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## Efficient Matlab (I): pairwise distances

I want to share some tricks for making Matlab function more efficient and robust. I decide to write a series of blog posts. This is the first one of this series, in which I want to show a simple function for computing pairwise Euclidean distances between points in high dimensional vector space.

The most important thing for efficient Matlab code is VECTORIZATION! Here I will take distance function as an example to demonstrate how accelerate your code by getting ride of “for-loops”.
Say, we have two sets of points X and Y in d-dimensional space. We want to compute the distances between any point in X and any other point in Y. This function is very useful. For example, if you want to find the k-nearest-neighbor of a point x in a database Y, an naive method is to first compute the distances between the point x and all points in Y and then sort the distances to find the k smallest ones.
We can naively implement the function by nested for loops:
function D = dumDistance(X,Y)
n1 = size(X,2);
n2 = size(Y,2);
D = zeros(n1,n2);
for i = 1:n1
for j = 1:n2
D(i,j) = sum((X(:,i)-Y(:,j)).^2);
end
end

Here, each column of matrices X and Y is the vector representation of a point in d-dimensional space. The nested for loop is hurting performance so much. Actually, this is how the pdist function in Matlab is implemented basically (That is how dumb sometime Matlab can be). We wanna get ride of the for-loops and vectorize the code as much as possible. Before write any Matlab code, a good practice is to first write your algorithm down on a paper using matrix notation.
We denote $x_i$ as a vector in X and $y_j$ as a vector in Y. The square of distance $d_{ij}$ between $x_i$ and $y_j$ is
$d_{ij}^2 = \|x_i-y_j\|^2=\|x_i\|^2+\|y_j\|^2-2,$
where $<\cdot,\cdot>$ is dot product. $d_{ij}^2$ can be considered as an entry of matrix D. Therefore, we can write the formula in matrix form as:
$D = \bar x1'+1\bar y'-2X'Y$

where, $\bar x$ is column vector of squared norms of all vectors in X. We can directly implement this by a one-liner:

function D = sqDistance(X, Y)
D = bsxfun(@plus,dot(X,X,1)',dot(Y,Y,1))-2*(X'*Y);

In my rough tests, the new code is about 60+ times faster than the old one for random dense matrices. For sparse matrices, the gap is even larger. That is just the power of vectorization.

If you want to compute the pairwise distances between all point pairs in a point set, you can simply replace the Y with X in the function. Matlab will do extra optimization for the matrix product like X’*X, where only half of the computation is done. A by-product is that the result matrix D is ensured to be symmetric, which usually cannot be guaranteed due to numerical representation of float number.

The takeaway of this post might be:
(1) Verctorizaton!
(2) Matrix multiplication like M=X’*X is efficient (cost half time of ordinary matrix multiplication) and D is guaranteed to be symmetric.
(3) Write your algorithm in Matrix form before writing the code.

Statistical inference
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### 17 Responses to Efficient Matlab (I): pairwise distances

1. Jotaf says:

Thanks for the useful post! Your code is quite concise. I actually collected a few different ways to do it from around the web, and yours turned out to be faster due to the particular order of the operations. For reference, here it is, along with 2 other slower implementations commented out (due to my data’s format I changed the code so the matrices are transposed).

% x2 = sum(X.^2, 2);
% y2 = sum(Y.^2, 2);
% D = x2 * ones(size(Y,1), 1)’ + ones(size(X,1), 1) * y2′ – 2 * X * Y’;

% D = bsxfun(@plus, sum(X .* X, 2), -2 * (X * Y’));
% D = bsxfun(@plus, sum(Y .* Y, 2)’, D);

D = bsxfun(@plus, dot(X,X,2), dot(Y,Y,2)’) – 2 * (X * Y’);

• statinfer says:

Actually, the code you posted seems to be the code I wrote and posted on matlab file exchange long time ago.

2. Tam T. Le says:

Thank you so much for your code. I can save time so much. It is indeed very fast!!!

3. Aviator says:

Could you also post any fast code for finding paiwise distance in a matrix(p x n), p: no. of points; n: dimension of space? Any help appreciated.
Thanks

4. Rundong says:

Reblogged this on Simplex Signum Veri and commented:
It seems that I had a serious bias towards Matlab and I didn’t got the real spirit of it. Now I need to learn the right way to use Matlab.

5. Reblogged this on jakirkpatrick and commented:
Changed some code from impossibly shore to merely slow!

6. Junjie Guan says:

Why “D is guaranteed to be symmetric.”, I though D is n1*n2

• statinfer says:

If you compute the distance between X and X, D is symmetric.

7. Peter says:

Thanks for posting this nice fast code. However, I had some questions. Firstly, there seems to be something strange in the code supplied.. I used it to calculate the smallest distance between two particles with central coordinates of (3,3) and (5,5). Surprisingly the distance between (1-1) and (2-2) were not 0. Moreover, the distance between (1-2) and (2-1) are not identical; the matrix looks like
1.99 e-06 3.99
4.00 4.82e-07
While the the code seems fast, there is some difference in the precision with the PDist function?
In addition, it is not clear to me from the example how you in the end can identiy which pair is closest together??

8. Dario says:

It is just what I was looking for!!! Thank you very much. Your code is completely concise and the explanation goes right to the point to understand the code. Thanks again.

9. Dy Shi says:

A rough estimation on the complexity: suppose each data point has D dimensions, and each scalar using K bits to be represented, two scalars’ multiplication roughly approximated as K summations.
the dum one has complexity: (2+K)*D*n1*n2
the smarter one has complexity: (K+1)*D*n1*n2+(K+1)*D*(n1+n2)+2n1*n2
It is quicker, but only linearly less. So the major difference comes from the inner-product trick of the Matlab implementation.
As for “all point pairs in a point set”, usually we can only compute the upper (or lower) part of the matrix, copy it to the other part if necessary, and leaves the diagonals all 0. This can sure the matrix’s symmetry. The dum one can be easily changed to:
for i = 1:(n1-1)
for j = (i+1):n2
D(i,j) = sum((X(:,i)-X(:,j)).^2); D(j,i) = D(i,j);
end
end
The smarter one, well, can “(X’*Y)” part be more efficient?
Anyway, thanks a lot on this.

10. hinna says:

Thanks a lot! You explained it so clearly…

11. Hello,
This is a very simplified and super fast code; I stumbled upon this site trying to vectorise my code in order to obtain all the pairwise distances in a set of vectors. However in my problem my distance is euclidean weighted where there is a series of weights applied to every component. i. e:

Given a set of weights w = [w1,w2,w3], and vectors x = [x1,x2,x3], and y = [y1,y2,y3]
how do we vectorize a calculation of all pair wise distances in a set of vectors
My distance is calculated as : w1(x1-y1)^2 + w2(x2-y2)^2 + w3(x3-y3)^3

Any suggestions?

Thanks

12. Davis says:

Do you need unlimited articles for your page
? I am sure you spend a lot of time writing articles, but you can save it for other tasks, just type in google: kelombur’s favorite tool

13. Mirkes says:

Many thanks for your code. I slightly modify it to even more fast calculation:
D = bsxfun(@plus,sum(X.^2)’,sum(Y.^2))-2*(X’*Y);
I do not understand why but it is faster.

