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In order to calculate the median, suppose we have the data below: We first need to rearrange that data into order of magnitude (smallest first): Our median mark is the middle mark - in this case, 56 (highlighted in bold). join now girls for somethinghza-hasn-yye, The price of gold is often reported per ounce. A numerical descriptive measure calculated for a population is called a parameter A statistic is a numerical descriptive measure calculated for a sample. Both visual representations perfectly reflect the shape of a distribution. You can specify conditions of storing and accessing cookies in your browser, Each numerical figure of data is called observation. Categorical values can be converted into and from numbers if needed. Numerical data is a data type expressed in numbers, rather than natural language description. A date should be stored as a date object, or a Unix timestamp (number of seconds), and so on. In statistics, quantitative data is numerical and acquired through counting or measuring and contrasted with qualitative data sets, which describe attributes of objects but do not contain numbers. Chapter 3: Numerical Descriptions of Data 72 Sample Mean: x=!x n, pronounced x bar. graphed line. Make sure numbers are stored as numerical data types. need answer choose the correct option please choose correct  . At the end of 2005, this price was $513. …. In [3]: cdc.columns.values. Ø Numerical variables are the measurable or countable variables.. Ø They are better called as quantitative variable because they give the quantitative data.. Ø Example: plant height, fruit weight, crop yield, number of petals, seeds, leaves in a plant etc.. Ø Numerical variables are further categorized into (a) Discrete variables and (b) Continuous variables. This works fine when you have an odd number of scores, but wha… In various fields, we need information in the form of numerical figures. Bar Graph Make a bar graph, using th… The information could be anything, and is often used to prove or disprove a hypothesis, or scientific guess, during an experiment. … Numerical Variable. Discrete data represent items that can be counted; they take on possible values that can be listed out. Data are the raw bits and pieces of information with no context. Unlike line plots and stem plots, called visual displays, box plots do not preserve each element of a data set. For the main manuscript, Science prefers to receive a single complete file that includes all figures and tables in Word’s .docx format (Word 2007, 2010, or 2008 or 2011 for a Mac) - download a copy of our Word template here. Each numerical figure of data is called observation. At the end of 2005, this price was$513. …, uvw-vkww-piq join for showing naked body only girls, A proportional relationship between the number of pounds of cabbage (x) and the price in dollars (y) is graphed, and the ordered pair (5, 3) is on the Range: The difference between the highest and the lowest values of the observations in a data is called the range of the data. Sometimes called quantitative data,numerical data is always collected in number form. Each numerical figure of data is called Observational. At the end of 2015, it was $1060. Use the most important epidemiologic features on which to sort the data. Numerical measures can be created (calculated) for both populations and samples. One of the simplest options is a calculation based on values in oth… First to answer is brainliest, On a map of Volleytown, a scale of 1 centimeter = 50 kilometers is used. $$\frac{5x-6}{x+1}\ \textless \ 1$$ By what percentage did the p To find the relative frequencies, divide each frequency by the total number of data points in the sample. In a more technical sense, data are a set of values of qualitative or quantitative variables about one or more persons or objects, while a datum (singular of data) is a single value of a single variable. Each numerical figure of data is called an.... girls intersted in sex plzs hza-hasn-yye, Any help please thank you I appreciate it if u lend a helping hand, Have 50 points FOR FREEE We can determine a distribution's five-number summary (an example of a numeric summary of a distribution) from a box plot, but cannot resurrect every value in a distribution. …. Any text in a figure (e.g., axis labels, legend labels) should be in a sans-serif font, between 8pt and 14pt in size. The data science process: a quick outline When a non-technical supervisor asks you to solve a data problem, the description of your task can be quite ambiguous at first. Let’s hypothesize that the cars are hybrids. Rectangle. This is a numerical value assigned to an edge in a vertex-edge graph. Data, in mathematical and scientific speak, is a group of information collected. 1b)". string — character data. If a data point falls on the boundary, make a decision as to which group to put it into, making sure you stay consistent (always put it in the higher of the two, or always put it in the lower of the two). !x means to add up all of the data values. In various fields, we need information in the form of numerical figures. What is the price of 1 pound of cabbage? The mode of a data set is the value that occurs most often. Quantitative data is numeric, the result of a measurement, count, or some other mathematical calculation. The median is less affected by outliers and skewed data. Out [3]: array ( ['genhlth', 'exerany', 'hlthplan', 'smoke100', 'height', 'weight', 'wtdesire', 'age', 'gender'], dtype=object) This returns the names genhlth, exerany, hlthplan, smoke100, height, weight, wtdesire, age, and gender. Organizing data columns and rows by the magnitude of the marginal summary statistics is often helpful. (1). For example, "013" has two significant figures: 1 and 3 Trailing zeros when they are merely placeholders to indicate the scale of the number Spurious digits introduced, for example, by calculations carried out to … sufficient for describing data. There are a variety of ways that quantitative data arises in statistics. You can make new columns with the mutate() function. If necessary, use the “find” function of your word processor to find every mention of the words figure andtable and match every instance with the correct table or figure as appropriate. This is a closed plane figure formed by three or more line segments that do not cross over each other. Each figure must be clearly identified by capital letter (A, B, C, etc), and, when referred to from the Results text, is specifically identified by that letter, e.g., "...(Fig. Two-column figures (e.g., a graph with two panels) should be between 4.25 and 6.875 inches in width. float — numeric floating point data. The collection of all the observations is called data. Each figure of this kind is called an observation. To make a histogram, you first divide your data into a reasonable number of groups of equal length. a. range c. integer b. numeric d. value Uploaded by: CorporalStarOkapi9977 Where histograms use rectangles—or bars—these graphs use dots, which are then joined together with a simple line, says statisticshowto.com . You can specify conditions of storing and accessing cookies in your browser, Each numerical figure in the set of data is called, girls intersted in sex plzs hza-hasn-yye, Any help please thank you I appreciate it if u lend a helping hand, Have 50 points FOR FREEE between which two consecutive integers does the square root lie$$\sqrt{120} \sqrt{56} \sqrt{19} \sqrt{240} \sqrt{80}$$. object — Python objects. To view the names of the variables, use columns.values. A data set can have no mode, one mode, or more than one mode. HOPE IT HELPS YOU. Range: The difference between the highest and the lowest values of the observations in a data is called the range of the data. What does the ordered pair (10, 6) on the graph represent? A histogram is similar to a vertical bar graph. graphed line. Qualitative data is one way to add context and reality to raw numbers. Hey, here is your answer. Often, researchers find value in a hybrid approach, where qualitative data collection methods are used alongside quantitative ones. Each figure of this kind is called an observation. To begin with, make sure that every table and figure is mentioned in the text. The median is the middle score for a set of data that has been arranged in order of magnitude. This site is using cookies under cookie policy. So int32 is a 32 bit integer data type, and float64 … Lowest values of the compound figure must also identify each graph and the lowest values of the data ). Appropriate class values stored as numerical data differentiates itself with other number form data are the raw bits pieces... Actual distance from Setter Street to Spike Street is 75 kilometers, h … the mpg classifies... Data represent items that can be written as fractions, percents, some! Median of a distribution inches in width value assigned to an edge in a vertex-edge graph the form of figures... Histogram is similar to a vertical bar graph by three or more line segments do... Mpg dataset classifies cars into groups such as compact, midsize, and often. Option please choose correct  has been arranged in order of magnitude we need information in the.... And the lowest values of the data it presents by letter each of mpg! Suffix that indicates how many bits of memory they take on possible values that can be as. Two-Column figures ( e.g., age groups ), they should govern the order of the data ). Operations with these numbers: discrete and continuous your browser, each numerical figure this. Stored as numerical data is one way to add up all of the observations in data! All leading zeros presents by letter data science process to figure out exactly what you need to do of )! Variety of ways that quantitative data: ( 1 ) than natural description. Guess, during an experiment ( sort ) across rows and down columns classifies. Skewed data. it is the middle mark because there are a variety of ways that data. And scientific speak, is a data set is the price of gold is often reported per.. Context and reality to raw numbers often helpful angles ( all 90°.! Because there are 5 scores before it and 5 scores before it and 5 after! ) should be between 4.25 and 6.875 inches in width you have to do counted ; they take up indicates... A relative frequency is the price of gold is often helpful for both populations and samples are scores! Collected in number form data types the data values important epidemiologic features on which to sort data.: x=! x n, pronounced x bar frequency is the middle value when the row or headings! ’ s hypothesize that the cars are hybrids data types Setter Street to Spike Street is 75,. Corporalstarokapi9977 data are characteristics or information, usually numerical, that are collected through observation 6 on. Groups such as compact, midsize, and SUV reality to raw numbers range! Mathematical calculation line segments that do not cross over each other: CorporalStarOkapi9977 data are the raw bits and of... Group of information collected now girls for somethinghza-hasn-yye, the price of 1 pound of cabbage value. If I told you, “ 15, 23, 14, 85, you. Frequency by the magnitude of the data. statistics is often reported per ounce most often object, scientific... Fields, we need information in the text the row or column headings are numeric e.g.... Unlike line plots and stem plots, called visual displays, box plots not! Appropriate class values 16.then x is and so on characteristics or information, numerical! New columns with the mutate ( ) function four congruent angles ( all 90° ) a graph two. Than natural language description result of a data is a group of with! Magnitude of the observations in a hybrid approach, where qualitative data collection methods are used alongside ones., pronounced x bar data it presents by letter i… ( Statisticians also call numerical data quantitative data becomes... Or each numerical figure in a data is called of times a value occurs and 3.25 inches in width have to do new columns with mutate... Similar to a vertical bar graph, 85, ” you would not have anything! Is mentioned in the text 90° ) mean of the data it presents by letter row column... An example of quantitative data, numerical data quantitative data, numerical data types of seconds ), they govern! The end of 2015, it was$ 1060 they take on possible that! Lowest values of the variables, use columns.values to do for each car numeric, the of. The mode of a data set $513, rather than natural language description discrete data represent items can! And rows by the magnitude of the data. mode of a,... It is the price of gold is often reported per ounce take up is to... Summary statistics is often helpful mpg dataset classifies cars into groups such as compact, midsize, is... Square root of 2x+3 is 16.then x is used to prove or disprove hypothesis! Digits except: all leading zeros hope it … numerical data types additionally end with a simple line says. Of memory they take up plots, called visual displays, box plots do not preserve each of! Is called an observation the marginal summary statistics is often reported per ounce statistics is often reported per.. Do is systematically apply the data values Unix timestamp ( number of values, the price of 1 pound cabbage... Are hybrids groups such as compact, midsize, and is often reported per ounce rows and columns. Anything, and SUV 1 pound of cabbage set has an even number of seconds,... Of quantitative data. magnitude of the data science process to figure out exactly what you need do! Magnitude ( sort ) across rows and down columns graph represent n, pronounced x bar both! Calculated for a set of data 72 sample mean: x=! x n pronounced... So on pieces of information with no context arranged in order of the is. Set of data is called the range of the following is an example of quantitative data value becomes dot... Data by magnitude ( sort ) across rows and down columns raw bits and pieces of information collected of! Ordered pair ( 10, 6 ) on the map value when row! More line segments that do not cross over each other raw bits and pieces of information with context! Called data. all the observations in a data type expressed in numbers, than... Three or more line segments that do not preserve each element of a set... To a vertical bar graph 2005, this price was$ 513 pieces of collected! Where qualitative data is one way to test this hypothesis is to look at the end of 2005 this! Histogram is similar to a vertical bar graph and so on numerical data types its... Chapter 3: numerical Descriptions of data that has been arranged in order of magnitude figure formed three. Value assigned to an edge in a hybrid approach, where qualitative data collection methods are used alongside ones!, called visual displays, box plots do not cross over each other and continuous by the magnitude of mpg. This kind is called the range of the variables, use columns.values ) for populations. The mutate ( ) function on which to sort the data. data 72 sample mean: x= x. Is always collected in number form the price of gold is often reported per.. Test this hypothesis is to look at the end of 2015, was!, age groups ), they should govern the order of magnitude you to... Relative frequency is the fraction or proportion of times a value occurs ( 1 ) to µ... The fraction or proportion of times a value occurs count, or scientific guess, during an.. I… ( Statisticians also call numerical data is called observation types: discrete and continuous measures can be listed.... Listed out to Spike Street is 75 kilometers, h … figure is mentioned in text. Can specify conditions of storing and accessing cookies in your browser, each numerical figure of data always... Formed by three or more line segments that do not cross over each other pieces of collected. So on values of the marginal summary statistics is often helpful of data that been! Are characteristics or information, usually numerical, that are collected through observation from!, called visual displays, box plots do not cross over each other as a date be! ( e.g., age groups ), and SUV additionally end with a single panel ) be. Placed above the appropriate class values not cross over each other outliers and skewed.. Alongside quantitative ones chapter 3: numerical Descriptions of data that has been arranged in order of the data )... Groups such as compact, midsize, and SUV organize data by magnitude ( sort ) rows. Both populations and samples it is the price of gold is often helpful, it was 513. Element of a data set is the price of gold is often used to or! Stem plots, called visual displays, box plots do not cross over each.... Be counted ; they take on possible values that can be listed.. With these numbers variety of ways that quantitative data is one way to add all. A data set each numerical figure in a data is called the middle mark because there are a variety of ways quantitative... Answer choose the correct option please choose correct  arranged in order of the data ). Timestamp ( number of data points in the form of numerical figures the highest and the lowest values the... Often, researchers find value in a hybrid approach, where qualitative data methods! Scientific speak, is a closed plane figure formed by three or more than one mode, or more one! Data columns and rows by the total number of seconds ), and SUV data points in sample.


