000004
3.000000000000000
-8.000000000000005
-24.000000000000004
14.000000000000009
-8.000000000000000
-3.999999999999999
4.000000000000004
-18.000000000000007
-18.000000000000011

The new data in Xcentered is the original ingredients data centered by subtracting the column means from corresponding columns.

Visualize both the orthonormal principal component coefficients for each variable and the principal component scores for each observation in a single plot.

load hald
[coeff,score,latent] = pca(ingredients);
Xcentered = score*coeff';

biplot(coeff(:,1:2),'scores',score(:,1:2),'varlabels',{'v_1','v_2','v_3','v_4'});

fig2plotly(gcf);

All four variables are represented in this biplot by a vector, and the direction and length of the vector indicate how each variable contributes to the two principal components in the plot. For example, the first principal component, which is on the horizontal axis, has positive coefficients for the third and fourth variables. Therefore, vectors v3 and v4 are directed into the right half of the plot. The largest coefficient in the first principal component is the fourth, corresponding to the variable v4.

The second principal component, which is on the vertical axis, has negative coefficients for the variables v1, v2, and v4, and a positive coefficient for the variable v3.

This 2-D biplot also includes a point for each of the 13 observations, with coordinates indicating the score of each observation for the two principal components in the plot. For example, points near the left edge of the plot have the lowest scores for the first principal component. The points are scaled with respect to the maximum score value and maximum coefficient length, so only their relative locations can be determined from the plot.

## T-Squared Statistic

Find the Hotelling’s T-squared statistic values.

load hald

The ingredients data has 13 observations for 4 variables.

Perform the principal component analysis and request the T-squared values.

load hald

[coeff,score,latent,tsquared] = pca(ingredients);
tsquared

tsquared =

5.680340841490963
3.075837035211974
6.000232793912607
2.619763232047949
3.368139445336655
0.566796674771214
3.481840731756965
3.979397901620162
2.608586242833562
7.481756329327633
4.183022234152594
2.232718720505655
2.721567817032091

Request only the first two principal components and compute the T-squared values in the reduced space of requested principal components.

load hald

[coeff,score,latent,tsquared] = pca(ingredients,'NumComponents',2);
tsquared

tsquared =

5.680340841490963
3.075837035211974
6.000232793912607
2.619763232047949
3.368139445336655
0.566796674771214
3.481840731756965
3.979397901620162
2.608586242833562
7.481756329327633
4.183022234152594
2.232718720505655
2.721567817032091

Note that even when you specify a reduced component space, pca computes the T-squared values in the full space, using all four components.

The T-squared value in the reduced space corresponds to the Mahalanobis distance in the reduced space.

load hald
[coeff,score,latent,tsquared] = pca(ingredients,'NumComponents',2);

tsqreduced = mahal(score,score)

tsqreduced =

3.317920809184487
2.007906801224289
0.587427300774747
1.738161538935849
0.295525889978777
0.422793560910442
3.245670063829736
2.691429716874080
1.361859279884830
2.990295498144075
2.437109148953178
1.378818611291408
1.525081780014099

Calculate the T-squared values in the discarded space by taking the difference of the T-squared values in the full space and Mahalanobis distance in the reduced space.

load hald
[coeff,score,latent,tsquared] = pca(ingredients,'NumComponents',2);
tsqreduced = mahal(score,score);

tsqdiscarded =

2.362420032306477
1.067930233987685
5.412805493137860
0.881601693112100
3.072613555357879
0.144003113860771
0.236170667927229
1.287968184746082
1.246726962948732
4.491460831183558
1.745913085199415
0.853900109214247
1.196486037017992

## Percent Variability Explained by Principal Components

Find the percent variability explained by the principal components. Show the data representation in the principal components space.

load imports-85

Data matrix X has 13 continuous variables in columns 3 to 15: wheel-base, length, width, height, curb-weight, engine-size, bore, stroke, compression-ratio, horsepower, peak-rpm, city-mpg, and highway-mpg.

Find the percent variability explained by principal components of these variables.

load imports-85

[coeff,score,latent,tsquared,explained] = pca(X(:,3:15));
explained

explained =

64.342913356114238
35.448392662688399
0.154993220067716
0.037863222808984
0.007807396888358
0.004771746794250
0.001304050511812
0.001056438605476
0.000529182750646
0.000186613509121
0.000158017147436
0.000017269286049
0.000006822827515

The first three components explain 99.95% of all variability.

Visualize the data representation in the space of the first three principal components.

load imports-85
[coeff,score,latent,tsquared,explained] = pca(X(:,3:15));

scatter3(score(:,1),score(:,2),score(:,3))
axis equal
xlabel('1st Principal Component')
ylabel('2nd Principal Component')
zlabel('3rd Principal Component')

fig2plotly(gcf);

The data shows the largest variability along the first principal component axis. This is the largest possible variance among all possible choices of the first axis. The variability along the second principal component axis is the largest among all possible remaining choices of the second axis. The third principal component axis has the third largest variability, which is significantly smaller than the variability along the second principal component axis. The fourth through thirteenth principal component axes are not worth inspecting, because they explain only 0.05% of all variability in the data.

To skip any of the outputs, you can use ~ instead in the corresponding element. For example, if you don’t want to get the T-squared values, specify

load imports-85

[coeff,score,latent,~,explained] = pca(X(:,3:15));


text:
# Checkerboard Infection Variant

There is an insidious virus (less contagious than C. Coli, but still dangerous) which plagues the squares of checkerboards. This virus spreads through social interaction: whenever the majority of a friend group becomes sick, the entire group becomes sick.

Specifically, any four squares which meet at a common vertex are mutual friends. Whenever three out of four squares in a friend group are sick, the remaining friend becomes sick the next day. This is the only way the infection can spread.

Initially, 14 squares of a checkerboard are infected. Is it possible for the infection to spread to the entire board?

The answer would be yes if 14 was replaced with 15, as shown below.

Hint:

See the solution to Checkerboard Infection. This puzzle also has a simple answer.

I know it is a checkerboard but this problem may need a bit of chess:

Say that a set of infected cells $C$ is rookable if a chess rook could travel from any square in $C$ to any other by a series of rook moves that all end on a cell in $C$. For example, two opposite corners of the board would not be rookable, but adding a third corner would be rookable. Clearly, if the whole board is infected, that forms a rookable set. Notice that if you run the infection backwards in time it will always still be a rookable set. Why? Well, removing one corner of a $2 \times 2$ still allows the rook to move around the other side of the $2 \times 2$ and access any row or column that it could before. Therefore, the initial infected cells formed a rookable set.