14. Mali says:

hi,

I want to ask how to calculate the distances in a multidimensinal matrices. let say A is a 20×8 and columns are described by [X1 Y1 X2 Y2 X3 Y3 X4 Y4] (4 points point1: (X1,Y1), point2(X2, Y2). point3:(X3, Y3), point4:(X4, Y4)). how can I calculate the all distances(for instance: distance1 is the distance between point1 and point2 distance2; point1 and point3 distance3; point1 and point4 etc. ) could you please help me?

format: plain


text:
This function plots a simple normal distribution or a normal distribution with shaded areas that indicate at which value a significant p-level is reached.

dist_norm(
norm = NULL,
mean = 0,
sd = 1,
p = NULL,
xmax = NULL,
geom.colors = NULL,
geom.alpha = 0.7
)

Arguments

norm Numeric, optional. If specified, a normal distribution with mean and sd is plotted and a shaded area at norm value position is plotted that indicates whether or not the specified value is significant or not. If both norm and p are not specified, a distribution without shaded area is plotted. Numeric. Mean value for normal distribution. By default 0. Numeric. Standard deviation for normal distribution. By default 1. Numeric, optional. If specified, a normal distribution with mean and sd is plotted and a shaded area at the position where the specified p-level starts is plotted. If both norm and p are not specified, a distribution without shaded area is plotted. Numeric, optional. Specifies the maximum x-axis-value. If not specified, the x-axis ranges to a value where a p-level of 0.00001 is reached. user defined color for geoms. See 'Details' in plot_grpfrq. Specifies the alpha-level of the shaded area. Default is 0.7, range between 0 to 1.

Examples

# a simple normal distribution
dist_norm()

# a simple normal distribution with different mean and sd.
# note that curve looks similar to above plot, but axis range
# has changed.
dist_norm(mean = 2, sd = 4)

# a simple normal distribution
dist_norm(norm = 1)

# a simple normal distribution
dist_norm(p = 0.2)


text:
### Unix

If you’re new to the empowering world of Unix and would like to dive in, you should start with checking out why learning Unix is valuable and run through the 5 short tutorials that comprise the Unix crash course. Other than that, feel free to check out the other pages under the Unix pull-down menu above and listed below 🙂

# Some terminology

Here are some terms that are often used interchangeably – not because it’s important to remember them or any differences (it’s not for most of us), but just to have them laid out somewhere.

Term What it is
shell what we use to talk to the computer; anything where we are pointing and clicking with a mouse is a Graphical User Interface (GUI) shell; something with text only is a Command Line Interface (CLI) shell
command line a text-based environment capable of taking input and providing output
Unix a family of operating systems (we also use the term “Unix-like” because one of the most popular operating systems derived from Unix is specifically named as not being Unix)
bash the most common programming language used at a Unix command-line


text:
Solution: Write structural formulas for all the alkene products that c...

Question

Write structural formulas for all the alkene products that could reasonably be formed from each of the following compounds under the indicated reaction conditions. Where more than one alkene is produced, specify the one that is the major product. (b) 1-Methylcyclopentyl chloride (sodium ethoxide, ethanol, 70°C)


text:
# Turbine

A turbine is a rotary engine that extracts energy from a fluid flow. Claude Burdin (1788-1873) coined the term from the Latin "turbo", or vortex, during an 1828 engineering competition. Benoit Fourneyron (1802-1867), a student of Claude Burdin, built the first practical water turbine.

The simplest turbines have one moving part, a rotor assembly, which is a shaft with blades attached. Moving fluid acts on the blades, or the blades react to the flow, so that they rotate and impart energy to the rotor. Early turbine examples are windmills and water wheels.

Gas, steam, and water turbines have a casing around the blades that contains and controls the working fluid. Credit for invention of the modern steam turbine is given to British Engineer Sir Charles Parsons (1854 - 1931).

A device similar to a turbine but operating in reverse is a compressor or pump. The axial compressor in many gas turbine engines is a common example.

Theory of operation

A working fluid contains potential energy (pressure head) and kinetic energy (velocity head). The fluid may be compressible or incompressible. Several physical principles are employed by turbines to collect this energy:

; Impulse turbines : These turbines change the direction of flow of a high velocity fluid jet. The resulting impulse spins the turbine and leaves the fluid flow with diminished kinetic energy. There is no pressure change of the fluid in the turbine rotor blades. Before reaching the turbine the fluid's "pressure head" is changed to "velocity head" by accelerating the fluid with a nozzle. Pelton wheels and de Laval turbines use this process exclusively. Impulse turbines do not require a pressure casement around the runner since the fluid jet is prepared by a nozzle prior to reaching turbine. describes the transfer of energy for impulse turbines.

; Reaction turbines : These turbines develop torque by reacting to the fluid's pressure or weight. The pressure of the fluid changes as it passes through the turbine rotor blades. A pressure casement is needed to contain the working fluid as it acts on the turbine stage(s) or the turbine must be fully immersed in the fluid flow (wind turbines). The casing contains and directs the working fluid and, for water turbines, maintains the suction imparted by the draft tube. Francis turbines and most steam turbines use this concept. For compressible working fluids, multiple turbine stages may be used to harness the expanding gas efficiently. describes the transfer of energy for reaction turbines.

Turbine designs will use both these concepts to varying degrees whenever possible. Wind turbines use an airfoil to generate lift from the moving fluid and impart it to the rotor (this is a form of reaction). Wind turbines also gain some energy from the impulse of the wind, by deflecting it at an angle. Crossflow turbines are designed as an impulse machine, with a nozzle, but in low head applications maintain some efficiency through reaction, like a traditional water wheel. Turbines with multiple stages may utilize either reaction or impulse blading at high pressure. Steam Turbines were traditionally more impulse but continue to move towards reaction designs similar to those used in Gas Turbines. At low pressure the operating fluid medium expands in volume for small reductions in pressure. Under these conditions (termed Low Pressure Turbines) blading becomes strictly a reaction type design with the base of the blade solely impulse. The reason is due to the effect of the rotation speed for each blade. As the volume increases, the blade height increases, and the base of the blade spins at a slower speed relative to the tip. This change in speed forces a designer to change from impulse at the base, to a high reaction style tip.

Classical turbine design methods were developed in the mid 19th century. Vector analysis related the fluid flow with turbine shape and rotation. Graphical calculation methods were used at first. Formulas for the basic dimensions of turbine parts are well documented and a highly efficient machine can be reliably designed for any fluid flow condition. Some of the calculations are empirical or 'rule of thumb' formulae, and others are based on classical mechanics. As with most engineering calculations, simplifying assumptions were made.

Velocity triangles can be used to calculate the basic performance of a turbine stage. Gas exits the stationary turbine nozzle guide vanes at absolute velocity "V"a1. The rotor rotates at velocity "U". Relative to the rotor, the velocity of the gas as it impinges on the rotor entrance is "V"r1. The gas is turned by the rotor and exits, relative to the rotor, at velocity "V"r2. However, in absolute terms the rotor exit velocity is "V"a2. The velocity triangles are constructed using these various velocity vectors. Velocity triangles can be constructed at any section through the blading (for example: hub , tip, midsection and so on) but are usually shown at the mean stage radius. Mean performance for the stage can be calculated from the velocity triangles, at this radius, using the Euler equation:

:$Delta;h = ucdot Delta;v_w$

Whence:

:$left \left(frac\left\{Delta;h\right\}\left\{T\right\} ight\right) = left\left(frac\left\{u\right\}\left\{sqrt\left\{T ight\right)cdotleft\left(frac\left\{Delta;v_w\right\}\left\{sqrt\left\{T ight\right)$

where: :$Delta;h =,$ specific enthalpy drop across stage:$T =,$ turbine entry total (or stagnation) temperature:$u =,$ turbine rotor peripheral velocity:$Delta;v_w =,$ change in whirl velocity

The turbine pressure ratio is a function of $left\left(frac\left\{Delta;H\right\}\left\{T\right\} ight\right)$ and the turbine efficiency.