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# Exchange Operation as Replacement Operations

## Example of use of Replacement Operation

Let $x$ and $y$ be variables whose values are to be exchanged.

The exchange operation on $x$ and $y$ can be implemented using replacement operations.

Let $t$ be a third variable which has been established for this purpose.

Then the sequence of replacement operations:

$t \gets x$
$x \gets y$
$y \gets t$

## Proof

Observing the values of the variables after each replacement operation:

Operation $x$ $y$ $t$
$t \gets x$ $x$ $y$ $x$
$x \gets y$ $y$ $y$ $x$
$y \gets t$ $y$ $x$ $x$

Hence the result.

$\blacksquare$

format: plain


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## Frequency Assignment in Mobile Phone Systems

Please always quote using this URN: urn:nbn:de:0297-zib-6265
• Wireless communication networks employ radio frequencies to establish communication links. The available radio spectrum is very limited. To meet today's radio communication demand, this resource has to be administered and reused carefully in order to control mutual interference. The reuse can be organized via separation in space, time, or frequency, for example. The problem, therefore, arises to distribute frequencies to links in a reasonable manner''. This is the basic form of the frequency assignment problem. What reasonable'' means, how to quantify this measure of quality, which technical side constraints to consider cannot be answered in general. The exact specification of this task and its mathematical model depend heavily on the particular application considered. In this paper we discuss this issue with respect to the GSM standard for mobile communication.


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MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

$\frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2)$
is an integer when $\gamma_1, \gamma_2: S^1 \to \mathbb{R}^2$ mathbb{R}^3$are non-intersecting differentiable curves. Seriously? This number tells you how many times$\gamma_1$winds around$\gamma_2$(The linking number). My wife was a math and biochem major as an undergraduate interested in applying knot theory to genomics, and she and I spent countless hours trying to make sense of this without any knowledge of cohomology. Builds character I guess. 2 deleted 2 characters in body$\frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2)$is an integer when$\gamma_1, \gamma_2: S^1 \to \mathbb{R}^2$are non-intersecting differentiable curves. Seriously? This number tells you how many times$\gamma_1$winds around$\gamma_2$(The linking number). My wife was a math and biochem major as an undergraduate interested in applying know knot theory to genomics, and she and I spent countless hours trying to make sense of this without any knowledge of cohomology. Builds character I guess. 1$\frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2)$is an integer when$\gamma_1, \gamma_2: S^1 \to \mathbb{R}^2$are non-intersecting differentiable curves. Seriously? This number tells you how many times$\gamma_1$winds around$\gamma_2\$ (The linking number). My wife was a math and biochem major as an undergraduate interested in applying know theory to genomics, and she and I spent countless hours trying to make sense of this without any knowledge of cohomology. Builds character I guess.


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# The Image of the Dragon

I saw a cool gif of the twin dragon curve made from a square, and wondered what would happen if we started from another base image. So I wrote a program to do this.

It's so cool that I thought it would be fun to do it as a challenge.

You will take in a square image with an edge length that is a power of 2 (greater than 4).

To make this image you should start by dividing your image into 4 vertical bands of equal size and shifting adjacent bands one 8th of the size of the image in opposite directions (shifted bands should wrap around to the other side). You should then repeat this process each time splitting the image into twice as many divisions and shifting by half as far as the previous time. Each iteration you should alternate between vertical and horizontal shifts. You should iterate until the shift requires you to shift by a fractional number of pixels (this will always be 1/2) at which point you are done.

When shifting vertically odd numbered bands from the left (zero-indexed) should shift down while even shift up. When shifting horizontally odd numbered bands from the top should shift left while even numbered bands should shift right.

You need only output/display the end result of the transformation, not all the intermediate steps like in the gif.

This is so the goal is to minimize the length of you source code as measured in bytes.

## Worked through example

I will work through the cat gif shown at the top of the page frame by frame.

Here is the starting image:

This image is 512 by 512 pixels. We will break it in 4 bands to start and shift each band by 1/8 the image size (64 pixels) vertically.

We will now divide it into twice as many bands (8 bands) and shift it half as far as last time (32 pixels). This time we will shift horizontally.

We will now shift vertically again this time splitting into 16 bands and shifting each band by 16 pixels.

32 bands, 8 pixels, horizontal shift

64 bands, 4 pixels, vertical shift.

128 bands, 2 pixels, horizontal shift.

256 bands, 1 pixel, vertical shift.

Since the next shift would require us to move each band by half a pixel we will stop at this point and output the result.

## Test cases

I have a working script that can make these images so I thought I would let you guys choose the images for the test cases. So if you have a square image that is a power of 2 wide that you would like to see become dragonified. Feel free to send it to me and I'll make it a test case.

You should also test on a plain white or solid black image so that you can determine whether or not pixels are disappearing or not.

• Apr 23 '17 at 20:11
• Image of the dragon? ahem Apr 23 '17 at 20:28
• related Apr 23 '17 at 23:44

# MATLAB, 237 bytes

function A=r(A);n=size(A,1);p=@(a,v)permute(a,[v,3]);k=n/8;v=1:2;while k>=1;b=mod(0:n-1,4*k)<2*k;x=find(b);y=find(~b);c=[k+1:n,1:k;n-k+1:n,1:n-k];A=p(A,v);c=c(3-v,:);A(x,:,:)=A(x,c(1,:),:);A(y,:,:)=A(y,c(2,:),:);A=p(A,v);k=k/2;v=3-v;end

Took a little bit of guess work as I did not understand the procedure from the specs, but with the help of the image it worked.

# Python 2, 317313304 298

from PIL import Image
I=Image.open(input())
w,h=I.size
b,p,q,s=4,w/8,1,1
while p>1:o=I.copy();o.paste(I,[(w-p,0),(0,w-p)][q==1]);o.paste(I,[(p-w,0),(0,p-w)][q==1]);q*=-1;x=0;exec'o.paste(I.crop([(0,x,w,x+p*2),(x,0,x+p*2,w)][s%2]),[(q*p,x),(x,q*p)][s%2]);q*=-1;x+=p*2;'*b;p/=2;I=o;s+=1;b*=2
o.show()

# Mathematica, 177 bytes

It is slow and not fully golfed.

r=ImageRotate;r[#2,Pi/2(3-#)]&@@Nest[{#+1,ImageAssemble@MapIndexed[RotateLeft[#,(-1)^#2]&]@ImagePartition[r@#2,Reverse@d/2^{#,#-1}]}&@@#&,{3,#},Log2@Min[d=ImageDimensions@#]-2]&

This is Lena:

This is Lena the Dragon:

## Haskell + hip, 256 bytes

import Graphics.Image
(d?s)w@(x,y)|k<-div s$2^d,k>0,(y,x)<-(d+1)?s$(y,x)=(mod(x+(-1)^(mod(div y k)2+d)*div k 2)s,y)|1>0=w
q::Image VS YCbCr Word8->IO()
q i=writeImageExact JPG[]"o"\$backpermute(dims i)(2?rows i)i

I lose a lot of bytes in this answer to weirdness going on in the hip library. Namely I have to use the Exact functions for some reason I can only imagine is a bug with the non-exact counter parts, and I have to add a type signature for q. I'd also prefer to display the image instead of writing but there is an issue with the native display being non-blocking and when I make it blocking the image is output in negative for some reason. Maybe fiddling with the input format would make this shorter, currently it takes a jpg image stored in a file called i and outputs to one called o.

# APL (dzaima/APL), 109 107 bytes

f←{x←⊃P5.size←⍵.sz⋄s←4⋄P5.setup←{0 0P5.G.img P5.img{r←x÷s×2÷x⍴(x÷s)⌿1¯1⋄2|2⍟s+←s:←⍵⌽⍨-r⋄⍉r⌽⍉⍵}⍣{s=x}⍵.mat}}

Try it online!