Notice also that the initial cells also must occupy every row and column. What is the smallest rookable set that also hits all the rows and columns? Well, imagine the rook starting somewhere and then having to hit the seven other columns and seven other rows. Any given move can enter a new row, or a new column, but not both. There for, there must be at least $7+7$ other initial cells besides the one the rook starts in, for a total of at least $15$.

• Wow, that's a brilliant argument! – Deusovi Mar 21 '18 at 7:05
• This is super surprising and beautiful, better than my solution! This is also way more general, working whenever the "friend groups" are all of the form {a,b} x {c,d}, instead of just {a,a+1} x {b,b+1}. Thank you for sharing! – Mike Earnest Mar 21 '18 at 7:30
• @MikeEarnest I'm curious about what you had - would you mind posting your solution? – Deusovi Mar 21 '18 at 7:38
• @Deusovi done :) – Mike Earnest Mar 21 '18 at 8:35

No, it is impossible to make it with less than 15.

Because

Imagine this problem was asked for 3x3, then you need to have at least 3 square filled next to each other at some place randomly at least to start to spread

and it will become like below

But with this,

you need to put another two square which should be anywhere except the middle. Otherwise there will be no solution.

Likewise

With 4x4, the same principle,

As shown below:

So

to spread it out everywhere, you just need to put two infected square not at the same square you had before every time the dimension increases.

As a result, We can conclude that to find it with any kind of dimension how many infected square you need, the formula will be as below:

$n_i=3+2(i-2)$

Note: This may not be the best way or the actual way to show it is

impossible with less than $n_8$ but by induction, i found this result.

• How do you know it's impossible with fewer? You've given a solution with $2i-1$ squares for each $i\times i$ board, but you haven't shown that it's impossible with fewer squares. – Deusovi Mar 21 '18 at 0:07
• (Put another way, you've given a nice strategy - but the point of the question is to show it can't be done with fewer, not that it can be done with this many. You've brushed off the meat of the question with the phrase "by induction, you can easily see".) – Deusovi Mar 21 '18 at 1:14
• @Deusovi i agree :) – Oray Mar 21 '18 at 5:02
• I think this is pretty close to a proof: basically the infection starts with three initial cells in a $2 \times 2$, and in order to spread to a new row or column, you need an additional initial cell (since otherwise it'll be quarantined to some $n \times m$ rectangle). So to go from $2 \times 2$ to $8 \times 8$ you need $6 + 6$ additional initial cells. However, to really turn this into a proof would require some work, especially since the infection can start in multiple places at once. I wonder if there is a simpler method ... – Tyler Seacrest Mar 21 '18 at 5:07

In the original problem, there was the handy observation that

as the infection spreads, the perimeter of the infected cells does not increase.

That still holds true in this problem, but it is no longer enough to prove what we want. Fortunately, the same argument can be salvaged with a tweak:

Let P be the perimeter of the infected cells, and let N be the number of contiguous groups of infected cells. As the infection spreads, the quantity Q = P – 2N does not increase. This is because any merging of two groups must be accompanied by a decrease in perimeter by 2 (and when 3 groups merge, perimeter decreases by 4).

Furthermore, when $k$ cells are initially infected, the initial value of Q is at most $2k$. Indeed, suppose these initial infections form N contiguous groups with sizes $k_1,k_2,\dots,k_N$. Each group has a perimeter of at most $4k_i - 2(k_i-1) = 2k_i + 2$; each cell contributes 4 edges, but we must subtract out the $k_i-1$ overlaps. Adding these all up, the total perimeter P of all these groups satisfies $P\le 2k+2N$, implying $Q\le 2k$.

Therefore, when you initially infect 14 cells, Q is at most 28, but to infect the whole board, Q needs to end at 32 – 2 • 1 = 30.

To infect an $m \times n$ board, you need at least

$m+n-1$

Proof:

Note: This proof is incorrect, and I don't think it can be easily fixed. Nevertheless I'm leaving this answer up because it shows the difficulty of this straightforward approach.

The proof is by induction.

Base case: It is obviously true that you need $1$ infected square to infect a $1 \times 1$ board. That may seem too trivial to work as a base case (though it is perfectly valid), so if you prefer you can use the fact that a $2\times2$ board needs at least $3$ infected squares to also infect the fourth.

Induction step: Suppose for the sake of induction the hypothesis holds for a particular board size $m \times n$. Add a new row or column to the side of that board. Every newly added square has only a single neighbour that is part of the old board, so if none of the added squares are infected, they can never have two infected neighbours and so will remain healthy. Therefore, to infect the newly added row/column, at least one of the new squares must start off sick. Furthermore, you cannot infect any of the new squares before its adjacent cell on the original board is infected. This means that the original board must be infected without any help from the squares in the new row/column (*). The original board needed at least $m+n-1$ infected squares, so the expanded board must therefore need at least one more, i.e. $(m+n-1)+1$ infected squares.
You can rewrite this as $(m+1)+n-1$ or $m+(n+1)-1$ depending on whether you added a row or a column to the board.

By induction the result follows: $m+n-1$ infected squares are needed on an $m \times n$ board.

The flaw at (*) is:

If you place two or more adjacent infected squares in the extra row/column, then the extra row/column helps infect the rest of the board, and maybe that allows the rest of the board to have fewer initially infected squares to compensate. The argument I used does not disprove the possibility of having an extra infected square in the extra column which then saves 2 or more initial infected squares from the rest of the board.