Modern turbine design carries the calculations further. Computational fluid dynamics dispenses with many of the simplifying assumptions used to derive classical formulas and computer software facilitates optimization. These tools have led to steady improvements in turbine design over the last forty years.

The primary numerical classification of a turbine is its "specific speed". This number describes the speed of the turbine at its maximum efficiency with respect to the power and flow rate. The specific speed is derived to be independent of turbine size. Given the fluid flow conditions and the desired shaft output speed, the specific speed can be calculated and an appropriate turbine design selected.

The specific speed, along with some fundamental formulas can be used to reliably scale an existing design of known performance to a new size with corresponding performance.

Off-design performance is normally displayed as a turbine map or characteristic.

Types of turbines

*Steam turbines are used for the generation of electricity in thermal power plants, such as plants using coal or fuel oil or nuclear power. They were once used to directly drive mechanical devices such as ship's propellors (eg the Turbinia), but most such applications now use reduction gears or an intermediate electrical step, where the turbine is used to generate electricity, which then powers an electric motor connected to the mechanical load.
*Gas turbines are sometimes referred to as turbine engines. Such engines usually feature an inlet, fan, compressor, combustor and nozzle (possibly other assemblies) in addition to one or more turbines.
*Transonic turbine. The gasflow in most turbines employed in gas turbine engines remains subsonic throughout the expansion process. In a transonic turbine the gasflow becomes supersonic as it exits the nozzle guide vanes, although the downstream velocities normally become subsonic. Transonic turbines operate at a higher pressure ratio than normal but are usually less efficient and uncommon. This turbine works well in creating power from water.
*Contra-rotating turbines. Some efficiency advantage can be obtained if a downstream turbine rotates in the opposite direction to an upstream unit. However, the complication may be counter-productive.
*Statorless turbine. Multi-stage turbines have a set of static (meaning stationary) inlet guide vanes that direct the gasflow onto the rotating rotor blades. In a statorless turbine the gasflow exiting an upstream rotor impinges onto a downstream rotor without an intermediate set of stator vanes (that rearrange the pressure/velocity energy levels of the flow) being encountered.
*Ceramic turbine. Conventional high-pressure turbine blades (and vanes) are made from nickel-steel alloys and often utilise intricate internal air-cooling passages to prevent the metal from melting. In recent years, experimental ceramic blades have been manufactured and tested in gas turbines, with a view to increasing Rotor Inlet Temperatures and/or, possibly, eliminating aircooling. Ceramic blades are more brittle than their metallic counterparts, and carry a greater risk of catastrophic blade failure.
*Shrouded turbine. Many turbine rotor blades have a shroud at the top, which interlocks with that of adjacent blades, to increase damping and thereby reduce blade flutter.
*Shroudless turbine. Modern practise is, where possible, to eliminate the rotor shroud, thus reducing the centrifugal load on the blade and the cooling requirements.
* Bladeless turbine uses the boundary layer effect and not a fluid impinging upon the blades as in a conventional turbine.
*Water turbines
**Pelton turbine, a type of impulse water turbine.
**Francis turbine, a type of widely used water turbine.
**Kaplan turbine, a variation of the Francis Turbine.
**Voith, water turbine.
*Wind turbine. These normally operate as a single stage without nozzle and interstage guide vanes. An exception is the Éolienne Bollée, which has a stator and a rotor, thus being a true turbine.

Other

*Velocity compound "Curtis". Curtis combined the de Laval and Parsons turbine by using a set of fixed nozzles on the first stage or stator and then a rank of fixed and rotating stators as in the Parsons, typically up to ten compared with up to a hundred stages, however the efficiency of the turbine was less than that of the Parsons but it operated at much lower speeds and at lower pressures which made it ideal for ships. Note that the use of a small section of a Curtis, typically one nozzle section and two rotors is termed a "Curtis Wheel"
*Pressure Compund Multistage Impulse or Rateau. The Rateau employs simple Impulse rotors separated by a nozzle diaphragm. The diaphragm is essentially a partition wall in the turbine with a series of tunnels cut into it, funnel shaped with the broad end facing the previous stage and the narrow the next they are also angled to direct the steam jets onto the impulse rotor.

Uses of turbines

Almost all electrical power on Earth is produced with a turbine of some type. Very high efficiency turbines harness about 40% of the thermal energy, with the rest exhausted as waste heat.

Most jet engines rely on turbines to supply mechanical work from their working fluid and fuel as do all nuclear ships and power plants.

Turbines are often part of a larger machine. A gas turbine, for example, may refer to an internal combustion machine that contains a turbine, ducts, compressor, combustor, heat-exchanger, fan and (in the case of one designed to produce electricity) an alternator. However, it must be noted that the collective machine referred to as the turbine in these cases is designed to transfer energy from a fuel to the fluid passing through such an internal combustion device as a means of propulsion, and not to transfer energy from the fluid passing through the turbine to the turbine as is the case in turbines used for electricity provision etc.

Reciprocating piston engines such as aircraft engines can use a turbine powered by their exhaust to drive an intake-air compressor, a configuration known as a turbocharger (turbine supercharger) or, colloquially, a "turbo".

Turbines can have very high power density (ie the ratio of power to weight, or power to volume). This is because of their ability to operate at very high speeds. The Space Shuttle's main engines use turbopumps (machines consisting of a pump driven by a turbine engine) to feed the propellants (liquid oxygen and liquid hydrogen) into the engine's combustion chamber. The liquid hydrogen turbopump is slightly larger than an automobile engine (weighing approximately 700 lb) and produces nearly 70,000 hp (52.2 MW).

Turboexpanders are widely used as sources of refrigeration in industrial processes.

Turbines could also be used as powering system for a remote controlled plane that creates thrust and lifts the plane of the ground. They come in different sizes and could be as small as soda can, still be strong enough to move objects with a weight of 100kg.

hrouded tidal turbines

An emerging renewable energy technology is the shrouded tidal turbine enclosed in a venturi shaped shroud or duct producing a sub atmosphere of low pressure behind the turbine, allowing the turbine to operate at higher efficiency (than the Betz limit [ [http://www.ndsu.nodak.edu/ndsu/klemen/Perfect_Turbine.htm Betz Limit] ] of 59.3%) and typically 3 times higher power output [ [http://www.cyberiad.net/library/pdf/bk_tidal_paper25apr06.pdf Paper by Brian Kirke] ] than a turbine of the same size in free stream.

As shown in the CFD generated figure [ [http://tidalenergy.net.au/ Tidal Energy] ] , it can be seen that a down stream low pressure (shown by the gradient lines) draws upstream flow into the inlet of the shroud from well outside the inlet of the shroud. This flow is drawn into the shroud and concentrated (as seen by the red coloured zone). This augmentation of flow velocity corresponds to a 3-4 times increase in energy available to the turbine. Therefore a turbine located in the throat of the shroud is then able to achieve higher efficiency, and an output 3-4 times the energy the turbine would be capable of if it were in open or free stream. For this reason shrouded turbines are not subject to the properties of the Betz limit.

Considerable commercial interest has been shown in recent times in shrouded tidal turbines as it allows a smaller turbine to be used at sites where large turbines are restricted. Arrayed across a seaway or in fast flowing rivers shrouded tidal turbines are easily cabled to a terrestrial base and connected to a grid or remote community. Alternatively the property of the shroud that produces an accelerated flow velocity across the turbine allows tidal flows formerly too slow for commercial use to be utilised for commercial energy production.

While the shroud may not be practical in wind, as a tidal turbine it is gaining more popularity and commercial use. A shrouded tidal turbine is mono directional and constantly needs to face upstream in order to operate. It can be floated under a pontoon on a swing mooring, fixed to the seabed on a mono pile and yawed like a wind sock to continually face upstream. A shroud can also be built into a tidal fence increasing the performance of the turbines.