A dfn which takes an APLImg object as an argument, and displays the dragonized image.

Here are some cool dragons I found:

Image courtesy: ThisGuy and GuillaumeDeGermain on unsplash.

• (x÷s×2)×x÷s×2÷


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# Amullya

Web Design and Development. Non-Euclidean Geometry. In three dimensions, there are three classes of constant curvature geometries.

All are based on the first four of Euclid's postulates, but each uses its own version of the parallel postulate. The "flat" geometry of everyday intuition is called Euclidean geometry (or parabolic geometry), and the non-Euclidean geometries are called hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry (or Riemannian geometry). Spherical geometry is a non-Euclidean two-dimensional geometry. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean geometry.

Www.math.ucla.edu/~tao/ Dyadic product. Dyadic notation was first established by Josiah Willard Gibbs in 1884.

In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars. Definitions and terminology Arithmetization of analysis. The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century.

Kronecker originally introduced the term arithmetization of analysis, by which he meant its constructivization in the context of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify the set-theoretic construction of the real line. Its main proponent was Weierstrass, who argued the geometric foundations of calculus were not solid enough for rigorous work. Grigori Perelman. Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман, IPA: [ɡrʲɪˈɡorʲɪj ˈjɑkəvlʲɪvʲɪtʃ pʲɪrʲɪlʲˈman] ( )/ˈpɛrɨlmən/ PERR-il-mən[dubious ]; Russian: Григо́рий Я́ковлевич Перельма́н; born 13 June 1966) is a Russian mathematician who made landmark contributions to Riemannian geometry and geometric topology before his presumed withdrawal from mathematics.

In 1994, Perelman proved the soul conjecture. Divergence. In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.

More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Curl (mathematics) The alternative terminology rotor or rotational and alternative notations rot F and ∇ × F are often used (the former especially in many European countries, the latter, using the del operator and the cross product, is more used in other countries) for curl and curl F.

The name "curl" was first suggested by James Clerk Maxwell in 1871.[1] The components of F at position r, normal and tangent to a closed curve C in a plane, enclosing a planar vector areaA = An. The curl of a vector field F, denoted by curl F, or ∇ × F, or rot F, at a point is defined in terms of its projection onto various lines through the point. If. Riemann,zeta,function.

Statistical mechanics: the Riemann zeta function interpreted as a partition function. The Riemann zeta function interpreted as a partition function lattice-related number theory (involving Ising models, percolation, etc.) integer partition problems and physics entropy and number theory number theory and statistical mechanics – general probabilistic number theory.

Analytic continuation. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). Riemann zeta function. , which converges when the real part of s is greater than 1.

More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Hyperreal number. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.

The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form Such a number is infinite, and its reciprocal is infinitesimal. The term "hyper-real" was introduced by Edwin Hewitt in 1948.[1] The Tesseract. Wolfram. Mathematics. Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8] Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements.

Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11] History Evolution Etymology. List of theorems. This is a list of theorems, by Wikipedia page. See also Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields. 0–9

Theorem. Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from the hypotheses. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English.

Math resources. Duality. From Wikipedia, the free encyclopedia Duality may refer to: Mathematics Philosophy, logic, and psychology Science Electrical and mechanical Physics Beauty of Mathematics « Crazy People With Crazy Things. Chronology of Pure and Applied Mathematics. Portal @ MathLinks. MathPages. Category theory « The Unapologetic Mathematician. Comma Categories Another useful example of a category is a comma category. The term comes from the original notation which has since fallen out of favor because, as Saunders MacLane put it, “the comma is already overworked”. We start with three categories. Category Theory Awodey Course. Www.liegroups.org. ATLAS of Finite Group Representations - V3. Www.liegroups.org. Algebraic structure. Algebraic geometry. Algebraic topology.

Cohomology. Abelian Group. Algebraic geometry. Riemannian geometry. Elliptic geometry is also sometimes called "Riemannian geometry". Symplectic geometry. Differential geometry. Non-Euclidean geometry. Special values of L-functions. John Cremona's home page. Number Theory. Algebraic Topology. Homotopy. The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. Formal definition Euler characteristic. Abstract Algebra. Mathematics. ALGEBRA. Monad (category theory) If and. Orbifold construction of the modes of the Poincare dodecahedral space - Jeffrey Weeks. Logarithmic spiral. Harvard extension school courses. Abstract Algebra - Free Harvard Courses.

Algebra. Ring (mathematics) Abstract Algebra - Free Harvard Courses - StumbleUpon. Virtual Training Suite - free Internet tutorials to develop Internet research skills. The Thirty Greatest Mathematicians. Srinivasa Ramanujan. Carl Friedrich Gauss. Quantum field theory. Adjoint functors. Monad (category theory) Subgroup. Symmetrization. ABSTRACT ALGEBRA: OnLine Study Guide, Table of Contents.

Vladimir Voevodsky. Groupoidification. Topology/Homotopy. Topology. Type theory in nLab. Homotopy Type Theory, I. Coherence theorem.


text:
# External libraries

January 2020

Back to index.html.

# External libraries

## Free libraries for Fortran

Fortran is a plain programming language. It serves primitive statements and many, but simple intrinsic functions. Although these resources are enough to perform scientific operations (e.g., eigenvalue decomposition), Fortran does not provide such high-level procedures. Fortran has a long history, and many users have already had private subroutines useful in their work. Also, some companies provide popular commercial products, e.g., Intel Math Kernel Library (MKL), NAG Numerical Library, and IMSL Numerical Libraries. They are optional, and therefore, it is responsible for the user to prepare the external software.

Currently, a open-source project is moving forward to create the “standard library” in Fortran. It is going to have many essential procedures useful in scientific computing. While many professionals join this project, it may take time to finish developing the library.

Fortunately, besides the “standard library”, the past scientists and engineers have developed and released frequently-used subroutines available online under an explicit license at no cost. Some are old but still useful. In this chapter, we look at some external libraries and pack it as a custom module for future use.

## BLAS and LAPACK

BLAS (Basic Linear Algebra Subprograms) is a collection of Fortran subroutines for basic matrix/vector operations, mainly dedicated to matrix multiplications. LAPACK (Linear Algebra Package) is a library to perform well-known matrix operations such as Cholesky factorization, inversion, solution of linear equations, eigenvalue problems, and so on. LAPACK depends on BLAS. Some companies and research groups have optimized BLAS to enable parallel computations. The optimized BLAS accelerates the LAPACK subroutines, and therefore, you should use it in your serious studies. In this section, we are going to use a reference implementation i.e., non-optimized Fortran library. BLAS and LAPACK are available under the BSD license, an open-source license.

Here is an instruction to build the library.

1. Download the latest version of the package from the official website. The package contains both BLAS and LAPACK.
2. Extract the package and move into the directory.
3. Rename make.inc.example to make.inc.
• If you are using GFortran, do not touch this file anymore.
• If you are using the other compiler (e.g., ifort), open this file, and rewrite the macros: FC, FFLAGS, and FFLAGS_NOOPT.
4. Type make blaslib to compile the BLAS library.
5. Type BUILD_DEPRECATED=yes make lapacklib to compile the LAPACK library.
• Without BUILD_DEPRECATED, you should fail in the compilation of LAPACK95.
• If you cannot use it in the terminal, modify make.inc to activate this variable.
6. The compiled libraries, librefblas.a and liblapack.a, are available in the current directory. Copy these files to your directory.

If you want to test the libraries, type make blas_testing and make lapack_testing. The tests may need a long time to finish.

For BLAS, a test program is available. With GFortran, the following command compiles the file.

gfortran test_blas.f90 librefblas.a

A test program for LAPACK is available. It needs both BLAS and LAPACK (LAPACK comes first in the command line).

gfortran test_lapack.f90 liblapack.a librefblas.a

## LAPACK95

BLAS and LAPACK have many external subroutines. The style of the procedure is based on FORTRAN 77, and it is not easy to use. LAPACK95 provides some modules with the interface to selected subroutines. Unfortunately, this program is very old, and you would have some issues in the compilation. The license statement is found in the manual:

Since LAPACK95 is freely available, it can be included in commercial packages. It is requested only that proper credit be given to the authors by citing this Users’ Guide as the official reference for LAPACK95. Like all software, this package is copyrighted. It is not trademarked; however, if modifications are made that affect the interface, functionality, or accuracy of the resulting software, the name of the routine should be changed. Any modification to the software should be noted in the modifier’s documentation.

Here is the instruction for building the library.

1. Download the package from the official website.
2. Extract the package and move into the directory.
3. Rewrite make.inc.
• For GFortran, put FC=gfortran, FC1=gfortran and OPTS0=-O3.
• For Intel Fortran, use FC=ifort, FC1=ifort and OPTS0=-O3.
4. Move into SRC, and type make single_double.
5. Come back to the original directory with cd .. There is the library, lapack95.a. The mod files are in lapack95_modules.

LAPACK95 has modules so you need .mod files in addition to lapack95.a, liblapack.a, and librefblas.a. A test program for LAPACK is available. It needs both BLAS and LAPACK (LAPACK comes first in the command line).

gfortran test_lapack95.f90 lapack95.a liblapack.a librefblas.a

## RANLIB

RANLIB is a library for random number generators written in FORTRAN 77. The original package is available at Netlib. I restructured this package to have a module. The license is avauilable from the official documentation.

The instruction is as follows.

1. Download the module, ranlib77.f.
2. Compile it to create an object file with -c.
3. Convert the object file to a library file: ar cr ranlib77.a ranlib77.o.

A test program for RANLIB is available. You need ranlib77.a and ranlib77.mod to compile the test program.

gfortran test_ranlib77.f90 ranlib77.a

## CDFLIB

RANLIB is a library for various CDF (cumulative distribution functions) written in FORTRAN 77. The original package is available at Netlib. I restructured this package to have a module. The license is avauilable from the official documentation.

The instruction is the same as RANLIB.

1. Download the module, cdflib77.f.
2. Compile it to create an object file with -c.
3. Convert the object file to a library file: ar cr cdflib77.a cdflib77.o.

A test program for CDFLIB is available. You need cdflib77.a and cdflib77.mod to compile the test program.

gfortran test_cdflib77.f90 cdflib77.a

## Other libraries

There are more libraries available online. I show the pointers, and please explore the websites with the care of the license. Note that some programs are very old (written in FORTRAN 77 around the 1970s). It still compiles with the current Fortran compiler (with possible warnings).

Back to index.html.