• "This means that the original board must be infected without any help from the squares in the new row/column." I don't see how that follows - why can't, say, adding 2 squares to the new row/column let us "close in on" the center somehow, letting us need less sick squares overall? Sure, getting the old board sick is sufficient, but I don't think you've quite shown it's necessary. – Deusovi Mar 21 '18 at 7:05
• @Deusovi: You're right. If the new column/row contains just non-adjacent infected squares then it is true, but with two or more adjacent ones the argument fails. – Jaap Scherphuis Mar 21 '18 at 7:15


text: 0000002179 00000 n When an electron completes a cycle around the Dirac point (a particular location in graphene's electronic structure), the phase of its wave function changes by Ï. ) of graphene electrons is experimentally challenging. 0000036485 00000 n Moreover, in this paper we shall an-alyze the Berry phase taking into account the spin-orbit interaction since this interaction is important for under- Active 11 months ago. Regular derivation; Dynamic system; Phase space Lagrangian; Lecture notes. 0000014889 00000 n In a quantum system at the n-th eigenstate, an adiabatic evolution of the Hamiltonian sees the system remain in the n-th eigenstate of the Hamiltonian, while also obtaining a phase factor. : Colloquium: Andreev reflection and Klein tunneling in graphene. 0000002704 00000 n The reason is the Dirac evolution law of carriers in graphene, which introduces a new asymmetry type. Recently introduced graphene13 ï¿¿hal-02303471ï¿¿ The U.S. Department of Energy's Office of Scientific and Technical Information @article{osti_1735905, title = {Local Berry Phase Signatures of Bilayer Graphene in Intervalley Quantum Interference}, author = {Zhang, Yu and Su, Ying and He, Lin}, abstractNote = {Chiral quasiparticles in Bernal-stacked bilayer graphene have valley-contrasting Berry phases of ±2Ï. Its connection with the unconventional quantum Hall effect in graphene is discussed. Berry phases,... Berry phase, extension of KSV formula & Chern number Berry connection ? In graphene, the quantized Berry phase Î³ = Ï accumulated by massless relativistic electrons along cyclotron orbits is evidenced by the anomalous quantum Hall effect4,5. In quantum mechanics, the Berry phase is a geometrical phase picked up by wave functions along an adiabatic closed trajectory in parameter space. 0000013594 00000 n 37 0 obj<> endobj Chiral quasiparticles in Bernal-stacked bilayer graphene have valley-contrasting Berry phases of ±2Ï. A (84) Berry phase: (phase across whole loop) B 77, 245413 (2008) Denis 0000013208 00000 n Berry phase in graphene: a semiâclassical perspective Discussion with: folks from the Orsaygraphene journal club (Mark Goerbig, Jean Noel Fuchs, Gilles Montambaux, etc..) Reference : Phys. 0000003989 00000 n Symmetry of the Bloch functions in the Brillouin zone leads to the quantization of Berry's phase. (For reference, the original paper is here , a nice talk about this is here, and reviews on â¦ Berry's phase, edge states in graphene, QHE as an axial anomaly / The âhalf-integerâ QHE in graphene Single-layer graphene: QHE plateaus observed at double layer: single layer: Novoselov et al, 2005, Zhang et al, 2005 Explanations of half-integer QHE: (i) anomaly of Dirac fermions; Our procedure is based on a reformulation of the Wigner formalism where the multiband particle-hole dynamics is described in terms of the Berry curvature. 125, 116804 â Published 10 September 2020 39 0 obj<>stream Unable to display preview. This is a preview of subscription content. These phases coincide for the perfectly linear Dirac dispersion relation. When considering accurate quantum dynamics calculations (point 3 on p. 770) we encounter the problem of what is called Berry phase. In gapped Bernal bilayer graphene, the Berry phase can be continuously tuned from zero to 2Ï, which offers a unique opportunity to explore the tunable Berry phase on physical phenomena. Phys. x�bf�ae`Z� �� @16� If an electron orbit in the Brillouin zone surrounds several Dirac points (band-contact lines in graphite), one can find the relative signs of the Berry phases generated by these points (lines) by taking this interaction into account. This is because these forces allow realizing experimentally the adiabatic transport on closed trajectories which are at the very heart of the definition of the Berry phase. monolayer graphene, using either s or p polarized light, show that the intensity patterns have a cosine functional form with a maximum along the K direction [9â13]. 0000001446 00000 n 0000018971 00000 n Rev. startxref Tunable graphene metasurfaces by discontinuous PancharatnamâBerry phase shift Xin Hu1,2, Long Wen1, Shichao Song1 and Qin Chen1 1Key Lab of Nanodevices and Applications-CAS & Collaborative Innovation Center of Suzhou Nano Science and Technology, Suzhou Institute of Nano-Tech and Nano-Bionics, Chinese Academy of Sciences 0000016141 00000 n Viewed 61 times 0 $\begingroup$ I was recently reading about the non-Abelian Berry phase and understood that it originates when you have an adaiabatic evolution across a â¦ Phys. Keywords Landau Level Dirac Fermion Dirac Point Quantum Hall Effect Berry Phase Sringer, Berlin (2003). the Berry phase.2,3 In graphene, the anomalous quantum Hall e ect results from the Berry phase = Ëpicked up by massless relativistic electrons along cyclotron orbits4,5 and proves the existence of Dirac cones. However, if the variation is cyclical, the Berry phase cannot be cancelled; it is invariant and becomes an observable property of the system. 14.2.3 BERRY PHASE. In graphene, the quantized Berry phase Î³ = Ï accumulated by massless relativistic electrons along cyclotron orbits is evidenced by the anomalous quantum Hall effect4,5. Here, we report experimental observation of Berry-phase-induced valley splitting and crossing in movable bilayer-graphene pân junction resonators. Advanced Photonics Journal of Applied Remote Sensing This process is experimental and the keywords may be updated as the learning algorithm improves. These phases coincide for the perfectly linear Dirac dispersion relation. Tunable graphene metasurfaces by discontinuous PancharatnamâBerry phase shift Xin Hu1,2, Long Wen1, Shichao Song1 and Qin Chen1 1Key Lab of Nanodevices and Applications-CAS & Collaborative Innovation Center of Suzhou Nano Berry's phase is defined for the dynamics of electrons in periodic solids and an explicit formula is derived for it. Berry's phase is defined for the dynamics of electrons in periodic solids and an explicit formula is derived for it. %PDF-1.4 %���� pp 373-379 | CONFERENCE PROCEEDINGS Papers Presentations Journals. Another study found that the intensity pattern for bilayer graphene from s polarized light has two nodes along the K direction, which can be linked to the Berryâs phase [14]. A A = ihu p|r p|u pi Berry connection (phase accumulated over small section): d(p) Berry, Proc. 0000001366 00000 n xref 0000028041 00000 n in graphene, where charge carriers mimic Dirac fermions characterized by Berryâs phase Ï, which results in shifted positions of the Hall plateaus3â9.Herewereportathirdtype oftheintegerquantumHalleï¬ect. The Berry phase in this second case is called a topological phase. By reviewing the proof of the adiabatic theorem given by Max Born and Vladimir Fock , in Zeitschrift für Physik 51 , 165 (1928), we could characterize the whole change of the adiabatic process into a phase term. Trigonal warping and Berryâs phase N in ABC-stacked multilayer graphene Mikito Koshino1 and Edward McCann2 1Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan 2Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom Received 25 June 2009; revised manuscript received 14 August 2009; published 12 October 2009 Phys. Abstract. 