Cabled to the mainland they can be grid connected or can be scaled down to provide energy to remote communities where large civil infrastructures are not viable. Similarly to tidal stream open turbines they have little if any environmental or visual amenity impact.

* RMS Lusitania
* Rotordynamics
* Secondary flow in turbines
* Turbinia
* Turbo-alternator

Notes

* [http://www.du.edu/~jcalvert/tech/fluids/turbine.htm Turbine introductory math]
* [http://a330.over-blog.com Turbines jet powered 1/16 scale Airbus A330]
* [http://www.howstuffworks.com/turbine3.htm Turbine working process]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Turbine — Turbine …   Deutsch Wörterbuch

• turbine — [ tyrbin ] n. f. • 1822; « tourbillon, roue de fuseau » 1532; lat. turbo, inis « tourbillon, toupie » ♦ Dispositif rotatif, destiné à utiliser la force vive d un fluide et à transmettre le mouvement au moyen d un arbre (pour qu il soit utilisé ou …   Encyclopédie Universelle

• turbiné — turbine [ tyrbin ] n. f. • 1822; « tourbillon, roue de fuseau » 1532; lat. turbo, inis « tourbillon, toupie » ♦ Dispositif rotatif, destiné à utiliser la force vive d un fluide et à transmettre le mouvement au moyen d un arbre (pour qu il soit… …   Encyclopédie Universelle

• Turbine (TB) — Turbine Banderas Historial …   Wikipedia Español

• Turbine — Tur bine, n. [L. turbo, inis, that which spins or whirls round, whirl.] 1. A water wheel, commonly horizontal, variously constructed, but usually having a series of curved floats or buckets, against which the water acts by its impulse or reaction …   The Collaborative International Dictionary of English

• turbine — Turbine: A type of engine in which all the parts that are in regular motion are rotating, making for very smooth operation. The basic gas turbine operates as follows: air enters the compressor and is compressed. It is then delivered to the… …   Dictionary of automotive terms

• Turbine — Banderas Historial …   Wikipedia Español

• Turbine — Turbine, Inc. Turbine, Inc. (прежде Turbine Entertainment Software, Second Nature, и изначально CyberSpace, Inc.) компания разработчик компьютерных игр, была основана Джоном Монсаррэтом, Джереми Гэффни, Кевином Лэнджевином, и Тимоти Миллером.… …   Википедия

• Turbine — Sf erw. fach. (19. Jh.) Entlehnung. Entlehnt aus frz. turbine, einer Neubildung zu l. turbo ( inis) m. Wirbel; alles, was sich im Kreis dreht .    Ebenso nndl. turbine, ne. turbine, nfrz. turbine, nschw. turbin, nisl. túrbína; turbulent. ✎ Cottez …   Etymologisches Wörterbuch der deutschen sprache

• turbine — UK US /ˈtɜːbaɪn/ noun [C] PRODUCTION ► a type of machine in which a wheel is driven by the movement of liquid or gas so that power is produced: »A one megawatt turbine can produce enough electricity annually to meet the needs of 700 households.… …   Financial and business terms

• turbine — (n.) 1838, from Fr. turbine, from L. turbinem (nom. turbo) spinning top, eddy, whirlwind, related to turba turmoil, crowd (see TURBID (Cf. turbid)). Originally applied to a wheel spinning on a vertical axis, driven by falling water. Turbo in… …   Etymology dictionary


text:
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text:
1. ## ode

Can you explain how to linearise

Thank you

Can you explain how to linearise

Thank you
To "linearize" a function about x= a, replace it by its tangent line approximation there.

For example, $(x^3)'= 3x^2$ so, in particular, at x= 1, its derivative is 3. The tangent approximation at x= 1 is y= 3(x- 1)+ 1= 3x- 2.

The linearization of x"+ 2ax'- x- x^3= F cos(bt) about x= 1 is x"+ 2ax'- x- 3x- 2= x"+ 2ax'- 4x- 2= F cos(bt) or x"+ 2ax'- 4x= 2+ F cos(bt).

Do the same at x= -1. At x= -1, the tangent approximation is y= 3(x+1)-1= 3x+ 2.

3. sorry I don't understand tangent approximation. can u explain and the how are we getting the linearised equation.
I think it is x''+2ax'-4x=Fcos(bt)
am I right?


text:
# Cost-optimal single-qubit gate synthesis in the Clifford hierarchy

1School of Physics, University of Melbourne, VIC, Parkville, 3010, Australia
2School of Mathematics and Statistics, University of Melbourne, VIC, Parkville, 3010, Australia

### Abstract

For universal quantum computation, a major challenge to overcome for practical implementation is the large amount of resources required for fault-tolerant quantum information processing. An important aspect is implementing arbitrary unitary operators built from logical gates within the quantum error correction code. A synthesis algorithm can be used to approximate any unitary gate up to arbitrary precision by assembling sequences of logical gates chosen from a small set of universal gates that are fault-tolerantly performable while encoded in a quantum error-correction code. However, current procedures do not yet support individual assignment of base gate costs and many do not support extended sets of universal base gates. We analysed cost-optimal sequences using an exhaustive search based on Dijkstra’s pathfinding algorithm for the canonical Clifford+$T$ set of base gates and compared them to when additionally including $Z$-rotations from higher orders of the Clifford hierarchy. Two approaches of assigning base gate costs were used. First, costs were reduced to $T$-counts by recursively applying a $Z$-rotation catalyst circuit. Second, costs were assigned as the average numbers of raw (i.e. physical level) magic states required to directly distil and implement the gates fault-tolerantly. We found that the average sequence cost decreases by up to $54\pm 3\%$ when using the $Z$-rotation catalyst circuit approach and by up to $33\pm 2 \%$ when using the magic state distillation approach. In addition, we investigated observed limitations of certain assignments of base gate costs by developing an analytic model to estimate the proportion of sets of $Z$-rotation gates from higher orders of the Clifford hierarchy that are found within sequences approximating random target gates.

Performing large scale complex quantum algorithms will require the use of robust error-correction protocols over encoded qubits. However, these protocols present unique challenges that need to be overcome in order to achieve universal quantum computation. In general, many logic gate operations used in quantum algorithms cannot be directly applied to encoded quantum states. Logic gate operations that can be directly applied transversally over the encoded qubits are considered to be easy to implement because they are inherently fault tolerant and hence require relatively small numbers of physical gates. In addition, there are gates that can be fault-tolerantly implemented, but require resource intensive magic state distillation procedures. These gates, together with the transversal gates, form universal sets of base gates that can be used to approximate any unitary gate up to arbitrary precision by using gate synthesis algorithms to generate corresponding sequences of base gates. Within the quantum compilation literature, gate synthesis primarily focuses on the Clifford+T set of universal base gates where the T gate is implemented through magic state distillation. However, other universal sets of base gates exist that can be implemented fault-tolerantly in error correction codes. For instance, by including gates from higher orders of the Clifford hierarchy, where the resource cost of gate implementation increases for increasing orders of the hierarchy.

Although single-qubit synthesis algorithms have been well studied within the quantum compilation literature, they typically do not yet support the variety of available base gates and do not optimise with respect to individual gate costs of implementation. In this work we use an exhaustive search algorithm, based on Dijkstra’s pathfinding algorithm, to find cost-optimal sequences for random target single-qubit gates. It is compatible with arbitrary sets of universal base gates with individually assigned costs, enabling us to explore potential cost benefits from including higher order Clifford hierarchy base gates. We calculate the synthesis cost against error scaling relations and compare between the canonical Clifford+T set of base gates and the sets resulting from accumulatively including Z-rotation gates from increasing orders of the Clifford hierarchy. Two gate implementation approaches were used to assign resource costs to gates from each order of the hierarchy. The first approach reduces the gate costs to T-counts by using a Z-rotation catalyst circuit to implement the gates. This resulted in the scaling factor decreasing by up to 54±3%. The second approach assigns the costs as the average number of raw magic states required to directly distil and implement the gates fault-tolerantly. This approach decreased the scaling factor by up to 33±2%.