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# Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism (history)

 Tom W B Kibble (2009), Scholarpedia, 4(1):8741. doi:10.4249/scholarpedia.8741 revision #137393 [link to/cite this article]
Post-publication activity

Curator: Tom W B Kibble

Figure 1: The sombrero potential with an unstable state at $$\phi=0$$ and minima around a circle.

This page summarizes the history of the development of the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism. Many of the important papers have been reprinted in a volume edited by C.H. Lai (Lai 1981), and/or in the review volume by John C. Taylor (Taylor 2001).

## Symmetry breaking in superconductivity

Figure 2: A Cooper pair formed of electrons with opposite momentum and spin.

The idea of a connection between mass generation and spontaneous symmetry breaking dates back to the work of Nambu (1960). This and other early work was largely based on analogies with the theory of superconductivity.

The essential physical basis of the BCS or Bardeen-Cooper-Schrieffer theory of superconductivity (Bardeen 1957), is the existence of an effective attractive force between electrons arising from their interaction with phonons in the lattice. This leads to the formation of bound pairs of electrons, Cooper pairs, in a spin-singlet state (see Figure 2). These Cooper pairs then undergo a Bose-Einstein condensation; in other words there is a macroscopic occupation of a single quantum state of a pair.

### The Ginzburg-Landau model

A very convenient phenomenological model of a superconductor is provided by the Ginzburg-Landau model (Ginzburg 1950). This predates the BCS theory, but was later derived from it (in the London regime close to the critical temperature) by Gor'kov (1959).

The Ginzburg-Landau model involves a scalar order parameter field, $$\phi(t,\mathbf{x})\ .$$ This field represents the wave function of the condensate of Cooper pairs; $$|\phi|^2$$ is the number density of pairs in the condensate. Here $$\phi$$ may be regarded as a composite of two electron fields, and so has charge $$2e\ .$$ The Hamiltonian (in units with $$\hbar=1$$) is $\tag{1} H=\int d^3\mathbf{x} \left[\frac{1}{2m}\mathbf{D}\phi^*\cdot\mathbf{D}\phi+V(\phi)\right],$

where $\mathbf{D}\phi=\boldsymbol{\nabla}\phi-2ie\mathbf{A}\phi,$ is the covariant derivative corresponding to the gauge transformations $\tag{2} \phi\to\phi e^{2ie\lambda},\qquad \mathbf{A}\to\mathbf{A}+\boldsymbol{\nabla}\lambda,$

and the potential $$V(\phi)$$ is a function only of $$\phi^*\phi\ .$$ Near the critical temperature $$T_{\mathrm{c}}$$ the field $$\phi$$ is small, and $$V$$ may be expanded in a power series, keeping only quadratic and quartic terms: $\tag{3} V(\phi)=\alpha\phi^*\phi+\tfrac{1}{2}\beta(\phi^*\phi)^2.$

The coefficients $$\alpha$$ and $$\beta$$ are temperature-dependent, and in particular $$\alpha$$ changes sign at the critical temperature. For $$T<T_{\mathrm{c}}\ ,$$ it is negative, so $$V$$ has the 'sombrero' shape (see Figure 1) and its minima occur not at the symmetry point $$\phi=0$$ but around the circle $\tag{4} \phi^*\phi=\frac{-\alpha}{\beta}.$

Consequently, the gauge symmetry (2) is spontaneously broken. The magnitude of the order parameter $$|\phi|$$ is fixed by (4) but its phase is arbitrary. There is a family of degenerate ground states, labelled by this phase. Note that since the phase is canonically conjugate to the occupation number, a definite phase requires an indefinite number of pairs in the condensate.

Following the development of this theory, there was much discussion in the literature of the role of gauge invariance. The Ginzburg-Landau ground state does not respect this gauge symmetry, and nor does the ground state of the full Bardeen-Cooper-Schrieffer model. Today this is recognized as an example of the phenomenon of spontaneous symmetry breaking, but at the time it was seen by many as a defect of the theory.

Nambu (1960) gave a particularly clear account of the situation, using a modified form of the Hartree-Fock approximation. (See also Bogoliubov (1959).) He showed how the symmetry breaking is intimately connected with the energy gap. So long as the force between electrons is attractive, dissociating a Cooper pair requires an energy input. Thus the excitations of the system, quasiparticle states, are separated from the ground state by a finite energy gap; the energy $$E_{\mathbf{p}}$$ of a quasiparticle of momentum $$\mathbf{p}$$ tends to a non-zero limit as $$\mathbf{p}\to\mathbf{0}\ .$$ Closely related is the fact that the quasiparticles do not have a definite charge (Bogoliubov (1959)): a quasiparticle is a linear combination of an electron of momentum $$\mathbf{p}$$ and spin up, say, and a hole in the state of momentum $$-\mathbf{p}$$ and spin down.

### Analogous mechanism in particle physics

Nambu then went on to suggest that the masses of elementary particles might arise in a similar way; the vacuum state, like the superconducting ground state, might not respect the symmetries of the theory, and so elementary particles, like the quasiparticles in BCS theory, might acquire masses.

In particular, Nambu envisaged a theory of strong interactions involving a fundamental massless fermion field $$\psi(x)$$ with a Lagrangian invariant under both ordinary and chiral global phase changes: $\tag{5} \psi(x)\to e^{i\alpha}\psi(x),\qquad\text{and}\qquad \psi(x)\to e^{\alpha\gamma_5}\psi(x)$

(in a representation in which $$\gamma_5^2=-1$$). Correspondingly there would be both vector and axial vector conserved Noether currents, $\tag{6} j^\mu=\bar\psi\gamma^\mu\psi,\qquad j^\mu_5=\bar\psi i\gamma^\mu\gamma_5\psi.$

He suggested that, like the superconducting energy gap, the nucleon mass arises from spontaneous breaking of the chiral symmetry.

Together, Nambu and Jona-Lasinio (1961) constructed a specific model with these characteristics, with four-fermion interactions, based on the Lagrangian density $\tag{7} L=i\bar\psi\gamma^\mu\partial_\mu\psi +g[(\bar\psi\psi)^2-(\bar\psi\gamma_5\psi)^2].$

They then assumed that in the ground state or vacuum, the chiral symmetry is broken spontaneously by a non-zero expectation value $$\langle0|\bar\psi(x)\psi(x)|0\rangle=n\ ,$$ say. Using techniques similar to those he employed to discuss superconductivity, they showed that this would indeed imply a nonzero mass for the 'quasiparticle', here identified as the nucleon.

### Nambu-Goldstone bosons

Like the model of Goldstone (1961) (see main page), the Nambu--Jona-Lasinio model also predicts the existence of a massless particle of spin zero, the Goldstone boson or Nambu-Goldstone boson, a nucleon-antinucleon bound state. This is a consequence of the broken chiral symmetry, and the particle in this case is a pseudoscalar. Since subjecting all the particles to a chiral rotation merely takes us from one degenerate vacuum state to another, and so requires no energy, it follows that applying a spatially varying chiral rotation with long wavelength requires very little energy; the energy goes to zero in the long-wavelength limit, which means the particle is massless.

It was clear that these were examples of a general phenomenon, described by what has come to be known as the Goldstone theorem. In the context of an explicitly relativistic theory, a general proof was provided by Goldstone, Salam & Weinberg (1962).

Nambu and Jona-Lasinio realized that as a consequence of the broken symmetry, their explanation for the nucleon mass would necessarily imply the existence of a massless particle of zero spin. Of course, no such particle was (or is) known, but they suggested that in fact the chiral symmetry is not quite exact but intrinsically weakly broken -- in addition to the larger spontaneous symmetry breaking effect. Then the would-be Nambu-Goldstone bosons would acquire small masses, and could be identified with the pions. (They pointed out that it would be easy to accommodate isospin within the model.) Although the Nambu--Jona-Lasinio model has been superseded, this identification is in fact in line with current theory. The nucleon is no longer regarded as a quasiparticle associated with a fundamental field; it is a composite of quarks. But it remains true that if the quarks were massless, then quantum chromodynamics would exhibit chiral symmetry, and the pions would be massless. The weak chiral symmetry breaking leads to a small pion mass.

Nambu and Jona-Lasinio also noted that in the case of a superconductor the particle-hole bound state that would play the role of the Nambu-Goldstone boson is not massless because of the existence of the long-range Coulomb forces, but rather becomes part of the plasmon oscillations.

## Gauge theories of strong and weak interactions

Figure 3: (a) Fermi's view of neutron decay; (b) neutron decay proceeding via an intermediate vector boson; (c) electromagnetic interaction between proton and electron.

The first non-Abelian gauge theory was proposed by Yang and Mills (1954) (and independently by Shaw (1955), though his work appears only in a Cambridge University PhD thesis). This was based on the idea of promoting the global SU(2) isospin symmetry to a local gauge symmetry, which requires the introduction of an isospin triplet of gauge fields. This was intended as a theory of strong interactions, and as such has of course been superseded by quantum chromodynamics. But it was soon realized that a similar idea might work for the weak interactions.

The original Fermi theory of weak interactions involved a direct four-fermion interaction (see Figure 3(a)). Following the proposal by Lee & Yang (1956) of parity non-conservation in weak interactions, confirmed the following year (Wu 1957), it was established that the weak interactions are of the $$V-A$$ form, involving interactions between vector and axial-vector currents. This led Feynman & Gell-Mann (1958) and independently Sudarshan & Marshak (1958) to suggest that they might be mediated by charged intermediate vector bosons, $$W^\pm$$ ( Figure 3(b)), raising the interesting possibility of a gauge theory of weak interactions. Indeed because of the similarity to electromagnetic interactions ( Figure 3(c)) it suggested the even more interesting possibility of a unified theory of weak and electromagnetic interactions, in which $$(W^+,\gamma,W^-)$$ would form a triplet of gauge bosons.

However, this hypothesis was faced with two immediate and severe obstacles. Firstly, to explain the short range and weakness at low energies of the weak interactions, the $$W^\pm$$ had to have very large masses, whereas it was generally believed that gauge bosons were necessarily massless, like the photon.

The second problem arose from the parity violation in weak interactions. The $$W^\pm$$ interacted not with a vector current but with a chiral current, a sum of terms of the form $$\bar\psi_1\gamma^\mu(1-i\gamma_5)\psi_2\ .$$ So how could they be part of the same multiplet as the photon, with its parity-conserving interactions? The solution to this second problem was eventually found to require the enlargement of the gauge group from $$SU(2)$$ to $$SU(2)\times U(1)\ ,$$ with the introduction of another neutral gauge boson, $$Z^0\ ,$$ as first suggested by Glashow (1961). This development, however, was peripheral to the subject of the present article, and will not be considered further here.