0 Berry phase in graphene: a semiâclassical perspective Discussion with: folks from the Orsaygraphene journal club (Mark Goerbig, Jean Noel Fuchs, Gilles Montambaux, etc..) Reference : Phys. Berry phase in graphene. Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations, vol. The phase obtained has a contribution from the state's time evolution and another from the variation of the eigenstate with the changing Hamiltonian. It is usually thought that measuring the Berry phase requires the application of external electromagnetic fields to force the charged particles along closed trajectories3. Abstract: The Berry phase of \pi\ in graphene is derived in a pedagogical way. 0000007386 00000 n 0000018422 00000 n The relationship between this semiclassical phase and the adiabatic Berry phase, usually referred to in this context, is discussed. Mod. 0000005982 00000 n 0000007703 00000 n It can be writ- ten as a line integral over the loop in the parameter space and does not depend on the exact rate of change along the loop. It is usually believed that measuring the Berry phase requires applying electromagnetic forces. This service is more advanced with JavaScript available, Progress in Industrial Mathematics at ECMI 2010 Some flakes fold over during this procedure, yielding twisted layers which are processed and contacted for electrical measurements as sketched in figure 1(a). The Berry phase in graphene and graphite multilayers. The ambiguity of how to calculate this value properly is clarified. As indicated by the colored bars, these superimposed sets of SdH oscillations exhibit a Berry phase of indicating parallel transport in two decoupled â¦ Bohm, A., Mostafazadeh, A., Koizumi, H., Niu, Q., Zwanziger, J.: The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics. On the left is a fragment of the lattice showing a primitive unit cell, with primitive translation vectors a and b, and corresponding primitive vectors G 1, G 2 of the reciprocal lattice. 192.185.4.107. 0000023643 00000 n In addition a transition in Berry phase between ... Graphene samples are prepared by mechanical exfoliation of natural graphite onto a substrate of SiO 2. 0000003452 00000 n 0000019858 00000 n Our procedure is based on a reformulation of the Wigner formalism where the multiband particle-hole dynamics is described in terms of the Berry curvature. (Fig.2) Massless Dirac particle also in graphene ? 0000001879 00000 n The same result holds for the traversal time in non-contacted or contacted graphene structures. Nature, Progress in Industrial Mathematics at ECMI 2010, Institute of Theoretical and Computational Physics, TU Graz, https://doi.org/10.1007/978-3-642-25100-9_44. Highlights The Berry phase in asymmetric graphene structures behaves differently than in semiconductors. Rev. The Berry phase, named for Michael Berry, is a so-called geometric phase, in that the value of the phase depends on the "space" itself and the trajectory the system takes. Electrons in graphene â massless Dirac electrons and Berry phase Graphene is a single (infinite, 2d) sheet of carbon atoms in the graphitic honeycomb lattice. discussed in the context of the quantum phase of a spin-1/2. Phys. �x��u��u���g20��^����s\�Yܢ��N�^����[� ��. 0000003090 00000 n © 2020 Springer Nature Switzerland AG. 37 33 10 1013. the phase of its wave function consists of the usual semi- classical partcS/eH,theshift associated with the so-called turning points of the orbit where the semiclas- sical â¦ 0000001804 00000 n On the left is a fragment of the lattice showing a primitive A direct implication of Berryâ s phase in graphene is. 0000000956 00000 n This property makes it possible to ex- press the Berry phase in terms of local geometrical quantities in the parameter space. We derive a semiclassical expression for the Greenâs function in graphene, in which the presence of a semiclassical phase is made apparent. Ever since the novel quantum Hall effect in bilayer graphene was discovered, and explained by a Berry phase of $2\ensuremath{\pi}$ [K. S. Novoselov et al., Nat. The relationship between this semiclassical phase and the adiabatic Berry phase, usually referred to in this context, is discussed. %%EOF Berry phase Consider a closeddirected curve C in parameter space R. The Berryphase along C is deï¬ned in the following way: Î³ n(C) = I C dÎ³ n = I C A n(R)dR Important: The Berry phase is gaugeinvariant: the integral of â RÎ±(R) depends only on the start and end points of C â for a closed curve it is zero. Nature, Nature Publishing Nature, Nature Publishing Group, 2019, ï¿¿10.1038/s41586-019-1613-5ï¿¿. Beenakker, C.W.J. Graphene (/ Ë É¡ r æ f iË n /) is an allotrope of carbon consisting of a single layer of atoms arranged in a two-dimensional honeycomb lattice. : Strong suppression of weak localization in graphene. Lecture 1 : 1-d SSH model; Lecture 2 : Berry Phase and Chern number; Lecture 3 : Chern Insulator; Berryâs Phase. Mod. 0000003418 00000 n Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K. When a gap of tunable size opens at the conic band intersections of graphene, the Berry phase does not vanish abruptly, but progressively decreases as the gap increases. Graphene is a really single atom thick two-dimensional Ëlm consisting of only carbon atoms and exhibits very interesting material properties such as massless Dirac-fermions, Quantum Hall eÅ ect, very high electron mobility as high as 2×106cm2/Vsec.A.K.Geim and K. S. Novoselov had prepared this Ëlm by exfoliating from HOPG and put it onto SiO Not logged in We derive a semiclassical expression for the Greenâs function in graphene, in which the presence of a semiclassical phase is made apparent. pseudo-spinor that describes the sublattice symmetr y. Rev. Fizika Nizkikh Temperatur, 2008, v. 34, No. Phase space Lagrangian. Berry phase in solids In a solid, the natural parameter space is electron momentum. Basic deï¬nitions: Berry connection, gauge invariance Consider a quantum state |Î¨(R)i where Rdenotes some set of parameters, e.g., v and w from the Su-Schrieï¬er-Heeger model. Lond. The electronic band structure of ABC-stacked multilayer graphene is studied within an effective mass approximation. 