The results of our paper demonstrate that the resource requirements of synthesis algorithms could be considerably improved by including higher orders of the Clifford hierarchy as base gates and optimising with respect to individually assigned gate costs.

### ► References

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[25] Earl T Campbell and Mark Howard. Magic state parity-checker with pre-distilled components. Quantum, 2: 56, 2018. 10.22331/Q-2018-03-14-56.
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### Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2022-05-17 13:59:19). On SAO/NASA ADS no data on citing works was found (last attempt 2022-05-17 13:59:20).


text:
User ivane - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:10:59Z http://mathoverflow.net/feeds/user/2196 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/7293#7293 Answer by ivane for What should be offered in undergraduate mathematics that's currently not (or isn't usually)? ivane 2009-11-30T16:55:47Z 2012-01-08T07:32:00Z
Once students have been exposed to linear algebra and vector calculus, build calculus on manifolds using many examples; i.e. go from real $\mathbb{R}$-abstract multilinear to the de Rham complex, illustrating in $\mathbb{R}^3$. All that easen differential geometry, differential topology Riemannian geometry, etc.
 http://mathoverflow.net/questions/68090/finite-sums-with-binomial-and-catalan-inverses Finite sums with Binomial and Catalan inverses ivane 2011-06-17T20:22:14Z 2011-06-18T01:21:39Z
In a recent failed-post about some partial sums with respect to the Central Binomial and Catalan number the formulas $$\sum_{k=0}^n\frac{4^k}{B_k}=\frac{4^{n+1}(2n+1)}{3 B_{n+1}}+\frac{1}{3}$$ $$\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{4^{n+1}(2n^2+5n+2)}{5 B_{n+1}}+\frac{1}{5}$$ were mention, here in MO, and I forgot to ask, so let me do it now:

Is This just one instance of some broader well known pattern?

Here, consider $B_m={2m \choose m}$ and $C_m=\frac{B_m}{m+1}$ for those set of numbers.
 http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Finite sum of integers inverses ivane 2011-06-14T17:53:22Z 2011-06-15T17:37:01Z
Recently, we learned from Renzo Sprugnoli that $\sum_{k=0}^n \frac{4^k}{B_k} =\frac{2n+1}{3}\frac{4^{n+1}}{B_{n+1}}+\frac{1}{3}$, where $B_n$ are the famous central binomial coefficients, $B_n={2n \choose n}$.

Nowadays it is posible to find; using W|A, The Wolframalpha Calculator, that:

$$\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{2n^2+5n+2}{5}\frac{4^{n+1}}{B_{n+1}}+\frac{1}{5}.$$

My "begs" are for someone to help me to pursuit some interesant generalizations or applications about the involved Catalan numbers.

Ref: R. Sprugnoli, "Sum of reciprocals of the central binomial coefficients", INTEGERS: Electronic journal of combinatorial number theory 6 (2006), #A27.

Update:

Is the formula $\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{2n^2+5n+2}{5}\frac{4^{n+1}}{B_{n+1}}+\frac{1}{5}$ well known?
 http://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web/22560#22560 Answer by ivane for Most helpful math resources on the web ivane 2010-04-26T05:04:27Z 2010-04-26T17:22:54Z
People: consider http://www.digizeitschriften.de/ tons of classical papers in english...

I think it is worth to check the 39 journals collection on world class referee-ed mathwork.

One paper on Mathematische Annalen (which is the very amusing): "On the holymorphic flow with an isolated singularity", is the famous GSV, gives you an index formula...
 http://mathoverflow.net/questions/11307/what-do-you-call-the-product-of-a-circle-and-an-annulus/14786#14786 Answer by ivane for What do you call the product of a circle and an annulus? ivane 2010-02-09T17:29:05Z 2010-02-09T17:50:50Z
that corresponds to the complement of a trivial (but essencial) torus knot in a open solid torus. For those -Fico had mention- they are called cable spaces and have nice foliation into circles. Its name is CS(1,0). Can you see what is CS(2,1)?

Edit at: utc-6 = 11:50 approx

you could also say the trivial I-bundle over the torus
 http://mathoverflow.net/questions/9147/branched-coverings-over-orbifolds-with-reflector-lines Branched coverings over orbifolds with reflector lines ivane 2009-12-17T02:15:57Z 2009-12-24T23:16:55Z
It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum_i^r\frac{a_i-1}{a_i})$, where $r$ is the number of cones with stabilizer of orders $a_1,...,a_r$ respectively.

Now, I don't know if everyone would like to know what is the corresponding relation when $B$ has reflector intervals or reflector circles, as I'd rather... so I dare to question:

Is there a generalization in this direction?
 http://mathoverflow.net/questions/7746/periodic-mapping-classes-of-the-genus-two-orientable-surface Periodic mapping classes of the genus two orientable surface ivane 2009-12-04T04:14:19Z 2009-12-17T20:31:26Z
Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and reintrepreting them as a circle bundles over orbifolds.

In the http://web.archive.org/web/20070316045651/http://www.smm.org.mx/SMMP/html/modules/Publicaciones/AM/Cm/35/artExp08.pdf -work you would like see the cases $O_1$, among $N_1$ and $N_2$, solved. Any feedback on the results and conjectures, some of them obviously false, will bring a lot of happiness :)
 http://mathoverflow.net/questions/8223/reducible-3d-torus-bundles Reducible 3d torus bundles ivane 2009-12-08T18:11:45Z 2009-12-17T20:30:33Z
Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,

could anyone give me a hint to classify them?

In contrast, do you agree that -in the sense of connected sum- all these bundles are irreducible?
 http://mathoverflow.net/questions/7968/two-solid-n-3-glued-by-its-boundary Two solid N_3 glued by its boundary ivane 2009-12-06T05:31:59Z 2009-12-17T19:56:32Z
Let $N_3$ be the genus three non orientable surface. Do we have an analogous 3d manifold as the solid torus and the solid Klein bottle for $N_3$? I don't see how to extend the ideas related to the 3d lens spaces. Any feedback would be super-welcome
 http://mathoverflow.net/questions/8445/learning-topology/8471#8471 Answer by ivane for Learning Topology ivane 2009-12-10T18:40:52Z 2009-12-12T03:42:43Z
Books on geometric group theory, nowaday are talking and applying algebra and topology technologies to algorithms, solvability of word problems, membership problems, context free languages, etc...

All these together with the notes of Peter May refered above, will almost show you the state-of-the-art.