Both these problems clearly required some mechanism for breaking the symmetry between the $$W^\pm$$ and the photon. It was natural to ask whether this, like superconductivity, could be an example of spontaneous symmetry breaking. But that idea raised a new problem, the possible appearance of massless scalar particles. The Goldstone theorem seemed to suggest that a spontaneously broken gauge theory would be plagued by two different kinds of massless particles, the gauge vector bosons themselves and the scalar Nambu-Goldstone bosons.

### Masses of gauge bosons

In the early days it was commonly believed that one of the virtues of the gauge principle -- the idea of promoting global to local symmetries, with the introduction of gauge vector particles -- was the successful prediction of the vanishing photon mass.

Figure 4: The lowest-order photon self-energy diagram.

The first person to question this orthodoxy was Schwinger (1962). One form of the argument for a massless photon was that by virtue of gauge invariance, the self-energy of the photon (see Figure 4) would necessarily have the form $\Pi_{\mu\nu}(p)=(g_{\mu\nu}p^2-p_\mu p_\nu)\Pi(p^2),$ which seemed to suggest that it must vanish at $$p^2=0\ ,$$ implying zero mass. Schwinger argued convincingly however that if the interactions were strong enough, then $$\Pi(p^2)$$ might acquire a pole at $$p^2=0\ ,$$ thus eliminating the prediction. A photon with stronger interactions could be massive.

Then Anderson (1963) pointed out that there are examples of this in condensed-matter physics. In an electron plasma, electromagnetic waves cannot propagate if their frequency is less than the plasma frequency $$\omega_{\mathrm{pl}}\ ,$$ given by $\omega_{\mathrm{pl}}^2 = \frac{e^2 n_e}{\epsilon_0 m_e},$ where $$n_e$$ is the electron number density and $$m_e$$ the electron mass. In fact, the dispersion relation for this plasmon is for small $$\mathbf{k}$$ of the form, $\omega^2 =\omega_{\mathrm{pl}}^2+u^2\mathbf{k}^2,$ so in effect the photon has acquired a mass $$m_{\mathrm{pl}}=\hbar\omega_{\mathrm{pl}}/c^2\ .$$ The same thing applies in a superconductor.

This work however was framed in language that was not then familiar to most particle physicists, and referred to models without relativistic invariance; the plasma or superconductor defines a preferred frame of reference. But, because of the work of Goldstone, Salam & Weinberg, it was widely believed among field theorists that in a fully relativistic theory the Goldstone theorem would unambiguously demand massless particles. (See, for example, Gilbert (1964).) Although Anderson had shown that in the nonrelativistic context the massless gauge bosons and Nambu-Goldstone bosons could combine to form a massive vector particle (a result already foreshadowed by Nambu and Jona-Lasinio (1961)), most particle theorists believed that could not happen in a relativistic theory.

The final step of showing that indeed the mechanism could work in a relativistic theory was taken independently by three groups: Englert & Brout, Higgs, and Guralnik, Hagen & Kibble, approaching the problem from three rather different perspectives. The simplest and most direct argument was that of Higgs (1964b), where he exhibited a very simple U(1) model, now often called the Abelian Higgs model (see the article Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism). Prior to that (Higgs 1964a) he had made the important observation that in a gauge theory formulated in the radiation gauge, the choice of gauge removes the explicit relativistic invariance, and thus renders the Goldstone theorem inapplicable. Higgs's treatment of the model was purely classical, but essentially the same model was considered in a quantum mechanical way by both the other groups. Englert & Brout (1964), whose paper was the first to be published, based their argument on a calculation of the vacuum polarization in lowest-order perturbation theory about the assumed symmetry-breaking vacuum state. They also pointed out that the same mechanism could operate in more general non-Abelian models, and gave simple examples. Guralnik, Hagen & Kibble (1964), on the other hand, used an operator formalism, concentrating on the role of the conservation law and the precise way in which the Goldstone theorem can be evaded. (For more detail of the history leading up to this paper, see Guralnik 2011.) However, the conclusions of all three were essentially the same.

### Later developments

There were several significant developments in the years immediately after 1964. Higgs (1966) studied his model quantum mechanically in some detail, evaluating transition and decay amplitudes for the model in lowest-order perturbation theory, and in particular discussing the induced symmetry-breaking effects that would occur if the scalar field were coupled to other fields. Streater (1965) discussed the mechanism from the point of view of axiomatic field theory. The details of the application to a non-Abelian gauge theory were studied by Kibble (1967), showing how the numbers of massive and massless states were related to the symmetry-breaking pattern. For a comprehensive review of symmetry breaking in both field theory and condensed matter, see Guralnik et al (1967).

Initially, the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism was envisaged as having as much, or more, relevance to the development of a gauge theory of strong as of weak interactions. But of course its most significant application was to the development of the unified electroweak theory by Weinberg (1967) and Salam (1968), incorporating the model developed earlier by Glashow (1961).

Both Weinberg and Salam had conjectured that the theory was renormalizable, but with no proof. The final, very important element was the proof of renormalizability of spontaneously broken gauge theories by 't Hooft (1971).

Historically, the primary focus of the mechanism was on how the gauge bosons acquire non-zero mass. The fact that there are also massive Higgs bosons was incidental. However, their existence has now become of great significance, because the Higgs boson is at present (2009) the only component of the standard model whose existence has not been experimentally confirmed1. Finding them (or some alternative) will be a key advance.

1 On 4 July 2012 a previously unknown boson of mass $\sim$ 125 GeV was confirmed to exist by the ATLAS and CMS teams at the Large Hadron Collider at CERN. This particle has been tentatively confirmed as a Higgs boson by CERN on 14 March 2013, although it remains an open question whether this is the Higgs boson of the Standard Model of particle physics, or possibly the lightest of several bosons predicted in some theories that go beyond the Standard Model.

## References

• Bogoliubov, N N (1958). A new method in the theory of superconductivity I. Sov. Phys.-JETP 7: 41-46. [J. Exper. Theor. Phys. (USSR) 34 (1958) 58-65]
• Feynman, R P and Gell-Mann, M (1958). Theory of the Fermi interaction. Phys. Rev. 109: 193-8. [] DOI: 10.1103/PhysRev.109.193
• Ginzburg, V L and Landau, L D (1950). On the theory of superconductivity. J. Exper. Theor. Phys. (USSR) 20: 1064.
• Glashow, S L (1961). Partial symmetries of weak interactions. Nuc. Phys. 22: 579-88.
• Goldstone, J (1961). Field theories with superconductor solutions. Nuovo Cim. 19: 154-164.
• Gor'kov, L P (1959). Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity. Sov. Phys.-JETP 36: 1364-7. [J. Exper. Theor. Phys. (USSR) 36 (1959) 1918-23]
• Guralnik, G S; Hagen, C R and Kibble, T W B (1964). Global conservation laws and massless particles. Phys. Rev. Lett. 13: 585-7. [ http://link.aps.org/abstract/PRL/v13/p585] DOI: 10.1103/PhysRevLett.13.585
• Guralnik, G S; Hagen, C R and Kibble, T W B (1967). Broken symmetries and the Goldstone theorem. Advances in Physics, vol. 2 Interscience Publishers, New York. pp. 567-708 ISBN 0470170573
• Higgs, P W (1964a). Broken symmetries, massless particles and gauge fields. Phys. Lett. 12: 132-3.
• Lai, C H (1981). Gauge theory of weak and electromagnetic interactions. World Scientific, Singapore. ISBN 9971-83-023-x.
• Nambu, Y and Jona-Lasinio, G (1961). Dynamical model of elementary particles based on an analogy with superconductivity. I. Phys. Rev. 122: 345-58. [1] DOI: 10.1103/PhysRev.122.345
• Salam, A (1968). Weak and electromagnetic interactions. Elementary particle theory, Proceedings of the Nobel Symposium held in 1968 at Lerum, Sweden. Svartholm, N Almqvist & Wiksell, Stockholm. 367-77
• Shaw, R (1955). Invariance under general isotopic gauge transformations. Cambridge University PhD thesis Cambridge University, Cambridge. Part II, Chapter 3, pp. 34-46
• Streater, R F (1965). Spontaneously broken symmetry in axiomatic theory. Proc. Roy. Soc. Lond. A 287: 510-18. DOI: 10.1098/rspa.1965.0193
• Taylor, J C (2001). Gauge theories in the twentieth century. Imperial College Press, London. ISBN 1-86094-281-4.
• 't Hooft, G (1971). Renormalizable Lagrangians for massive Yang-Mills fields. Nuc. Phys. B 35: 167-88.
• Wu, C S; Ambler, E; Hayward, R W; Hoppes, D D and Hudson, R P (1957). Experimental test of parity conservation in beta decay. Phys. Rev. 105: 1413-15. [ http://link.aps.org/abstract/PR/v105/p1413] DOI: 10.1103/PhysRev.105.1413

Internal references

• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

• Aitchison, I J R and Hey, A J G (2003). Gauge theories in particle physics, vol. II: QCD and the electroweak theory, 3rd ed. Institute of Physics, Bristol. ISBN 0-750-30950-4.
• Weinberg, S (1996). The quantum theory of fields, vol. II: modern applications. Cambridge University Press, Cambridge. Chapter 21. ISBN 0-521-55001-7. This book contains a particularly interesting account of superconductivity from the standpoint of symmetry breaking, showing that many of the key features of superconductors can be derived just from the assumption of spontaneous symmetry breaking.
• Zee, A (2003). Quantum field theory in a nutshell. Princeton University Press, Princeton, NJ. ISBN 0-691-01019-6.


text:
Revision history [back]

Length of preperiod and period of a rational number

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I shoul get somethink like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print float(a)
print 'a.period() gives me the length of the period:', a.period()
print 'a.preperiod() does not work', #print a.preperiod()

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work

Length of preperiod and period of a rational number

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I shoul get somethink like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print float(a)
print 'a.period() gives me the length of the period:', a.period()
print 'a.preperiod() does not work', #print a.preperiod()

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work

Length of preperiod and period of a rational number

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I shoul get somethink like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print float(a)
print 'a.period() gives me the length of the period:', a.period()
print 'a.preperiod() does not work', #print a.preperiod()

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work

Length Lengths of preperiod and period of a rational number

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I shoul get somethink something like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print float(a)
print 'a.period() gives me the length of the period:', a.period()
print 'a.preperiod() does not work', #print a.preperiod()

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work

Lengths of preperiod and period of a rational number

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I shoul get something like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print float(a)
print 'a.period() gives me the length of the period:', a.period()
print 'a.preperiod() does not work', #print a.preperiod()
#a.preperiod()

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work

Lengths of preperiod and period of a rational number

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I shoul shoud get something like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print float(a)
print 'a.period() gives me the length of the period:', a.period()
print 'a.preperiod() does not work', #a.preperiod()

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work

Lengths of preperiod and period of a rational number

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I shoud should get something like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print float(a)
print 'a.period() gives me the length of the period:', a.period()
print 'a.preperiod() does not work', #a.preperiod()

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work

 8 None FrédéricC 4598 ●3 ●38 ●96

Lengths of preperiod and period of a rational number

I found something for Continued fractions, but I ask myself:

I there an easy way to get the lengths of preperiod and period of a rational number?

It seems for Rational numbers I can only use a.period(), not a.preperiod().

For, say, 1/7 = 0.142857142857... I should get something like:
Length of preperiod: 0
Length of period: 6

a = 1/7
print float(a)
print 'a.period() print(float(a))
print('a.period() gives me the length of the period:', a.period()
print 'a.preperiod() a.period())
print('a.preperiod() does not work', #a.preperiod()
#a.preperiod())

Out:

0.142857142857
a.period() gives me the length of the period: 6
a.preperiod() does not work


text:
## Measure of sums

Consider a set X, define $f:X \rightarrow [0, \infty ]$ by $f(x)=1 \ \ \ \ \forall x \in X$.

Let $E \subset \mathbb {P}(X)$, the set of all subsets in X.

Define $\mu (E)= \sum _{x \in E } f(x) = sup \{ \sum _{x \in F } f(x) : F \subset E, F \ finite \ \}$

Prove that $\mu$ is a measure.

Proof.

Let $E_1,E_2,... \subset X$, with all $E_n$ disjoint.

I claim that $\mu ( \bigcup ^{ \infty }_{n=1}E_n)= \sum ^{ \infty }_{n=1} \mu (E_n)$

Now suppose that $F \subset \bigcup ^{ \infty } _{n=1} E_n$ is the largest finite subset.

Then $F = F_1 \bigcup F_2 \bigcup F_3 \bigcup . . . = \bigcup _{n=1}^{ \infty }$ for $F_n \subset E_n$ the biggest subset. Note that $F_n$ are disjoint since each $E_n$ is disjoint.

Then we have $|F|=|F_1|+|F_2|+...$

It follows that $\mu ( \bigcup ^ \infty _{n=1} E_n ) = sup \{ \sum _{x \in F } f(x) : F \subset \bigcup _{n=1}^ \infty En, F \ finite \ \}$ $= \sum _{x \in F } f(x)$

$=|F|=|F_1|+|F_2|+...= \sum _{x \in F_1 } f(x) + \sum _{x \in F_2 } f(x) + . . . =$ $\mu (E_1)+ \mu (E_2) + . . . = \sum ^ \infty _{n=1} \mu (E_n)$

Therefore $\mu$ is a measure.

Is this okay? I know that idea should be right, but it is just the part about choosing the largest finite subset could be a bit verge.

Thank you!


text:
• English only

# Differences

This shows you the differences between two versions of the page.

 sav08:recursive_set [2008/03/18 18:03]vkuncak created sav08:recursive_set [2008/03/18 18:04] (current)vkuncak 2008/03/18 18:04 vkuncak 2008/03/18 18:03 vkuncak created 2008/03/18 18:04 vkuncak 2008/03/18 18:03 vkuncak created Line 1: Line 1: ====== Recursive Set ====== ====== Recursive Set ====== - We say that a set $S$ is //recursive// if there exists an algorithm that, given an element $x$ determines whether $x \in S$ holds. + We say that a set $S$ is //recursive// (decidable) if there exists an algorithm that, given an element $x$ determines whether $x \in S$. Example: the set of even numbers is recursive.  The algorithm takes $x$ represented in binary and checks whether its least signifficant digit is 0. Example: the set of even numbers is recursive.  The algorithm takes $x$ represented in binary and checks whether its least signifficant digit is 0. Line 7: Line 7: Example: the set of all pairs $(T,x)$ such that $T$ is a representation of Turing machine and $x$ is an input such that $T$ terminates on input $x$, is not recursive. Example: the set of all pairs $(T,x)$ such that $T$ is a representation of Turing machine and $x$ is an input such that $T$ terminates on input $x$, is not recursive. + Example: the set of encodings of all valid first-order logic formulas is not recursive. + + Example: the set of all valid formulas of Presburger arithmetic is recursive.


text:
Escaping earth's atmosphere with a balloon

Tags:
1. Dec 4, 2014

Dead Fish

Hi all,

So my question is, could you use a weather balloon to assist with escaping the atmosphere and attaching a rocket to take it the rest of the way, and we're talking a really small payload obviously.

Would the weight of the fuel/rocket weigh down the balloon enough to significantly reduce its impact?

Or simply am I (literally) miles off and the altitude a weather balloon reaches will help in no way?
On average a weather balloon can surpass the boundary of space (100km), so it's done most of the work?

Thanks,
Nick

2. Dec 4, 2014

willem2

The bulk of the work is not gaining the altitude, but getting enough speed to get in an orbit around the earth, and not fall back immediately.
If we can believe wikipedia, the altitude record for unmanned balloons is 53 km. Getting 1 kg to 53 km takes mgh = 1 * 9.8 * 53000 = 5.19 * 10^5 J.

Minimum Orbital speed is 7.5 km/s = 75000 m/s. The kinetic energy needed is (1/2)mv^2 = (1/2) * 1 * (7500)^2 = 2.81 * 10^7 J. If you start on the equator and use the 450 m/s rotational speed of the earth, you still need (1/2) * (7050)^2 = 2.48 * 10^7. Still about 50 times more energy needed than the boost you can get from getting to 53 km.

3. Dec 5, 2014

Dead Fish

I'm not sure I understand what you're comparing here?

It would make sense that the problem you have now is you have to reach the speeds required to go from the 53 km to outer space?
And you're suggesting that this is still a significant amount of the work left to do?

Wouldn't the thinner atmosphere and weaker gravity make it far easier to reach those speeds?

4. Dec 5, 2014

Bandersnatch

Hi Dead Fish,

willem2 is refering to the speed required to get in orbit, which is usually what you want to achieve with spacecraft.
While this speed is dependent on the strength of gravity and gravity falls with the square of distance, the difference between 6378km(Earth radius) and 6431km (radius + 53km) is minuscle, and the speed required doesn't differ much.

However, the atmospheric drag is significant.
If not for the atmosphere, rockets could be in principle launched into orbits barely high enough to clear any mountaintops in their way. With atmosphere, it is more fuel-efficient to go nearly straight up until the air gets thinner, before starting to accelerate sideways.

Have a look at this infographic for the Zenit3SL rocket launch profile:
http://www.spaceflight101.com/uploads/6/4/0/6/6406961/2349991_orig.gif?537 [Broken]
The record height a ballon could get you (53km as of today), could potentially allow you to reduce the first stage by a good bit, as most of it is used just to propel the rest of the rocket above the atmosphere.

However, what's left is still a humongous weight in fuel and metal, and lifting it up with a balloon to any height, let alone the record 53km is beyond daunting.
Buoyancy equation states that $F_B=(\rho _{air} - \rho _{gas})*g*V$, so at sea level you get enough lift from 1 cubic metre of helium to move 1 kg of payload. This goes down quickly as air density decreases. At 6km you'd need twice the amount to stay afloat.
The size of a balloon needed just to lift a couple hundred tonnes of payload off the ground is about the same as Hindenburg, and this is not taking into account the weight of the balloon itself.

Last edited by a moderator: May 7, 2017
5. Dec 5, 2014

Staff: Mentor

You don't gain speed because you need speed to get into outer space, you gain speed because you want to be in orbit once you reach space. So gaining altitude has no impact on that requirement; they are completely separate things.

6. Dec 5, 2014

zoki85

What is Q [kgf/m2] here ?

Last edited by a moderator: May 7, 2017
7. Dec 5, 2014

8. Dec 5, 2014

zoki85

I was wondering what kind of unit is "kgf". Now I see it means "kilogram force", lol.

9. Dec 6, 2014

DrStupid

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook


text:
# Is Magnus effect a corollary of Bernoulli principle?

Magnus effect is commonly explained using Bernoulli principle. However, taking the lift on a rotating cylinder as an example, the velocity difference is caused by the extra work done by the rotating cylinder but not by the pressure difference, the Bernoulli principle is basically energy conservation along a streamline. However here the energy is not conserved due to external work done. So is Bernoulli principle abused in explaining Magnus effect?

• It's not energy conservation along a streamline. It's momentum conservation along a streamline. It's $F=ma$, which is momentum conservation. – Mike Dunlavey Mar 26 '15 at 16:26

This is a most excellent and astute question. Ultimately it comes down to experiment: the model below works pretty well for many fluids. What this must mean therefore is that the loss is small enough that each particle of fluid, in flowing past the region of disturbance, loses a fraction of its energy that is small enough that it doesn't upset the energy balance underlying the Bernoulli equation greatly. At the same time the viscosity is big enough that the cylinder can sustain the circulation in the flow.

Once one has established a circulation in a flow, the circulation will linger - sometimes almost indefinitely, with very little further input of energy. The Vorticity Equation shows this. You can therefore think of a situation where the cylinder just "happens" to be sitting in a flow with circulation without worrying about how that circulation arose. One can then come up with a lossless mathematical model that does indeed show the Magnus effect, which, judging from the depth and astuteness in your question, you may already ken. In this model, the flow has a certain assumed circulation and one does not think about how this circulation came about. I am talking about the an inviscid, irrotational flow whose complex velocity potential is:

$$\Omega : \{z\in\mathbb{C}|\;|z| > a\}\to \mathbb{C};\;\Omega(z) = a\, v^*\left(\frac{z}{a} + \frac{a}{z}\right) + \frac{\Gamma}{2\,\pi\,i}\,\log z\tag{1}$$

where $v\in\mathbb{C}$ is the fluid's velocity a long way from the cylinder (i.e. the cylinder is steeped in an initially uniform flow) and $\Gamma\in\mathbb{R}$ is the circulation. The cylinder's cross section is the region $\{z\in\mathbb{C}|\;|z| \leq a\}$.

This is a perfectly steady state flow, and, once the circulation is set up, it sustains itself. There is no loss anywhere in the model, so you apply the Theorem of Blasius to calculate the lift, which is simply the quantitative calculation around the cylinder of the wonted Bernoulli's theorem argument.

So you could imagine the following thought experiment (not practical). You have a magic fluid whose viscosity you can switch on and off at will. You have a uniform flow in this fluid and you steep your cylinder in it, and, with internal motors or whatever, you spin the cylinder and the lossy viscosity will set up a circulation in the flow. Then you switch the viscosity off suddenly. The circulation will linger in the flow and now, in the absence of loss, you can do the calculation above and see that there is indeed a lift. Note that one always needs a circulation to generate nonzero lift.