0000005342 00000 n Part of Springer Nature. Thus this Berry phase belongs to the second type (a topological Berry phase). We discuss the electron energy spectra and the Berry phases for graphene, a graphite bilayer, and bulk graphite, allowing for a small spin-orbit interaction. built a graphene nanostructure consisting of a central region doped with positive carriers surrounded by a negatively doped background. The change in the electron wavefunction within the unit cell leads to a Berry connection and Berry curvature: We keep ï¬nding more physical This nontrivial topological structure, associated with the pseudospin winding along a closed Fermi surface, is responsible for various novel electronic properties. Berry phase Consider a closeddirected curve C in parameter space R. The Berryphase along C is deï¬ned in the following way: X i âÎ³ i â Î³(C) = âArg exp âi I C A(R)dR Important: The Berry phase is gaugeinvariant: the integral of â RÎ±(R) depends only on the start and end points of C, hence for a closed curve it is zero. These keywords were added by machine and not by the authors. Berry phase in metals, and then discuss the Berry phase in graphene, in a graphite bilayer, and in a bulk graphite that can be considered as a sample with a sufficiently large number of the layers. Local Berry Phase Signatures of Bilayer Graphene in Intervalley Quantum Interference Yu Zhang, Ying Su, and Lin He Phys. In gapped Bernal bilayer graphene, the Berry phase can be continuously tuned from zero to 2ï°, which offers a unique opportunity to explore the tunable Berry phase on the physical phenomena. 0000000016 00000 n The influence of Barryâs phase on the particle motion in graphene is analyzed by means of a quantum phase-space approach. Springer, Berlin (2002). PHYSICAL REVIEW B 96, 075409 (2017) Graphene superlattices in strong circularly polarized ï¬elds: Chirality, Berry phase, and attosecond dynamics Hamed Koochaki Kelardeh,* Vadym Apalkov,â  and Mark I. Stockmanâ¡ Center for Nano-Optics (CeNO) and Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA and Berryâs phase in graphene Yuanbo Zhang 1, Yan-Wen Tan 1, Horst L. Stormer 1,2 & Philip Kim 1 When electrons are conï¬ned in two-dimensional â¦ The relative phase between two states that are close I It has become a central unifying concept with applications in fields ranging from chemistry to condensed matter physics. Not affiliated Novikov, D.S. : Elastic scattering theory and transport in graphene. trailer [30] [32] These effects had been observed in bulk graphite by Yakov Kopelevich , Igor A. Luk'yanchuk , and others, in 2003â2004. Rev. Massless Dirac fermion in Graphene is real ? 0000020974 00000 n Berry phase in quantum mechanics. The emergence of some adiabatic parameters for the description of the quasi-classical trajectories in the presence of an external electric field is also discussed. 0000017359 00000 n 0000004745 00000 n Berry phase in graphene within a semiclassical, and more speciï¬cally semiclassical Greenâs function, perspective. <]>> Ghahari et al. Preliminary; some topics; Weyl Semi-metal. Second, the Berry phase is geometrical. Rev. B, Zhang, Y., Tan, Y., Stormer, H.L., Kim, P.: Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Contradicting this belief, we demonstrate that the Berry phase of graphene can be measured in absence of any external magnetic ï¬eld. In Chapter 6 wave function (6.19) corresponding to the adiabatic approximation was assumed. Morozov, S.V., Novoselov, K.S., Katsnelson, M.I., Schedin, F., Ponomarenko, L.A., Jiang, D., Geim, A.K. 8. 0000050644 00000 n TKNN number & Hall conductance One body to many body extension of the KSV formula Numerical examples: graphene Y. Hatsugai -30 When electrons are confined in two-dimensional materials, quantum-mechanically enhanced transport phenomena such as the quantum Hall effect can be observed. Over 10 million scientific documents at your fingertips. Electrons in graphene â massless Dirac electrons and Berry phase Graphene is a single (infinite, 2d) sheet of carbon atoms in the graphitic honeycomb lattice. : The electronic properties of graphene. graphene rotate by 90 ( 45 ) in changing from linearly to circularly polarized light; these angles are directly related to the phases of the wave functions and thus visually conï¬rm the Berryâs phase of (2 ) Berry phase in graphene within a semiclassical, and more speciï¬cally semiclassical Greenâs function, perspective. Berry phase of graphene from wavefront dislocations in Friedel oscillations. But as you see, these Berry phase has NO relation with this real world at all. Cite as. Rev. Soc. Because of the special torus topology of the Brillouin zone a nonzero Berry phase is shown to exist in a one-dimensional parameter space. The Dirac equation symmetry in graphene is broken by the Schrödinger electrons in â¦ It is usually thought that measuring the Berry phase requires It is known that honeycomb lattice graphene also has . Graphene, consisting of an isolated single atomic layer of graphite, is an ideal realization of such a two-dimensional system. This effect provided direct evidence of graphene's theoretically predicted Berry's phase of massless Dirac fermions and the first proof of the Dirac fermion nature of electrons. Graphene as the first truly two-dimensional crystal The surprising experimental discovery of a two-dimensional (2D) allotrope of carbon, termed graphene, has ushered unforeseen avenues to explore transport and interactions of low-dimensional electron system, build quantum-coherent carbon-based nanoelectronic devices, and probe high-energy physics of "charged neutrinos" in table-top â¦ The influence of Barry’s phase on the particle motion in graphene is analyzed by means of a quantum phase-space approach. 0000046011 00000 n In this approximation the electronic wave function depends parametrically on the positions of the nuclei. Symmetry of the Bloch functions in the Brillouin zone leads to the quantization of Berry's phase. B 77, 245413 (2008) Denis Ullmo& Pierre Carmier (LPTMS, Université ParisâSud) Phys. 6,15.T h i s. Because of the special torus topology of the Brillouin zone a nonzero Berry phase is shown to exist in a one-dimensional parameter space. Download preview PDF. Roy. Lett. Rev. This so-called Berry phase is tricky to observe directly in solid-state measurements. When a gap of tunable size opens at the conic band intersections of graphene, the Berry phase does not vanish abruptly, but progressively decreases as â¦ Now, please observe the Berry connection in the case of graphene: $$\vec{A}_B \propto \vec{ \nabla}_{\vec{q}}\phi(\vec{q})$$ The Berry connection is locally a pure gauge. @article{osti_1735905, title = {Local Berry Phase Signatures of Bilayer Graphene in Intervalley Quantum Interference}, author = {Zhang, Yu and Su, Ying and He, Lin}, abstractNote = {Chiral quasiparticles in Bernal-stacked bilayer graphene have valley-contrasting Berry phases of ±2Ï. 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### Theory:

It is evident from the excavations of the Indus valley civilisation that people had good knowledge about measurement.
In ancient Egypt and Rome, units like cubit and yard were in practice, respectively.
A cubit is a measure of length from elbow to tip of finger, whereas, a yard is the distance between the end of the outstretched arm and the chin.
Example:
Even today, we can notice that garland sellers use cubit for measuring flowers, and, yard is used for measuring cloths.

In ancient India, people used angul (finger) or a mutthi (fist) as units for small measurements.

In all these methods, people used their body parts like hand or foot for measuring length. But, it was inconsistent and lacked uniformity since it varies from person to person.

To attain uniformity in measurement, French scientists created a standard unit of measurement called the metric system. Nowadays, the International System of Units (SI units) is used as a standard unit.
Length
Length, in general, is the measurement of distance. The SI unit of length is a $$metre$$ ($$m$$).
Units of length
One $$metre$$ ($$m$$) is divided into $$100$$ equal parts called $$centimetre$$ ($$cm$$).
$$1$$ $$m = 100$$ $$cm$$
One $$centimetre$$ ($$cm$$) has $$10$$ equal parts called $$millimetre$$ ($$mm$$).
$$1$$ $$cm = 10$$ $$mm$$
For measuring longer distances, we use a larger unit of length called, $$kilometre$$ ($$km$$).
$$1$$ $$km = 1000$$ $$m$$
Measuring the length of an object
In general, we use a metre scale or rod to measure length.
1. Take the metre scale and keep it along with the length of the box to measure.

One should never slide the scale across the box as it will produce an incorrect reading.

2. Check the markings of the scale. If the end of the scale ($$0\ cm$$) is broken, then we should consider the next marking of the scale (say, $$1.0\ cm$$) as one reading and measure the length.

Here, the reading at one end is $$1.0\ cm$$, and the reading at other end is $$12.0\ cm$$ then, we should subtract the two readings. Now, the correct reading is $$11.0\ cm$$.

3. Also, the position of the eye plays an important role in measurements. The eye should be placed vertically above the point where the measurement has to be taken. If you view from either side of the vertical position, then the readings will be viewed differently.

Steps to measure a curved line
Length of a curved line can be found using a thread.

1. Tie a knot at one end and place it along the curved line.
2. Now, mark the points where the curved line starts ($$A$$) and ends ($$B$$) on the thread.
3. Then, keep the thread on the metre scale and measure the length between the marks of the thread. Thus, the length of a curved line is measured.
Reference:
https://media.cheggcdn.com/study/5b1/5b185b8a-9b14-4fde-a78e-4d2660c7272e/202573-10.1-16IPS1.png
https://www.ekshiksha.org.in/chapter/54/images_of_Motion_and_Measurement_of_Distances_VI/8.png
https://i.redd.it/e8z5sqfgsoz11.jpg
https://www.ekshiksha.org.in/chapter/54/images_of_Motion_and_Measurement_of_Distances_VI/7.png
https://miro.medium.com/max/2116/1*xHnkSecupuspX8WaNlYh3A.jpeg
https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcS9abCoMbdDdsG3XnKKV6O1BKSI3o6rLc6mgg&usqp=CAU


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# Gyroscopes curving space by their energy

1. Dec 22, 2015

### Doug Brown

many properties of gyroscopic motion are non-intuitive because I think the standard physical features are not correctly identified by observers. Some who know little quickly claim that this or that law of physics is violated and some new discovery is afoot. I tend to think that all laws of physics are being obeyed but the features are not properly identified by most observers who are too quick to claim knowledge of the underlying dynamics

While I know from certain posts I have seen on this GR forum that I have lot to learn about GR and curvature of space and that mass/energy causes curvature of space from the non-vanishing of the energy-stress tensor alone, wouldn't it be the case that a spinning disc or gyroscope or flywheel is concentrating energy in a small space in such a way as to curve space? Wouldn't this explain precession motion as resulting from the center of mass of objects following curved space lines made curved by the increase in energy density local to the gyroscope? So the spinning top falls due to gravity not directly toward the center of the earth but rather in a helical path around the axis pointing to the center of the earth, and this helical path is the result of the energy concentration which is in the energy-stress tensor curving the space in the first place?

Therefore, can or has a full mathematical GR analysis of gyroscopes been done to explain behaviours and derive behaviours from gyroscopic motion and combinations of gyroscopes in relative motion?

2. Dec 22, 2015

### Staff: Mentor

It's possible in principle, sure. And a spinning disc or gyroscope that concentrates enough energy in a small enough space will have a stress-energy tensor that causes the appropriate spacetime curvature. (It's spacetime curvature, btw, not space curvature.)

In practice, however, no ordinary spinning disc or gyroscope has anywhere near enough stress-energy to cause measurable spacetime curvature. So this can't be the explanation of precession.

You don't need GR to analyze an ordinary gyroscope; precession of a gyroscope can be analyzed just fine using Newtonian gravity. (Of course "Newtonian gravity" here is really the Newtonian approximation to GR, but the point is that there are no uniquely GR effects involved.)

3. Dec 23, 2015

### Staff: Mentor

I doubt that we have any materials which could handle the stresses involved in rotating so fast that relativistic effects became important.

4. Dec 23, 2015

### Doug Brown

thank you for clarifications

Now, given the role normalized velocity plays in the Lorentz factor $\gamma = {1\over \sqrt{ 1 - \beta^2 }}$ where $\beta = v/c$ is the light-speed normalized velocity making clear when relativistic effects are important in Special Relativity, what is this equivalent factor for General Relativity?

Now, given this equivalent factor for GR, which I assume is some value of Energy density, what value must the energy density of our toy gyroscope (just a spinning disc) be greater than in order for curvature of space to be achieved at some macroscopic level noticable to any observer in our laboratory?

In gyroscopic physics discussions it is often stated that "there is a lot of energy stored in this particular gyro right now". Well can we put together this calculation here in this thread and see how big must the disc be, how fast must its rotation be in order for an observably macroscopic curvature of spacetime to be detected in the laboratory even if that laboratory has to be much bigger than CERN's LHC.