Further, check Graham NIblo work at http://www.personal.soton.ac.uk/gan/Welcome.html

UPDATE: Follow books on geometric-group theory to a more especific titles mentioned here in MO.
 http://mathoverflow.net/questions/8361/hnn-extensions-which-are-free-products HNN extensions which are free products ivane 2009-12-09T16:03:21Z 2009-12-11T18:46:08Z
which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles...
 http://mathoverflow.net/questions/3858/introductory-text-on-geometric-group-theory/8588#8588 Answer by ivane for Introductory text on geometric group theory? ivane 2009-12-11T18:35:22Z 2009-12-11T18:35:22Z
What about this wonder:

Peter Scott, Terry Wall, Topological methods in group theory, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press (1979) 137-203.

simply beauty and useful
 http://mathoverflow.net/questions/8521/nice-proof-of-the-jordan-curve-theorem/8532#8532 Answer by ivane for Nice proof of the Jordan curve theorem? ivane 2009-12-11T05:01:00Z 2009-12-11T05:01:00Z
You should compare with: "Geometric Topology in Dimensions 2 and 3", Moise, Edwin E. (1977). Springer-Verlag and tell
 http://mathoverflow.net/questions/8361/hnn-extensions-which-are-free-products/8519#8519 Answer by ivane for HNN extensions which are free products ivane 2009-12-11T02:07:31Z 2009-12-11T02:16:15Z
Let me add an explicit partial solution to the question above: for a torus bundle $E$ over the circle, the fundamental group of $E$ can't be a free product of groups, because if it were, the fundamental subgroup of the fiber would be a free product (by the Kurosch's theorem) which is impossible for the torus, then $E$ isn't a connected sum, hence irreducible.

At least the HNN extensions which are free products can't be torus bundles, and in fact, no other surface bundles unless the surface be the 2-sphere
 http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/8466#8466 Answer by ivane for books well-motivated with explicit examples ivane 2009-12-10T17:48:43Z 2009-12-10T18:10:04Z
Characteristic classes by Milnor-Stasheff, 1974. This book from Princeton marks (i think) the synthesis of several years of maturation for the real beginnings of modern topology, the next years that came...

In their 20 chapters, preface, 3 appendices, bibliograph and index, anyone gonna see a jewel master piece of math
 http://mathoverflow.net/questions/8216/less-elementary-group-theory/8239#8239 Answer by ivane for less elementary group theory ivane 2009-12-08T19:57:14Z 2009-12-08T19:57:14Z
re-develop your course employing category theory vision: Once you construct objects construct arrows. Example, if you have defined groups inmediately define homomorphisms, if you define subgroups and quotients, define kernels...etc, arrive to the homomorphism's fundamental theorems...
 http://mathoverflow.net/questions/5499/which-mathematicians-have-influenced-you-the-most/8062#8062 Answer by ivane for Which mathematicians have influenced you the most? ivane 2009-12-07T01:49:10Z 2009-12-07T01:54:37Z
Consider looking forward: Grisha Perelman, his strange history  wiki G.Perelman. However the first millenium prize awarded. Even note this: Terence Tao said... "well, it's amazing"
 http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/7967#7967 Answer by ivane for books well-motivated with explicit examples ivane 2009-12-06T05:05:58Z 2009-12-07T00:53:59Z
"Differential Topology", Guillemin-Pollack, 1974.
 http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/7958#7958 Answer by ivane for books well-motivated with explicit examples ivane 2009-12-06T04:19:54Z 2009-12-06T04:19:54Z
"Riemannian Geometry", Gallot-Hulin-Lafontaine, 1987, plenty of examples and exercises and the motivation: the own one helps...
 http://mathoverflow.net/questions/7834/undergraduate-differential-geometry-texts/7948#7948 Answer by ivane for Undergraduate Differential Geometry Texts ivane 2009-12-06T02:35:22Z 2009-12-06T02:35:22Z
commentaries like without much formalism like those to the Thorpe's book, I think, are really discouraging for everyone, then students don't want to do complex calculations 'cuz they are ugly... Nah!

I believe that the std covariant derivative of $\mathbb{R}^3$ and the induced connection to a surface, via the Gauss equation (to quick deduce a formula for the gaussian curvature), paves the way to grasp better thing like the geodesic curvature and the Gauss-Bonnet must-do's, for: O'Neil is suitable!
 http://mathoverflow.net/questions/3951/memorizing-theorems/7937#7937 Answer by ivane for Memorizing theorems ivane 2009-12-06T01:00:00Z 2009-12-06T01:00:00Z
I would advise to study the proofs until plainly pure memory, BUT each time you repeat the reasons involved, the understanding switch always in ON position... that allows you to gain real and great intuitions, believe-me...
 Ah! if the repetitions are in front of someone else -who is stuying the same- there'll be a lot of more FUN... put attention in that Gowers is telling us: ...eventualy you will be able of finding quick proofs of everything you masters!
 http://mathoverflow.net/questions/3242/canonical-examples-of-algebraic-structures/7301#7301 Answer by ivane for Canonical examples of algebraic structures ivane 2009-11-30T18:06:27Z 2009-12-06T00:25:00Z
study presentations, zum beispiel:

$\langle a,b\mid a^2=b^3=e, ab=b^2a \rangle$

vs

$\langle a,b\mid a^2=b^3=e, ab=ba \rangle$ or $\langle a\mid a^6=e \rangle$

compare nuances...

Other: the $Out\pi_1(F)$ of any surface. Btw this is the famous mapping class group of the surface $\cal{MCG}(F)$.

To get a real modern grasp on the subjet you should include topological techniques: homotopy, cohomotopy, homology and cohomology, K-theory... you might enter in contact with a types of structures like free groups, free products, amalgamated products, HNN extensions of groups, Grothendieck groups and the like...
 http://mathoverflow.net/questions/7308/n-3-and-n-4-periodic-and-pseudo-anosov-auto-homeomorphisms N_3 and N_4 periodic and pseudo Anosov auto-homeomorphisms ivane 2009-11-30T19:41:44Z 2009-12-04T06:49:18Z
It is well know that the genus three non orientable surface, N3, has only periodic and reducible auto-homeomorphisms, meanwhile the surface N4 is the first non orientable surface with pseudo Anosov maps. Also, recently profesor B.Szepietowski gave the MCG presentation of N4, from where, I calculated that there are seven periodic mapping classes. The question is: are there more?

Curiously, the torus and N3 show also seven periodic mapping classes each but we would like to understand better why N3 loses pseudo Anosov maps which contrast with the fact that N3 is got from the 2-torus via an one-point blow up...
 http://mathoverflow.net/questions/6142/circle-bundles-over-rp2/7288#7288 Answer by ivane for Circle bundles over $RP^2$ ivane 2009-11-30T16:34:16Z 2009-12-02T19:48:07Z
I think that they have Seifert fiber space presentation as: $(On,1|(1,b))$.

Or $(On,1|(1,b),(a_1,b_1),...,(a_r,b_r))$, if you allow an orbifold with cone points in $RP^2$.

You can look at the cases by decomposing $RP^2=Mo\cup_{\partial}D$, so the orientable 3-manifold will be the
 1) orientable $Q=Mo\tilde{\times}S^1$, the twisted circle bundle over the mobius band, very well known being equivalent to the orientable I-bundle over the Klein bottle, with boundary a torus $T$,
 2) and a Dehn-filling in the remaining disk $D$, with a whichever fibered solid torus or tori.