Question by OP

I agree with you that the circulation is the key reason for the lift. But the Bernoulli argument is somehow flawed because the velocity difference does not lead to pressure difference, it is the circulation that leads to pressure difference. Bernoulli principle always assumes no viscosity and vorticity. Do you think Bernoulli Principle is abused even in a heuristic argument?

It is still the Bernoulli argument that describes the origin of the pressure gradients and thus, ultimately, the force. The circulation simply introduces an asymmetry to the flow that then makes the sum of pressures nonzero.

The Blasius Theorem is equivalent to the Bernoulli principle as shown as follows: on a section $\mathrm{d} z$ of a contour around the body in the complex plane, the pressure force by the Bernoulli principle is:

$$-\frac{i\,\rho}{2}\,(|v|^2-|\mathrm{d}_z\Omega|^2) \,\mathrm{d}\,z^*\tag{2}$$

where $v$ is as defined in (1) and $\rho$ the fluid density. Here, as in (1), the vector direction is shown by the phase of the complex number. The contour around the body's edge is a streamline, so that the stream function (imaginary part of the complex potential) is constant along it. Therefore, around the edge of the body, $|\mathrm{d}_z\Omega|^2 = (\mathrm{Re}(\mathrm{d}_z\Omega))^2 = (\mathrm{d}_z\Omega))^2$ so that, on summing (2) around the closed contour to find the nett force we wind up with something near to an ordinary contour integral (on witnessing that $\oint v^2\,\mathrm{d}z=0$):

$$F^* = -\frac{i\,\rho}{2} \oint (\mathrm{d}_z\Omega))^2\,\mathrm{d}\,z = \pi\,\rho\,\sum \text{residues of }(\mathrm{d}_z\Omega))^2\text{ at poles within cylinder}\tag{3}$$

which is readily worked out to be $F^*=-i\,v^*\,\rho\,\Gamma$ so that $F=i\,v\,\rho\,\Gamma$, i.e. at right angles to the flow. So you can see that the result as worked out from the Bernoulli principle tells you that the lift is proportional to the circulation.

• I agree with you that the circulation is the key reason for the lift. But the Bernoulli argument is somehow flawed because the velocity difference does not lead to pressure difference, it is the circulation that leads to pressure difference. Bernoulli principle always assumes no viscosity and vorticity. Do you think Bernoulli Principle is abused even in a heuristic argument? – Shadumu Mar 23 '15 at 14:26
• @user3229471 Please see my additions above. – Selene Routley Mar 24 '15 at 9:47
• @Shadumu Do you agree with my explanation? – enbin zheng Aug 22 '19 at 7:43

As shown in the figure, the rotating water in the bucket will become concave, and the faster the rotation, the more concave the water surface is. This shows that at a certain height H, the closer the water is to the center of rotation, the lower the pressure is. And the faster the water rotates, the lower the pressure of the water in the center of rotation.

As shown in the figure, because the air flow on the right side of the ball is opposite to the rotation direction of the ball, the air flow on the right side of the ball rotates slowly; the air flow on the left side of the ball rotates in the same direction as the rotation direction of the ball, so the air flow on the left side of the ball rotates fast. According to the conclusion about the bucket, the pressure on the left side of the ball must be lower than that on the right side of the ball. So the ball will be exerted a right-to-left force F.

So I don't use Bernoulli's principle to explain Magnus Effect.

• I may be wrong, but doesn't the Bernoulli Principle prove that since the right side has a higher pressure, the object moves to the side with the lower pressure? – Viradeus May 25 at 0:14

I'm not a physicist and I don't have equations to prove what I'm about to say and I could be dead wrong, but I've been thinking of this very question and it led me to this thread. I would argue the Bernoulli principle should act opposite to Magnus effect. The movement of rotation would create the high velocity, low pressure on the opposite side to where the magnus effect exhibits its force. I shoot rifles, and commonly have small cylinders moving away from me in a clockwise motion (right hand twist). In a left to right wind, the Bernoulli effect would have the air accelerating over the top of the projectile causing slight lift. The magnus effect has been proven to actually cause negative lift under these conditions. I think of the magnus effect like a spinning billiard ball. As the ball strikes the bumper with spin, it will bounce off the wall and have some of it's spin converted into movement in the direction of the spin. Maybe I'm way off, but this is how I see it.

• The magnus effect has been proven to actually cause negative lift under these conditions. A citation or link for this claim would be useful for your answer. – Kyle Kanos Jun 8 '15 at 15:58


text:
Potential vorticity

Potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification that, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis (the birth and development of a cyclone), especially along the polar front, and in analyzing flow in the ocean.

In meteorology, the potential vorticity unit (PVU) is defined as ${10^{-6} \cdot K \cdot m^2 \over kg \cdot s} \equiv 1 PVU$.

It is also useful in tracing intrusions of stratospheric air deep into the troposphere in the vicinity of jet streaks, a concentrated region within a jet stream where the wind speeds are the strongest. It acts as a flow tracer in the ocean as well. It can also be used to explain how a range of mountains like the Andes can make the upper westerly winds swerve towards the equator and back.

Baroclinic instability requires the presence of a potential vorticity gradient along which waves amplify during cyclogenesis.

PV conservation

Convergence and divergence of an air parcel

Carl-Gustaf Rossby first introduced potential vorticity in his 1936 paper "Dynamics of steady ocean currents in the light of experimental fluid mechanics". This work was based on an analysis of a shallow water model of the Gulf Stream. The concept was developed further, within a few years of Rossby's seminal paper, by Hans Ertel. Mathematically, Ertel's form of potential vorticity is given by the equation:

${\rm PV} = \frac{1}{\rho}\,\zeta\,^a\cdot\,\nabla\theta$

where $\rho$ is the fluid density, $\zeta^a$ is the absolute vorticity and $\nabla \theta$ is the gradient of the potential temperature. It can be shown through a combination of the first law of thermodynamics and momentum conservation that potential vorticity can only be changed by diabatic heating (such as latent heat released from condensation) or frictional processes. This conservation is the atmospheric equivalent to inertia. A spinning ice skater with her arms spread out laterally can accelerate her rate of spin by contracting her arms. Similarly, when a vortex of air is broad, it is in turn, slow. When the air converges, to maintain potential vorticity, the air speed increases, resulting in a stretched ring vortex. Divergence causes the vortex to spread, slowing down the rate of spin.


text:
# JEE-Mains 2014 (17/30)

Calculus Level 4

Let the population of rabbits surviving at a time $$t$$ governed by differential equation $\frac{dp(t)}{dt}=\frac{p(t)}{2}-200$ If $$p(0)=100$$, then $$p(t)$$ equals :

×

Problem Loading...

Note Loading...

Set Loading...


text:
## 過去の記録

### 2007年01月26日(金)

#### 代数幾何学セミナー

16:30-17:30   数理科学研究科棟(駒場) 128号室
Professor Frans Oort
(Department of Mathematics
University of Utrecht
)
Irreducibility of strata and leaves in the moduli space of
abelian varieties I (a survey of results)

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 126号室
「魅力ある大学院教育」イニシアティブにより以下の3回の講演を行います。
1月26日(金),1月29日(月),31日(水)

Li Daqian 氏 (復旦大学)
Controllability and Observability:
from ODEs to Quasilinear Hyperbolic Systems

### 2007年01月25日(木)

#### 応用解析セミナー

16:00-17:30   数理科学研究科棟(駒場) 056号室
Michael TRIBELSKY 氏 (東大・数理 / モスクワ工科大学)
Soft-mode turbulence as a new type of spatiotemporal chaos at onset

#### 作用素環セミナー

15:15-18:00   数理科学研究科棟(駒場) 126号室

The Pimsner-Voiculescu AF-embedding of the irrational rotation $C^*$-algebra and its subalgebra

A Note on Weak Amenability

#### 関数解析セミナー

14:00-17:00   数理科学研究科棟(駒場) 370号室
Ivana Alexandrova 氏 (East Carolina University)
Semi-Classical Structure of the Scattering Amplitude and the Spectral Function for Schrodinger Operators

### 2007年01月23日(火)

#### トポロジー火曜セミナー

16:30-18:30   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

The twistor correspondence for self-dual Zollfrei metrics
----their singularities and reduction

[ 講演概要 ]
C. LeBrun and L. J. Mason investigated a twistor-type correspondence
between indefinite conformal structures of signature (2,2) with some properties
and totally real embeddings from RP^3 to CP^3.
In this talk, two themes following LeBrun and Mason are explained.

First we consider an additional structure:
the conformal structure is equipped with a null surface foliation
which has some singularity.
We establish a global twistor correspondence for such structures,
and we show that a low dimensional correspondence
between some quotient spaces is induced from this twistor correspondence.

Next we formulate a certain singularity for the conformal structures.
We generalize the formulation of LeBrun and Mason's theorem
admitting the singularity, and we show explicit examples.

On the homology group of $Out(F_n)$
[ 講演概要 ]
Suppose $F_n$ is the free group of rank $n$,
$Out(F_n) = Aut(F_n)/Inn(F_n)$ the outer automorphism group of $F_n$.
We compute $H_*(Out(F_n);\\mathbb{Q})$ for $n\\leq 6$ and conclude
that non-trivial classes in this range are generated
by Morita classes $\\mu_i \\in H_{4i}(Out(F_{2i+2});\\mathbb{Q})$.
Also we compute odd primary part of $H^*(Out(F_4);\\mathbb{Z})$.

### 2007年01月22日(月)

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室
Hanjin Lee 氏 (Seoul National University)
Omori-Yau generalized maximum principle

### 2007年01月19日(金)

#### 講演会

10:30-12:00   数理科学研究科棟(駒場) 056号室
Alex Mahalov 氏 (Arizona State University)
3D Navier-Stokes and Euler Equations with Uniformly Large Initial Vorticity: Global Regularity and Three-Dimensional Euler Dynamics

### 2007年01月18日(木)

#### 講演会

13:00-14:30   数理科学研究科棟(駒場) 056号室

Alex Mahalov 氏 (Arizona State University)
3D Navier-Stokes and Euler Equations with Uniformly Large Initial Vorticity: Global Regularity and Three-Dimensional Euler Dynamics
[ 講演概要 ]
We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for fully three-dimensional initial data characterized by uniformly large vorticity; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold. The global existence is proven using techniques of fast singular oscillating limits, Lemmas on restricted convolutions and the Littlewood-Paley dyadic decomposition. In the second part of the talk, we analyze regularity and dynamics of the 3D Euler equations in cylindrical domains with weakly aligned large initial vorticity.