5. Dec 23, 2015

### Staff: Mentor

That means "a lot of energy for a mechanical device". It's not enough for relativistic effects to even be a rounding error.
For the rotational energy of the disk to produce detectable curvature effects, it must be order-of-magnitude comparable to the rest energy of the disk. That rest energy is given by the famous $E=mc^2$ where $m$ is the rest mass of the disk; that mass must be large enough to be detectable by an Eotvos-style experiment. We're talking about something like a disk with a radius of fifteen centimeters turning at a few billion revolutions per minute and storing as much energy as several hundred multi-megaton nuclear weapons.

Last edited: Dec 24, 2015
6. Dec 24, 2015

### Doug Brown

ok great, thank you very much for nailing down those magnitudes. Now I'd like to try to use the provided orders of magnitude in some mathematics and then plot some relationships so your 15 cm is just one point on a curve in a plot, and your few billion rpm is one point on a curve. Does PhysicsForums (PF) allow insertion of figures & plots like it allows insertions of latex equations? if someone knows how to insert a graph of some sort, I'd like to write down some equations now and then add some plots based on some of the equations. Any PF graphics insertion help would be super. Thanks

Formalism to deduce laboratory values for parameters

Given a disc of uniform mass m, radius r (any thickness) spinning an angle $\alpha$ in $s$ seconds at angular velocity $\omega_{\alpha} = {\alpha \over s}$ its moment of inertia around its center axis is $I = {1\over 2} m r^2$ giving it an angular momentum of $L=I \omega_{\alpha}={1\over 2} { \omega_{\alpha} m r^2}$.

If this spinning disc is now placed at the end of a rod of length $x$ (along an $x$ axis) fixed at the origin of a coordinate system, a torque $\tau = x \times F$ is created by the downward force of gravity $F=mg$. In this coordinate system $\theta=0$ defines the $z$ axis, while $\theta=\pi/2$ defines the azimuthal $x-y$ plane where $\phi=0$ defines the $x$ axis.

The disc's angular momentum $L$, gravity $g$ and the constraint that the disc is fixed a length $x$ from an origin then causes a precession about the polar axis with an azimuthal angular velocity $\omega_{\phi} = {\tau \over L}= 2 {x g \over { \omega_{\alpha} r^2 }}$ assuming $\omega_{\alpha} \gg \omega_{\phi}$.

The rotational energy of this system is then $E_R= E_{\alpha} + E_{\phi} = {1\over 2} I ({\omega_{\alpha}}^2 + {\omega_{\phi}}^2)$. Since $\omega=2\pi f$ where $f$ is the frequency in rpm the rotational energy becomes $E_R=\pi^2 m r^2 (f_{\alpha}^2 + f_{\phi}^2)$.

So now if the rotational energy $E_R$ is chosen to rival its rest mass energy $E_0=mc^2$ then $E_R=E_0$. This equality is $\pi^2 m r^2 (f_{\alpha}^2 + f_{\phi}^2) = mc^2$.

But now it looks like $m$ drops out and we are left with some relation from which we can solve for $r$ as some function $h(f_{\alpha},f_{\phi})$ or either of the $f$'s as some function $j(r)$ but that seems like barking up the wrong tree.

Rather a total energy $E_T=E_R+E_0$ seems to be more relevant and this is what must be set to your value of "as several hundred multi-megaton nuclear weapons".

In any event, if the $f$s are in hertz = cycles per $s$, $r$ in $cm$, $c$ in $cm/s$ do these units come out right? what can be done with this equality? what's the next step in the thought experiment of deriving actual laboratory values for $m$, $r$, $x$, $f_\alpha$ and $f_\phi$ to force this gyroscope system to curve spacetime explicitly?

Last edited: Dec 24, 2015
7. Dec 24, 2015

### Staff: Mentor

Where does the gravity come from? Are you trying to figure out the precession of the gyroscope, or how much it curves spacetime? An external source of gravity is irrelevant for the latter question, and how much the gyroscope curves spacetime is irrelevant for the former question. Even if the gyroscope does contain enough energy to produce non-negligible spacetime curvature, that curvature won't make the gyroscope precess. Precession is caused by some externally applied torque; there's no way a gyroscope can apply a torque to itself.

The $f_{\phi}$ term is irrelevant here, for the reasons given above. Also, it looks like you've made a slight mistake in deriving the $f$ version. If $I = \frac{1}{2} m r^2$ and $E_R = \frac{1}{2} I \omega_{\alpha}^2 = \frac{1}{4} m r^2 \omega_\alpha^2$ and $\omega_\alpha = 2 \pi f_\alpha$, then I get $E_R = \frac{1}{4} m r^2 4 \pi^2 f_\alpha^2 = \pi^2 m r^2 f_\alpha^2$.

$x$ is irrelevant, for the reasons given above. So is any "rotational energy" from precession (what you are calling $\omega_{\phi}$); "energy" due to the motion of the gyroscope (or any other object) relative to something else doesn't contribute to the spacetime curvature produced by the object.

The spacetime curvature produced by the gyroscope is due to its stress-energy tensor, as I said in post #2 above. Nothing else contributes. So you need to figure out the stress-energy tensor of the gyroscope. I don't have an explicit expression handy for this, but the rest energy $m c^2$ and the rotational energy $\frac{1}{2} I \omega_{\alpha}^2$ will definitely both contribute to it. Other contributions will be the angular momentum and the pressure and stresses within the gyroscope, caused by its rotation.

Also, note that the rule of thumb Nugatory gave (rotational energy same order of magnitude as rest energy) was the condition for the rotational energy of the gyroscope to cause non-negligible spacetime curvature compared to the rest energy. But this assumes that the rest energy itself is large enough to cause non-negligible spacetime curvature. For an ordinary-sized gyroscope, you would need extremely sensitive measurements to detect the spacetime curvature produced by its rest energy $m c^2$ (for example, you could do a Cavendish-style experiment with torsion balances to measure the gravitational attraction of the gyroscope). But even for an ordinary-sized gyroscope, say 10 kg, the rest energy $m c^2$ is enormous: about $10^{18}$ Joules for a 10 kg gyroscope, or about 250 megatons of TNT. In other words, that enormous amount of rest energy produces a very tiny amount of spacetime curvature, that requires extremely sensitive measurements to detect.

Further, the rotational energy can't actually be equal to the rest energy, because that would make the outer edge of the gyroscope move faster than light. The linear velocity of the gyroscope's outer edge is $\omega_\alpha r$, and that must be less than $c$, so we must have $\omega_\alpha < c / r$. That sets a limit on the rotational energy of $E_R = \frac{1}{4} m r^2 \omega_\alpha^2 < \frac{1}{4} m c^2$, or 1/4 of the rest energy. So for the above 10 kg gyroscope, the maximum rotational energy would be about $2.5 \times 10^{17}$ Joules, or about 62 megatons of TNT. (Note that this fraction is independent of the radius of the gyroscope--if we increase the radius, we decrease the maximum allowed angular velocity in the same proportion.)

Last edited: Dec 24, 2015
8. Dec 24, 2015

### Doug Brown

Thank you for additional comments. Still digesting your comments but I can see at outset some of my context is not clear to you. I am thinking of a real disc in my hands in my lab and the Formalism I wrote is for a table top gyroscope set up where I have a) a spinning disc, and b) it is has a rod through its center one end of which is suspended by a string from the ceiling or it could be in a frame with acting gimbals.

Now, if I want to consider $only$ the disc alone I can take $x=0$ (which implies $f_\phi=0$) but if I take $x\ne0$ then there will be gravity because we are in a real lab, there will be torque by gravity and $that$ will cause precession (again here on earth with my hands nearby). This is the system I want to consider and make calculations about which is why I set up that formalism to make concrete all the values we are now going to be getting explicit about.

So while it might not serve other aspects of the calculation you are thinking about, I need those to be included in the system I am calculating about because this is related to a device I can actually build which helps me think about it. Now, I will go back to your comments and continue digesting them. Thank you for your detailed specific input!

9. Dec 24, 2015

### Staff: Mentor

I just assumed that $E_R$ and $E_0$ would both be on the order of a kilogram or so to get that result. That's getting down towards the lower limit of detectable gravitational effects.

There are enough practical problems here that any calculation you do is only going to be accurate within an order of magnitude or so. For example, no real energy-storing gyro is a uniform disk; the classical formulas for moment of inertia and rotational kinetic energy cannot be trusted at relativistic speeds (although for values of $\gamma=1/\sqrt{1-v^2/c^2}$ not too far from one the error will be not be too far from the value of $\gamma$); and probably some other stuff that I haven't though about.

Because it is completely impossible to build such a thing (just for grins, you might calculate the centrifugal forces at the rim of a 15cm radiius object rotating at a few billions of RPMs; and do remember that eventually all that energy is going to be released explosively unless you can find a way of safely dissipating the equivalent of a medium-large nuclear war) order-of-magnitude calculations around some reasonably chosen parameters will deliver most of the physical insight here.

10. Dec 24, 2015

### Doug Brown

The gravity comes from Earth gravity since my discs are in my lab and I want to hang the disc from a string from the ceiling of the lab. Therefore, this will be considered to be in the already curved spacetime of my lab space, but still I think we can still talk about the curvature coming from the spinning disc and I want to calculate the actual values for all the parameters to make this curvature non-negligible

11. Dec 24, 2015

### Doug Brown

This comment seems to be more to my initial query. I have been assuming that if an enormous amount of energy were stored in the spinning disc then that energy would indeed be capable of curving spacetime. If that is not the case at all, then I need to learn more here on this very point.

12. Dec 24, 2015

### Doug Brown

oh yes you're right, I see my error of not including the 2π in the square, thank you for the correction!

13. Dec 24, 2015

### Doug Brown

Ok I see your point. I will now build $T^{\mu\nu}$ for the gryoscope element by element (this will take some time, and perhaps not soon, but I do want to come back to this and fill out this matrix in full -- and will do so in this thread in LaTeX). I still want to add plots of the various quantities still also so if anyone knows how to insert plots into threads on PF please inform.

14. Dec 24, 2015

### Doug Brown

while $E_0$ is a constant, $E_R$ can take on any values we tell it to take on, so it does not have to be assumed to be any particular value like a kilogram or so. In fact I want to set it to whatever the minimum value it must be in order to curve spacetime, and I hope the rest of this analysis assigning specific functional forms to the tensor elements of $T^{\mu\nu}$ will take us there. We will have all of $m$, $r$, $f_\alpha$ as knobs to set getting us there and if $x\ne0$ then $f_\phi$ as well.

Your guidance is very helpful and crucial, thank you very much for contributing it.

Last edited: Dec 24, 2015
15. Dec 24, 2015

### Doug Brown

I expect after we get further in this analysis we will certainly be calculating what any Lorentz factor value we end up with. If we want to keep it near 1 then by all means we will try. As stated in post #13 I will be building out all the elements of $T^{\mu\nu}$ inserting anywhere I can the various functional quantities I have described in the formalism above and hopefully some of those will be in the tensor elements of $T^{\mu\nu}$

Last edited: Dec 24, 2015
16. Dec 24, 2015

### Doug Brown

Thank you for these very crucial contributions. This nails exactly the point Nugatory made about the spinning disc not satisfying all the non-relativistic relations assumed and provides conversions to some of the relevant units I need. This is the heart of the matter. Need time for further digesting :)

17. Dec 24, 2015

### Staff: Mentor

Yes, and in this calculation, the fact that the gyroscope is hanging from a string is irrelevant. All you need is the gyroscope's rest mass, radius, and rate of rotation. (Actually, in general, you would need its energy density as a function of position, and you would not be able to assume that it was a perfect sphere, but we can assume that its density is constant and that it is perfectly spherical as a reasonable idealized case.)

As the numbers I gave show, an "enormous amount of energy" by everyday standards can still be negligible (or almost so) as far as producing spacetime curvature is concerned.

18. Dec 24, 2015

### Staff: Mentor

Any non-zero values will curve spacetime to some extent so there is no minimum - but it sounds as if you're looking for an effect that is large enough to detect. For that, you need kilograms (as opposed to, for example, grams or micrograms), so you might as well assume a kilogram-sized mass as you're considering the general scale of the mechanical system. If you decide later that you want the relativistic effects to be five times as large, you can dial the mass up by a factor of five.

19. Dec 24, 2015

### Staff: Mentor

As the discussion progresses, it sounds more as if you are looking for a general understanding of the stress energy tensor of a rotating body and how it affects the curvature of spacetime around it. For that, you can look at the relativistic gyroscopes that nature has already helpfully scattered throughout the universe - rotating collapsed stars. Google for "Kerr metric" to find the details of the solution and see how the various physical properties of the rotating body contribute.