We could say that $(On,1\mid (1,b))=Q\cup_T W(1,b)$, for a fibered $(1,b)$ solid torus $W$
 http://mathoverflow.net/questions/2340/what-is-the-first-interesting-theorem-in-insert-subject-here/7362#7362 Answer by ivane for What is the first interesting theorem in (insert subject here)? ivane 2009-12-01T03:49:09Z 2009-12-01T03:49:09Z
3-manifolds, existence and uniqueness of decomposition into irreducible pieces in the connected sum (Kneser, Milnor-Wallace)
 http://mathoverflow.net/questions/2851/algorithmic-combinatorics-resources/7303#7303 Answer by ivane for Algorithmic Combinatorics resources? ivane 2009-11-30T18:47:52Z 2009-11-30T19:47:00Z
a key ingredient in modern software like mathematica is using hypergeometric functions. Let me recall the Knuth's statement in his Concrete Mathematics... "with hypergeometric functions you are likely to embrace each elementary functions studied in a technological career"... well, sort of
 http://mathoverflow.net/questions/2479/de-rham-cohomology-of-surfaces/7298#7298 Answer by ivane for de Rham Cohomology of surfaces ivane 2009-11-30T17:45:31Z 2009-11-30T17:45:31Z
check these of J. Harrison and the ivancevics bros

1)http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.4991v3.pdf

and

2)http://arxiv.org/PS_cache/math-ph/pdf/0501/0501001v2.pdf

these skip a little Mayer-Vietoris... (but believe me: it is unwise avoid Mayer-Vietoris :)
 http://mathoverflow.net/questions/2949/great-mathematical-figures-and-or-diagrams/7289#7289 Answer by ivane for Great mathematical figures and/or diagrams? ivane 2009-11-30T16:47:19Z 2009-11-30T16:47:19Z
see my starting effort to help see 3-manifolds at http://commons.wikimedia.org/wiki/Category:3-manifolds
 http://mathoverflow.net/questions/11307/what-do-you-call-the-product-of-a-circle-and-an-annulus/11315#11315 Comment by ivane ivane 2012-10-12T01:48:13Z 2012-10-12T01:48:13Z @Ryan: $M\ddot{o}\times S^1$ is the non-orientable twisted $I$-bundle over the torus. http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/7293#7293 Comment by ivane ivane 2012-08-30T18:24:46Z 2012-08-30T18:24:46Z on the beginning missing duality of finite dimensional vector spaces (preferable over the real field) garanties from minor to almost no understanding :P http://mathoverflow.net/questions/68090/finite-sums-with-binomial-and-catalan-inverses/68104#68104 Comment by ivane ivane 2011-06-18T01:04:57Z 2011-06-18T01:04:57Z you are right, i make a mistake because i should write the next formulas: $\sum_{k=0}^n\frac{4^k}{B_k}=\frac{4^{n+1}(2n+1)}{3B_{n+1}}+\frac{1}{3}$ and $\sum_{k=0}^n\frac{4^k(k+1)}{B_k}=\frac{4^{n+1}(2n^2+5n+2)}{5B_{n+1}}+\frac{1}{5}$ http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-15T17:55:51Z 2011-06-15T17:55:51Z ooh!, i found that Witula and Slota have: FINITE SUMS CONNECTED WITH THE INVERSES OF CENTRAL BINOMIAL NUMBERS AND CATALAN NUMBERS in the AEJM. From the abstract, can anyone infer that Sprugnoli's method is different? http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-15T17:26:56Z 2011-06-15T17:26:56Z Yemon, i'm strucked to know that this formulas correspond to someone which already saw them. I would like to see the ref, please. "Begs" or requests are the same, if i use a ridicule tone nothing distorts the math significance, to, connect wisdom. And to the bead about w|a, it is easy to check they and i can do the calculazhons http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-14T23:16:27Z 2011-06-14T23:16:27Z does it hurts that someone tells you that your theorems can be reached by W|A? http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-14T20:29:54Z 2011-06-14T20:29:54Z Excellent ref, just what i was looking for, Lot of Thanx http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-14T18:23:42Z 2011-06-14T18:23:42Z a simple one: how W|A did it? he.he http://mathoverflow.net/questions/67784/finite-sum-of-integers-inverses Comment by ivane ivane 2011-06-14T18:09:34Z 2011-06-14T18:09:34Z Even more complex are the relations inter $\sum_{k=0}^n\frac{1}{{2k\choose k}}$ and $\sum_{k=0}^n\frac{k+1}{{2k\choose k}}$... http://mathoverflow.net/questions/18570/why-are-tensors-a-generalization-of-scalars-vectors-and-matrices Comment by ivane ivane 2010-05-31T15:47:01Z 2010-05-31T15:47:01Z would like to add that the grasping of the fundamental sense for these objects and properties, are implanted around the generalization of calculus: differential forms and its applications... http://mathoverflow.net/questions/21235/how-to-shown-that-the-tangent-bundle-of-a-vector-space-is-a-vector-bundle Comment by ivane ivane 2010-04-13T19:02:01Z 2010-04-13T19:02:01Z you're going to see better if the normed vector space is taken the fiber $F$ in your definition of vector bundle AND for the tangent bundle you use this type functions: $(x,v)\to(x,J(\varphi_j^{−1}\circ\varphi_i)(x)v)$ where $J$ is for jacobian and $v\in F=\mathbb{R}^n$ http://mathoverflow.net/questions/11307/what-do-you-call-the-product-of-a-circle-and-an-annulus Comment by ivane ivane 2010-02-09T16:56:44Z 2010-02-09T16:56:44Z at commons.wikimedia.org/wiki/Category:3-manifolds you could see some propose. Would be a place to start proposal. Personaly I like "moxi" for $M\ddot{o}\times I$ and twisted-moxi to $M\ddot{o}^\stackrel{\sim}\times I$... http://mathoverflow.net/questions/8717/reducible-3d-n-3-bundles Comment by ivane ivane 2009-12-14T01:31:46Z 2009-12-14T01:31:46Z i.e. in the complement of the invariant multicurve? http://mathoverflow.net/questions/8223/reducible-3d-torus-bundles/8241#8241 Comment by ivane ivane 2009-12-14T00:58:35Z 2009-12-14T00:58:35Z @Sam: 10-4, gonna check, super-thanks http://mathoverflow.net/questions/8717/reducible-3d-n-3-bundles Comment by ivane ivane 2009-12-14T00:36:55Z 2009-12-14T00:36:55Z @Sam: but N_3 don't have pA according to: R.C.Penner. "A construction of pseudo-Anosov homeomorphisms", Trans.Amer.Math. Soc. 310 (1988) No 1, 179-197.


text:
missautonomy97 Group Title Evaluate for a = 4 and b = -6. Select the simplest form. a/2b Please Explain how you got the answer so I can successfully complete the rest (: Thanks! I do online and it's Christmas break and I have no one to explain this. 2 years ago 2 years ago

1. myininaya Group Title

$\frac{4}{2(-6)}$ a is 4 b is -6 i just plugged them in

2. henkjan Group Title

substitute values for a and b: a/2b = 4/(2*-6) = -4/12 Simplify by dividing both -4 and 12 by the same number.. They can both be divided by 4 so: -4/12 = (-4/4)/ (12/4) =-1/3

3. myininaya Group Title

now 4 is 2*2 does anything cancel?

4. myininaya Group Title

and then something else will cancel you can do it henkjan's way too


text:
# Thread: Applications of Derivatives help

1. ## Applications of Derivatives help

A Ferris wheel with a 30 ft radius makes one revolution every 10 sec:
a) Assume that the center of the Ferris wheel is located at the point (0,40), and write parametric equations to model its motion.
b)At t=0 the point P on the Ferris wheel is located at (30,40). Find the rate of horizontal movement, and the rate of vertical movement of the point P when t=5sec and t=8 sec.

For a, I think the answer is x=30cost, and y=30sint+40, since the radius is 30ft and it's center is at (0,40), but i think I'm wrong.
For b, I'm not too sure. The concept of horizontal and vertical movement isn't clear to me, especially when t is a value.
Please show work and have the answers, so I won't have to keep guessing, thank you.

A Ferris wheel with a 30 ft radius makes one revolution every 10 sec:
a) Assume that the center of the Ferris wheel is located at the point (0,40), and write parametric equations to model its motion.
b)At t=0 the point P on the Ferris wheel is located at (30,40). Find the rate of horizontal movement, and the rate of vertical movement of the point P when t=5sec and t=8 sec.