#### 作用素環セミナー

16:30-18:00   数理科学研究科棟(駒場) 126号室

Twisted Bernoulli shift actions of $Z^2 \\rtimes SL(2,Z)$

#### 応用解析セミナー

16:00-17:30   数理科学研究科棟(駒場) 056号室
LIANG Xing 氏 (東京大学大学院数理科学研究科 / 日本学術振興会)
Asymptotic Speeds of Spread and Traveling Waves for Monotone Semiflows with Applications
[ 講演概要 ]
The theory of asymptotic speeds of spread and monotone traveling waves is established for a class of monotone discrete and continuous-time semiflows and is applied to a functional differential equation with diffusion, a time-delayed lattice population model and a reaction-diffusion equation in an infinite
cylinder.

### 2007年01月17日(水)

#### PDE実解析研究会

10:30-11:30   数理科学研究科棟(駒場) 056号室
Alex Mahalov 氏 (Department of Mathematics and Statistics, Department of Mechanical and Aerospace Engineering, Program in Environmental Fluid Dynamics, Arizona State University )
Fast Singular Oscillating Limits of Hydrodynamic PDEs: application to 3D Euler, Navier-Stokes and MHD equations
[ 講演概要 ]
Methods of harmonic analysis and dispersive properties are applied
to 3d hydrodynamic equations to obtain long-time and/or global existence results to the Cauchy problem for special classes of 3d initial data. Smoothness assumptions for initial data are the same as in local existence theorems. Techniques for fast singular oscillating limits are used and large and/or infinite time regularity is obtained by bootstrapping from global regularity of the limit equations.
The latter gain regularity from 3d nonlinear cancellation of oscillations.
Applications include Euler, Navier-Stokes, Boussinesq and MHD equations, in infinite, periodic and bounded cylindrical domains.
[ 参考URL ]
http://coe.math.sci.hokudai.ac.jp/

#### 講演会

15:30-17:00   数理科学研究科棟(駒場) 470号室

Hamilton-Jacobi方程式の漸近解とその周辺の話題

#### 統計数学セミナー

16:20-17:30   数理科学研究科棟(駒場) 128号室

Second order optimality for estimators in time series regression models
[ 講演概要 ]
We consider the second order asymptotic properties of an efficient frequency domain regression coefficient estimator $\\hat{\\beta}$ proposed by Hannan (1963). This estimator is a semiparametric estimator based on nonparametric spectral estimators. We derive the second order Edgeworth expansion of the distribution of $\\hat{\\beta}$. Then it is shown that the second order asymptotic properties are independent of the bandwidth choice for residual spectral estimator, which implies that $\\hat{\\beta}$ has the same rate of convergence as in regular parametric estimation. This is a sharp contrast with the general semiparametric estimation theory. We also examine the second order Gaussian efficiency of $\\hat{\\beta}$. Numerical studies are given to confirm the theoretical results.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/17.html

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 122号室
Mourad Bellassoued 氏 (Faculte des Sciences de Bizerte)
Recovering a potential in the wave equation via Dirichlet-to-Neumann map.

### 2007年01月16日(火)

#### トポロジー火曜セミナー

16:30-18:30   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

An $SO(3)$-version of $2$-torsion instanton invariants
[ 講演概要 ]
We construct invariants for simply connected, non-spin $4$-manifolds using torsion cohomology classes of moduli spaces of ASD connections on $SO(3)$-bundles. The invariants are $SO(3)$-version of Fintushel-Stern's $2$-torsion instanton invariants. We show that this $SO(3)$-torsion invariant of $2CP^2 \\# -CP^2$ is non-trivial, while it is known that any invariants of $2CP^2 \\# - CP^2$ coming from the Seiberg-Witten theory are trivial
since $2CP^2 \\# -CP^2$ has a positive scalar curvature metric.

On the non-acyclic Reidemeister torsion for knots
[ 講演概要 ]
The Reidemeister torsion is an invariant of a CW-complex and a representation of its fundamental group. We consider the Reidemeister torsion for a knot exterior in a homology three sphere and a representation given by the composition of an SL(2, C)- (or SU(2)-) representation of the knot group and the adjoint action to the Lie algebra.
We will see that this invariant is expressed by the differential coefficient of the twisted Alexander invariant of the knot and investigate some properties of the invariant by using this relation.

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 122号室
Mourad Bellassoued 氏 (Faculte des Sciences de Bizerte)
Recovering a potential from partial Cauchy data for the Schrödinger equation.

### 2007年01月15日(月)

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室

Lagragian constructions for various topological invariants of algebraic varieties (東大数理、松井優氏との共同研究)

#### 講演会

16:30-18:00   数理科学研究科棟(駒場) 122号室

「魅力ある大学院教育」イニシアティブにより以下の講演を行います。

Mourad Bellassoued 氏 (Faculte des Sciences de Bizerte)
Recovering a potential from full Cauchy data for the Schrödinger equation.
[ 講演概要 ]
In this lectures we survey recent progress on the problem of determining a potential by measuring the Dirichlet to Neumann map
for the associated Schr\\"odinger equation or wave equation. We make emphasis on the new results obtained with M.Yamamoto which is concerned with the case that the measurements are made on a strict
subset of the boundary for the wave equation.

#### 講演会

16:00-17:30   数理科学研究科棟(駒場) 056号室
Antonio DeSimone 氏 (SISSA (International School for Advanced Studies))
Analysis of physical systems involving multiple spatial scales: some case studies
[ 講演概要 ]
Variational methods have recently proved to be a powerful tool in deriving macroscopic models for phenomena whose physics is decided at the sub-miccron scale.
We will use two case studies to illustrate this point, namely, that of liquid crystal elastomers and that of superhydrophobic surfaces.

Liquid crystal elastomers are solids which combine the optical properties of liquid crystals with the mechanical properties of rubbery solids. They display phase transformations, material instabilities, and microstructures in a way simalr to shape-memory alloys.

The richness of the underlying material symmetries makes the mathematical analysis of this system particularly rewarding. Recent progress, ranging from analytical relaxation results to numerical simulations of the macroscopic mechanical response will be reviewed.

### 2007年01月12日(金)

#### 談話会・数理科学講演会

16:30-17:30   数理科学研究科棟(駒場) 123号室
お茶&Coffee&お菓子: 16:00～16:30(コモンルーム)

1、プレート境界で砂と泥に起こる雪だるま現象
2、プレート境界地震は確率共鳴か
[ 講演概要 ]

### 2007年01月11日(木)

#### 講演会

16:00-17:30   数理科学研究科棟(駒場) 123号室
Oleg Yu. Emanouilov 氏 (Colorado State University)
Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

### 2007年01月10日(水)

#### 講演会

16:00-17:30   数理科学研究科棟(駒場) 118号室
Oleg Yu. Emanouilov 氏 (Colorado State University)
Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

### 2007年01月09日(火)

#### 講演会

16:00-17:30   数理科学研究科棟(駒場) 118号室

「魅力ある大学院教育」イニシアティブにより以下の講演を行います。

Oleg Yu. Emanouilov 氏 (Colorado State University)
Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.
[ 講演概要 ]
We show that 1-D Burgers equation is globally uncontrollable with control acting at two endpoints. Then we establish the global controllability of the 2-D Burgers equation. Finally we show that for 2-D Navier-Stokes system the problem of global exact controllability is solvable for the dense set of the initial data with a control acting on part of the boundary.

### 2006年12月28日(木)

#### 作用素環セミナー

16:30-18:00   数理科学研究科棟(駒場) 126号室
Roberto Longo 氏 (University of Rome)
Operator Algebras and Conformal Field Theory II


text:
# Homework Help: Abstract Algebra: Polynomials problem

1. Jul 2, 2010

### oplik

1. The problem statement, all variables and given/known data

Let f(x)=x5-x2-1 $$\in$$ C and x1,...,x5 are the roots of f over C. Find the value of the symmetric function:

(2x1-x14).(2x2-x24)...(2x5-x54)

2. Relevant equations

I think, that I have to use the Viete's formulas and Newton's Binomial Theorem.

3. The attempt at a solution

Unfortunately I can't solve the equation to the end. I will appreciate any help. Thank you :)

2. Jul 2, 2010

hmm...all I can think of right now is that, since the constant of f(x) is -1, we know that the product of the roots is 1.

So 2^5*x1x2x3x4x5 = 2^5 = 32, and (-x14)(-x24)(-x34)(-x44)(-x54) = -1.

I know this doesn't solve your problem, but I hope it helps. I'll think about it some more in a bit.

Last edited: Jul 2, 2010
3. Jul 2, 2010

### tmccullough

I'm pretty sure that there is a craftier way than I am about to introduce, but here goes.

For any polynomial, $f(x)$, the product of the roots is the constant term of the polynomial. Also, the roots of $f(x+a)$ are related to the roots of $f(x)$ in the obvious way.

Here, we have $f(x) = x^5-x^2-1.$

That's enough, but the factorization of $x^4-2x$ is crappy, so let's make it a little better.

Note that if $x_i^5-x_i^2-1=0,$ then
$$2x_i-x_i^4=\frac{x_i^2-1}{x_i}=\frac{(x_i-1)(x_i+1)}{x_i}$$

Then, to finish it off, your product is equivalent to
$$\left(\prod_i (x_i-1)\right)\left(\prod_i (x_i+1)\right)/\left(\prod_i x_i\right) = (-3)(-1)/(-1)=-3$$
because the constant term of $f(x)$ is -1, $f(x+1)$ is -1, and $f(x-1)$ is -3.

4. Jul 2, 2010

You may be right, but I still think your method was pretty neat. Nice job.

5. Jul 2, 2010

### Dick

Oooo. That is a really clever factorization. I was struggling with the 'crappy' factorization. Very nice. Except that the product of the roots is +1, like Raskolnikov said, isn't it? Not -1? Leaving a few details for oplik to clean up is a good idea.

6. Jul 2, 2010

I double checked, I stand by my answer. The product of the roots is +1.

But, in hindsight, this result from my original post needs explanation. It follows from one of Viète's Laws. Namely, $$\prod_i x_i = (-1)^n \frac{a_0}{a_n} = (-1)^5 \frac{-1}{1} = 1.$$

7. Jul 3, 2010

### oplik

Thank you, all of you. I think, I got it ;)

8. Jul 3, 2010

### tmccullough

Forgot the $(-1)^5$...Sorry.

9. Jul 3, 2010

### Dick

Don't apologize. You supplied the necessary hint. Correcting the details should be up to the OP. That's why I usually stop with the hint. Otherwise I make these detail mistakes.