For a, I think the answer is x=30cost, and y=30sint+40, since the radius is 30ft and it's center is at (0,40), but i think I'm wrong.
For b, I'm not too sure. The concept of horizontal and vertical movement isn't clear to me, especially when t is a value.
Please show work and have the answers, so I won't have to keep guessing, thank you.
you have to take the angular velocity, $\omega$ , into account ...

$x = 30\cos(\omega t)$

$y = 30\sin(\omega t) + 40$

where $\omega = \frac{2\pi}{10} = \frac{\pi}{5}$

when t = 0 , point P is at x = 30 and y = 40

when t = 2.5, point P should be at the very top of the wheel ... check it.

to find the rates of horizontal and vertical motion, find $\frac{dx}{dt}$ and $\frac{dy}{dt}$ respectively and evaluate them at the requested times.

3. Can you explain to me how W equals pi/5?

$\frac{1 \, revolution}{10 \, seconds} = \frac{2\pi \, radians}{10 \, seconds} =$ ?


text:
# Proof Without Words – Part 6

Algebra Level 4

The image above illustrates which of the following identities?

×


text:
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USF Home > College of Arts and Sciences > Department of Mathematics & Statistics

Mathematics & Statistics

# Colloquia — Fall 2002

## Tuesday, December 17, 2002

Title
Speaker

Time
Place

Another Measure of Uncertainty
M. Rao
University of Florida
3:00pm-4:00pm
PHY 130
Arunava Mukherjea

Abstract

Information Theory originates from two path breaking papers of Claude E. Shannon (1948) in which he proposed a quantitative measure of information and uncertainty in a random phenomenon based on the classical Boltzmann Entropy of Statistical Physics. Shannon proposed that a measure of information in the occurance of an uncertain event of probability $$p$$ is $$-\log p$$. It is difficult to overestimate the influence of this work on modern information theory.

In this paper we use the cumulative distribution of a random variable to define the information content in it and thereby develop a novel measure of information that parallels Shannon entropy, which we call cumulative residual entropy (CRE). The salient features of CRE are,

1. it is more general than the Shannon entropy in that its definition is valid in the continuous and discrete domains,
2. it possesses more general mathematical properties than the Shannon entropy and
3. it can be easily computed from sample data and these computations aymptotically converge to the true values.

We discuss some applications of our new information measure to image processing and point out its advantages over the use of traditional Shannon entropy. We also give a precise formula relating CRE and Shannon entropy.

## Wednesday, December 4, 2002

Title
Speaker

Time
Place

Extension of Slater's List: New Generalizations of Rogers-Ramanujan Type Identities
Tina Garrett
Carleton College
Northfield, MN
3:00pm-4:00pm
PHY 118

Abstract

In this talk we will review several known identities of Rogers-Ramanujan Type. We will outline a method for extending these identities to give more general theorems and give several examples from L. J. Slater's 1950 list of theorems that can be improved.

## Friday, November 22, 2002

Title
Speaker

Time
Place

An Algorithmic Approach to Rogers-Ramanujan Type Identities
Andrew Sills
Penn State University
4:15pm-5:15pm
PHY 108

Abstract

108 years after their initial discovery by L. J. Rogers, the Rogers-Ramanujan identities continue to stimulate research in numerous areas of the mathematical sciences including the theory of partitions, Lie algebras, statistical physics, and symbolic computation.

I will discuss a method for producing polynomial generalizations of Rogers-Ramanujan type identities via an algorithmic method using nonhomogeneous $$q$$-difference equations. Next, I will discuss some of the implications of this method for algorithmic proof theory and statistical mechanics.

## Friday, November 22, 2002

Title

Speaker

Time
Place

Integrability Characteristics and Coherent Structures of Integrable Two-Dimensional Generalizations of NLS Type Equations
S. R. Choudhury
University of Central Florida
3:00pm-4:00pm
LIF 262
Wen-Xiu Ma

Abstract

A recent algorithmic procedure based on truncated Painlevè expansions is used to derive Lax Pairs, Darboux Transformations, Hirota Tau Functions, and various soliton solutions for integrable $$(2+1)$$ generalizations of NLS type equations [1]. In particular, diverse classes of solutions are found analogous to the dromion, instanton, lump, and ring soliton solutions derived recently for $$(2+1)$$ KdV Type Equations, the Nizhnik-Novikov-Veselov Equation, and the Broer-Kaup system. If time permits and results are available, possible applications of these solutions, as well as possible applications of the same techniques to non-isospectral integrable hierarchies, may be considered.

[1] A. V. Mikhailov and R. I. Yamilov, On Integrable Two-Dimensional Generalizations of NLS Type Equations, Phys. Letters A 230 (1997), no. 5-6, 295-300.

## Wednesday, November 20, 2002

Title
Speaker

Time
Place

Inversion of Bilateral Basic Hypergeometric Series
Michael Schlosser
University of Vienna
Vienna, AUSTRIA
3:00pm-4:00pm
TBA

Abstract

We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised $${}_6\psi_6$$ summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.

## Friday, November 15, 2002

Title
Speaker

Time
Place

The Generalized Chebyshev Polynomials
Yang Chen
Imperial College
London, UK
3:00pm-4:00pm
LIF 262

Abstract

We construct explicitly using theta functions on a particular Riemann surface polynomials orthogonal over a union of several disjoint intervals.

## Tuesday, November 12, 2002

Title
Speaker

Time
Place

A Generalization of Polynomials: Nichols Algebras
Matias Grana
M.I.T.
Cambridge, MA
3:30pm-4:30pm
PHY 118
Masahiko Saito

Abstract

In the 70's two physicists, Yang (doing quantum mechanics) and Baxter (doing statistical mechanics) arrived to the same equation, the now famous Yang-Baxter equation. In the 80's two mathematicians, Drinfeld and Jimbo, constructed an enormous family of solutions to that equation, opening a branch of algebra (and physics) now known as Quantum Groups.

In a completely unrelated fashion, in his Ph.D. thesis in 1978, Warren Nichols defined a family of algebras which he called Bialgebras of Type One. By results proved in the 90's, we now know how to produce one of these algebras from any solution of the Yang-Baxter equation. We call then these algebras by “Nichols algebras”. Examples of them are polynomial algebras, exterior algebras and “quantum Borel algebras”, which are a main part of the Quantum Groups defined by Drinfeld and Jimbo.

I will define Nichols algebras and give some examples based on quandles. (Quandles are a nice tool for topologists, but this is another story). The talk is intended for a general audience; no knowledge of physics, Yang-Baxter equations, Quantum Groups, Quandles or any other “mysterious” thing is required, as I'll define every used object.

Title
Speaker

Time
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text:
# Which if the following sentences uses commas correctly?

Which if the following sentences uses commas correctly?

## This Post Has 4 Comments

1. Expert says:

$Which topic sentence best fits the passage? a. child labor was a serious problem during the industr$

2. Expert says:

idk i just need the points

3. emily0259 says:

D: JANUARY 9, 1983, IS MY BIRTHDAY .

This sentence correctly uses commas. There needs to be a comma after the specific day and the year. The other sentences are missing commas or have too many. In A: i was born in bryan texas . There should be a comma between Bryan and Texas. B: IN JUNE, 1826 MY GREAT-GRANDFATHER CAME TO AMERICA . There should not be a comma after the month. C: PATRICIA HAMMOND MD WILL BE SPEAKING ABOUT SKIN CANCER . There should be a comma between Hammond and MD.

4. alanflores40 says:

We need the sentences to answer.


text:
# Algebra Examples

Graph f(x)=2^x
Exponential functions have a horizontal asymptote. The equation of the horizontal asymptote is .
Horizontal Asymptote: