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MathGroup Archive: March 1997 [00150]
[Date Index] [Thread Index] [Author Index]
How can I handle Operator Algebra ?
• To: mathgroup at smc.vnet.net
• Subject: [mg6482] How can I handle Operator Algebra ?
• From: "Koichiro Yamaguchi" <tigercat at da.mbn.or.jp>
• Date: Mon, 24 Mar 1997 21:38:28 -0500 (EST)
• Organization: Media,Tokyo,JAPAN
• Sender: owner-wri-mathgroup at wolfram.com
I am a beginner user of Mathematica version 2, so following question is
based on what I have done on Mathematica version 2.
I have a question about operator algebra. I think this question might be a
FAQ. But I could find no "answers for FAQ" of this newsgroup on my
newsserver, and also no explanation in the S.Wolfram's "Mathematica book
for version 3". So excuse me for asking this.
According to "Mathematica book" we can treat pure functions such as those
include some differential operators using "# and &" or "Function". Since it
seemed both of them gave same answers, only the results using "# and &" are
written below.
I tried to make an angular momentum operator as follows.
And here I've got what I expected.
{-(z (D[#1, y] & )) + y (D[#1, z] & ),
z (D[#1, x] & ) - x (D[#1, z] & ),
-(y (D[#1, x] & )) + x (D[#1, y] & )}
Then I applied this operator to the length r.
rl=Sqrt[ DotProduct[r,r] ];
-(z (D[#1, y] & )) + y (D[#1, z] & )
rl // lx
But Mathematica did not accept multiple "&".
(-(z (D[#1, y] & )) + y (D[#1, z] & ))[Sqrt[x + y + z ]]
In case of single "&", I could get a correct answer.
lx2= (-z D[#,y] + y D[#,z])&
-(z D[#1, y]) + y D[#1, z] &
rl // lx2
However, if I use single "&" to define differential operator vector like
Mathematica rejects to recognize this as a vector. How should I I handle
Mathematica to do some elementary operator algebra like this ?
Thanks in advance for any comments and any answers.
Koichiro Yamaguchi
E-mail: tigercat at da.mbn.or.jp
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Programing in "C" ---help
can someone help me ,,i am beginner in this,,,
this is my home work:
Write an algorithm that will be of three integers find the maximum and minimum, and calculate the difference between the two. Write an algorithm for the programming language C and draw a diagram.
Post what you've got so far and what you're having trouble with.
with alll man ,,,i say i am beginner in this areaaa
You also say it's your homework.
yes i say,,i am first year on univerzity electrotehnical ing ,,,,or explain some....i write some but i donot know how to continueee..........i dont know is it ok
#include <stdio.h>
#include <stdlib.h>
int main()
int a;
int b;
int c;
int min;
int max;
max = b, min = a;
min = b, max = a;
max = c, min = a;
max = a, min = c;
max = c, min = b;
max = b, min = c;
Where you have commas, those should actually be on separate lines that each end with semicolons.
i donot know men,,,, imposible to get a result (max - min ) i donot know have tooo do this ,,,,if you know you write me some exampes or help me in some way....
will you help me ,,,,say me that i know what to do or wait you
how to calculate diference bettwen two max and min
Same as always; max - min.
cout << " Diff " << (max-min) << endl;
Topic archived. No new replies allowed.
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How to Calculate the Area of an Equilateral Triangle | The Classroom | Synonym
An equilateral triangle is a triangle with all three sides of equal length. The surface area of a two dimensional polygon such as a triangle is the total area contained by the sides of the polygon.
The three angles of an equilateral triangle are also of equal measure in Euclidean geometry. Since the total measure of the angles of a Euclidean triangle is 180 degrees, this means that the angles
of an equilateral triangle all measure 60 degrees. The area of an equilateral triangle may be calculated when the length of one its sides is known.
Calculating the Area
Step 1
Determine the area of a triangle when the base and height are known. Take any two identical triangles with base s and height h. We can always form a parallelogram of base s and height h with these
two triangles. Since the area of a parallelogram is s x h, the area A of a triangle is therefore ½ s x h.
Step 2
Form the equilateral triangle into two right triangles with the line segment h. The hypotenuse of one of these right triangles length s, one of the legs has length h and the other leg has length s/2.
Step 3
Express h in terms of s. Using the right triangle formed in step 2, we know that s^2 = (s/2)^2 + h^2 by the Pythagorean formula. Therefore, h^2 = s^2 -- (s/2)^2 = s^2 -- s^2/4 = 3s^2/4, and we now
have h = (3^1/2)s/2.
Step 4
Substitute the value of h obtained in step 3 into the formula for a triangle's area obtained in step 1. Since A = ½ s x h and h = (3^1/2)s/2, we now have A = ½ s (3^1/2)s/2 = (3^1/2)(s^2)/4.
Step 5
Use the formula for area of an equilateral triangle obtained in step 4 to find the area of an equilateral triangle with sides of length 2. A = (3^1/2)(s^2)/4 = (3^1/2)(2^2)/4 = (3^1/2).
Style Your World With Color
Photo Credits
• Mainland High School, MathisFun
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Convert mile/hour second to inch/square second - Conversion of Measurement Units
›› Convert mile/hour second to inch/square second
›› More information from the unit converter
How many mile/hour second in 1 inch/square second? The answer is 0.0568181818182.
We assume you are converting between mile/hour second and inch/square second.
You can view more details on each measurement unit:
mile/hour second or inch/square second
The SI derived unit for acceleration is the meter/square second.
1 meter/square second is equal to 2.23693629205 mile/hour second, or 39.3700787402 inch/square second.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between miles/hour second and inches/square second.
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ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data.
Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres
squared, grams, moles, feet per second, and many more!
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How to simplify/solve this differential equation
December 29th 2012, 08:43 AM #1
Junior Member
Jun 2010
How to simplify/solve this differential equation
I have the following equation
$\frac{\partial}{\partial y}\left(y\frac{dm}{dx}+m\frac{dy}{dx}\right)-\frac{dm}{dx}=0$
where $m$ is a function of $y$ (say $m=f\left(y\right)$) and $y$ is a function of $x$ (say $y=g\left(x\right)$). Are there any conditions under which $\frac{dm}{dx}$ becomes identically zero and
hence this equation can be reduced to the follwoing form which is easier to solve:
$\frac{\partial}{\partial y}\left(m\frac{dy}{dx}\right)=0$
If such condtions do not exist, what is the best and easiest method to solve the original equation?
Re: How to simplify/solve this differential equation
If m= f(y) and y= g(x), then $\frac{dm}{dx}= \frac{dm}{dy}\frac{dy}{dx}$ so the answer to your question is "No". That product will be 0 if and only if at least one of $\frac{dm}{dy}$ or $\frac
{dy}{dx}$ is 0- in other words if m is NOT a function of y or y is NOT a function of x. Your real problem is that you have two "unkowns", m as a funciton of y and y as a function of x, but only
one equation.
Re: How to simplify/solve this differential equation
Many thanks!
I have two equations not just one because I know f(y) and I want to find g(x) which is the function of interest to me.
Re: How to simplify/solve this differential equation
I also wish to know if this equation can be solved numerically for g(x) if analytical solution is not possible.
December 29th 2012, 08:49 AM #2
MHF Contributor
Apr 2005
December 29th 2012, 08:56 AM #3
Junior Member
Jun 2010
December 29th 2012, 09:26 AM #4
Junior Member
Jun 2010
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Fractal superconductivity near localization threshold
We develop a theory of a pseudogap state appearing near the superconductor-insulator transition in strongly disordered metals with attractive interaction. We show that such an interaction combined
with the fractal nature of the single particle wave functions near the mobility edge leads to an anomalously large single particle gap in the superconducting state near SI transition that persists
and even increases in the insulating state long after the superconductivity is destroyed. We give analytic expressions for the value of the pseudogap in terms of the inverse participation ratio of
the corresponding localization problem.
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MathGroup Archive: March 2007 [00393]
[Date Index] [Thread Index] [Author Index]
Re: PN junction Simulation with Mathematica
• To: mathgroup at smc.vnet.net
• Subject: [mg74137] Re: PN junction Simulation with Mathematica
• From: "Michael Weyrauch" <michael.weyrauch at gmx.de>
• Date: Mon, 12 Mar 2007 22:04:20 -0500 (EST)
• References: <et36tn$qjb$1@smc.vnet.net>
unfortuately you do not provide proper Mathematica code
that one could easily run through, but looking at your equations
I am pretty sure that the problem rests with Nd[x] and Na[x]
in the equation for the electric field. If it is a fixed distribution
of charges then it this should go into the boundary conditions.
Mathematica tries to solve for that, and there is no differential
equation for that....
Regards Michael
<e-touch at libero.it> schrieb im Newsbeitrag news:et36tn$qjb$1 at smc.vnet.net...
> Dear MathGroup
> I'm a student and I'm trying to simulate the bahaviour of a p-n junction =
> by solving the continuity equations for charge carriers and the poisson e=
> quation.
> It is a system of coupled PDEs
> 1)d/dt P[x,t]=-k*P[x,t]*d/dx El[x,t]
> -k*El[x,t]*d/dx P[x,t]
> +k2 dd/dx2 P[x,t];
> 2)d/dt N[x,t]=+k*N[x,t]*d/dx El[x,t]
> +k*El[x,t]*d/dx N[x,t]
> +k2 dd/dx2 N[x,t];
> 3)d/dx El[x,t]=P[x,t]-N[x,t]+Nd[x]-Na[x]
> Where I want to find P[x,t],N[x,t] and El[x,t].
> Nd[x] and Na[x] are functions wich define the distribution of fixed charg=
> es(and the initial conditions for P and N)
> The first two equations without the terms with El[x,t] are just simply di=
> ffusion equations and I have no problems to solve them...but when I try t=
> o couple them with the Electric field I get only error messages:
> "NDSolve::pdord: Some of the functions have zero differential order so th=
> e \
> equations will be solved as a system of differential-algebraic equations.=
> "
> "NDSolve::bcart: Warning: An insufficient number of boundary conditions h=
> ave \
> been specified for the direction of independent variable x. Artificial \
> boundary effects may be present in the solution."
> "LinearSolve::sing: Matrix SparseArray[<180030>,<<1>>] is singular."
> "NDSolve::icfail: Unable to find initial conditions which satisfy the res=
> idual \
> function within specified tolerances. Try giving initial conditions for =
> both \
> values and derivatives of the functions."
> I insert in NDSolve all the initials conditions wich have physically sens=
> e...
> -P[x,0]=UnitStep[-x]
> -P[100,t]=0
> -P[-100,t]=1
> -N[x,0]=UnitStep[x]
> -N[100,t]=1
> -N[-100,t]=0
> -El[x,0]=0
> -El[100,t]=0 should be not necessary
> -El[-100,t]=0 should be not necessary
> I saw in the archive that somebody already tried to solve his kind of pro=
> blem, but it didn't get help from those posts...
> I'm prety new of Mathematica, so, if someone could help me or tell if the=
> re is another way to solve this problem it would be really helpful.
> Thank you
> Matteo=0A=0A=0A------------------------------------------------------=0AP=
> assa a Infostrada. ADSL e Telefono senza limiti e senza canone Telecom=0A=
> http://click.libero.it/infostrada=0A
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Math Forum Discussions
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Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
Topic: Re: The Prime Directive
Replies: 8 Last Post: Oct 1, 2012 3:57 PM
Messages: [ Previous | Next ]
Re: The Prime Directive
Posted: Sep 28, 2012 5:38 PM
Wayne Bishop wrote:
> [...] such as the Oklahoma School for Sciences and Mathematics,
> a competitive two year, highly competitive boarding school
> for kids across Oklahoma, this written by a guy with a PhD
> in mathematics from MIT:
** ...the highest average ACT score in the state. I also heard
** that it has the highest % of PhD's on its faculty of any
** high school. For example, the physics teacher got his PhD
** from Princeton, worked at Lawrence Livermore Labs, and has
** taught at Columbia. None of these guys, myself included,
** can teach in the public schools because we don't have the
** teaching certification. The school is specifically exempted
** from the dictates of the state dept. of ed., which allows
** it to assemble such a faculty.
When I taught at OSSM's "sister school" in Louisiana (Aug. 1996
to May 1999), I was in regular contact with one of the math guys
at OSSM [1], and I see he's still there. At the time, and perhaps
even now, he was the only high school math teacher I knew of who
was fairly active in professional level mathematical pursuits
(research papers, writing reviews for Mathematical Reviews and
Zentralblatt für Mathematik, etc.). That, and the fact that the
couple of years before I began at LSMSA I had a lot of contact
with some Polish mathematicians (the OSSM guy I'm talking about
is from Poland), was what initially led me to write him shortly
after I began at LSMSA. I notice the OSSM web page no longer has
the following statement, which was on their main web page for
many years (back as far as 1996, I think):
"The most rigorous academic program of its kind in the nation."
- - Julian C. Stanley, Ph.D.
O-K, I found the statement. They've moved it from the front page
to here [2].
[1] http://web.archive.org/web/20001011025754/http://www.ossm.edu/~abucki/
[2] http://www.ossm.edu/about/awesome-facts/
Dave L. Renfro
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Abundances in H II Regions and Planetary Nebulae -
G. Stasinska
1.2. Heating and cooling
During the photoionization process, the absorption of a photon creates a free electron which rapidly shares its energy with the other electrons present in the gas by elastic collisions, and thus
heats the gas. The energy gains are usually dominated by photoionization of hydrogen atoms, although photoionization of helium contributes significantly. Intuition might suggest that T[e] will
decrease away from the ionizing source, since the ionizing radiation field decreases because of geometrical dilution and absorption in the intervening layers. This is actually not the case. The total
energy gains per unit volume and unit time at a distance r from the ionizing source are schematically given by:
If ionization equilibrium is achieved in each point of the nebula, one has (in the "on-the-spot case")
Therefore, G can be written
Thus < E > can be seen as the average energy gained per photoionization, and is roughly independent of r. It can be shown (see e.g. Osterbrock 1989), that when the ionization source is a blackbody of
temperature T[], one has < E > kT[]. Therefore:
meaning that the energy gains are roughly proportional to the temperature of the ionizing stars.
Thermal losses in nebulae occur through recombination, free-free radiation and emission of collisionally excited lines. The dominant process is usually due to collisional excitation of ions from
heavy elements (with O giving the largest contribution, followed by C, N, Ne and S). Indeed, these ions have low-lying energy levels which can easily be reached at nebular temperatures. The
excitation potentials of hydrogen lines are much higher, so that collisional excitation of H^0 can become important only at high electron temperatures.
For the transition l of ion j of an element X^i, in a simple two-level approach and when each excitation is followed by a radiative deexcitation, the cooling rate can be schematically written as
where [ijl] is the collision strength, [ijl] is the statistical weight of the upper level, and [ijl] is the excitation energy.
If the density is sufficiently high, some collisional deexcitation may occur and cooling is reduced. In the two-level approach one has:
So, in a first approximation, one can write that the electron temperature is determined by
where G is given by Eq. (1.18) and L[coll]^ijl by Eq. (1.20).
The following properties of the electron temperature are a consequence of the above equations:
- T[e] is expected to be usually rather uniform in nebulae, its variations are mostly determined by the mean energy of the absorbed stellar photons, and by the populations of the main cooling ions.
It is only at high metallicities (over solar) that large T[e] gradients are expected: then cooling in the O^++ zone is dominated by collisional excitation of fine structure lines in the ground level
of O^++, while the absence of fine structure lines in the ground level of O^+ forces the temperature to rise in the outer zones (Stasinska 1980a, Garnett 1992).
- For a given T[] , T[e] is generally lower at higher metallicity.
- For a given metallicity, T[e] is generally lower for lower T[] .
- For a given T[] and given metallicity, T[e] increases with density in regions where n is larger than a critical density for collisional deexcitation of the most important cooling lines (around 5 ×
10^2 - 10^3 cm^-3).
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How can pressure and velocity be inversely related in fluid flow as per bernoullie? [Archive] - Ask Me Help Desk
View Full Version : How can pressure and velocity be inversely related in fluid flow as per bernoullie?
We know Force F is directly proportional to pressure and to change in velocity w.r.t to time as per Newton 2nd law then how can pressure and velocity can be inversely proportional during fluid flow
@ jcaron2:- Your description is simply outstanding. Clearly proves that you have understood the concept very well.
What is the effect of magnetic field on fluid flow through porous medium if conductive fluid is injected from one side of it?
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Meeting Details
For more information about this meeting, contact Fei Wang, Hope Shaffer, Toan Nguyen, Mark Levi, Victor Nistor, Jinchao Xu, Ludmil Zikatanov.
Title: Algorithms for anisotropic mean curvature flow of networks.
Seminar: Computational and Applied Mathematics Colloquium
Speaker: Selim Esedoglu, University of Michigan
Abstract Link: http://www.math.lsa.umich.edu/~esedoglu/
ABSTRACT: Threshold dynamics is an algorithm for moving an interface (e.g. a surface in 3D) by mean curvature motion. It was proposed by Merriman, Bence, and Osher in 1989, and also extended to
networks of surfaces in the same paper. This dynamics arises as gradient flow for the sum of the areas of the surfaces in the network, and plays a prominent role in materials science applications
where it describes the motion of grain boundaries in polycrystals (such as most metals) under heat treatment. Further extension of the algorithm to weighted mean curvature flow of networks, where the
surface tension of each interface in the network may be different and may depend on the direction of the normal, is of great interest for applications, but has remained elusive. In fact, previous
attempts at even the simpler case where the area of each surface in the network is weighted by a different constant turn out to be flawed, mainly due to the difficulty of ensuring that certain
natural angle conditions are satisfied along triple curves (where three surfaces meet). We describe how to extend threshold dynamics, first to unequal but constant surface tensions (joint work with
Felix Otto), and then to unequal and anisotropic (normal dependent) surface tensions (joint work with Matt Elsey and Felix Otto).
Room Reservation Information
Room Number: MB106
Date: 10 / 14 / 2013
Time: 02:30pm - 03:30pm
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Here's the question you clicked on:
what is the answer to 12.8033 if you rounded it
• one year ago
• one year ago
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please help
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rounded to the nearest what? 100, 10, 1, 10, 1/100, etc.?
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12.8000, right?
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YAY! I am right
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can u help me with another one im 11 i need help with my math
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Sure! Im 11 too, but in 6th grade
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Put in school at 4
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oh im going to 6th grade now
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Oh ok Thats cool
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wat about 0.24
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0.20. because it is closest to the tenth
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Medal? XD
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Anything else?
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yes 43.382
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What do I round it too?
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is it to the nearest hundredth, pretty?
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the thousand
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Hmm, leme see
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40.380? Maybe?
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tenths | hundredths | thousandths <---- those are your columns to the right of the decimal place. 2 is in the thousandth place, right, because it's in the third column. 2 is closer to 0 than 10
so you round down.
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Do you understand?
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thanks wat about 4.6666
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to the thousandth again?
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so which number is in the tenths column?
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yup, so now look to the hundredths place, which is the number just to the right of the 6, and if that one is higher closer to 10 we round up the tenths place and if it's closer to 0 we round
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is replying to Can someone tell me what button the professor is hitting...
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Preston, WA Math Tutor
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cryptographically sound Dolev Yao style security proof of the Otway-Rees protocol
Results 1 - 10 of 21
, 2005
"... Abstract. We examine various indistinguishability-based proof models for key establishment protocols, namely the Bellare & Rogaway (1993, 1995), the Bellare, Pointcheval, & Rogaway (2000), and
the Canetti & Krawczyk (2001) proof models. We then consider several variants of these proof models, ident ..."
Cited by 44 (8 self)
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Abstract. We examine various indistinguishability-based proof models for key establishment protocols, namely the Bellare & Rogaway (1993, 1995), the Bellare, Pointcheval, & Rogaway (2000), and the
Canetti & Krawczyk (2001) proof models. We then consider several variants of these proof models, identify several subtle differences between these variants and models, and compare the relative
strengths of the notions of security between the models. For each of the pair of relations between the models (either an implication or a non-implication), we provide proofs or counter-examples to
support the observed relations. We also reveal a drawback with the original formulation of the Bellare, Pointcheval, & Rogaway (2000) model, whereby the Corrupt query is not allowed. 1
- IN PROC. IEEE SYMPOSIUM ON SECURITY AND PRIVACY , 2004
"... We investigate the relation between symbolic and cryptographic secrecy properties for cryptographic protocols. Symbolic secrecy of payload messages or exchanged keys is arguably the most
important notion of secrecy shown with automated proof tools. It means that an adversary restricted to symboli ..."
Cited by 41 (9 self)
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We investigate the relation between symbolic and cryptographic secrecy properties for cryptographic protocols. Symbolic secrecy of payload messages or exchanged keys is arguably the most important
notion of secrecy shown with automated proof tools. It means that an adversary restricted to symbolic operations on terms can never get the entire considered object into its knowledge set.
Cryptographic secrecy essentially
- In Proc. 19th IEEE CSFW , 2006
"... We describe a faithful embedding of the Dolev-Yao model of Backes, Pfitzmann, and Waidner (CCS 2003) in the theorem prover Isabelle/HOL. This model is cryptographically sound in the strong sense
of reactive simulatability/UC, which essentially entails the preservation of arbitrary security proper ..."
Cited by 26 (7 self)
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We describe a faithful embedding of the Dolev-Yao model of Backes, Pfitzmann, and Waidner (CCS 2003) in the theorem prover Isabelle/HOL. This model is cryptographically sound in the strong sense of
reactive simulatability/UC, which essentially entails the preservation of arbitrary security properties under active attacks and in arbitrary protocol environments. The main challenge in designing a
practical formalization of this model is to cope with the complexity of providing such strong soundness guarantees. We reduce this complexity by abstracting the model into a sound, light-weight
formalization that enables both concise property specifications and efficient application of our proof strategies and their supporting proof tools. This yields the first tool-supported framework for
symbolically verifying security protocols that enjoys the strong cryptographic soundness guarantees provided by reactive simulatability/UC. As a proof of concept, we have proved the security of the
Needham-Schroeder-Lowe protocol using our framework.
- Information and Computation , 2007
"... We define reactive simulatability for general asynchronous systems. Roughly, simulatability means that a real system implements an ideal system (specification) in a way that preserves security
in a general cryptographic sense. Reactive means that the system can interact with its users multiple times ..."
Cited by 23 (4 self)
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We define reactive simulatability for general asynchronous systems. Roughly, simulatability means that a real system implements an ideal system (specification) in a way that preserves security in a
general cryptographic sense. Reactive means that the system can interact with its users multiple times, e.g., in many concurrent protocol runs or a multiround game. In terms of distributed systems,
reactive simulatability is a type of refinement that preserves particularly strong properties, in particular confidentiality. A core feature of reactive simulatability is composability, i.e., the
real system can be plugged in instead of the ideal system within arbitrary larger systems; this is shown in follow-up papers, and so is the preservation of many classes of individual security
properties from the ideal to the real systems. A large part of this paper defines a suitable system model. It is based on probabilistic IO automata (PIOA) with two main new features: One is generic
distributed scheduling. Important special cases are realistic adversarial scheduling, procedure-call-type scheduling among colocated system parts, and special schedulers such as for fairness, also in
combinations. The other is the definition of the reactive runtime via a realization by Turing machines such that notions like polynomial-time are composable. The simple complexity of the transition
functions of the automata is not composable. As specializations of this model we define security-specific concepts, in particular a separation between honest users and adversaries and several trust
models. The benefit of IO automata as the main model, instead of only interactive Turing machines as usual in cryptographic multi-party computation, is that many cryptographic systems can be
specified with an ideal system consisting of only one simple, deterministic IO automaton without any cryptographic objects, as many follow-up papers show. This enables the use of classic formal
methods and automatic proof tools for proving larger distributed protocols and systems that use these cryptographic systems.
- Computer Security, Proceedings of ESORICS 2005, number 3679 in Lecture Notes in Computer Science , 2005
"... The abstraction of cryptographic operations by term algebras, called Dolev-Yao models, is essential in almost all tool-supported methods for proving security protocols. Recently significant
progress was made in proving that such abstractions can be sound with respect to actual cryptographic reali ..."
Cited by 16 (5 self)
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The abstraction of cryptographic operations by term algebras, called Dolev-Yao models, is essential in almost all tool-supported methods for proving security protocols. Recently significant progress
was made in proving that such abstractions can be sound with respect to actual cryptographic realizations and security definitions. The strongest results show this in the sense of reactive
simulatability/UC, a notion that essentially means retention of arbitrary security properties under arbitrary active attacks and in arbitrary protocol environments, with only small changes to both
abstractions and natural implementations.
- Proc. 11th European Symp. on Research. in Comp. Sec , 2006
"... Abstract We present a computational analysis of basic Kerberos with and without its public-key extension PKINIT in which we consider authentication and key secrecy properties. Our proofs rely on
the Dolev–Yaostyle model of Backes, Pfitzmann, and Waidner, which allows for mapping results obtained sym ..."
Cited by 15 (4 self)
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Abstract We present a computational analysis of basic Kerberos with and without its public-key extension PKINIT in which we consider authentication and key secrecy properties. Our proofs rely on the
Dolev–Yaostyle model of Backes, Pfitzmann, and Waidner, which allows for mapping results obtained symbolically within this model to cryptographically sound proofs if certain assumptions are met. This
work was the first verification at the computational level of such a complex fragment of an industrial protocol. By considering a recently fixed version of PKINIT, we extend symbolic correctness
results we previously attained in the Dolev– Yao model to cryptographically sound results in the computational model.
- In Proc. 13th CCS , 2006
"... A large body of work exists for machine-assisted analysis of cryptographic protocols in the formal (Dolev-Yao) model, i.e., by abstracting cryptographic operators as a free algebra. In
particular, proving secrecy by typing has shown to be a salient technique as it allowed for elegant and fully autom ..."
Cited by 13 (2 self)
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A large body of work exists for machine-assisted analysis of cryptographic protocols in the formal (Dolev-Yao) model, i.e., by abstracting cryptographic operators as a free algebra. In particular,
proving secrecy by typing has shown to be a salient technique as it allowed for elegant and fully automated proofs, often
- PROCEEDINGS OF IFIP SEC 2006 , 2006
"... Symbolic secrecy of exchanged keys is arguably one of the most important notions of secrecy shown with automated proof tools. It means that an adversary restricted to symbolic operations on
terms can never get the entire key into its knowledge set. Cryptographic key secrecy essentially means comput ..."
Cited by 7 (1 self)
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Symbolic secrecy of exchanged keys is arguably one of the most important notions of secrecy shown with automated proof tools. It means that an adversary restricted to symbolic operations on terms can
never get the entire key into its knowledge set. Cryptographic key secrecy essentially means computational indistinguishability between the real key and a random one, given the view of a much more
general adversary. We analyze the cryptographic key secrecy for the strengthened Yahalom protocol, which constitutes one of the most prominent key exchange protocols analyzed symbolically by means of
automated proof tools. We show that the strengthened Yahalom protocol does not guarantee cryptographic key secrecy. We further show that cryptographic key secrecy can be proven for a slight
simplification of the protocol by exploiting recent results on linking symbolic and cryptographic key secrecy in order to perform a symbolic proof of secrecy for the simplified Yahalom protocol in a
specific setting that allows us to derive the desired cryptographic key secrecy from the symbolic proof. The proof holds in the presence of arbitrary active attacks provided that the protocol is
relying on standard provably secure cryptographic primitives.
"... The abstraction of cryptographic operations by term algebras, called Dolev-Yao models or symbolic cryptography, is essential in almost all tool-supported methods for proving security protocols.
Recently significant progress was made – using two conceptually different approaches – in proving that Dol ..."
Cited by 7 (1 self)
Add to MetaCart
The abstraction of cryptographic operations by term algebras, called Dolev-Yao models or symbolic cryptography, is essential in almost all tool-supported methods for proving security protocols.
Recently significant progress was made – using two conceptually different approaches – in proving that Dolev-Yao models can be sound with respect to actual cryptographic realizations and security
definitions. One such approach is grounded on the notion of simulatability, which constitutes a salient technique of Modern Cryptography with a longstanding history for a variety of different tasks.
The other approach strives for the so-called mapping soundness – a more recent technique that is tailored to the soundness of specific security properties in Dolev-Yao models, and that can be
established using more compact proofs. Typically, both notions of soundness for similar Dolev-Yao models are established separately in independent papers. In this paper, the two approaches are
related for the first time. Our main result is that simulatability soundness entails mapping soundness provided that both approaches use the same cryptographic implementation. Interestingly, this
result does not dependent on details of the simulator, which translates between cryptographic implementations and their Dolev-Yao abstractions in simulatability soundness. Hence, future research may
well concentrate on simulatability soundness whenever applicable, and resort to mapping soundness in those cases where simulatability soundness is too strong a notion.
- Proc. ACISP 2005 , 2005
"... Abstract. We observe that the definitions of security in the computational complexity proof models of Bellare & Rogaway (1993) and Canetti & Krawczyk (2001) require two partners in the presence
of a malicious adversary to accept the same session key, which we term a key sharing requirement. We then ..."
Cited by 5 (3 self)
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Abstract. We observe that the definitions of security in the computational complexity proof models of Bellare & Rogaway (1993) and Canetti & Krawczyk (2001) require two partners in the presence of a
malicious adversary to accept the same session key, which we term a key sharing requirement. We then revisit the Bellare–Rogaway three-party key distribution (3PKD) protocol and the Jeong–Katz–Lee
two-party authenticated key exchange protocol T S2, which carry claimed proofs of security in the Canetti & Krawczyk (2001) model and the Bellare & Rogaway (1993) model respectively. We reveal
previously unpublished flaws in these protocols where we demonstrate that both protocols fail to satisfy the definition of security in the respective models. We present a new 3PKD protocol as an
improvement with a proof of security in the Canetti & Krawczyk (2001) model and a simple fix to the specification of protocol T S2. We also identify several variants of the key sharing requirement
and present a brief discussion. 1
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Marlborough, MA Algebra 2 Tutor
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The Naked Pair in Sudoku
Copyright © University of Cambridge. All rights reserved.
This article describes a particular strategy useful in solving Sudoku puzzles known as "the naked pair". The discussions will be based on the Sudoku shown in Figure 1, and which can also be
downloaded here .
The game of Sudoku consists of a grid of nine rows and nine columns subdivided into nine 3x3 subgrids. It has two basic rules:
1. Each column, each row, and each box (3x3 subgrid) must have the numbers 1 to 9.
2. No column, row or box can have two squares with the same number.
To help locate boxes (squares of 9 cells that contain each of the digits 1-9), I have numbered them from 1-9 as shown in Figure 2.
Individual cells are located using the notation (row, column), that is row number followed by the column number. In Figure1 (1,5) contains the number 9.
In solving any puzzle, the first thing we can do is find all the available single candidates in the rows, columns and boxes. That is, individual cells that are the only place where particular numbers
can be put.
To explain this a little more, let's look at Figure 3 where we have listed all the possible numbers that can fit into each cell (taking into account what already appears in the rows, columns and
boxes). We can now find four single candidates by the method of "force" .
These single candidates are:
6 in cell (2,4) because there is nowhere else in this block (and column) that the 6 can go,
9 in cell (5,4) because this is the only cell in this block (and row) where the 9 can be placed,
4 in cell (6,9), the only cell in this block, row and column that 4 can be placed and, similarly,
4 in cell (8,5).
Figure 3
We can now remove:
all the other 6's in row 2 and all the other 9's in column 4 (see Figure 4).
After finding all the available single candidates, we can start tackling all the naked pairs.
Looking at Figure 4, the cells (4,1) and (5,1) in column 1 have the same two candidates 2 and 6, forming a naked pair. This means that the two cells can be the only place for 2 and 6 in the same
column and the same box. As a result, the options 2 and 6 can be removed from the candidates of the other cells in the same column and the same box. So 2 can be removed from cell (1,1), and 2 and 6
can be removed from cell (8, 1). As cells (4,1) and (5,1) belong to box 4, the candidates 2 and 6 can also be removed from cells (5,2) and (5,3). Likewise, the candidate 2 can be removed from cell
Figure 4
The naked pair with the candidate numbers of 2 and 6 was easy to spot. However, a naked pair can often be found hiding as a "hidden pair" among other redundant candidate numbers. There is one such
example in Figure 4. We can find the numbers 1 and 7 in box 7 and box 9 and in both the last two rows. This means that no other empty cell in the last 3 rows, except the cells (7,4) and (7,5) can
contain the naked pair of 1 and 7 as a solution.
When we are in a hurry, especially during a competition, we tend to pencil in redundant options. For example in Figure 3, the two cells (7,4) and (7,5) contain 2, 3 and 9 as well as 1 and 7 which, as
shown, are the only two candidate numbers. This results in a change from 1 and 7 being a naked pair to a hidden pair. Hence it is a good practice to make a note, for example by circling the redundant
clue numbers (Figure 4). The same rule can be applied to instances of more than two cells, e.g. "naked triplets" and "naked quads".
In row 6, the only position possible for a 2 is (6, 4). See the paragraph above Figure 4 if you cannot see why the 2 in (6,2) cannot be used. This means that the three cells (6,2), (6,6) and (6,8) in
row 6 form a naked triplet with the candidate numbers 5, 7 and 8.
The three cells (7,7), (7,9) and (9,9) in box 9 form another naked triplet with the candidate numbers 2, 3 and 9. Hence the redundant options 2 and 3 can be removed from cell (9,7). Similarly, the
redundant options 3 and 9 can be removed from cell (8,8). As a result, the cells (9,7) and (8,8) form a new naked pair with the candidate numbers 5 and 6.
Finally, the three cells (1,8), (2,8) and (3,9) in box 3 form a naked triplet with the candidate numbers 1, 3 and 9. This means the redundant option 3 can be removed from cells (1,7) and (2,7)
forming a naked pair with the candidate numbers 4 and 8.
A puzzle consisting of only single candidates and naked pairs should be classified under the easy category. After all the redundant candidates in the empty cells are removed by the technique of
"naked pair" new single candidates begin to appear in the puzzle.
The rest of the puzzle can be easily solved by basic techniques. I leave the rest of the solution to the readers.
A second sudoku article can be found here.
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Find an efficient procedure to compute the discrete cosine & sin transforms using fft
October 5th 2011, 05:36 PM
Find an efficient procedure to compute the discrete cosine & sin transforms using fft
Can't figure this out to save my life. Please help if possible
Consider the discrete cosine and sine transforms of a real sequence of numbers
${x_j, j=0,....,2N-1}$
$C_k = \sum_{j=0}^{2N-1} x_j cos(\frac{jk\pi}{N}),$
$S_k = \sum_{j=0}^{2N-1} x_j sin(\frac{jk\pi}{N})$
Find an efficient procedure to compute $C_k, S_k$ using a Fast Fourier Transform of $\{x_j, j=0,...2N-1\}$.
Can someone give me a hint/explain to me the way to go about doing this problem? I am completely lost. Thanks!
October 6th 2011, 10:56 PM
Re: Find an efficient procedure to compute the discrete cosine & sin transforms using
Can't figure this out to save my life. Please help if possible
Consider the discrete cosine and sine transforms of a real sequence of numbers
${x_j, j=0,....,2N-1}$
$C_k = \sum_{j=0}^{2N-1} x_j cos(\frac{jk\pi}{N}),$
$S_k = \sum_{j=0}^{2N-1} x_j sin(\frac{jk\pi}{N})$
Find an efficient procedure to compute $C_k, S_k$ using a Fast Fourier Transform of $\{x_j, j=0,...2N-1\}$.
Can someone give me a hint/explain to me the way to go about doing this problem? I am completely lost. Thanks!
What is the real part of the DFT of a real sequence? What is the imaginary part?
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A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence
Advances in Numerical Analysis
Volume 2013 (2013), Article ID 687382, 7 pages
Research Article
A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence
Department of Environmental Health Science, University of Swaziland, P.O. Box 369, Mbabane H100, Swaziland
Received 25 October 2012; Revised 25 December 2012; Accepted 17 February 2013
Academic Editor: Michele Benzi
Copyright © 2013 Ababu Teklemariam Tiruneh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to
the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly
large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the
iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method
is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without
resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on
Newton method has better convergence characteristics than the traditional Newton method.
1. Introduction
Iterative procedures for solutions of equations are routinely employed in many science and engineering problems. Starting with the classical Newton methods, a number of methods for finding roots of
equations have come to exist, each of which has its own advantages and limitations. The Newton method of root finding is based on the iterative formula: Newton’s method displays a faster quadratic
convergence near the root while it requires evaluation of the function and its derivative at each step of the iteration. However, when the derivative evaluated is zero, Newton method stalls. For low
values of the derivative, the Newton iteration offshoots away from the current point of iteration and may possibly converge to a root far away from the intended domain. For certain forms of
equations, Newton method diverges or oscillates and fails to converge to the desired root. In addition, the convergence of Newton method can be slow near roots of multiplicity although modifications
can be made to increase the rate of convergence [1].
Modifications of the Newton method with higher order convergence have been proposed that require also evaluation of a function and its derivatives. An example of such methods is a third order
convergence method by Weerakoon and Fernando [2] that requires evaluation of one function and two first derivatives. A fourth order iterative method, according to Traub [3] also requires evaluation
of one function and two derivative evaluations. Grau-Sánchez and Díaz-Barrero [4] gave a compositing of function evaluation at a point and its derivative to improve the convergence of Newton’s method
from 2 to 4. Recently other methods of fifth, sixth, seventh, and higher order convergence have been proposed [5–11]. While higher order Newton methods ensure faster convergence, their stability for
certain equation forms may have the same problem similar to that of the traditional Newton method.
The secant method does not require evaluation of derivatives. However, the rate of convergence is about 1.618, the convergence may be a problem for some forms of equations, and the secant method may
fail to converge in those cases. Muller’s method is an extension of the secant method to a quadratic polynomial [12]. It requires three functional evaluations to start with but continues with one
function evaluation afterwards. The method does not require derivatives and the rate of convergence near the root is superlinear, that is, about 1.84. However, Muller’s method can converge to a
complex root from an initial real number [13]. Muller’s method also requires the three points to be distinct. If two of the points coincide, the method degenerates to secant method.
Hybrids of methods are also used to provide stability and ensure convergence to a desired root. For example, Newton or secant methods can be combined with bisection to bracket the root by a small
interval so that a good initial guess is available for applying Newton method with quadratic convergence. Dekker’s method [14] combines bisection method with that of secant method. The method starts
by bracketing the root between two initial points that have functional values opposite in sign. The secant estimate of is compared with the bisection of the interval, the one estimate resulting in
small interval with the point of smaller magnitude functional value is chosen, and the iteration continues. Brent’s method [15] is a root finding algorithm that combines root bracketing, bisection,
and inverse quadratic interpolation. It is a modification of Dekker’s method to avoid slow convergence when the difference between consecutive estimates of is arbitrarily small. In such cases
bisection is used for the next root estimate.
The Leap-frogging Newton method [16] uses the Newton method as an intermediate step for the estimation of the root followed by the application of the secant method using the initial point and the
intermediate point found by Newton method. The method has cubic convergence and works in certain pathological cases where Newton method may fail. However, being dependent on Newton method for the
intermediate step, the method may suffer from the same drawbacks of using the traditional Newton method mentioned above.
2. Method Development
It will be shown that the iterative formula for the two-point method will take the form The method starts by selecting two points lying on a curve , namelyand. A line connecting the two points
is drawn from the pointto the pointas shown in Figure 1. A new variable(the cotangent of the angle between this line and the vertical) is defined so that In general, for any other point
lying on the curve, the variableis defined so that: Next, Newton method of finding roots will be applied takingas the independent variable and as the dependent variable. Denoting that is the
estimate of the root for whichfrom the Newton method with the corresponding value of and applying Newton method gives Substituting for and the equivalent expressions in terms of and values,
namely, into (5) above gives the following expression: The derivative is evaluated from and using the formula The evaluation of the derivative in turn gives Therefore, can now be written in terms of
, and as follows: Substituting the above expression for in the equation gives Solving for the root estimate and further rearranging results in Continuing the iteration using the above formula, for
the th step of the iteration, theth estimate of the root will take the form Denoting by will result in the expression
3. Proof of Super-Quadratic Convergence
Recalling the iteration formula of (14), Defining the error at the th iteration to be , whereis the root of the equation desired, the errors at the th and th iteration are also defined similarly
as follows: The iteration formula in (14) can now be rewritten in terms of the error terms as follows: Expanding the , and , terms about the root using Taylor series expansion, where for.
The Taylor series expansions ofand about are formulated similarly as follows: The algebraic expression of (18), after substituting the previous Taylor series forms, was simplified using MATLAB
program. After, the fourth order error terms in and were discarded and the error terms of the denominator were also discarded compared to the dominant term of , the resulting expression will be as
follows: Therefore, the error sequence has the simplified expression Defining positive real terms and so that Near the root, the above expression approaches the constant terms in. Therefore, the
power of the error term shall approach zero: The positive solution ofis the one for which the error sequence converges to zero. Therefore, the iteration formula of (14) near the root has a
convergence of order 2.414 which is super-quadratic.
4. Stability of the Proposed Method
The proposed method shows stability near points where application of the traditional Newton method may result in oscillation, divergence, or offshooting away from the desired root or offshooting to a
possibly invalid domain where the function may not be defined. This stability characteristic may be studied by examining the iteration formula again which takes the following form: where the
variableis defined as The estimate for the root at the th iteration, , therefore, can be taken as the weighted sum of the and values at the th and th iteration, respectively. The weighing factors
are () for and () for . When the method converges to the root, the value of approaches unity. This is found by examining the behavior of near the root and noting that approaches 0 near the root; that
is, The iterative formula will, near the root, approach the following expression: The iteration, therefore, moves away from with a weighing factor approaching zero and gives weighing factor of 1 to .
Near points where the derivative of the function may approach zero, thevalue approaches infinity and the root estimate weighs heavily in favor of rather than . This can be seen from the following
limit evaluation of : The iteration, therefore, moves away from with a weighing factor approaching zero and gives weighing factor of near to 1 to .
Near points where the derivative of the function may approach zero, Newton method typically displays oscillation or offshooting behavior. The proposed two-point modified Newton procedure however
shows stability without displaying oscillation and offshooting tendencies. For example, for the function , the derivative of(i.e., ) is zero at . Starting Newton method near this point, say ,
will result in offshooting of the iteration to a different domain.
As shown in Figure 2, for a starting pointwhere the function has near zero derivative, the traditional Newton method of shoots to a root further from the nearest root of . The Newton method as
such converges to . On the other hand, the proposed two-point Newton method shows stability and converges to the nearest root of .
Another example of the stability of the proposed method is when Newton method diverges instead of converging to the desired root. Figures 3 and 4 show comparison of the iterative values offor the
proposed method and the traditional Newton method for the function which is often used to illustrate the pathological condition with respect to application of Newton’s method which leads to an
oscillating divergence to infinity. By contrast the proposed method displays an oscillating convergence to the desired root albeit with slower rate of convergence of order 1.0.
5. Application Examples
Equations used to test efficiency of root finding methods are used here to evaluate the number of iterations required to reach to a specified level of convergence (Table 1). The stopping criterion
used for the iteration process is given by The rate of convergence towards the root for each step of the iteration is evaluated using the formula Table 1 shows comparison of the proposed two-point
Newton method with the Newton and secant methods for a number of equations used to test efficiency of root finding methods elsewhere. A super-quadratic convergence with which the proposed method
converges to the root is mostly evident withvalues being close to 2.414 during most of the iterations. It can also be seen from Table 1 that a less number of iterations are required to reach
convergence for the proposed method than those required for Newton and secant methods. For example, for equations with multiple roots such as , shown in Table 1, Newton method displays linear
convergence while the proposed method converges super-quadratically with an order of 2.414. Similar order of convergence is also observed for the equation .
5.1. Examples Where the Proposed Method Works While Newton Method Fails
The advantage of the use of the proposed two-point Newton method is shown for cases where the Newton method and in several cases also the secant method fail to converge to the root. Table 2 shows the
results of the iteration for several examples of equations. In all of the examples listed, Newton method fails to converge whereas the proposed two-point Newton method converges. Secant method also
fails to converge in several of the examples cited in Table 2. For example, in the case of, Newton method starting with fails with the second iteration because the estimated value is a negative
number whose logarithm is undefined. For the caseNewton method always diverges to increasingly large number for any startingvalue while the proposed two-point Newton method converges
super-quadratically near the root. For, Newton method also diverges whereby each iteration gives an estimate of the root which is twice the previous value and with alternate signs leading to
oscillating divergence. On the other hand application of the proposed two-point modification of Newton method leads to oscillating convergence at somehow reduced rate of convergence (of order 1 as
explained above).
For the equation, both of Newton and secant methods continue to diverge to increasingly large values while the proposed two-point Newton method shows stability of convergence. For the polynomials
of different degrees cited in the Table 2, Newton and secant methods display oscillation for the starting values shown in the Table 2 while the proposed two-point Newton method displays stability and
super-quadratic convergence for iteration near the root.
6. Conclusion
A numerical procedure of root finding using two-point modification of Newton method has been presented. It is proved that the method has a super-quadratic convergence of order about 2.414. The method
is based on application of Newton iteration formula by taking as the independent variable the cotangent of the angle between the line connecting the two successive points of iteration with the
vertical and as the dependent variable the given function . The resulting iteration formula for root estimation is shown to be the weighted sum of the estimates of the two previous iterations with a
weighing factor that penalizes the iteration point having undesirable characteristics such as a near zero derivative. For example, near a point where the derivative is zero, the weighing factor for
that point will be near zero effectively moving the iteration away from that undesirable point.
Application examples have been given to demonstrate that the proposed method requires fewer number of iterations for convergence to a root than the traditional Newton and secant methods. The method
offers a particular advantage for cases where the traditional Newton method and its variants of various order convergence may not converge. A number of examples are given where the proposed method
converges to a root in a stable manner without oscillation, divergence, or offshooting from the desired domain of the root, whereas the Newton method and in several of the cases also the secant
method display undesirable behaviors which prevent convergence to the desired root.
In terms of computational effort, the proposed method requires one function evaluation and one derivative evaluation at each step of the iteration except for the first step where two functional
evaluations are required. The proposed method, therefore, requires essentially the same number of functional evaluations as the traditional Newton method while offering a super-quadratic convergence
of order 2.414.
1. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, 5th edition, 1994. View at Zentralblatt MATH
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4. M. Grau-Sánchez and J. L. Díaz-Barrero, “A technique to composite a modified Newton's method for solving nonlinear equations,” Annals of the University of Bucharest, vol. 2, no. 1, pp. 53–61,
2011. View at Zentralblatt MATH · View at MathSciNet
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Scholar · View at Zentralblatt MATH · View at MathSciNet
7. J. Kou and X. Wang, “Sixth-order variants of Chebyshev-Halley methods for solving non-linear equations,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1839–1843, 2007. View at
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13. W. R. Mekwi, Iterative methods for roots of polynomials [M.S. thesis], University of Oxford, 2001.
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London, UK, 1969. View at Zentralblatt MATH
15. R. P. Brent, Algorithms for Minimization without Derivatives, chapter 4, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973. View at MathSciNet
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twin clocks-is it acceleration?
Thanks. Do you have some links on Fermi-normal coordinates in uniformly accelerated frames and Minkowski space-time? The wiki says it is just a generalization of Rindler for curved space-times, so I
assumed that for accelerated frames in flat space time they yield the same results.
For a beautiful derivation that Rindler coordinates are radar coordinates, our very own George Jones recently produced:
From the wikipedia article, the metric for Rindler coordinates is:
ds^2 = - g^2 x^2 dt^2 + dx^2 ...
I don't know of a good online reference for Fermi-Normal coordinates of a uniformly accelerated observer in SR. However, section 6.6 of MTW derives this (without calling it that). This section is all
flat spacetime. The metric becomes:
ds^2 = -(1+ g x)^2 dt^2 + dx^2 ...
Note how this form approaches Minkowski metric for x=0. For Rindler coordinates x=0 represents the horizon. Also note that if you just transform Rindler to have its x=1 line become new x=0, you still
don't get Fermi Normal; you get:
ds^2 = -(g+gx)^2 dt^2 + dx^2 ...
What you need is x' = x - 1/g to get from Rindler to Fermi-Normal.
[Edit: On further thought, it appears that Fermi-Normal coordinates as above, for a uniformly accelerating observer would still have the same simultaneity surfaces as Rindler and Radar - just labeled
differently. Thus, it seems, you need
in acceleration - e.g. the classic twin with slightly rounded turnaround - to expose the difference between Radar simultaneity and Born Rigid simultaneity (which is what Fermi-Normal uses).]
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Regular boundary point
From Encyclopedia of Mathematics
A point Dirichlet problem in the sense of Wiener–Perron (see Perron method) takes the boundary value
The regular boundary points of thin set; the set Irregular boundary point) is a polar set of type
For barrier (a function
and let capacity of the set
diverges, or for
diverges, where
(Wiener's criterion).
[1] M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" Uspekhi Mat. Nauk , 8 (1941) pp. 171–232 (In Russian)
[2] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[3] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)
The polarity of the set of irregular boundary points is contained in the Kellogg–Evans theorem. See, e.g., [a1] for irregular boundary points in abstract potential theory.
[a1] J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a2] H. Lebesgue, "Sur des cas d'impossibilité du problème de Dirichlet ordinaire" C.R. Séances Soc. Math. France , 41 (1913) pp. 17
[a3] H. Lebesgue, "Conditions de régularité, conditions d'irrégularité, conditions d'impossibilité dans le problème de Dirichlet" C.R. Acad. Sci. Paris , 178 (1924) pp. 349–354
[a4] N. Wiener, "The Dirichlet problem" J. Math. Phys. , 3 (1924) pp. 127–146
[a5] M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)
[a6] J. Wermer, "Potential theory" , Lect. notes in math. , 408 , Springer (1981)
How to Cite This Entry:
Regular boundary point. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Regular_boundary_point&oldid=15319
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098
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Portability GHC
Stability unstable
Maintainer stephen.tetley@gmail.com
Approximate glyph size calculations for Label's and their bounding boxes.
Calculations are based on metrics derived from the Courier font. As Courier is a monospaced font, applying these metrics to other font families will usually produce over-estimates (bounding boxes
will be longer than the true visual length of the text). Furthermore, even italic or bold Courier will have different metrics.
This is a deficiency of Wumpus, and limits its text handling capabilities - for example, text cannot be reliably centered or right aligned as its true length is not known. However, more powerful
alternatives would need access to the metrics embedded within font files. This would require a font loader and add significant implementation complexity.
Type synonyms
data AfmUnit Source
Wrapped Double representing 1/1000 of the scale factor (Point size) of a font. AFM files encode all measurements as these units.
Eq AfmUnit
Floating AfmUnit
Fractional AfmUnit
Num AfmUnit
Ord AfmUnit
Real AfmUnit
RealFloat AfmUnit
RealFrac AfmUnit
Show AfmUnit
Tolerance AfmUnit
Scaling values derived from Courier
mono_x_height :: AfmUnitSource
The ratio of x height to point size of a letter in Courier.
This is also known as the "body height".
mono_x_height = 426
mono_left_margin :: AfmUnitSource
The left margin for the bounding box of printed text as a ratio to point size for Courier.
mono_left_margin = -46
Courier metrics
textWidth :: FontSize -> CharCount -> DoubleSource
textWidth : font_size * char_count -> PtSize
Text width at the supplied font_size. It is expected that the char_ount has been calculated with the charCount function.
NOTE - this does not account for any left and right margins around the printed text.
Size calculation
textBounds :: FontSize -> DPoint2 -> String -> BoundingBox DoubleSource
textBounds : font_size * baseline_left * text -> BBox
Find the bounding box for the character count at the supplied font-size.
The supplied point represents the baseline left corner of the a regular upper-case letter (that is without descenders). The bounding box adds a margin around all sides of the text.
The metrics used are derived from Courier - a monospaced font. For proportional fonts the calculated bounding box will usually be too long.
charCount :: String -> CharCountSource
charCount : string -> CharCount
Count the characters in the supplied string, escaping the string as necessary.
Escapes count as one character - for instance, the length of this string:
... is 6.
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MathGroup Archive: April 2010 [00310]
[Date Index] [Thread Index] [Author Index]
Re: beginner question about syntax
• To: mathgroup at smc.vnet.net
• Subject: [mg109088] Re: beginner question about syntax
• From: Mike Bryniarski <melondisco at gmail.com>
• Date: Mon, 12 Apr 2010 22:59:42 -0400 (EDT)
• References: <hputp3$lh7$1@smc.vnet.net>
For map to work you must be mapping over the elements of an
expression, be it a list like {a,b,c} or a generic expression like
in your first example :
Map[Function[x, x^2], a]
a is an atomic symbol, Map normally tries to apply the function to
level one of the expression but "a" has no level 1 this fails, what
you would want to use is:
Function[x, x^2][a]
In your second example what is going on is that a+b+c is really seen
by Mathematica as Plus[a,b,c],
Map[Function[x, x^2], a + b + c]
is really
Map[Function[x, x^2],Plus[a,b,c]]
and the function is applies to each element of the Plus statement
while keeping Plus as the "head" of the expression
an intermediate step might look something like:
Plus[Function[x, x^2][a],Function[x, x^2][b],Function[x, x^2][c]]
which becoms:
On Apr 12, 6:48 am, AK <aaa... at googlemail.com> wrote:
> Hi,
> I'm a fairly seasoned user of another system who's just started using
> Mathematica. Although at the moment I'm just playing around with
> Mathematica (without any specific task at hand), trying to figure out
> the Mathematica way of doing things from the documentation
> (particularly the examples) there are some things I can't seem to wrap
> my head around. For example, can anyone explain the outputs for the
> inputs below:
> In[1]:= Map[Function[x, x^2], a]
> Out[1]:= a
> In[2]:=Map[Function[x, x^2], a + b + c]
> Out[2]:= a^2 + b^2 + c^2
> If I enclose the second argument of Map[] inside a list, I get the
> expected output, but I don't understand what the operations given in
> the example above represent and why the outputs are what they are.
> Would appreciate an explanation for what's going here... thank you in
> advance.
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[IPython-User] questions about latex rendering in ipython with sympy
[IPython-User] questions about latex rendering in ipython with sympy
Fernando Perez fperez.net@gmail....
Mon Jan 9 20:41:28 CST 2012
On Mon, Jan 9, 2012 at 6:27 PM, Comer Duncan <comer.duncan@gmail.com> wrote:
> Hi,
> I am a recent user who wants to use ipython .12 to write notebooks. I have
> ipython installed and working on my macbook pro. I am beginning to write a
> notebook and like very much the flexibility it allows. Thanks to the
> developers.
> The notebook I am working on uses sympy. I note that the sympy example
> renders equations. However, in my humble view the quality of the rendered
> equations is minimal. Mathjax does a much better job I think. I have
> cribbed the sympy example as a starter for building the notebook. I am
> wondering whether ipython works with CDN Mathjax rather than the currently
> working (but not particularly pretty) equation renderings used in ipython?
> I have a separate blog which uses Mathjax and like its features.
> Thanks for any help you can provide. Sorry for the primitive nature of my
> concerns!
I'm sorry but I'm not sure what can be going on, as the notebook does
use mathjax to render its equations as well. The sympy objects simply
return a latex expression that is put into html, and we let mathjax
finish rendering it. So I don't quite understand what differences you
are seeing between the rendering of the notebook and mathjax output
Perhaps a screenshot of the rendering you consider sub-par would help
us figure it out.
More information about the IPython-User mailing list
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Recent Applications of Mathematics
up vote 31 down vote favorite
What are the recent and new applications of Mathematics in other Sciences ?
Let me try to be more precise about the question:
• By "recent" I mean the last 15 years.
• By "new" I want to exclude the standard answers like cryptography or finance
• By "applications" I mean a mathematical concept (or even a trick) successfully used in another field (preferably not Theoretical Physics) to solve a problem or to shed a new light on a
• I would prefer to see recent applications of modern Mathematics, but new applications of classical results should be considered as a valid answer too.
• the answer should not simply be "XXX was successfully applied to YYY". it should contain a short explanation of the mathematical concept involve, and a description of the problem/phenomenon it
The typical example I have in mind does not strictly answer the question (first it is an application of Theoretical Physics to condensed matter, and then I am not giving the required details): it
seems that some 2d quantum field theories which had a priori no physical meaning were successfully used to understand properties of graphene (I think this was really unexpected: 2d conformal field
theory was considered as a toy model to approach the understanding of more relevant field theories in higher dimension).
v2: I hope this version is better than the previous one.
soft-question big-list applications
6 From his web page, the OP knows a good deal more about this than the people he is asking. – Will Jagy Apr 24 '11 at 22:19
7 I cast a vote against closing - although I am instinctively not keen on the question, it seems a reasonable one with decent criteria for what would constitute a good answer. (It would help to give
some idea of what counts as "recent", though) – Yemon Choi Apr 24 '11 at 22:32
4 @Ryan, the example Damien mentions is actually a very dramatic and unusual case; not just what you might typically see picking up a random physics or chemistry journal. – Scott Morrison♦ Apr 24
'11 at 22:41
8 I also vote against closing (even though somehow the current phrasing was initially off-puting). Remember the new procedure; if you want to vote to close, you instead need to write a comment
cancelling one of Yemon's, André's or my "anti-votes". – Scott Morrison♦ Apr 24 '11 at 22:43
7 The question is undoubtedly interesting, but too vague, and I don't think that the OP has made the necessary effort to clarify the question. Usually, vague questions do not generate very good
answers. – Thierry Zell Apr 24 '11 at 23:16
show 8 more comments
12 Answers
active oldest votes
(Disclaimer: I am writing based on what I remember from one seminar that I heard given by Alain goriely, and so claim responsibilities for all inaccuracies!)
Basic differential geometry has been applied to the problem of protein folding and dynamics of protein in an interesting way (and this is fairly new work from what I gathered). Here I give
one simple example:
A little bit of remedial biology in case we forgot: a protein molecule is formed by one or more polypeptide chains. Each chain is made of building blocks called amino acids joined together
by peptide bonds. As its name suggests, a polypeptide chain is just a long string of amino acids chained end to end. What determines the shape of protein molecules is the individual amino
acids. Roughly speaking, each building block (amino acid) is formed by a backbone (something common to all amino acids), together with one or more things than hangs off the backbone. The
backbone gives the initial chain like structure of the polypeptide chain. The interaction between the things hanging off the backbone, and between the things and the surrounding
environment, is what drives the dynamical folding of the protein giving its final shape.
up vote 11
down vote For traditional protein dynamics, or for traditional storage of protein structures, what they do is they take the numerically computed (or experimentally observed) protein structure, and
accepted define a map $\pi$. The map $\pi(n)$ roughly gives the (relative) spatial position of the $n$th amino acid in the chain. Separately there is also a map $\nu(n)$ which gives the orientation
of the amino acid, and what is hanging off the backbone there.
For some less precise dynamical computations, going through the whole list of all positions can be computationally intensive (a protein can have upwards of tens or hundreds of thousands of
amino acids), without being particularly accurate. On the other hand, the basic larger-scale structure of polypeptide chains are fairly well known (classical in the biology literature),
and includes things like alpha-helices, beta-pleats, and turns. These three most common structures are all well-approximated by constant torsion and constant curvature space-curves.
Therefore, a computationally less demanding way of storing the approximate structure of the protein backbone would be to decompose the folded structure into its "secondary structures",
approximate each of those by these space-curves (each can be parametrized completely by the torsion, curvature, total length, starting position, and starting direction). This allows better
memory use and faster computations for certain numerical simulations of protein dynamics.
1 This is a remarkable breakthrough and I wish I could find some good review articles on the state of the art.The current research literature is very disorganized and difficult to ferret
out by the usual search engine means.Any suggestions? – Andrew L Apr 25 '11 at 3:11
UPDATE:I DID find this article at the archive-but as far as I know,it's the only one I could find so far:arxiv.org/PS_cache/arxiv/pdf/0809/0809.2079v1.pdf – Andrew L Apr 25 '11 at 3:22
7 FWIW I believe linking-to-the-arXiv orthodoxy is to link to the main page of the article and not the PS_cache copy of a particular version of it. arxiv.org/abs/0809.2079 – anon Apr 25
'11 at 7:49
Unfortunately, my shallow knowledge on the subject all came from attending one seminar. You can try e-mailing professor Goriely if you want to learn more. – Willie Wong Apr 25 '11 at
Thanks a lot for this very nice answer. – DamienC Apr 26 '11 at 7:25
add comment
The new area of discrete differential geometry is solving problems in computer graphics, such as creating more lifelike hair in animation. I became aware of this from an article in the New
York Times (see http://www.nytimes.com/2010/12/30/movies/30animate.html). An excerpt: "... the images on screen are not the result of a patchwork of technical tricks, but of precise
mathematical equations based on the way the world actually looks and operates — in a word, physics. They use what is known as discrete differential geometry, a field so new that the first
up vote textbook on the subject was published only two years ago. [...] The uses of discrete differential geometry go far beyond animation. Johns Hopkins Medical Center, for instance, is using Mr.
16 down Grinspun’s computer simulations to predict how needles move through human flesh, so that doctors can train to do laparoscopic surgery on virtual bodies instead of the real thing."
Here is a link to a book: http://www.amazon.com/Discrete-Differential-Geometry-Graduate-Mathematics/dp/0821847007.
add comment
The AMS article is pretty good: http://www.ams.org/happening-series/hap7-pixel.pdf
up vote 11 down vote Basically, some new math results show how to combine a lot of 1-pixel sensor readings into a complete picture of something that would normally be done with a sensor array in a
1 Agreed - the AMS article is indeed VERY good. – DamienC May 4 '11 at 10:23
add comment
Space-time codes (or why your router has two antennas): http://en.wikipedia.org/wiki/Space_time_code
up vote 10 down vote
Thanks a lot for your answer - it would be nice to explain a bit more what are the math involved. – DamienC May 4 '11 at 10:20
add comment
How about applications of discrete complex analysis to statistical physics? There was a surge of work this past decade on the subject, such as proofs of conformal invariance of 2-D models
(Ising, Potts, Spinglass, O(n),etc.). Before, there were mainly unrigorous physics arguments to prove the various facts involved, such as the value of the Honeycomb Constant. The machinery
up vote 9 of SLE and discrete complex analysis has been extremely insightful in the proofs involved. Much of the methodology is based on the foundational work done by Onsager and Baxter decades
down vote before.
add comment
There are applications of homology theory both to Topological Data Analysis and other parts of Applied Topology (work of Gunnar Carlson and his coworkers in Stanford), via the CHOMP project
to the structure of materials, dynamics of Evolution of Pattern Complexity during Phase Separation, etc. (The Stanford webpage mentions many more applications and they really merit a
up vote 8 mention but I leave the 'reader' the joy of browsing around the links there.) There is also work by Robert Ghrist again on Applied Topology.
down vote
The work of Ghrist might deserve its own separate answer, as some of it seems to tackle very different types of applications. – Thierry Zell Apr 25 '11 at 15:46
You are probably right, so why not add a summary as a separate answer. – Tim Porter Apr 26 '11 at 7:03
add comment
Perhaps the following fits the bill for recent / modern / etc.:
up vote 5 down vote Applications of Random Matrix Theory to Economis, Finance, Political Science
add comment
(From SIAM web site.) The behavior of a dynamical system is determined by its "skeleton," which consists of the different attractors (steady states, periodic solutions, or more complicated
sets), as well as saddle-type objects with their global stable and unstable manifolds. Global manifolds are complicated objects that must be found numerically. They are hypersurfaces
up vote consisting of infinitely many trajectories that end up (or come from) a saddle-type object. All other trajectories qualitatively follow the dynamics given by the "nearest" global manifolds.
5 down This feature was recently utilized in the Genesis Mission, which sent a spacecraft to a saddle-type periodic orbit around the Lagrange point between the earth and the sun to collect solar
vote dust particles. The spacecraft traveled on global manifolds to its destination and back to earth virtually without any fuel.
add comment
Applications of non-Archimedean analysis (a p-adic analog of the symmetric stable process and the corresponding heat-like equation) to dynamics of complex systems like proteins or spin
glasses. See http://www.worldscinet.com/brl/03/0303/S1793048008000836.html
or http://iopscience.iop.org/0305-4470/35/2/301
Applications of fractional calculus (a branch of classical analysis dealing with fractional derivatives and integrals) to diffusion phenomena in disordered systems. See a survey paper
up vote 4 with an interesting title "The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics":
down vote
Mathematically, this is (in the simplest case) a kind of a theory of parabolic equations with a fractional time derivative.
add comment
Statistics applied to microarray data in biology. And Bernd Sturmfels and his students have been applying algebraic geometry to this. He wrote a book titled Algebraic statistics for
computational biology. Biology is a field that will explode in coming decades. Advances in that field will probably capture the public imagination the way physics did in the 20th century.
The next Einstein could be a biologist.
up vote (Does any of the empirical data used in any research papers published in that field have any validity? That's something that could bear inspection. People put vast numbers of large data sets
4 down on the internet, and others base research papers on them. But before they do that they transform raw numbers by, for example, raising everything to the power 3/2. If you later ask them
vote specifically how they transformed it, they may be blindsided by the question and think you're making a strange unusual request that they'd never have expected in a million years. People
don't normally ask such weird things. At least such has been my impression.)
add comment
Gröbner bases give a nice way to solve many problems in computer vision and robotics. Often these problems involve solving systems of polynomial equations eg. those describing the
projection of a point in 3D on the 2D plane of a camera sensor or those describing how the position of a robot component depends on the details of a sequence of articulated joints. Here's a
up vote 3 paper that uses such methods to investigate the minimum amount of information to infer properties of cameras, among other things.
down vote
add comment
Pseudodifferential (PDO) and Fourier integral operators (FIO) and also Wick and antiWick operators have applications in signal processing, especially in analysis of non stationnary signals
like speech signals. Examples of such time frequency representation are spectrograms, scalograms which use wavelet theory and similar tools like Gaborets, chirplets, ridglets,...etc. This
up vote 0 tools are also used to solve PDEquations as they serve to almost diagonalization of a large class of PDO and FIO.
down vote
add comment
Not the answer you're looking for? Browse other questions tagged soft-question big-list applications or ask your own question.
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MathGroup Archive: October 2013 [00015]
[Date Index] [Thread Index] [Author Index]
Re: Error in slope and intercept
• To: mathgroup at smc.vnet.net
• Subject: [mg131781] Re: Error in slope and intercept
• From: Ray Koopman <koopman at sfu.ca>
• Date: Thu, 3 Oct 2013 00:36:00 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• Delivered-to: l-mathgroup@wolfram.com
• Delivered-to: mathgroup-outx@smc.vnet.net
• Delivered-to: mathgroup-newsendx@smc.vnet.net
On Wed, Oct 02, 2013 @ 02:53 AM, Gebbo <nicolasgebbo at googlemail.com> wrote:
> I was informing myself about linear regression with
> error in both observables and found this algorithm:
> (Weighted Orthogonal Regression by Alternating Least Squares)
> worals[x_, y_, sx_, sy_] := Block[
> {a,b,f,z, u = 1/sx, v = 1/sy, w = (sy/sx)^2},
> {a,b} = (y*v).PseudoInverse@{v,x*v}; f = #.#&[(a+b*x-y)v];
> While[f > (z = (x*w + (y-a)b)/(b^2 + w);
> {a,b} = (y*v).PseudoInverse@{v,z*v};
> f = #.#&@Join[(z-x)u,(a+b*z-y)v])];
> {f,{a,b}}]
> which gives me {chisquare, {intercept, slope}} as output.
> This works prefectly fine but I'd like to get the error on
> slope and intercept. I don't understand enough about Mathematica
> or the theory behind this algorithm, so i would pref the solution
> or hinds our i can write it myself.
> Thanks Gebbo
The model is {x = z + d, y = a + b*z + e}, where d & e are random errors,
and a, b, & z are unknowns for which values are to be found that minimize
the weighted sum of squares f = #.# & @ Join[(z-x)/sx,(a+b*z-y)/sy].
We start with z = x, then minimize f alternately with respect to either
{a,b} or z with the other held constant, until f no longer changes.
To estimate the error in {a,b}, I usually jackknife the solution.
n = Length@x; {f,ab} = worals[x,y,sx,sy];
{jab,jc} = {ab + (n-1)(ab-Mean@#), (n-1)^2/n Covariance@#}& @
jab is the jackknifed estimate of {a,b}, and jc is the jackknifed estimate
of its covariance matrix; Sqrt@Diagonal@jc gives the estimates of the two
standard errors.
See also http://forums.wolfram.com/mathgroup/archive/2009/Feb/msg00478.html
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We learn so early in life about +, −, × and ÷ that we tend to see these operations as the unique foundation stones of arithmetic, on which everything else must be built. But there are other operators
we can apply to pairs of numbers. In a paper recently posted on the arXiv, Shinji Tanimoto of Kochi Joshi University asks if there might be some operation that lies between addition and
Here’s what between means. Suppose the operation exists, and assign it the symbol ◊. (Tanimoto chooses a different symbol, but that one is harder to encode in HTML.) Now, for all positive a and b:
if a+b < a×b, a+b < a ◊ b < a×b;
if a+b > a×b, a+b > a ◊ b > a×b;
if a+b = a×b, a+b = a ◊ b = a×b;
Can we actually find an operation that has these properties? As a strategy for searching, Tanimoto suggests looking at the various definitions of a mean. Associated with addition is the arithmetic
mean, defined as (a+b)/2. Likewise multiplication has the geometric mean, √ab. The hypothetical new operation ◊ should also have an associated mean, which for all a and b would lie between the
arithmetic and the geometric means of a and b. And such a mean does exist! It was studied at length by Carl Friedrich Gauss, who called it the arithmetic-geometric mean, or AGM.
The AGM is defined as the limit of an iterative process:
function agm(a, b) =
if a == b return a
else return agm( (a+b)/2, sqrt(a*b) )
Viewed as a computer program, this is one of those weird malformed algorithms that ought to run forever but actually—if a and b are finite-precision floating-point numbers—returns a value quite
promptly. Gauss proved that the iterates of a and b converge on the same value; furthermore, that value is always between the arithmetic and the geometric means (in the sense of “between” given
So now we have a mean that lies between the arithmetic mean and the geometric mean. How do we get from there to a binary operator ◊ that interpolates between addition and multiplication? Consider the
following two identities, where AM is the arithmetic mean and GM is the geometric mean:
a+b = AM(a, b) + AM(a, b)
a×b = GM(a, b) × GM(a, b)
These equations can be taken as definitions of the + and × operators; in other words, we can define addition and multiplication as the unique operations that make the identities valid. And we can
write the same kind of equation for the ◊ operator and the AGM:
a◊b = AGM(a, b) ◊ AGM(a, b)
Again, the ◊ operation is to be defined as the unique operation that satisfies the identity. Tanimoto transforms this equation into the form:
AGM(1, (a◊b) ) = AGM(a, b),
which can be “solved for ◊” by iterative methods.
At this point a numerical example will help. Suppose a = 3 and b = 5. AGM(3, 5) evaluates to approximately 3.936, and so the ◊ operation needs to be defined in such a way that AGM(1, (3◊5) ) will
also have the value 3.936. It turns out that AGM(1, 9) yields the correct result, and so it follows that 3◊5 must be equal to 9. Note that 3◊5 = 9 lies between 3+5 = 8 and 3×5 = 15, as required.
Pretty cool, eh?
No doubt the ◊ operation will soon be added to the elementary-school curriculum, alongside the standard quartet of ambition, distraction, uglification and derision.
Update 2007-09-03: Please see the comments for important corrections.
5 Responses to Addiplication
1. Maybe I’m missing something, but isn’t there a trivial solution? Let me use . instead of your symbol, and AM for arithmetic mean. Then why not define: a.b = AM(a+b, a*b) ?
2. Another problem: Tanimoto shows that 1 ◊ x = x for all (positive) x, which violates your stipulation
if a+b > a×b, a+b > a ◊ b > a×b;
(here a=1, b=x).
3. Uh oh. As far as I can tell, the issues raised by Jonathan Katz and Barry Cipra are problems with my interpretation of Tanimoto’s paper, not problems with the paper itself.
To address Barry’s point first, it appears that the correct relations are as follows:
if a+b ≤ a×b, a+b ≤ a ◊ b ≤ a×b;
if a+b ≥ a×b, a+b ≥ a ◊ b ≥ a×b;
As for the much simpler construction by Jonathan Katz: Tanimoto writes “the binary operation can be regarded as an intermediate operation between addition and multiplication” but never claims it
is unique in this respect. For that matter, I made no claim of uniqueness either, but I certainly missed the point that constructing such an intermediate operation is not in itself particularly
remarkable. Katz’s example seems an especially nice one. Of course one could also use the geometric mean of a+b and a×b, or even the AGM of the sum and the product.
4. ∗ is a pretty close facsimile of Tanimoto’s star operator.
5. That’s HTML entity ampersand lowast semicolon.
This entry was posted in mathematics.
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Summary: An Effective Proof that Open Sets are Ramsey
Jeremy Avigad
January 22, 1996
Solovay has shown that if O is an open subset of P() with code S
and no infinite set avoids O, then there is an infinite set hyperarithmetic
in S that lands in O. We provide a direct proof of this theorem that is
easily formalizable in ATR0.
1 Introduction
A plausible generalization of Ramsey's theorem asserts that for every two-
coloring of the infinite subsets of there is an infinite homogeneous set, that is,
an infinite subset of every infinite subset of which has been assigned the same
color. Unfortunately, under the axiom of choice, this generalization is false: by
transfinite recursion along a well-ordering of the reals one can cook up a color-
ing with no infinite homogeneous set. On the other hand, the nonconstructive
nature of this counterexample suggests that perhaps the theorem might hold
true for colorings that are "well-behaved" or "easily definable."
To that end, we define a partition to be a subset of the power set of ,
with the understanding that the infinite subsets falling inside the partition are
colored, say, red, and those outside the partition are colored blue. If P is a
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OP-SF WEB
Extract from OP-SF NET
Topic #13 ----------------- OP-SF NET ------------------ May 17, 1995
From: Hans Haubold
Subject: New book on mathematical software
Authors: T. Watanabe, M. Natoni, and T. Oguni (Editors)
Title: Mathematical Software for the P.C. and Workstations:
A Collection of Fortran 77 Programs
Publisher:Elsevier Science B.V., North-Holland, Amsterdam,
The book is the translation of its Japanese version from 1989
and contains instructions and mathematical outline for
numerical computations focusing on (1) the Gamma function and
related functions, Bessel functions of first and second kind
and related functions, various orthogonal polynomials,
exponential and trigonometric integrals, the error function
and related functions, the Fresnel integrals, the complete
elliptical integrals and random number generation, (2)
algebraic equations, transcendental equations, ordinary
differential equations, simultaneous linear equations, inverse
matrices, eigenvalue problems of matrices, and (3) data
analysis, including least-square methods, optimization
methods, interpolations, and Fourier transforms. A diskette is
included with all the source code in double precision for real
values of the argument of the function.
The bibliographies attached to each chapter in the book refer
particularly to Japanese research work done in the field of
numerical analysis and supplements in this sense D.W. Lozier
and F.W.J. Olver's very useful paper that assists to locate
approximations and software for the numerical generation of
special functions.
Back to Home Page of
SIAM AG on Orthogonal Polynomials and Special Functions Page maintained by Martin Muldoon
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Posts by
Total # Posts: 1,086
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ENGLISH revised
BACKGROUND: Ten years ago, I started working as a clerk for DMD Medical Supplies. Six months ago, Liz, the human resources director, promoted me to office manager. I manage two employees: Jack and
Ruth. My office provides secretarial support for the four members of the executi...
ENGLISH; improving your writing
REPHRASED and punctuation corrected... For the past three months, I have noticed work related problems occurring within our company of DMD Medical Supplies. The quality of work has not been
satisfactory, causing employees to be under stress and the workflow has been uneven bet...
ENGLISH; improving your writing
BACKGROUND: Ten years ago, I started working as a clerk for DMD Medical Supplies. Six months ago, Liz, the human resources director, promoted me to office manager. I manage two employees: Jack and
Ruth. My office provides secretarial support for the four members of the executi...
BACKGROUND: Ten years ago, I started working as a clerk for DMD Medical Supplies. Six months ago, Liz, the human resources director, promoted me to office manager. I manage two employees: Jack and
Ruth. My office provides secretarial support for the four members of the execut...
Improving your writing
Sorry forgot to paste the paragraphs,, For the past three months, I have noticed work related problems occurring within our company of DMD Medical Supplies; the quality of work has not been
satisfactory, causing employees to be stressed, and the workflow has been uneven amongs...
Improving your writing
BACKGROUND: Ten years ago, I started working as a clerk for DMD Medical Supplies. Six months ago, Liz, the human resources director, promoted me to office manager. I manage two employees: Jack and
Ruth. My office provides secretarial support for the four members of the executi...
why does cobalt chloride strip turn pink or blue?
A machine is used to fill soda bottles in a factory. The bottles are labeled as containing 2.0 liters, but extra room at the top of the bottle allows for a maximum of 2.25 liters of soda before the
bottle overflows. The standard deviation of the amount of soda put into the bot...
The least integer of a set of consecutive integers is -25 if the sum of these integers is 26,how many integers are in this set?
The side length of square ABCD is 1 unit. Its diagonal, AC, is a side of square ACEF. Square ACEF is then enlarged by a scale factor of 2. What is the area of the enlargement, in square units?
Oops. You were correct, it is A, just read it in my text book :-) "By age 2, girls appear overwhelmed by boys' rambunctious behavior. When a girl and a boy are paired in a laboratory play session,
the girl is likely to stand idly by while the boy explores the toys.&qu...
child development associate
It's B.
Human Resources
What types of issues might indicate a need for training? From what sources would these issues be identified? If you were a training manager, how would you prioritize training needs from these
sources? My position I selected was Healthcare Administration.
Write a balanced reaction for the formation of Fe2O3 (s)?????
I am confuse how do you divided 60 into4,356 please help!
Suppose that a 200g mass (0.20kg) is oscillating at the end of a spring upon a horizontal surface that is essentially friction-free. The spring can be both stretched and compressed and have a spring
constant of 240N/m. It was originally stretched a distance of 12cm (0.12m) fro...
Suppose that a 200g mass (0.20kg) is oscillating at the end of a spring upon a horizontal surface that is essentially friction-free. The spring can be both stretched and compressed and have a spring
constant of 240N/m. It was originally stretched a distance of 12cm (0.12m) fro...
wich material is the most dense air water plastic or iron
A wooden pyramid, 12 inches tall, has a square base. A carpenter increases the dimensions of the wooden pyramid by a factor of 5 and makes a larger pyramid with the new dimensions. Describe in
complete sentences the ratio of the volumes of the two pyramids.
THERE WERE 390 ADULT AND CHILDREN TICKETS THAT WERE SOLD AT A PLAY. THE THEATRE SOLD 2.25 TIMES MORE ADULT TICKETS THEN CHILDREN TICKETS. IF ADULT TICKETS COST $8 AND CHILD TICKETS COST $3. WHAT WAS
social studies
I want to run away have any tips
I am not sure how to convert my Cartesian xyz-coordinates into meters. x = 0.666 y = 0.666 z = 0.327 a = 6,377,563.396 meters b = 6,356,256.910 meters
Charlene has 2 5/8 yards of fabric. The curtain she is making requires 3 3/4 yards. How much more fabric does she need?
consumer math
The following describes the status of the O'Daniel's budget. A) It is in surplus B) It is in deficit C) It is balanced
yeah I meant cm. so for F= 8.99*10^9(9.11*10^-31)/ .01m^2 ?
A free electron and a free proton are exactly 1.0 com apart. Find the magnitude and direction of (a) the acceleration of the proton and (b) the acceleration of the electron.
PHYSICS help please
Two small charged balls have a repulsive force of 0.12 N when they are separated by a distance of 0.85 m. The balls are moved closer together, until the repulsive force is 0.60 N. How far apart are
they now?
consumer math
The graph represents the purchasing power if the inflation is 12%. Estimate the purchasing power of a monthly salary of $3000?
You made $3000 on a part time job last year. If you expect to make the same amount this year and the inflation rate is 3%, what is your purchasing power?
When 16.3 g of magnesiumm reacts with 4.52 g of oxygen, how many grams of magnesium oxide will be performed? Identify the limiting and excess reactants.
Math: Word Problem/Percets
Math: Word Problem/Percets
Math: Word Problem/Percets
Stevie bought a stereo for $220 and put it on sale at his store at a 70% markup rate. What was the retail price of the stereo?
In how many ways can this be done?
Assume that a committee consists of 12 people including the chairperson and that the chairperson must select a committee of 3 people from the other 11 members of the committee.
College Chemistry
i have no idea. i was given a few conversion factors but i always end up with something that i dont know how to convert
College Chemistry
previous answer to problem before this one(relevant in 2nd half of this question): 294.472689amu what would be the mass of 1.36 moles of these atoms? Enter your answer here: ____ g
Assume that there are 11 girls and 9 boys in the neighborhood club, and a team of 6 is to be selected.How many different teams can be selected if each team must contain exactly 3 girls and 3 boys?
Assume that a committee consists of 12 people including the chairperson and that the chairperson must select a committee of 3 people from the other 11 members of the committee. In how many ways can
this be done?
Assume that the committee consists of 8 Republicans and 6 Democrats. A subcommittee consisting of 7 people is to be selected. How many such subcommittees are possible if each subcommittee must
contain at least 1 and no more than 3 Democrats?
Assume that you have 7 dimes and 5 quarters (all distinct), and you select 4 coins. In how many ways can the selection be made so that at least 3 coins are dimes?In how many ways can the selection be
made so that at least 3 coins are dimes?
divide 3 gallons by 4
divide 1 yd 4 in. by 5
How do i multiply 2 yards 1 foot by 8
Multiply 2 yards 1 foot by 8
Subtract 1 ft 7 in. from 2ft. 2in.
Subtract 1 foot 7 inch from 2 feet 2 inches
Last one i swear.... If the manatees were able to make 3 adaptions to their body or way of life for their survival, what would they be?
i still dont know..
Identify one external feature of the manatee that allows it to live in water. Please help me with this one too!?
They both eat plants?
How do i subtract 20 minutes 45 seconds from 1/2 hour
Can you please give me study links? For angle sum theory, reflection, rotation,translation,Dilation,Complementary,Supplementary,parrell lines with transversal, vertical,
cooresponding,alt,int,alt,ext? Do you have any tricks to remember how to do these? NEED TO PASS THIS TEST. ...
Find the area bounded by y=1/x, the x-axis and the lines y=1 and x=4
If the ball is tossed upward at 20 m/s, how long should it take before it reverses its direction (reaches its maximum height where its velocity is zero)?
A person pushes a 17.0 kg lawn mower at constant speed with a force of 74.0 N directed along the handle, which is at an angle of è = 48.0° to the horizontal (b) Calculate the horizontal retarding
force on the mower (c) Calculate the normal force exerted vertically u...
he added 15+16+17... not all even numbers
find the area enclosed by y=x^2 and y=sinx in the first quadrant.
40-24=16 16/40= 2/5 spokes
3 and 1.5
World Civilization
what was the fighting in russia and se europe known as?
what was the primary goal of president ford's WIN program?
Social Studies 7R
i need help with my homework ugh
A skateboarder, starting from rest, rolls down a 12.5-m ramp. When she arrives at the bottom of the ramp her speed is 6.45 m/s. (a) Determine the magnitude of her acceleration, assumed to be
consumer math
The following describes the status of the O'Daniel's budget. A) It is in surplus B) It is in deficit C) It is balanced
consumer math
The graph represents the purchasing power if the inflation is 12%. Estimate the purchasing power of a monthly salary of $3000?
7th Grade English
Bethany showed perserance by showing that even though she only has one arm she can still do pretty much everything a normal person can do with two arms.
Ray AB bisects angle CAD. Find the value of x. CAB= 15X-25 BAD= 4X+63
The bill for a family of four at a restaurant was $ 62. Simon paid $ 72 for the meal and tip. How much was the tip? What percent was the tip?
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Math Brain T
Can you figure out the value of the angles?
How many children were born?
Figure out what number comes next!
Number re-arrangement puzzle!
Cool Math Puzzle from Bilbao!
Challenging Math Problem!
A challenging math type puzzle/brain teaser!
What number does each character represent?
Moving Cards Lateral Thinking Puzzle!
math puzzle, things flying about
Hard Math Puzzle with coins!
Visual Spatial Challenge!
Math Brain Teaser, tell me our ages...
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Functions of Pseudodifferential Operators
up vote 7 down vote favorite
Suppose I have a self-adjoint pseudo-differential operator $A$ on $\mathbb{R}^n$ and a continuous function $f$ (possibly bounded, or Schwartz, or compactly supported) on its spectrum. Then I can
consider the operator $f(A)$ defined by functional calculous.
Is $f(A)$ again a pseudo-differential operator and if yes, how are the symbols related?
In what way does the type of operator or the type of function matter?
pseudo-differential-opera sp.spectral-theory
add comment
3 Answers
active oldest votes
Here is a good reference for this
Michael Taylor: Pseudodifferential Operators, Princeton University PRess, 1981
In Chapter 12 it explains how to construct $f(A)$ when $A$ is elliptic selfadjoint of order $1$, $A\geq 0$, and $f$ is a smooth symbol of order $m$, i.e., $f$ is smooth
up vote 6 down vote accepted
$$ f^{(k)}(\lambda)= O(|\lambda|^{m-k}),\;\;\lambda\to\infty $$,
for any nonnegative integer $k$
add comment
Another reference is
Dimassi-Sjöstrand: Spectral asymptotics in the semiclassical limit, Cambridge University Press (1999)
See in particular Chapter 8 ("Functional Calculus for pseudodifferentiql operators").
up vote 6
down vote The results are in the semiclassical setting. The result which may interest you is Theroem 8.7 which states that if $P$ is a selfsadjoint h-pseudodiff operator and $f\in C^\infty_0$,
then f(P) is again a pseudo with leading symbol $f(p((x,\xi))$ , where $p(x,\xi)$ is the leading symbol of $P$.
The method to prove this is based here on a standard Cauchy formula. (I think in the book of Taylor, cited by Liviu Nicolaescu a different strategy is used).
If $f$ has compact support, then $f(P)$ is smoothing so the principal symbol is $0$. Also, Theorem 8.7 that you mention does not seem to cover Seeley's construction of complex powers
of an elliptic operators. Taylor's approach does. Maybe the compact support condition is not needed? – Liviu Nicolaescu Feb 9 '12 at 13:45
By leading symbol I mean the first term of the semiclassical expansion, not the principal symbol. – Hans Feb 9 '12 at 15:47
add comment
Since the question posed is about the "In what way does the type of operator or the type of function matter?", I thought the following observation will be apt:
As pointed out by Liviu Nicolaescu in the comment above, Taylor's approach seems to have much wider applicability when it comes to functional calculus. In fact in page 295 of Taylor's book
it is mentioned that Seeley's results form a special case of the result.
up vote 2
down vote Moreover, these methods have gone beyond elliptic operators. For instance, Uhlmann, Melrose and Guillemin have developed a framework of distributions whose wavefront sets are in several
Lagrangian intersecting manifolds (pseudodifferential operators with singular symbols) for a functional calculus on real principal-type operators, operators of double characteristics, wave
operators. Principal symbols also have been computed for these operators and all the computations are purely symbolic.
add comment
Not the answer you're looking for? Browse other questions tagged pseudo-differential-opera sp.spectral-theory or ask your own question.
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Pulleys are mechanisms compost by wheel and rope used to lift heavy objects onto tall heights. They change the direction of an applied force and they can even reduce the force needed to lift a
weight. Pulleys systems are common used in constructions.
In the exhibit, systems of one to five pulleys are uses to lift a weight of 1lb onto a 1 ft height.
In order to lift the 1lb weight W you have to apply a force of F on the rope equal to the weight W. The rope is now under a tension T equal to the force F.
To lift this weight a distance of H=1 ft you will have to pull in a length L= H=1 ft of the rope.
The mechanical advantage M is one: M=W/F=1.
In order to lift the 1lb weight W you have to apply a force of F on the rope. Like in the simple case of the block and tackle the rope is under a tension T equal to the applied force F. But in this
case the weight is supported by twice the tension: 2T=W. The force F you have to apply then is only half of the weight W.
By having the second pulley you have decreased the force needed to lift the weight. The mechanical advantage M is now two: M=W/F=2T/T=2.
However, order to lift the weight a distance of H=1 ft you will have to pull in a length L = H x M =2 ft of the rope.
You gained by having to apply a smaller force, but had to compensate by having to pull a longer length of rope.
Here the mechanical advantage is M=3 and here it is M=4.
The mechanical advantage M is equal to number of ropes present at the weight end. The force needed to raise the weight is W/M. In order to lift the weight a distance H you will have to pull a
corresponding longer length of rope L = H x M.
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[Haskell-cafe] matrix computations based on the GSL
Jacques Carette carette at mcmaster.ca
Thu Jun 30 09:19:08 EDT 2005
David Roundy and Henning Thielemann have been arguing about the nature
of vectors and matrices, in particular saying:
>On Thu, Jun 30, 2005 at 02:20:16PM +0200, Henning Thielemann wrote:
>>On Thu, 30 Jun 2005, David Roundy wrote:
>>Matrices _and_ vectors! Because matrices represent operators on vectors
>>and it is certainly not sensible to support only the operators but not
>>the objects they act on ... Adding a vector type by a library that is
>>build on top of a matrix library seems to me like making the first step
>>after the second one.
>No, matrices operate on matrices and return matrices. This is the
>wonderful thing about matrix arithmetic, why it's unique, and why I'd like
>to have a library that supports matrix arithemetic.
The really funny thing about that exchange is that you are *both right*
! You're just using different interpretations of the same objects.
1) Matrices represent linear operators which naturally act (via
application) on vectors
2) Matrices of compatible sizes, almost form a non-commutative graded
ring. It does not matter what a matrix represents here, this is true
purely algebraically. [to be a proper ring, the ``compatible sizes''
condition would need to be dropped]
There is a problem that the _types_ associated with both interpretations
are quite different. But if you want 2 different products on vectors
(inner and outer) represented as matrices, you have no choice -- the
``inner product'' will return a 1x1 matrix, not a real.
The same thing is true for differential operators, BTW. They can either
represent actions on spaces of smooth-enough functions, or represent
elements of a Weyl Algebra (or Ore Algebra if you really want to be
algebraic). You end up having the dichotomy of algebraic D-modules
versus analytic D-modules, where they share a number of theorems, but
the ``corner'' cases behave quite differently.
More information about the Haskell-Cafe mailing list
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can anyone give me the exact solution of equation without using taylor expansion
September 29th 2010, 10:52 PM #1
Sep 2010
can anyone give me the exact solution of equation without using taylor expansion
$a^x + b^x = c$
here a , b , c , are constants x is only variable
I have solved using Taylor expansion and can easily be done be graphs or a computatonal code
Is there any algebraic method?????????
In general, no, you cannot solve that equation in terms of "elementary" functions.
September 30th 2010, 04:12 AM #2
MHF Contributor
Apr 2005
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Lakeside, CO Statistics Tutor
Find a Lakeside, CO Statistics Tutor
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math tutoring
In Math Lab, I showed a student how to find the slope of a line, using "rise over run" - subtracting the y's, subtracting the x's, then dividing.
I showed another student how to use a number line to find absolute value.
I worked with 3 students to find the areas and perimeters of a square, rectangle, and triangle in both inches and centimeters. They had fun measuring bricks on the wall, and tiles on the floor.
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Change of variable triple integral question.
December 15th 2011, 11:42 AM #1
Nov 2008
Change of variable triple integral question.
Question asks:
Find the volume of the solid bounded by: z = 4x^2 + y^2 and the cylinder y^2+z=2
In the solution they used change of variable by putting in x = r cosθ/sqrt 2 and
y = r sin θ, z = z
this looks like cylindrical co ordinates to me, but I thought that you always use
x = r cos θ
can someone explain why divide it by root 2? And how can you just look at questions like these and decide what change of variable to make? I'm really confused.
Re: Change of variable triple integral question.
The projection onto the $xy$ plane of the intersection of the surfaces $z=4x^2+y^2$ and $y^2+z=2$ is $4x^2+2y^2=2$ or equivalently $\frac{x^2}{(1/\sqrt{2})^2}+y^2=1$ (ellipse). The substitution
$x=(r\cos \theta)/\sqrt{2},\;y=r\sin \theta$ transforms the ellipse into the circle $r=1$ .
December 16th 2011, 12:07 AM #2
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How to play 3-over-2 and other complicated rhythms
This is what “3-over-2″ or “3 against 2″ or “2 against 3″ looks like:
It’s a complicated rhythm that takes a while to get the hang of. In the bass clef are straight quarter notes but in the treble cleff are tuplets, which means three quarter notes played in the time of
two regular ones.
So for every two quarter notes in the left hand, the right hand is supposed to play three quarter notes.
You might also encounter 3-over-2 with eighth notes:
The principle is the same: two hands playing in different rhythms.
To get a feel for this rhythm, I suggest you take a walk:
1. Count 1, 2, 3, 4, 1, 2, 3, 4, and so on on each step. Left foot is 1, right foot is 2, left foot is 3, right foot is 4, and so on. You’re counting a regular 4/4 beat.
2. Hold your arms by your side. Now slap both your hands on your legs as triplets: 123-123-123-123… So for each footstep, your hands hit your sides three times.
3. Now alternate your hands: RLR-LRL-RLR-LRL, …
4. Stopping hitting your leg with your left hand. Now your right hand is hitting the tuplets, i.e. the “3″ from “3-over-2″.
5. Hit your left hand twice for each foot step to play the duple notes, i.e the “2″ from “3-over-2″.
It can be rather tricky at first to coordinate this, but with some practice — a long walk — you should get the hang of it.
Another approach is to count out the rhythm:
1. First, count the triplets as one-and-two-and-three-and-, one-and-two-and-three-and-, …
2. Then count it as one-two-and-three, one-two-and-three, …
3. Play the left hand on one and the and between two and three.
4. Play the right hand on one, two and three.
I hope this picture will make it a little clearer:
Once you get a feel for this rhythm, it isn’t so hard to play anymore.
Read more articles on Piano Clues:
Basic Theory How to Record Piano
Chords and Harmony Software and Virtual Instruments
The Circle of Fifths Scales and Exercises
Arrangement, Improvisation and Composition Digital Pianos
Reading Music and Sheet Music Links and Other Stuff
1. In Chopin’s Fantasie-Impromptu, the difficulty starts to arise in the fifth measure where 8 notes in the right hand is against 6 in the left. In this case how do I play it?
2. It’s actually 3 against four.
3. I have been working on the sight reading studies at http://pianostudies.angelfire.com/ and there are four or five very difficult studies that have 3 against 4 and 4 against 5 and 3 against 2 etc.
4. Pingback: V’Shamru « Tiven Weinstock | Composer
5. Hi,
I’m a drummer, and i frequently struggle this kind of polyrythms …
My advice is to consider any polyrythms as a linear rythm. As far as rythm is concerned it’s really only maths and time division.
What i did for example to play 6 on my Left Hand, 5 on my Right hand and 4 on my right foot for example is quite simple :
I trace a line on a paper, make to marks. Then i subdivide this unit by 6, and 5 and 4.
This give you a visualisation of how the notes are place each other.
Then try to sing it.
Once you master it vocaly.
Apply it very slowly with your hands and feet. Once you master the sequence, mentally isolate each part of your body to make it very consistent.
This is very effective and the foundation of any independance exercices on drums.
6. This is a very informative and creative tutorial. I had figured out the rhythm on my own, but this confirmed what I thought it was. Personally, I think the second approach will be the easiest way
to understand it.
7. The second approach is really helpful…hope I can get the hang of 3 against 2 soon…Thank you so much!!
8. I appreciate what you are saying very much I learned 3/2 when I was fourteen, BUT and a very very big BUT, 3/4 is entirely different as the idea of dividing 4 and puting 3 in between or vis/versa
is and must be an illusion in order to be musical in other words human. I understand the urge to want to subdivide mathematically (I have it) but it is a crutch and not the truth for sure. This
reminds me of calculus, you know… the area of a wine barrel, a fantasy that works with in limited reality(the limit), but reaching musical time ala say Fantasie Impromptu, requires stretching the
boundaries of your psyche to reach ala metaphor towards the impossible goal of reconciling 3 against four, just like squaring the circle. I haven’t got it yet but I am sure this is the way
towards that reconciliation. Thank you Chopin for opening the doorway.
This app lets you find out which chords you are playing. You select the notes on a piano keyboard and Reverse Chord Finder tells you the names of the matching chords. The ideal app for songwriters,
musicians and music students. Get it from the iPhone App Store »
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Check of permuations
April 15th 2012, 06:21 PM #1
Nov 2011
Check of permuations
So I did this problem, but my answers seem way big, can anyone confirm or deny and help with these?
how many distinct 3 cycles are there in s12?
I got 440, doing 12!/(9!*3)
how many elements of order 35 in s12?
I got 13,685,760, doing (12!/5!*7)*5!/5 which is the same as 12!/35 obviously
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independence of four variables
January 24th 2010, 08:57 PM #1
Jan 2010
independence of four variables
I am wondering if anyone knows the rules regarding the independence of four variables. Or a means of deriving such properties.
p(A^B^C^D)= p(A)p(B)p(C)p(D), pairwise independence, etc?
I'm not sure what your question is.
Pairwise only refers to sets of two random variables.
You have four sets.
We must have :
P(A^B^C^D)= P(A)P(B)P(C)P(D)
In other words, you have to test the formula for any "subset" of {A,B,C,D}
January 24th 2010, 09:08 PM #2
January 24th 2010, 11:38 PM #3
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Physics Forums - View Single Post - Electric field at a distance from a charged disk
A disk of radius 2.4 cm carries a uniform surface charge density of 3.1 μ C/m2. Using reasonable approximations, find the electric field on the axis at the following distances.
I have used the equation E=(Q/ε[0])(1/(4*pi*r^2))
I also tried the equation E=(Q/2(ε[0]))(1-(z/(√(z^2)+(r^2)))
Thanks in advance for the help. Both equations have not led me to the correct answer.
*note, there is not a figure provided for this question*
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Videos for constant - Homework Help Videos - Brightstorm
52 Videos for "constant"
Understanding and solving problems using Planck's constant.
Understanding and solving problems using Planck's constant.
Overview of the mathematical constant "e".
Examples of factoring out a greatest common factor constant and an overview of the idea of factoring, or "undistributing"
Solving an equation of the form log x = constant
Overview of the mathematical constant "e".
How to use the constant multiple rule and the sum rule to differentiate combinations of power functions, radical functions, and reciprocal functions.
Solving an equation of the form log x = constant
How to solve quadratic equations by factoring if there is a factor that is a constant.
How to solve quadratic equations by factoring if there is a factor that is a constant.
How to define direct variation.
How to define indirect variation.
How to evaluate a polynomial in function notation.
How to use Pascal's triangle.
How to use Pascal's triangle.
How to solve an indirect variation problem using variables.
How to solve a direct variation problem using variables.
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2 Answers
Edited by
Azzi Abdelmalek
on 10 Sep 2013
Accepted answer
No products are associated with this question.
how to use a while loop to solve this?
Hello, I need help with while loop please help Use a while loop to add all of the even numbers up: 2 + 4 + 6 + 8 + · · · until the sum exceeds 5 million. What is the actual sum? How many even numbers
were added?
0 Comments
An example how the while loop works
If we have to do 1+2+3+4+...+10
while k<10
Try to adapt this to your problem
0 Comments
For homework questions it is recommended to show, what you have tried already. Then the forum can help to solve a specific problem.
Note that Gauss has found a formula to solve the addition in several seconds without a computer. You can re-order the terms of his formula to solve your problem also.
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Topic: Standards for K-4,5-8,9-12
Replies: 9 Last Post: Jul 11, 1995 9:02 PM
Messages: [ Previous | Next ]
Re: Standards for K-4,5-8,9-12
Posted: Jul 10, 1995 11:52 PM
Richard Fouchaux writes:
[[[[I am a student teacher, but I own a copy of the Standards, have attempted
read the whole thing, and I also subscribe to an NCTM journal.]]]]]
According to my mail.....there are some on this list that would say you have
no business posting in this arena if you have only attempted to read the
whole thing.....(I am not one of those people).
Harv B
Date Subject Author
7/9/95 Standards for K-4,5-8,9-12 Eileen Abrahamson
7/9/95 Re: Standards for K-4,5-8,9-12 Kreg A. Sherbine
7/9/95 Re: Standards for K-4,5-8,9-12 Michael Paul Goldenberg
7/10/95 Re: Standards for K-4,5-8,9-12 Kelly
7/10/95 Re: Standards for K-4,5-8,9-12 DoctorCHEK@aol.com
7/11/95 Re: Standards for K-4,5-8,9-12 Michael Paul Goldenberg
7/11/95 Re: Standards for K-4,5-8,9-12 Richard Fouchaux
7/11/95 Re: Standards for K-4,5-8,9-12 Howard L. Hansen
7/11/95 Re: Standards for K-4,5-8,9-12 DoctorCHEK@aol.com
7/11/95 Re: Standards for K-4,5-8,9-12 Michael Paul Goldenberg
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32 stone is how many pounds
You asked:
32 stone is how many pounds
Say hello to Evi
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Need to check this work
July 7th 2011, 07:18 AM #1
Oct 2010
Mumbai, India
Need to check this work
I am trying to answer the question which I have attached. Please tell me if the answers are correct.
a) let M(x,y) = x has forgiven y R(x)= x is a saint
U=universe of discourse= all humans
$\forall x \left[\exists y M(x,y)\Rightarrow R(x) \right]$
b)let R(x) = x is in calculus class ; Q(x) = x is in discrete math class
M(x,y)= x is smarter than y
U=universe of discourse = all humans
$\forall x \left[R(x)\Rightarrow eg\left(\forall y\left[Q(y)\RightarrowM(x,y)\right]\right)\right]$
c)let P(x) = x likes mary ; R(x) = x is mary
U=universe= all humans
$\left[\forall x \left(eg R(x)\Rightarrow P(x)\right)\right]\wedge\left[\forall x\left( R(x)\Rightarrow eg P(x)\right)\right]$
d)let P(x) = x is a police officer
Q(x) = x is jane
R(x)= x is roger
M(x,y) = x saw y
U=universe= all humans
$\forall x \left[ Q(x)\Rightarrow \exists y\left ( P(y)\wedge M(x,y) \right)\right]\wedge \forall x \left[ R(x)\Rightarrow \exists y \left( P(y) \wedge M(x,y) \right ) \right]$
e)let P(x) = x is police officer
M(x,y) = x saw y
let j=jane and r=roger
again U=universe of discourse = all humans
$\exists y \left( P(y)\wedge M(j,y)\wedge M(r,y) \right)$
Re: Need to check this work
can anybody help ?
Re: Need to check this work
I think this is correct. (In (b), you need to insert a space between \Rightarrow and M.)
I am not sure if your problem allowed defining individual constants along with predicates. In (e) you did define j and r. For example, (c) could be ∀x (L(x,m) <-> x ≠ m) where L(x,y) means x
likes y and m means Mary.
It is easier to read formulas if predicates are named mnemonically, e.g., C(x) for calculus, S(x,y) for saw, etc.
Re: Need to check this work
thanks makarov
so b) will be
$\forall x \left[R(x)\Rightarrow eg\left(\forall y\left[Q(y)\Rightarrow M(x,y)\right]\right)\right]$
c) becomes much simpler with your approach, thanks
July 7th 2011, 09:28 AM #2
Oct 2010
Mumbai, India
July 7th 2011, 02:54 PM #3
MHF Contributor
Oct 2009
July 7th 2011, 09:56 PM #4
Oct 2010
Mumbai, India
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Selected Publications of Sam Wagstaff
Publications listed in MathSciNet in 2006
Cryptanalysis of Number Theoretic Ciphers
Factorizations of b^n ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12, up to High Powers
Two papers on probable primes These articles propose an algorithm for selecting industrial-grade primes for use in cryptography. This algorithm is recommended in ANSI Standards X9.31 and X9-80. The
first paper states a famous challenge for primality testing.
Paper with Jason Gower on SQUFOF in Mathematics of Computation, 77 (2008), 551-588. (Corrected version.)
"Congruences for r[s](n) modulo 2s," Journal of Number Theory 127 (2007) 326-329.
Same paper also available here.
Paper on the Cunningham Project, In "High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowrie Williams," A. Stein and A. van der Poorten, editors, Fields Institute
Communications, v. 41 (2004), Amer. Math. Soc., pages 367--378. MR 2005m:11234.
An efficient time-bound hierarchical key management scheme for secure broadcasting, IEEE Transactions on Dependable and Secure Computing, vol. 5, no. 2, April-June, 2008, pages 65-299, joint paper
with E Bertino and N Shang.
Paper on primes with a fixed number of 0 or 1 bits
Is there a shortage of primes for cryptography? International Journal of Network Security, vol. 3, no. 3, pages 296-299.
Factors of Bernoulli and Euler Numbers
The Number Field Sieve on Many Computers by R.-M. Elkenbracht-Huizing, Peter L. Montgomery, R. D. Silverman, R. K. Wackerbarth and S. S. Wagstaff, Jr., Proceedings of the Fifth Conference of the
Canadian Number Theory Association, Rajiv Gupta and Kenneth S. Williams, editors. Centre de Recherches Mathematiques (CRM, Montreal) Proceedings and Lecture Notes Series, published by the American
Mathematical Society (AMS), volume 19 (1999), pages 81-85.
MPQS with three large primes, P. Leyland, A.K. Lenstra, B. Dodson, A. Muffett, S.S. Wagstaff, Jr., in Algorithmic Number Theory, Proceedings ANTS 2002, volume 2369 of Springer-Verlag Lecture Notes in
Computer Science, 2002, pages 448--462.
Factors of Bell Exponential Numbers
Paper on watermarking with quadratic residues
Implementing the hypercube quadratic sieve with two large primes, Brian Carrier and Samuel S. Wagstaff, Jr., in Proceedings of the International Conference on Number Theory for Secure Communications,
Srinivasa Ramanujan Centre, SASTRA Deemed University, Kumbakonam, India, 20 & 21 December, 2003, pages 51--64. The PostScript version of the same paper.
Computing Euclid's primes, Samuel S. Wagstaff, Jr., in Bulletin of the Institute for Combinatorics and its Applications 8 (1993), pages 23--32. MR 94e:11139.
The New Mersenne Conjecture Paper with Paul Bateman and John Selfridge, American Mathematical Monthly 96 (1989), 125-128. The table for this paper.
Send e-mail to Sam Wagstaff
(This page last modified July 18, 2013)
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[FOM] paradox and circularity
Stephen Yablo yablo at mit.edu
Fri Sep 13 11:52:29 EDT 2002
Thomas Forster writes (in a paper he mentioned recently here) "There
is a more or less received modern view of the paradoxes which one
associates with the names of Russell and Tarski to the effect that
paradox can be evaded if one recognizes that every sentence belongs
to a particular level in an infinite hierarchy of linguistic levels.
The idea being that truth-predicates for languages lower down in the
hierarchy are to be found only higher up. It seems to me that the
real significance of Yablo's observation is that this is not enough
by itself. Evidently the sentences in Yablo's example can be _typed_
in the appropriate sense...Nevertheless we still have a paradox."
It seems to be also part of this more or less received view that set
paradoxes can be evaded if one recognizes that every set belongs to a
particular level in an infinite hierarchy of types, with sets of type
n can allowed to belong only to sets of type n+1 (or alternative type
m higher than n). If that is right then, just as one would expect
there to be no semantical paradoxes in a Tarskian hierarchy typed by
the integers, one would expect there to be no set paradoxes in a
Russellian hierarchy typed by the integers. Here is an apparent
counterexample, or at least I am wondering if it is a
ccounterexample. It's a variant of Mirimanoff's paradox making no
play (apparently) with the idea of self-membership.
A set S of type k is well-founded if there are no sets S_(k-1),
S_(k-2), etc. such that S contains S_(k-1) contains S_(k-2) and so on
For each integer n, let G_n be the set of well-founded sets of type (n-1).
On the one hand, each G_n must be well-founded, because an infinite
descending membership chain starting from it would include an
infinite descending membership chain starting from one of its
members, and its members are one and all well-founded.
On the other hand, if each G_n is well-founded, then it belongs to
the set of well-founded sets one level up, that is G_n belongs to
G_(n+1). Since n here ranges over the integers this gives us an
infinite descending chain: each G_k contains G_(k-1) contains G_(k-2)
etc. So no G_n is well-founded. Contradiction.
Does this work?
More information about the FOM mailing list
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Understanding Engineers: Feasibility
In an internal blogpost inside Atlassian, I described a certain problem as being ‘very hard’, to explain why our efforts were better spent elsewhere. Later, as I was walking across the bridge into
work I had a moment to look at that statement through the eyes of a non-engineer. Atlassian prides itself on hiring really smart people. What use are they if they can't solve hard problems?
To that end, here's a quick lexicon of what computer programmers generally mean when they're talking about how hard some problem is, starting with the most extreme:
The man most commonly regarded as the 'father' of computer science is the English mathematician, Alan Turing. Turing did a lot of work in World War II helping the Allies break the German military
ciphers. To reward him, the British Government convicted him of gross indecency after the war (he was homosexual), took away his security clearance and put him on hormone therapy. His death not long
after is generally accepted as suicide.
Anyway. Turing's most famous contribution to computer science is the Church-Turing Thesis. This describes a theoretical device called a Universal Turing Machine that is both capable of solving any
computational problem that could be represented as an algorithm, and of emulating any other device that solves computational problems.
Anything that can be (deterministically) mechanically computed can be computed by a Turing machine. Anything that performs deterministic mechanical computations is really just a Turing machine.
Engineers speak of computer hardware or programming languages that have this full range of computation as being "Turing complete".
In this framework, the word 'impossible' has a definite meaning. A problem is impossible if its solution can not be computed by a Turing machine.
Admittedly, this isn't a very useful distinction. On one hand, Turing machines are a mathematical theory. They're infinitely fast and have unlimited storage, and you have an unbounded amount of time
to write your program for it. As such, many things that are theoretically possible are practically impossible (more on that later).
On the other hand (and this trips up engineers all the time), it doesn't ascribe any value to mostly solving a problem. No Turing machine can tell you if you're going to like a book or not, but
Amazon makes a lot of money out of coming out with a close-enough guess.
At the other end of the scale, we have problems that are trivial. This definition can be expressed in far fewer words:
I know how to solve this problem.
To a programmer, a problem is trivial if there is a clear solution, and the only thing that needs to be done is to implement it.
The only caveat is that triviality refers to how hard the problem is to solve, not how hard it is to implement the solution. So there is no necessary relation between a task being trivial, and how
long it takes. To the programmer, once the plans for the bridge have been drawn up, the materials chosen properly and the model tested for how it would survive wind, traffic and earthquakes, actually
building the bridge is trivial.
A problem is unfeasible if enough of the solution is known to determine that you don't have the resources to solve it. You might not have the free developer resources, or the available expertise, or
it might just need more hardware than you can ever afford.
The field of cryptography offers us a perfect example of a problem that is both trivial and unfeasible. The algorithms that encrypt and decrypt data are well-known and published. Assuming the
algorithm works ‘as advertised’, to decrypt some data you just need to write a program that implements that algorithm, and throw enough computing hardware at the problem to run that algorithm with
every possible key.
Cryptography works by choosing keys in a way that ensures there's not enough computing hardware available, practically speaking, to run that trivial program to completion. Decrypting such data is
An ex-Google non-engineer described 'non-trivial' thus in the Xooglers blog:
It means impossible. Since no engineer is going to admit something is impossible, they use this word instead. When an engineer says something is "non-trivial," it's the equivalent of an airline
pilot calmly telling you that you might encounter "just a bit of turbulence" as he flies you into a cat 5 hurricane.
This quote shows both the difference between the engineer and non-engineer's conception of non-triviality, and their different definitions of impossible (Flying through a hurricane isn't necessarily
impossible, you just really wouldn't want to do it).
In clear opposition to the meaning of trivial, non-trivial means:
I do not know how to solve this problem completely.
Non-trivial contains dangerous unknowns. Some part of it is not yet understood, or lies outside the range of things the programmer has done before, or can quickly imagine a workable solution to. The
more experienced the programmer who tells you a problem is non-trivial, the more concerned you should be.
Given time for research and experimentation, a non-trivial problem can be made trivial. Or it could be determined to be hard.
Hard, and Very Hard.
Hard problems are a class of non-trivial problem^2 where some of the unknowns, some of the problems the engineer would have to find a way to overcome, are known to be difficult to find solutions to.
These are problems the engineer has tried to solve in the past, or has seen the resources that other developers have had to throw at similar problems to solve them.
'Hard' can also be used to describe non-trivial problems with a big potential for 'unknown unknowns' - things the developer not only doesn't know the solution to, but doesn't even know what kind of
problems he's likely to face.
Scaling out a web application is hard. There are any number of thorny problems to do with performance, distribution, data access, caching and so on that take quite a bit of effort to solve, and are
often different enough from application to application for there to be no good, general solution. There are also inevitable thorny problems that you haven't even anticipated, lurking in the wings.
Very Hard is the extreme of hard problems. You'll often see both words capitalised for emphasis, even in the middle of a sentence. Indexing the entire World Wide Web and providing relevant search
results in millisecond response times is a Very Hard problem. Breaking commercial-grade encryption within practical hardware and time limitations is a Very Hard problem. Peace in the Middle East is a
Very Hard problem.
'Very Hard' is usually reserved for the class of problem that if you solved it, you could change the world. Or at least build a successful business on top of your solution.
^1 When I first posted this, I mentioned in an aside that the correct term was 'not feasible'. Further research suggests that unfeasible is also in common usage, but the vast majority of emails have
been in support of ‘infeasible’. Never rely on a single American dictionary as a source.
^2 Gary Capell pointed out to me in email that, in classic engineering understatement, hard, or even Very Hard problems can sometimes be referred to as ‘distinctly non-trivial’.
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Line Problem
Friedrich-Schiller-Universität Jena
Fakultät für Mathematik und Informatik
The No-Three-in-Line Problem
The no-three-in-line problem is a geometric arrange problem. Let a n × n grid be given in the Euclidean plane for any natural number n . The task is to mark as many of the intersection points of the
n^2 grid points as possible under the condition that no three of them lie in a straight line. One can obviously mark at most 2n points. The problem of finding for which n this upper bound is reached
is known as the no-three-in-line problem, introduced by Dudeney 1917.
A closely related question is: How many points can maximally be marked under this restriction? P. Erdös has shown that (1-eps)n points can be placed in a given n × n grid for all sufficient large n.
Hall, Jackson, Sudbery, and Wild made the substantial improvement to (3/2-eps)n in 1975. This is the best known asymptotic lower bound. But Guy and Kelly used a probabilitic argument to support their
conjecture that for large grids the limit does not reach 2n -- the main conjecture.
Until march 1992 I had computed for small values of n such no-three-in-line configurations. The last class I counted was for n=23 with one diagonal-reflection symmetry. This was the only missing
number in my 1992 JCT publication. All those old results are summarized in this table. To get an impression of the increasing number of examined cases have a look at the counted vertices in the
recursive total tree searchi which is subdivided in the symmetry classes. Look for a short explanation of symmetry classes and their conditions.
The whole old data is availible in coded form via anonymous ftp, but this old format which uses a nonalphanumeric alphabet is no longer recommanded. Better you have a look to both, old and new, data
as a collection of many files or as a list of coded configurations (1.6MB). To decypher this code use this C-program. For those who haven't access to a C-compiler: Each line represents a
configuration and each character in it a marker position, except the first one, which denotes the symmetry class. The positions are coded from left to right and from top to bottom in each
configuration. They are numbered by the alphabet 0,1,2,...,9,A,B,C...,Z,a,b,...,z only in their column positions.
For example here are all now known configurations with dia2- and full-symmetry. Also you can see this as a 44KB picture of 33+3 configurations.
Also an overview of the number of configurations is given with this hopefully up-to-date table. The used method to find these solutions is mainly a sophisticated branch and bound algorithm.
My program nothree needs about 4 n^4 + O(n^2) bytes storage and was first implemented in july 1988. Its last release was of november 1991 and consists of about 600 lines of C sourcecode.
A new version was just in april 1996 implemented. Aside the fact that computers nowadays seem to be by a factor 15-20 quicker, the refined algorithm itself gives a speedup of 1.5 - 8 depending on the
examined symmetry class. It is very important to traverse the tree of the possible configurations in a proper chosen order. And this depends stronly of the symmetry class.
Furthermore with a simple trick, the restriction of n <= 64 (due to the internal data structure) could be risen to n <= 90 without any runtime or memory increase. Currently I can check n = 72 for
configurations with full symmetry in the same cputime as those for n = 60 with the old algorithm used 1991.
Since 7th June 1996 a new format for the coded configurations is in use: Now each configuration is lead in by a symmetry-class character of the list (. : / - o c x + *) which indicates the symmetry
(iden rot2 dia1 ort1 rot4 near dia2 ort2 full) respectively. Further, the marked positions are numbered by the alphabet 0,1,2,...,9,A,B,C...,Z,a,b,...,z in their column position. And a code word must
be terminated by a newline or space or tab. As long as no configuration with n > 62 is known the more limited alphabet size doesn't matter in opposite to the old one (n > 96 would have caused
Future Aims
• Finding configurations for larger n
• Settling the Main Conjecture of No-Three-in-Line
There are only finite many different configurations.
If n tends to infinity, there are only (2/3PI^2)^(1/3)n points selectable such that any three of them do not lie in a straight line.
Conjecture I (<= 1952)
There are exactly 3 configurations having the full symmetry of the grid.
Conjecture II (<= 1968)
Each configuration having ort2 symmetry has the full symmetry of the grid.
Conjecture III (<= 1991)
There are exactly 5 configurations having reflection symmetry in the mid-perpendiculars.
Conjecture IV (<= 1996)
There are about 34 configurations having reflection symmetry in the main-diagonal and side-diagonal.
Conjecture III was disproved in June 1996 by the discovery of a further configuration having reflection symmetry!
To understand the relative number of solutions in the classes ort1, rot2, dia1 or the classes ort2, rot4, dia2 notice that in the orthogonal symmetry class ort1, resp. ort2, two independent selected
points force typically the blockade of n points for 2 of 6, respectively for 8 of 28, straight lines together with their 2, resp. 6, symmetry points. For the diagonal reflection classes dia1 and dia2
two independent selected points force typically the blockade of n/2 points ... . And in the rotation symmetry classes rot? two independent selected points force typically the blockade of about C log
(n). The effect of blockade of the remainding 4 of 6, respectively 20 of 28, straight lines seem roughly independent of the regarded symmetry class.
Therefore one should expect most different configurations for given n in the rotational symmetry classes but least different configurations in those with mid-perpendicular symmetry. For a precise
analysis of the asymmetric case, see the paper of Guy and Kelly 1968. Their idea can be applied to a special symmetry class to get probabilistic estimations for the number of different
configurations. The relation of the numbers for different symmetry class but fixed n should be at least the same as the counted ones for the generatable configurations.
Here are some remarks about symmetry conditions/effects in no-three-in-line configurations.
Due to a correspondence from John Selfridge, I checked the collected configurations whether they contain a no-3-in-line solution for smaller n than the given one. Besides the 3 known "diagonal
extentions" for o2423670617014535 , :3458148A130A7902692567 , :372845190A460A19562837 the only subsolutions I found is the 2x2 solution. Almost all of these configurations belongs to symmetry class
rot2 or rot4. Only 5 examples for n=12, 13, 14, are assymmetric and one with n=15 has reflection symmetry at its long diagonal. But there are more assymmetric for n >= 15. Those configurations having
2x2 blocks in class rot4 have the only block at their center, except the 2 configurations o2323676701014545 and o9OGHGHAM6M06OP34BFBTEQKL12128JALRSRS893F0IEIPQ45NT7N7JCDCD5K. There are centerblock
configurations in this symmetry class for n=2,10,14,22,28,30,32,...,42,44 and probably for more n >= 46.
• H.E. Dudeney, "Amusements in Mathematics", Nelson, Edinburgh 1917, pp. 94, 222
• K.F. Roth, Journal London Math. Society V.26 / 1951, pp. 204
• R.K. Guy, Bulletin Malayan Math. Society (1952-1953), E22
• Acland-Hood, Bulletin Malayan Math. Society (1952-1953), E82
• P.A. Kelly, Master's Thesis, University of Calgary, 1967
• R.K. Guy and P.A. Kelly, Canadian Mathematical Bulletin V.11 / 1968, pp. 527-531
• M.A. Adena, D.A. Holton and P.A. Kelly, Combinatorial Mathematics: Proceedings of the Second Australian Conference Lecture Notes in Mathematics, V.403 / 1974, pp 6-17.
• R.R. Hall, T.H. Jackson, A. Sudberry and K. Wild, Journal of Combinatorial Theory Series A, V.18/1975 pp. 336 - 341 [<=10]
• D. Craggs and R. Hughes-Jones, Journal of Combinatorial Theory Series A, V.20/1976 pp. 363 - 364 [11,12]
• M. Gardner, Scientific American V236 / March 1977, pp. 139 - 140 [13-16]
• T. Kløve, Journal of Combinatorial Theory Series A, V.24/1978 pp. 126 - 127 [14,16]
• T. Kløve, Journal of Combinatorial Theory Series A, V.26/1979 pp. 82 - 83 [18,20]
• D.B. Anderson, Journal of Combinatorial Theory Series A, V.27/1979 pp. 365 - 366 [22,24,26]
• H. Harborth, P. Oertel and T. Prellberg, Discrete Mathematic V73/1988 pp. 89-90 [17,19]
• Gerken, personal communication of H. Harborth 1990, [21,23]
• A. Flammenkamp, Journal of Combinatorial Theory Series A, V.60/1992 pp. 305 - 311 [...,44,46]
In the 1992 published enumeration table in JCT wrong numbers were given in column o for n >= 30. A program bug causes some missed solutions in symmetry class rot4, which were discovered now by a
recomputation. The correct numbers are 92 instead of 62 for n = 30 and 101 in place of 99 for n = 32.
Due to my unconcentration the value for n=20 in column : is a misscount. The correct value is 675 instead of 693, the value of all configuration with at least rot2 symmetry. Sorry, but I forgot
totally to indicate a computation of Benjamin Chaffin, Williamstown, MA, USA. In march 2006 he generated all solutions for n=17 and n=18 and he emailed me the corresponding total counts in the same
month. Moreover he set up this webpage of his research results up to size n=18. Any comments or questions please to my current Email-address
Achim Flammenkamp
2014-03-12 22:08 UT+1
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Research Areas
Algebra and Number Theory
Algebra, Algebraic Groups, Algebraic Geoemtry, Analytic Number Theory, Arithmetic Geometry
Analysis & Differential Geometry
Differential geometry, applied analysis, partial differential equations, complex analysis, geometric function theory, spectral theory, and mathematical physics.
Medical imaging, massive data processing, scientific visualization, grid computing, distributed biomedical informatics, decision support systems for system-level e-science, public health
informatics, metadata and data management, data privacy.
Data Management and Informatics
Data security, information retrieval, statistical analysis, and data integration in the context of medical, public health, and biological data managment. Computational biology and bioinformatics
(comparative genomics), computational infrastructure for scientific analysis.
Discrete Mathematics & Theoretical Computer Science
Ordered combinatorial and algebraic structures, theory of computation, circuit complexity, geometric algorithms, approximation algorithms, graph theory, hypergraphs, ramsey theory, combinatorics,
extremal graph theory, approximation algorithms, combinatorial optimization, polyhedral combinatorics, mathematical programming, combinatorial scientific computing.
Distributed Computing Laboratory
Metacomputing, distributed systems, collaboration technologies, and networking. Current projects are focusing on self-organizing frameworks for cooperative resource-sharing and collaboration group
communication in the context of peer-to-peer networks.
Numerical Analysis & Scientific Computing
Numerical Linear Algebra, Numerical Optimization, Numerical Partial Differential Equations, Numerical Methods for Inverse Problems, Image Processing, Geophysics, Computational Fluid Dynamics,
Numerical Methods for Markov Chains, Parallel and Distributed Computing.
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[SOLVED] Fourier series help
November 25th 2007, 06:14 AM #1
[SOLVED] Fourier series help
Let f(x) = sin x for 0 < x < pi . Let a0 + summation from n = 1 to infinity an cosnx be the fourier cosine series which represents f(x) find the value of the coefficient a4.
when i attempted this question when integrating from -pi to pi f(x) cosnx, i treated f(x) to be 0 from x= -pi to 0 since its not defined. however, in the solution, they extended f(x) to be an
even function on -pi to pi. why is it that they can do that?
Let f(x) = sin x for 0 < x < pi . Let a0 + summation from n = 1 to infinity an cosnx be the fourier cosine series which represents f(x) find the value of the coefficient a4.
when i attempted this question when integrating from -pi to pi f(x) cosnx, i treated f(x) to be 0 from x= -pi to 0 since its not defined. however, in the solution, they extended f(x) to be an
even function on -pi to pi. why is it that they can do that?
You want to exand $\sin x$ as a cosine series on $(0,\pi)$. Extend $\sin x$ to $(-\pi, \pi)$ in an even manner. So $f(-x) = \sin x$ for $x\in (0,\pi)$. Then this will be a cosine series if
exteneded periodically by $2\pi$. The coefficients are:
$\pi a_0 = 2\int_0^{\pi}\sin x dx, \ \pi a_n = 2\int_0^{\pi}\sin x \cos nx dx, \ b_n = 0 \mbox{ for }n\geq 1$.
November 25th 2007, 07:12 AM #2
Global Moderator
Nov 2005
New York City
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Amesos2::FunctionMap< ConcreteSolver, Scalar >
template<template< typename, typename > class ConcreteSolver, typename Scalar>
struct Amesos2::FunctionMap< ConcreteSolver, Scalar >
Passes functions to TPL functions based on type.
Helper class which passes on function calls to the appropriate Solver function based on the type of its scalar template argument.
Some Solvers have solver and matrix builder functions defined based on data type. One function for complex, one for double precision complex, another for float , and yet another for double. To work
elegantly with the Amesos2::Solver interface we want to be able to perform a single function call which is appropriate for the scalar type of the Matrix and MultiVectors that we are working with. The
FunctionMap class provides that capability.
The class template is specialized for each Solver and each data type that it supports, and errors are thrown for other data types.
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Topic: Matheology � 300
Replies: 2 Last Post: Jul 12, 2013 4:41 PM
Virgil Re: Matheology � 300
Posted: Jul 12, 2013 4:41 PM
Posts: 7,011
Registered: 1/6/11 In article <ceb27d70-9aa1-4aae-92d5-4b66e8e1317d@googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
> On Friday, 12 July 2013 01:12:23 UTC+2, Zeit Geist wrote:
> >>
> >
> >> Try to find some individuals that are not in one and the same line. Fail.
> >> Recognize - or, most probably, not.
> >
> > Ok, you win.
> There exist a line in the list, k, such that all n e |N, n e k.
> No!!! It is simply absurd and stupid to talk about all n in |N.
Then it must be equally stupid to claim proof by induction that
something is true for all n in |N.
But since , at least outside of WM's wild weird world of WMytheology,
induction produces valid proofs of statements of the form
"for all n in |N, f(n)"
WM's WMytheology does not hold outside of WM's WMytheology
> > Now, consider the line before k, m.
> We know m consists of each member k except the last element.
> Since k contians no last element, m has the same elements as k.
> Therefore, every line contains all the Natual Numbers.
> > Is that a valid proof?
> > I think it is.
> It is a valid proof. It proves that IF all naturals exist, THEN something
> goes wrong..
So that, forunately only in WM's wild weird world of WMytheology,
WM declares that all inductive proofs are invalid.
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Serre-Tate 1964 Woods Hole notes
up vote 6 down vote favorite
I am not sure if this is the right venue to ask this. Apologies in advance.
I would like to clarify the following. When people give as reference:
J.-P. SERRE and J. TATE.-Mimeographed notes from the 1964 A.M.S. Summer Institute in Algebraic Geometry at Woods Hole;
which notes do they exactly refer to? I downloaded a 1964 Woods Hole notes from Milne's site: http://www.jmilne.org/math/Documents/ but didn't find notes which were co-authored by Serre and Tate.
There is one written solely by Tate and another solely by Serre. Another part contains notes by the two of them together with Lubin.
reference-request arithmetic-geometry
add comment
1 Answer
active oldest votes
They are probably referring to the sections "Serre discussed..." and "Tate discussed..." in the Seminar report by Lubin, Serre, Tate, which outline what has become known as Serre-Tate
theory. But without knowing the context, I can only guess. It is quite likely the author didn't have access to the notes of conference, because only a very small number were produced,
up vote 6 down and they have become available on the internet only fairly recently.
vote accepted
Some works which cited the Serre-Tate notes (in the manner mentioned above) that I know of are: Katz' Serre-Tate Local Moduli, Katz-Mazur's Arithmetic Moduli of Elliptic Curves and
Peter Norman's Lifting Abelian Varieties. It is very likely that these authors have had direct access to the notes. Anyway, I do agree with you. We can only guess. Maybe it really
was the Lubin-Serre-Tate notes. – Octobris Jul 15 '13 at 13:29
Well, in that case, they were just being lazy (or assumed that the reader didn't have access). – abz Jul 15 '13 at 14:14
7 I think this answer is exactly right. – Lubin Jul 15 '13 at 19:14
add comment
Not the answer you're looking for? Browse other questions tagged reference-request arithmetic-geometry or ask your own question.
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A question about the existence of a specific extension of a character.
up vote 4 down vote favorite
general situation:
Let $ N \leq G $ be a subgroup,and let $ \chi \in Irr(G) $ be an irreducible character of G such that $\chi_N $ is not irreducible( i dont think that this is really needed) and let $ \psi \in Irr(N)
$ be an irreducible constituent of $\chi_N$. Assume there is an subgroup H of G with $ N \leq H \leq G $ to which $ \psi $ is extendible. Then there is a character $ \mu \in Irr(H) $ with $ \mu_N =\
psi$ and $[\chi_H , \mu] \neq 0.$ NOTE: the problem here is to find such an extension with $[\chi_H , \mu] \neq 0.$ Is this true in general?
I have the following specific situation: Let J be a nilpotent finitedimensional algebra over a finite field. Then G=1+J(called finite algebra group) is a p-group,$N=1+J^{2}$ is a normal subgroup. In
the paper "On characters and commutators of finite algebra groups" written by Halasi,he writes:
Lemma 3.1: "Let G=1+J be a finite Algebra group and $\chi \in $ Irr(G).Then the following properties are equivalent:
1.There exists a proper algebra group H and $\varphi \in Irr(H)$ such that $\varphi^{G}=\chi$.
2.$\chi_{1+J^{2}}$ is not irreducible.
Assume now that $\chi_{1+J^{2}}$ is not irreducible and let $\psi \in Irr(1+J^{2})$ be a constituent of $\chi_{1+J^{2}}$.Let H be a maximal algebra subgroup such that $\psi $ is extendible to H.Then
H $ \neq $G.We choose a $\varphi \in Irr(H) $such that $\varphi$ is an extension of $\psi$ and $\varphi$ is a constituent of $\chi_H$...."
I wonder why there is such a $\varphi$ .Is it true in the general situation or what properties of 1+J and $1+J^{2}$ or H(being maximal) are used here? Thanks for helping
gr.group-theory characters
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3 Answers
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Given $N \subseteq H \subseteq G$ and an irreducible character $\psi$ of $N$ that has an extension to $H$, it is NOT in general true that every irreducible character $\chi$ of $G$ that lies
over $\psi$ must lie over some extension of $\psi$ to $H$. Probably the easiest counterexample is to take $N = 1$ and $H = G$, where $G$ is any nonabelian group. Let $\psi$ be the principal
character of $N$. Then of course, $\psi$ extends to $H$, but if $\chi$ is any nonlinear irreducible character of $G$ then $\chi$ does not lie over any extension of $\psi$ to $H$.
up vote 5
down vote A case where it is true that $\chi$ must lie over an extension of $\psi$ to $H$ is where $N$ is normal in $H$ and $H/N$ is abelian. In that case, every character if $H$ lying over $\psi$ is
an extension of $\psi$, so $\mu$ can be taken to be an arbitrary irreducible constituent of $\chi_H$.
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Hi, I just saw a theorem in a paper of Isaacs which could be useful. Assume additional to the general case(with same notation)that N is normal in H and H/N is abelian. Then the lemma:
"Let N be a normal subgroup of H and H/N abelian.Let $\vartheta \in Irr(H) $and $\psi \in Irr(N)$ and $[\psi,\vartheta_N] \neq 0$.Then every Z $\in Irr(H)$ with $[Z_N,\psi] \neq 0$ has the
form $Z=\lambda \vartheta$ for a linear $\lambda \in Lin(H/N) $ "
tells us that every irreducible constituent of $ \psi^{H} $ has the form $\lambda \vartheta$,where $\lambda \in Lin(H/N)$ because if Z is an irreducible constituent of $ \psi^{H} $ then $
[Z,\psi^{H}]=[Z_N,\psi]$.We can then write $ \psi^{H} $ as a sum of all different $\lambda \vartheta$ with multiplicity one,because $[\psi^{H} , \lambda \vartheta]=[\psi, (\lambda \
up vote 3
down vote We have $[\chi_H , \psi^{H}]=[\chi,\psi^{G}]=[\chi_{N},\psi] \neq 0$ Choose an irreducible constituent of $ \psi^{H} $,called $\phi$ with $[\phi,\chi_H] \neq 0.$ On the one hand we have: $
(\psi^{H})_{N} = |H:(N)| \psi $ and on the other (since $[\phi,\psi^{H}]=[\phi|{N} , \psi]$):
$(\psi^{H})_{N} = a \phi +etc$ Comparing these 2 gives: $\phi|{N}=c \psi$ for natural a and c and etc as a combination of other caracters. we have to show that c=1 to finish the proof. $c=
[\phi|{N} , \psi]=[\phi,\psi^{H}]$,but we showed above that $\psi^{H}$ has only irreducible constituents with multiplicity one. I am thankful for proofreading this or giving hint to avoid
the additional assumptations.
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Let $N$ be a normal subgroup of a finite group $G$, $\chi\in\mathrm{Irr}(G)$, and $\psi$ be an irred constituent of $\chi_{N}$. Let $H$ be a subgroup containing $N$ such that $H/N$ is
abelian and $\psi$ is extendible to $H$. Then there exists a $\mu\in\mathrm{Irr}(H)$ such that $\mu_{N}=\psi$ and $[\chi_{H},\mu]\neq0$.
Edit: My previous statement was wrong as it left out the necessary hypothesis that $H/N$ be abelian. See Marty Isaacs's answer.
Below is my old argument. When $H/N$ is not abelian it does not go through because it is not necessarily true that $\mu_{N}=\psi$; in general $\mu_{N}$ will be a number of copies of $\psi$.
up vote When $H/N$ is abelian there is a much simpler argument, as given in Marty Isaacs's answer.
3 down
vote The natural context for this statement is Clifford theory for finite groups (see Section 6 in Isaacs's book). To see why this is true, first note that $H$ must be a subgroup of the
stabiliser $S$ of $\psi$ in $G$. Moreover, there exists a $\rho\in\text{Irr}(S)$ such that $\rho_{N}$ contains $\psi$ and $\rho^{G}=\chi$. Since $\rho_{N}$ contains $\psi$, there exists an
irred constituent $\mu$ of $\rho_{H}$ such that $\mu_{N}$ contains $\psi$. Since $\psi$ is extendible to $H$, a theorem of Gallagher's (Isaacs's book (6.17)) implies that $\mu_{N}=\psi$.
Since $\mu^{S}$ contains $\rho$, $\mu^{G}$ contains $\chi$, and so $\chi_{H}$ contains $\mu$.
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Cubic Lattices
Pearce, S.C. (1995) Cubic Lattices. Journal of Applied Statistics, 22 (3). pp. 355-373. ISSN 0266-4763. (The full text of this publication is not available from this repository)
In some experiments, the problem is to compare many unstructured treatments in small blocks, the classical example being the study of new plant varieties on variable land. A common method is to use
lattice designs, i.e. block designs based upon rows and columns of a square format, with further replicates being formed, if required, from orthogonal squares applied to the format It has been known
for some time that cubes can be used instead; this paper sets out to explore the possibilities. There are two cases. In one case, the blocks are formed from the planes of the cube and, in the other
case, from its lines. The cubic lattice basically has three replicates-one from each dimension-but, if two or four replicates are required, a design can be found by omitting or duplicating one of the
dimensions. Where standard treatments need to be introduced, a useful device is to reinforce, i.e. supplement each block with additional plots of standards, with each block of a replicate being
supplemented in the same way. These possibilities are examined. It emerges that cubic lattices with two or three replicates usefully extend the range of available designs, but that those with four
replicates are disappointing. However, there is the alternative of using designs based upon Latin cubes. This matter is not taken far but it is shown that, where the Latin cube exists, it gives a
better design. A quick way of calculating an approximate analysis of variance is given, which is applicable in a wide range of cases.
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On Maskin monotonicity of soluti
Rev. Econ. Design DOI 10.1007/s10058-008-0062-7 ORIGINAL PAPER
On Maskin monotonicity of solution based social choice rules
Claus-Jochen Haake · Walter Trockel
Received: 26 June 2007 / Accepted: 28 October 2008 © Springer-Verlag 2008
Abstract Howard (J Econ Theory 56:142–159, 1992) argues that the Nash bargaining solution is not Nash implementable, as it does not satisfy Maskin monotonicity. His arguments can be extended to other bargaining solutions as well. However, by defining a social choice correspondence that is based on the solution rather than on its realizations, one can overcome this shortcoming. We even show that such correspondences satisfy a stronger version of monotonicity that is even sufficient for Nash implementability. Keywords Maskin monotonicity · Social choice rule · Bargaining games · Nash program · Mechanism · Implementation JEL Classification 1 Introduction Hurwicz (1994) in contrast to large parts of the literature stresses the fundamental difference between games and mechanisms (game forms). The concept of a game form, that allows it to formally separate the rules of a game from players’ individual evaluations of the outcomes, is a cardinal tool for applications of game theory. The possibility to choose the outcome space of a mechanism C71 · C78 · D61
Dedicated to Leo Hurwicz whose conceptual rigor and clarity has become a treasured benchmark for the profession. C.-J. Haake (B · W. Trockel ) Institute of Mathematical Economics, Bielefeld University, Bielefeld, Germany e-mail: chaake@wiwi.uni-bielefeld.de W. Trockel e-mail: wtrockel@wiwi.uni-bielefeld.de
C.-J. Haake, W. Trockel
(or game form) according to the specific needs of the problem to be modeled makes implementation theory a powerful instrument. A key role in that theory is played by the property called Maskin monotonicity, that is a necessary property for a social choice rule to be implementable in Nash equilibrium (see Maskin 1999). While many specific applications almost naturally distinguish “the” suitable mechanism, thus outcome function, this is not the case when cooperative solutions are to be Nash implemented. The aim to relate cooperative solutions of coalitional games to Nash equilibria of non-cooperative games in strategic or extensive form goes back to Nash (1951, 1953) and is now commonly referred to as the “Nash program”. The exact relation between the Nash program and implementation theory has been addressed explicitly in the literature only in the last decade. Serrano (1997) states: “The Nash program and the abstract theory of implementation are often regarded as unrelated research agendas”, and Bergin and Duggan (1999) write: “. . . because the implementation-theoretic and traditional approaches both involve the construction of games and game forms whose equilibria have specific features, considerable confusion surrounds the relationship between them.” Several articles have recently tried to dispose of this confusion: Dagan and Serrano (1998), Serrano (1997, 2005a,b), Bergin and Duggan (1999), Trockel (2002a, 2003). At the heart of the problem lies the fact that a cooperative solution as a technical concept is distinct from a social choice rule. Consequently, Nash implementation of a cooperative solution is literally impossible, as it is not well defined. A crucial step in making solutions implementable is therefore the interpretation of a solution as a social choice rule. Formally, this means the suitable definition of a solution based social choice rule that carries the characteristic features of the underlying solution. While social choice rules are mappings associating certain outcomes to profiles of preferences or utility functions, solutions associate feasible (monetary or utility) payoffs of players to certain coalitional games. A basic task is it therefore to understand the relation between utility profiles and coalitional games. In their seminal paper, Bergin and Duggan (1999) explain this problem by use of the notions of “effectivity” and “supportability”. Supportability associates with a coalitional game an underlying profile of utility functions supporting it. Effectivity associates with any utility profile a coalitional form to describe the potential strategic effects on coalitional worths. While here and likewise in Trockel (2002a, 2003) the relation between social choice rules and solutions is formally analyzed, it is ignored in large parts of the literature, a fact that contributes to the “confusion” mentioned above. Nash in his non-cooperative foundation of the Nash bargaining solution left the supportability problem unsettled. Implementation in the sense of mechanism theory was not yet an issue for him. Howard (1992) and Moulin (1984) provided early implementations in subgame perfect equilibria of the Nash and the Kalai–Smorodinsky solutions, respectively. They both ignored, or better avoided, the effectivity problem by introducing solutions directly as social choice functions defined on a space of utility profiles. In order to implement bargaining solutions like those of Nash or Kalai–Smorodinsky one has to generate an outcome space and to define solution based social choice rules. This corresponds to solve in that context the supportability-effectivity problem.
On Maskin monotonicity of solution based social choice rules
There are obviously several possibilities to factorize a payoff vector function into an outcome function and a vector of utility functions. The two extreme cases are to take (a) the outcome space as identical to the strategy space, choosing the outcome function as the identity map and the utility functions as the payoff functions; (b) the outcome space to be the space of payoff vectors, choosing the utility functions as projections to payoffs and the outcome function as the payoff vector function. For different choices of outcome space and preferences on the outcome space one clearly gets different solution based social choice rules. And Maskin monotonicity may very well depend on the actually selected solution based social choice rule. Howard (1992) argues that, due to a lack of Maskin monotonicity, the Nash bargaining solution fails to be Nash implementable. That a suitably defined Nash bargaining social choice rule is in fact Nash implementable has been demonstrated by van Damme (1986), Naeve (1999), and Trockel (2000, 2002b). In the next section we shall revisit the example by Howard and show that by choosing a different outcome function we can define a Nash social choice rule that is Maskin monotonic. We shall extend this discrete context to its convexification where our reasoning remains true. In Sect. 3 we provide an alternative approach to Howard’s example that allows it to avoid the violation of Maskin monotonicity. Section 4 briefly sketches that the situation with some other Pareto efficient solutions is similar. We particularly focus on the Kalai–Smorodinsky solution. Again, the examples are discrete and chosen in such a way that bargaining solutions are well defined and unique but allow for straightforward extensions to the convexified bargaining sets. The key property for this conclusion is some symmetry property of the considered bargaining solutions. The concluding Sect. 5 considers essential monotonicity that for more than two players is sufficient for Nash implementability. For any Pareto efficient bargaining solution we establish essential monotonicity, hence Maskin monotonicity, of the induced solution based social choice rule in our setup.
2 Howard’s example We consider a bargaining problem, in which two agents negotiate over the alternatives Q, S, V . If they do not come to an agreement the outcome is the status quo alternative Q. The set of admissible utility profiles on A := {Q, S, V } is U := {u, u } ≡ {(u 1 , u 2 ), (u 1 , u 2 )} where u i , u i , i = 1, 2 are real valued (von Neumann–Morgenstern) utility functions that are defined on A as follows: u(Q) = u (Q) = (q1 , q2 ) ≡ (q1 , q2 ) = (0, 0), u(S) = u (S) = (s1 , s2 ) ≡ (s1 , s2 ) = (1, 1), u(V ) = (v1 , v2 ) = (0, 2), u (V ) = (v1 , v2 ) = (3/4, 2).
C.-J. Haake, W. Trockel
Fig. 1 Howard’s example for the Nash solution
ˆ A bargaining solution in this framework is a mapping λ : {u(A), u(B)} −→ u(A)∪ ˆ ˆ u (A) with λ(u(A)) ∈ u(A) and λ(u (A)) ∈ u (A). The Nash solution is the bargaining ˆ ˆ ˆ ˆ solution ν that solves maxλ λ1 (u(A)) λ2 (u(A)) and maxλ λ1 (u (A)) λ2 (u (A)). ˆ ˆ ˆ The Nash social choice rule in this model is given by the correspondence ϕ ν : ˆˆ ν (w) := argmax ˆ U ⇒ A with ϕ ˆ a∈A w1 (a) w2 (a). As depicted in Fig. 1, for the profile u, we obtain ϕ ν (u) = {S} because u 1 (S) u 2 (S) = ˆˆ ˆˆ s1 s2 maximizes the Nash product u 1 (a) u 2 (a) on A. For the profile u we get ϕ ν (u ) = {V }. Indeed, now u 1 (V ) u 2 (V ) = v1 v2 = 3/2 > 1 = u 1 (S) u 2 (S). Hence, the switch from profile u to profile u results in a different social optimum in A. In particular, outcome S drops out of the Nash correspondence. However, we see no preference reversal involving S that is induced by that switch: S remains the best outcome for player 1 and the second ranked outcome for player 2. Therefore, Maskin monotonicity is violated. The arguments do not change when we replace A by the mixture set generated by A. For instance, let Q, S, V be defined as (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively, and let A := convex hull(A). As any point in A is a convex combination of (i.e., a probability distribution over) Q, S, V , its utility is simply the expected value of u i or u i (i = 1, 2), respectively. Now, u and u —for convenience we denote their extensions to A again by u, u —map A onto different compact convex sets. Next, we present an alternative to Howard’s model in which Maskin monotonicity is satisfied. 3 Alternative model for Howard’s example Let B := u(A) = {(0, 0), (1, 1), (0, 2)} and B := u (A) = {(0, 0), (1, 1), (3/4, 2)} two bargaining games with status quo point (0, 0) and B := {B, B } be the set of feasible bargaining games. Let λ, ν : B −→ R2 be bargaining solutions defined by
On Maskin monotonicity of solution based social choice rules
Fig. 2 Alternative approach to Howard’s example
λ(B) = λ(B ) = (1, 1), ν(B) = (1, 1), ν(B ) = (3/4, 2). Obviously, ν is the Nash solution on B as it maximizes the Nash product on B and B . Observe that B and B are exactly the two bargaining problems considered in the previous section. ˜ Now, define the outcome space A to be the set of all bargaining solutions on B, i.e., ˜ A := {α : B −→ B ∪ B | α(B) ∈ B, α(B ) ∈ B }. ˜ ˜ On A we define profiles of utility functions u, u by setting for any α ∈ A, ˜ ˜ u(α) := α(B) ˜ and u (α) = α(B ). ˜
˜ ˜ Let U := {u, u }. The bijection between U and B associating u with B and u with ˜ ˜ ˜ ˜ B provides the effectivity/supportability of Bergin and Duggan (1999) in our specific context! ˜ ˜ Next, we define our Nash social choice rule ϕ ν : U ⇒ A by ϕ ν (u) := argmax u 1 (α) u 2 (α) ⊇ {ν, λ}, ˜ ˜ ˜ ˜ ˜ ˜ ϕ (u ) := argmax u 1 (α) u 2 (α) ⊇ {ν}.
˜ α∈ A ν ˜ α∈ A
Note that in particular λ ∈ ϕ ν (u ). Thus, when switching from u to u the former social ˜ ˜ ˜ optimum λ is no longer one at preferences u (see Fig. 2). But, now a preference reversal ˜ involving social optima is involved. Indeed, while at u we have u(λ) = u(ν) = (1, 1), ˜ ˜ ˜ ˜ ˜ ˜ ˜ we get u 1 (ν) = 3/4 < 1 = u 1 (λ) and u 2 (ν) = 2 > 1 = u 2 (λ) at profile u . Thus, ν ˜ is strictly better than λ for player 2. Hence, Maskin monotonicity is not violated. Note that the two examples capture the same situation; a socially desired outcome is no longer desirable after a switch of utility profiles. But due to a different choice of the outcome space, and hence, of the social choice rule, Maskin monotonicity may or may not be satisfied. As we demonstrate in Sect. 5, the solution based choice correspondence ϕ ν does satisfy Maskin monotonicity. In fact, weak and full Nash
C.-J. Haake, W. Trockel
Fig. 3 The Kalai–Smorodinsky solution
implementation of the Nash bargaining solution based on social choice rules have been established in Trockel (2000, 2002b). 4 Further examples Howard’s observation that the specific Nash social choice rule is not Maskin monotonic is not limited to the Nash bargaining solution. Figure 3 illustrates an example with five physical outcomes A = {Q, S, V, W 1 , W 2 } and two profiles of utility functions u, u given by u(S) = u (S) = (5/4, 3/4), u(V ) = u (V ) = (1, 1), u(W 1 ) = u (W 1 ) = (2, 0), u(W 2 ) = (0, 6/5), u (W 2 ) = (0, 2), u(Q) = u (Q) = (0, 0). Again, we consider the two bargaining problems B = u(A) and B = u (A). Analogously to the definition of the Nash social choice rule in Sect. 2, we define the ˆˆ Kalai–Smorodinsky social choice rule ϕ κ : {u, u } ⇒ A by ϕ κ (w) := argmaxa∈A ˆˆ wi (a) κ (u) = {S} and ϕ κ (u ) = {V }. ˆ mini=1,2 max . Immediate calculations reveal ϕ ˆ ˆˆ a ∈A wi (a ) Again, physical outcome S is no longer desirable, when moving from u to u , but the ˆˆ ranking of outcomes in A are identical in u and u . So, ϕ κ is not Maskin monotonic. As in Sect. 2, nothing is altered, when considering A and extensions of u and u . By a closer inspection of Fig. 3, it is straightforward that the lack of Maskin monotonicity can be replicated for any Pareto efficient and symmetric bargaining solution.1 However, for positive implementation results of the Kalai–Smorodinsky solution, we refer to van Damme (1987), Haake (2000), or Trockel (1999). In the next section, we show in general that any solution based social choice correspondence that stems from a Pareto efficient bargaining solution is Maskin monotonic.
1 Roughly, one has to define u, u such that u (A) is obtained from u(A) by exchanging coordinates, but without reversing preferences over A. With an appropriate choice of utilities of S and V the solution switches between these physical outcomes.
On Maskin monotonicity of solution based social choice rules
5 Monotonicity Trockel (2002a) shows that any solution based social choice rule stemming from a Pareto efficient bargaining solution does satisfy Maskin monotonicity—a necessary condition for Nash implementability. As we demonstrate in this section a solution based social choice correspondence in fact satisfies a stronger version of monotonicity: essential monotonicity. Yamato (1992, Theorem 2), shows that this version is sufficient for Nash implementation, when there are at least three players.2 We consider a population I := {1, . . . , n} of n players. An n-person bargaining game B consists of a closed and convex subset of Rn —the utility possibility set—and an interior point—the status quo point—such that the set of status quo dominating points is bounded. Let B be a non-empty set of (admitted) bargaining games for n persons. ˜ We define the outcome space A to be the set of all bargaining solutions on B, i.e., ˜ ˜ := {α : B −→ Rn | α(B) ∈ B, B ∈ B}. By U we denote the set of all (admitted) A ˜ such that there is a well defined one-to-one corresprofiles of utility functions on A ˜ pondence between U and B along the effectivity/supportability results in Bergin and ˜ ˜ Duggan (1999). To be precise, u = (u 1 , . . . , u n ) ∈ U if and only if there is B ∈ B ˜ ˜ ˜ we have u(α) = α(B), meaning that player i evaluates bargaisuch that for all α ∈ A ˜ ning solutions by the utility they assign to him in bargaining problem B. Therefore, we henceforth identify utility functions profile u with bargaining problem B or u ˜ ˜ with B . ˜ Let η ∈ A be a prespecified bargaining solution. Define a (solution based) social ˜ ˜ ˜ ˜ choice correspondence ϕ η : U ⇒ A by ϕ η (u) := {α ∈ A | α(B) = η(B)} = ˜ ˜ | u(α) = u(η)}. That means, ϕ η assigns to u ∈ U all bargaining solutions ˜ {α ∈ A ˜ ˜ ˜ ˜ in A that coincide with η on u (i.e., on B). Put differently, when defining ϕ η (u), ˜ the corresponding bargaining problem B is the only relevant one. Therefore, if η is supposed to be a desirable bargaining solution, then all solutions that coincide with η ˜ on B should be equally desirable and are therefore collected in ϕ η (u) as well. ˜ ˜ For i = I , u ∈ U and α ∈ A define i’s lower contour set of α at u by Li (u, α) := ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ {α ∈ A | u i (α ) ≤ u i (α)}. A social choice correspondence F : U ⇒ A is Maskin ˜ ˜ ˜ ˜ monotonic, if for all i ∈ I , u, u ∈ U , α ∈ F(u), Li (u, α) ⊆ Li (u , α) implies ˜ ˜ α ∈ F(u ). ˜ ˜ ˜ ˜ ˜ ˜ Let M be a subset of A and F : U ⇒ A. An outcome α ∈ M is F-essential for i ∈ ˜ ˜ ˜ if there exists u ∈ U with α ∈ F(u) and Li (u, α) ⊆ M. Denote by Essi ( M, F) ˜ ¯ I in M, ¯ ¯ ˜ F satisfies essential monotonicity, if for all the set of F-essential outcomes for i in M. ˜ ˜ ˜ ˜ ˜ i ∈ I , u, u ∈ U , and all α ∈ F(u), Essi (Li (u, α), F) ⊆ Li (u , α) implies α ∈ F(u ). ˜ ˜ Theorem (Yamato (1992), Theorem 2) Suppose n ≥ 3. If F satisfies essential monotonicity, then F is Nash implementable. ˜ Proposition Let η ∈ A be a Pareto efficient bargaining solution. Then ϕ η is essentially η is Nash implementable, if there are three or more players. monotonic. Hence ϕ
2 See also Danilov (1992). In Yamato’s work, this condition was originally termed strong monotonicity,
but is now more frequently, and more appropriately, found under the term we use.
C.-J. Haake, W. Trockel
Proof We start with two immediate observations. ˜ 1. For all u ∈ U we have η ∈ ϕ η (u). ˜ ˜ ˜ ˜ ˜ ˜ 2. For all u ∈ U , i ∈ I and β ∈ ϕ η (u), Li (u, β) = Li (u, η). ˜ ˜ Now, let u, u ∈ U and α ∈ ϕ η (u) be such that Essi (Li (u, α), ϕ η ) ⊆ Li (u , α) for ˜ ˜ ˜ ˜ ˜ ˜ all i ∈ I . We need to show α ∈ ϕ η (u ). ˜ ˜ ˜ First, for all i ∈ I , any β ∈ ϕ η (u) is ϕ η -essential in Li (u, α). To see this, take u as ˜ utility profile u in the definition of essential outcomes. Then, clearly, β ∈ ϕ η (u) and ¯ ˜ ˜ by the second observation Li (u, β) ⊆ Li (u, α). Hence, for all i ∈ I we have ϕ η (u) ⊆ Essi (Li (u, α), ϕ η ) ⊆ Li (u , α). ˜ ˜ ˜ With the first observation, η ∈ Li (u , α), and therefore u i (η) ≤ u i (α) (i ∈ I ), which ˜ ˜ ˜ is equivalent to η(B ) ≤ a(B ), where B is the bargaining problem identified with ˜ ˜ u . Since η is Pareto efficient, η(B ) = a(B ), i.e., u i (η) = u i (α) (i ∈ I ), implying ˜ ˜ α ∈ ϕ η (u ). It is easy to see that essential monotonicity implies Maskin monotonicity. We can therefore confirm the following result in Trockel (2002a), as it is a direct corollary of the Proposition. Corollary Any solution based social choice correspondence with underlying Pareto efficient bargaining solution is Maskin monotonic. References
Bergin J, Duggan J (1999) Implementation-theoretic approach to non-cooperative foundations. J Econ Theory 86:50–76 Dagan N, Serrano R (1998) Invariance and randomness in the Nash program for coalitional games. Econ Lett 58:43–49 Danilov V (1992) Implementation via Nash equilibria. Econometrica 60:43–56 Haake C-J (2000) Support and implementation of the Kalai–Smorodinsky bargaining solution. In: Inderfurth K, Schwödiauer G, Domschke W, Juhnke F, Kleinschmidt P, Wäscher G (eds) Oper. Research Proceedings (1999). Springer, Heidelberg, pp 170–175 Howard JV (1992) A social choice rule and its implementation in perfect equilibrium. J Econ Theory 56:142–159 Hurwicz L (1994) Economic design, adjustment processes, mechanisms and institutions. Econ Des 1:1–14 Maskin ES (1999) Nash equilibrium and welfare optimality. Rev Econ Stud 66:23–38 Moulin H (1984) Implementing the Kalai–Smorodinsky solution. J Econ Theory 33:32–45 Nash JF (1951) Non-cooperative games. Ann Math 54:286–295 Nash JF (1953) Two person cooperative games. Econometrica 21:128–140 Naeve J (1999) Nash implementation of the Nash bargaining solution using intuitive message spaces. Econ Lett 62:23–28 Serrano R (1997) A comment on the Nash program and the theory of implementation. Econ Lett 55:203–208 Serrano R (2005a) Fifty years of the Nash program, 1953–2003. Invest Econ 29:219–258 Serrano R (2005b) Nash program. In: Durlauf S, Blume L (eds) The new palgrave dictionary of economics, 2nd edn. McMillan, London Trockel W (1999) Unique implementation for a class of bargaining solutions. Int Game Theory Rev 1:267–272 Trockel W (2000) Implementation of the Nash solution based on its Walrasian characterization. Econ Theory 16:277–294
On Maskin monotonicity of solution based social choice rules Trockel W (2002a) Integrating the Nash program into mechanism theory. Rev Econ Des 7:27–43 Trockel W (2002b) A universal meta bargaining realization of the Nash solution. Soc Choice Wel 19: 581–586 Trockel W (2003) Can and should the Nash program be looked at as a part of mechanism theory? In: Sertel MR, Korey S (eds) Advances in economic design. Springer, Heidelberg, pp 153–174 van Damme E (1986) The Nash bargaining solution is optimal. J Econ Theory 38:78–100 van Damme E (1987) Stability and perfection of Nash equilibria. Springer, Berlin Yamato T (1992) On Nash implementation of social choice correspondences. Games Econ Behav 4: 484–492
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Abstract Heresies
I'm sure most people who read this blog know a bit about what this means, but I want to try to explain it in a way my mom could understand. I won't go into excruciating detail, and I'll probably
gloss over interesting points.
Let's start with an analogy. Suppose you go to the store to buy a present for your nephew. He likes jigsaw puzzles, so you want to get him one. He's going to be twelve years old, so you need to find
one that has the appropriate challenge. A simple one with four pieces might be great for a two year-old, but your nephew would find that boring. A big one with ten thousand pieces is probably too
much for him. You know he just finished one with five hundred pieces, so you pick one with six hundred pieces because it won't be too easy, but not too frustrating. What you are doing is estimating
the complexity of the puzzle.
When you get to the store you find that they are sold out of jigsaw puzzles. They have a variety of other puzzles, though. They have a Rubik's cube, a Sudoku book, some cryptograms, a Tower of Hanoi,
etc. They have models you can put together as well. But these things aren't jigsaw puzzles. You can't estimate the complexity the same way. A model with five hundred pieces would be quite a bit more
difficult to assemble than a jigsaw puzzle with the same number of pieces.
Perhaps you look at the age ratings for the various puzzles. The eight-piece Tower of Hanoi puzzle is rated for ages ten and up, so maybe a twelve piece? The standard 3x3x3 Rubik's cube is rated for
ages eight and up, but would the 4x4x4 be too hard or too easy? What about the 5x5x5?
The kind of rating system you use to describe the difficulty of a puzzle is analogous to the “complexity class” of a problem. The complexity class roughly describes how hard a problem will be to
solve, and more importantly, how much harder ‘big’ problems are compared to the ‘small’ ones. Here's what I mean: if I take a jigsaw puzzle, any jigsaw puzzle, and double the number of pieces in it,
I'll make it roughly twice as hard (more or less). If I take a cryptogram and double the number of words in it, I'll actually make it a lot easier. If I take the Tower of Hanoi and double the number
of discs, I'll make it incredibly harder, maybe even impossible. These puzzles are in different complexity classes.
As it turns out, a lot of puzzles are in the same complexity class as the jigsaw puzzle: the difficulty is reasonably proportional to the number of pieces. Adding a piece or two doesn't change things
that much, and it is easy to figure out how long it will take to solve if you've done a few of them. A lot of puzzles are more like the Tower of Hanoi. Adding a single piece will double the amount of
time it takes to solve them.
P and NP are complexity classes for problems you might want to solve on a computer. There are all sorts of rating systems, but everyone is familiar with the MPAA movie ratings. There are all sorts of
complexity classes, but most computer scientists are familiar with P and NP.
I'll pause here because the analogy is wearing thin. The main point is that “P ≠ NP” is a statement about complexity classes. In particular, it talks about two of the most popular complexity classes
and that is part of the reason it is interesting.
5 comments:
My wife's the jigsaw puzzlist in the family, and she's not sure (too many confounding factors), but my intuition says that the difficulty of jigsaw puzzles grows as O(N^2), because at least at
the beginning you have to find out which of N pieces matches with which of N-1 other pieces.
But of course quadratic growth is not really much worse than linear in this regime.
The brute force approach could indeed be N^2 but luckily we don't have to solve them that way. A 10k piece puzzle takes hours (for my mother and I anyway).
If you could do a 1000 piece puzzle in one hour and it grew at N^2 then the 10k piece puzzle should take 100 hours.
This comment has been removed by the author.
The Towers of Hanoi has at least one simple algorithm, so you can do it with any disks.
It is never impossible.
Towers of Hanoi is simple enough, but it is impractical. Although a tower of, say, thirty disks is possible to solve in some theoretic sense, it is impractical for a human being to carry out.
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Eigenvalue problem
February 23rd 2011, 08:06 PM
Eigenvalue problem
$m^2+\lambda=0\Rightarrow \exp(\pm xi\sqrt{\lambda})$
$\varphi=C_1\cos(x\sqrt{\lambda})+C_2\sin(x\sqrt{\l ambda})$
$\varphi_1(0): \ C_1=1$
$\varphi_1'(0): \ C_2=0$
$\varphi_2(0): \ C_1=0$
$\displaystyle\varphi_2'(0): \ C_2=\frac{1}{\sqrt{\lambda}}$
$\displaystyle\varphi_2=\frac{\sin(x\sqrt{\lambda}) }{\sqrt{\lambda}}$
$\displaystyle\varphi(x)=A\cos(x\sqrt{\lambda})+B\f rac{\sin(x\sqrt{\lambda})}{\sqrt{\lambda}}$
$\varphi'(0): \ B=0$
$\varphi(L): \ A\cos(L\sqrt{\lambda})=0$
$\displaystyle\lambda_n=\left(\frac{(2n+1)\pi}{2L}\ right)^2, \ \ n\in\mathbb{Z}$
$\displaystyle\varphi_n(x)=A_n\cos\left(\frac{(2n+1 )\pi x}{2L}\right)$
$\displaystyle f(x)=\sum_{n=0}^{\infty}A_n\cos\left(\frac{(2n+1)\ pi x}{2L}\right)$
Is this correct so far?
February 23rd 2011, 08:21 PM
Assuming the above is correct, I am now going to show that all the eigenvalues are real.
Let $L\sqrt{\lambda}=a+bi, \ \ a,b\in\mathbb{R}$
$\displaystyle\cos(a)\cosh(b)=0, \ \ \cosh(b)>0, \ \ \cos(a)=0\Rightarrow a=\frac{\pi}{2}+\pi k, \ \ k\in\mathbb{Z}$
$-\sin(a)\sinh(b)=0, \ \ \sinh(b)=0\Rightarrow b=0, \ \ \sin(a)=0\Rightarrow a=\pi k$
$\cos(a+bi)eq 0$
Thus, all eigenvalues are real.
February 24th 2011, 01:27 AM
Just commenting on the proof of the realness of the eigenvalues: you could save yourself a bit of effort if you could show that your original differential operator is self-adjoint. In this
context, that amounts to showing that the operator is of the Stürm-Liouville type.
February 24th 2011, 09:24 AM
Just commenting on the proof of the realness of the eigenvalues: you could save yourself a bit of effort if you could show that your original differential operator is self-adjoint. In this
context, that amounts to showing that the operator is of the Stürm-Liouville type.
I haven't arrived to that in the book yet. Is post 1 correct though?
February 24th 2011, 09:41 AM
Post 1 looks correct so far as it goes. You should probably, if you haven't yet, go ahead and show that there are no eigenvalues for either the $\lambda<0$ or $\lambda=0$ case.
February 24th 2011, 05:25 PM
Doesn't the fact that cosine can't be of form cos(a + bi) show that lambda can't be less than 0?
February 25th 2011, 01:52 AM
I've pored over your post # 2 now, and I think I can finally discern the logic you're employing there. Basically, it comes down to this: $b = 0$, or you get $a$ having to be both an odd-integer
multiple of $\pi/2$, and a multiple of $\pi$, which can't be. Therefore, $b=0$. I think your overall logic works, provided that the form of your solution hasn't already assumed that $\lambda>0.$.
You have not shown that lambda can't be zero. In order to do that, you have to re-solve the DE with that assumption in mind (the solutions you get for that case are not obtainable with any
selection of the integration constants for the lambda not zero case), and show that there are no eigenvectors.
I would probably solve the problem this way: break it up into three cases, according to the dichotomy law: $\lambda<0,\;\lambda=0,\;\lambda>0.$ For $\lambda<0,$ let $\lambda=-\alpha^{2}.$ For $\
lambda=0,$ do the obvious. And for $\lambda>0,$ let $\lambda=\alpha^{2}.$ In each case, you get a different form of the solution, with which you work to see if it's an allowed case. For $\lambda
<0$, you get exponentials. For $\lambda=0,$ you get straight lines. For $\lambda>0,$ you get sinusoids. Remember that, by definition, eigenvectors cannot be identically zero. That fact, in this
case, rules out the $\lambda<0$ and $\lambda=0$ cases.
Make sense?
February 25th 2011, 12:27 PM
I've pored over your post # 2 now, and I think I can finally discern the logic you're employing there. Basically, it comes down to this: $b = 0$, or you get $a$ having to be both an odd-integer
multiple of $\pi/2$, and a multiple of $\pi$, which can't be. Therefore, $b=0$. I think your overall logic works, provided that the form of your solution hasn't already assumed that $\lambda>0.$.
You have not shown that lambda can't be zero. In order to do that, you have to re-solve the DE with that assumption in mind (the solutions you get for that case are not obtainable with any
selection of the integration constants for the lambda not zero case), and show that there are no eigenvectors.
I would probably solve the problem this way: break it up into three cases, according to the dichotomy law: $\lambda<0,\;\lambda=0,\;\lambda>0.$ For $\lambda<0,$ let $\lambda=-\alpha^{2}.$ For $\
lambda=0,$ do the obvious. And for $\lambda>0,$ let $\lambda=\alpha^{2}.$ In each case, you get a different form of the solution, with which you work to see if it's an allowed case. For $\lambda
<0$, you get exponentials. For $\lambda=0,$ you get straight lines. For $\lambda>0,$ you get sinusoids. Remember that, by definition, eigenvectors cannot be identically zero. That fact, in this
case, rules out the $\lambda<0$ and $\lambda=0$ cases.
Make sense?
For lambda = 0, wouldn't it be easier to just show:
$\cos(L\sqrt{\lambda})=0\Rightarrow 1=0$ which is never true.
When lambda < 0, the term inside the cosine is complex, and I have already shown that complex numbers aren't eigenvalues of the solution.
February 25th 2011, 12:35 PM
It is not only not easier to show it this way, it is impossible! In even writing down the $\cos$ function at all, you've already assumed that that is the form of the solution when $\lambda=0,$
which simply isn't true. Instead, you must re-solve the DE from scratch (it's quite straight-forward, really), and then apply the boundary conditions. There is no other way that I know of to show
that $\lambda=0$ is not an eigenvalue.
When lambda < 0, the term inside the cosine is complex, and I have already shown that complex numbers aren't eigenvalues of the solution.
Like I said in my previous post, as long as you haven't already assumed a form of the solution that is only applicable when $\lambda>0,$ then your proof works out fine.
I would, incidentally, put more English in your proof of post # 2. It's a bit hard to follow what you're doing. Don't write so that you can be understood! Write so that you can't be
February 25th 2011, 12:46 PM
It is not only not easier to show it this way, it is impossible! In even writing down the $\cos$ function at all, you've already assumed that that is the form of the solution when $\lambda=0,$
which simply isn't true. Instead, you must re-solve the DE from scratch (it's quite straight-forward, really), and then apply the boundary conditions. There is no other way that I know of to show
that $\lambda=0$ is not an eigenvalue.
Like I said in my previous post, as long as you haven't already assumed a form of the solution that is only applicable when $\lambda>0,$ then your proof works out fine.
I would, incidentally, put more English in your proof of post # 2. It's a bit hard to follow what you're doing. Don't write so that you can be understood! Write so that you can't be
From my understand, plugging in lambda = 0 is fine.
The example in my book has:
$\displaystyle \varphi(L)=\frac{\sin(L\sqrt{\lambda})}{\sqrt{\lam bda}}$
$\displaystyle \frac{\sin(L\sqrt{\lambda})}{\sqrt{\lambda}}=0$
Then states: We return now to the problem of finding all all the eigenvalues, that is, all the solutions of the equation. If lambda = 0 the left member is to be interpreted as
$\displaystyle\lim_{\lambda\to 0}\frac{\sin(L\sqrt{\lambda})}{\sqrt{\lambda}}=L eq 0$
I used what was obtained $\varphi(L)$ as well so I don't see what the difference is besides the example is different.
February 25th 2011, 01:01 PM
Well, that method could well be valid: I don't know. To me it seems a bit strange to write down the sin or cosine, which isn't the solution to the $\lambda=0$ case, and then turn around and use
that form of the solution to show that $\lambda=0$ is not an eigenvalue.
Here's what I would do: $\lambda=0$ implies $\varphi''=0,$ and so $\varphi(x)=mx+b.$ The $\varphi'(L)=0$ condition implies $m=0,$ and thus $\varphi(x)=b.$ But $\varphi(0)=0$ implies $b=0,$ and
hence $\varphi(x)=0,$ which is not allowed, because eigenfunctions can't be identically zero by definition. Done. Is that not fairly intuitive?
February 25th 2011, 02:10 PM
Now, I am asked to show that eigenfunctions are orthogonal.
$\displaystyle\int_0^L\varphi_n(x)\varphi_m(x) \ dx=0, \ \ meq n$
$\displaystyle\int_0^L(\varphi_n(x))^2 \ dx>0$
$\displaystyle\int_0^L\cos\left(\frac{(2n+1)\pi x}{2L}\right)\cdot\cos\left(\frac{(2m+1)\pi x}{2L}\right) \ dx=0, \ \forall meq n$
$\displaystyle \int_0^L\left[\cos\left(\frac{(2n+1)\pi x}{2L}\right)\right]^2 \ dx=\frac{L}{2}$
To save space, I haven't shown the steps for orthogonality but it does hold.
Now, I am supposed to use everything in this thread to solve:
$\text{D.E.}=u_t=ku_{xx}, \ \ \ t>0, \ \ \ 0<x<L$
$\displaystyle\text{B.C.}=\begin{cases} u_x(0,t)=0\\u(L,t)=0\end{cases}, \ \ \ t>0$
$u(x,0)=L-x, \ \ \ 0<x<L$
Not sure where to begin.
February 25th 2011, 02:47 PM
Now, I am asked to show that eigenfunctions are orthogonal.
$\displaystyle\int_0^L\varphi_n(x)\varphi_m(x) \ dx=0, \ \ meq n$
$\displaystyle\int_0^L(\varphi_n(x))^2 \ dx>0$
$\displaystyle\int_0^L\cos\left(\frac{(2n+1)\pi x}{2L}\right)\cdot\cos\left(\frac{(2m+1)\pi x}{2L}\right) \ dx=0, \ \forall meq n$
$\displaystyle \int_0^L\left[\cos\left(\frac{(2n+1)\pi x}{2L}\right)\right]^2 \ dx=\frac{L}{2}$
To save space, I haven't shown the steps for orthogonality but it does hold.
Now, I am supposed to use everything in this thread to solve:
$\text{D.E.}=u_t=ku_{xx}, \ \ \ t>0, \ \ \ 0<x<L$
$\displaystyle\text{B.C.}=\begin{cases} u_x(0,t)=0\\u(L,t)=0\end{cases}, \ \ \ t>0$
$u(x,0)=L-x, \ \ \ 0<x<L$
Not sure where to begin.
Now you need to separate the PDE
Assume $u(x,t)=T(t)X(x)$ this gives
$u_{t}=\dot{T}X \text{ and } u_{xx}=TX''$
This gives
$\displaystyle \dot{T}X=TX'' \iff \frac{\dot{T}}{T}=\frac{X''}{X}=-\lambda$
Now the $X$ equation is what you have already solved
$X''-\lambda X=0$
Now solve for $T(t)=e^{-\lambda t}$ and you will have the general form of the solution to your equation.
Using your initial condition gives this
$\displaystyle u(x,0)=\sum_{n=0}^{\infty}a_ne^{-n(0)}\varphi_n(x)=L-x$
Now use your innerproduct and the orthogonality relationships to solve for the $a_n$
Now integrate bothsides from 0 to L and see what happens!
February 25th 2011, 04:00 PM
Now you need to separate the PDE
Assume $u(x,t)=T(t)X(x)$ this gives
$u_{t}=\dot{T}X \text{ and } u_{xx}=TX''$
This gives
$\displaystyle \dot{T}X=TX'' \iff \frac{\dot{T}}{T}=\frac{X''}{X}=-\lambda$
Now the $X$ equation is what you have already solved
$X''-\lambda X=0$
Now solve for $T(t)=e^{-\lambda t}$ and you will have the general form of the solution to your equation.
Using your initial condition gives this
$\displaystyle u(x,0)=\sum_{n=0}^{\infty}a_ne^{-n(0)}\varphi_n(x)=L-x$
Now use your innerproduct and the orthogonality relationships to solve for the $a_n$
Now integrate bothsides from 0 to L and see what happens!
What about the k?
$\displaystyle u_t=ku_{xx}\Rightarrow\frac{\dot{T}}{T}=k\frac{X'' }{X}=-\lambda\text{?}$
February 25th 2011, 04:27 PM
I didn't see your k
I would separate like this
$\displaystyle \frac{\dot{T}}{kT}=\frac{X''}{X}=-\lambda$
This won't change the X equation and
$T(t)=e^{-k\lambda t}$ and just keep going
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We present results of a search for anomalous production of events containing a charged lepton (either electron or muon) and a photon, both with high transverse momentum, accompanied by additional
signatures, X, including missing transverse energy (MET) and additional leptons and photons. We use the same kinematic selection criteria as in a previous CDF search, but with a substantially larger
data set, 929 pb-1, a ppbar collision energy of 1.96 TeV, and the upgraded CDF II detector. We find 163 Lepton+Photon+MET events versus a standard model expectation of 150.6 +- 13.0 events. The level
of excess observed in Run I, 16 events with an expectation of 7.6 +- 0.7 events (corresponding to a 2.7 sigma effect), is not supported by the new data. In the signature of (ee/&mu&mu)+Photon+X we
observe 74 events versus an expectation of 65.1 +- 7.7 events. In this sample we find no events with an extra photon and so find no events like the one ee+&gamma&gamma+MET event observed in Run I. In
the signature of Electron+Muon+Photon+X we find no events versus an expectation of 1.0 +- 0.3 events. In the signature of Multi-Photon+Lepton+X we find no events versus an expectation of 0.62 +- 0.15
Selection Cuts for Lepton + Photon + X
The Selection Cuts for Lepton+Photon+X. The kinematic selection is set a priori - it is the same as criteria used in the previous Run I CDF search (hep-ex/0202044), and the same as used in recent
CDF Run II publication on 305 pb-1(hep-ex/0605097)
PS or PDF, or JPG,
Lepton + Photon + Missing Transverse Energy Plots
The distributions for events in the Lepton+Photon+Missing Transverse Energy sample (points) in a) the E_T of the photon; b) the E_T of the lepton, c) the Missing Transverse Energy, and d) the
transverse mass of the Lepton+Photon-Missing Transverse Energy system. The histograms show the expected SM contributions, including estimated backgrounds from misidentified photons and leptons. We
find 163 Lepton+Photon+MET events versus a standard model expectation of 150.6 +- 13.0 events. The level of excess observed in Run I, 16 events with an expectation of 7.6 +- 0.7 events
(corresponding to a 2.7 sigma effect), is not supported by the new data.
PS or PDF, or JPG,
Multi-Lepton + Photon Plots
The distributions for events in the Multiple Leptons + Photon sample (points) in a) the E_T of the photon; b) the E_T of the lepton, c) the 2-body mass of the dilepton system, and d) the 3-body
invariant mass of Leptons + Photon. The histograms show the expected SM contributions. In the signature of Multi-Lepton + Photon + X we observe 74 events versus an expectation of 65.1 +- 7.7 events
. In this sample we find no events with an extra photon and so find no events like the one ee+gg+MET event observed in Run I.
PS or PDF, or JPG,
Multi-Lepton + Photon Plots: missing transverse energy
The distributions in Missing Transverse Energy, observed in the Multiple Leptons + Photon inclusive search for muon pairs (Left) and electron pairs (Right). The histograms show the expected SM
contributions. In the Multi-Lepton + Photon + X with MET > 25 GeV 3 events observed are consistent with the SM expectation of 0.6 +- 0.1 events.
PS or PDF, or JPG,
Lepton+Photon+MET Summary table
A comparison of the numbers of events predicted by the Standard Model and the observations for the Lepton+Photon+Missing Transverse Energy search. The SM predictions for the search are dominated by
W+gamma and Z+gamma production. Other contributions come from the tri-boson processes W+gamma+gamma and Z+gamma+gamma, leptonic tau decays, and misidentified leptons, photons, or Missing Transverse
PS or PDF, or JPG,
Multi-Lepton + Photon Summary table
A comparison of the numbers of events predicted by the Standard Model and the observations for the Multiple Leptons + Photon search. The SM predictions for the search are dominated by Z+gamma
production. Other contributions come from the Z+gamma+gamma, and misidentified leptons and photons.
PS or PDF, or JPG,
Electron+Muon+Photon Summary table
A comparison of the numbers of events predicted by the Standard Model and the observations for the Electron+Muon+Photon search. The SM predictions for the search are dominated by Z+gamma
production. Other contributions come from the W+gamma, Z+gamma+gamma, W+gamma+gamma, and misidentified leptons and photons.
PS or PDF, or JPG,
Multi-Photon + Lepton Summary table
A comparison of the numbers of events predicted by the Standard Model and the observations for the Multiple Photons + Lepton search. Dominant contribution comes from misidentified photons. The SM
predictions are dominated by Z+gamma+gamma production.
PS or PDF, or JPG,
Additional Plots for Lepton+Photon+MET Search
Lepton+Photon+MET. The distributions for events in a) H_T, the sum of the transverse energies of the lepton, photon, jets and MET; b) the distance in eta-phi space between the photon and lepton; c)
the angular separation in phi between the lepton and MET; and d) the invariant mass of the Lepton+Photon system.
PS or PDF, or JPG,
Additional Plots for Lepton+Photon+MET Search in the Muon channel
Muon+Photon+MET. The distributions for events in in a) the E_T of the photon; b) the P_T of the muon; c) the missing transverse energy; and d) the transverse mass of the Muon+Photon+MET.
PS or PDF, or JPG,
Muon+Photon+MET. The distributions for events in a) H_T, the sum of the transverse energies of the muon, photon, jets and MET; b) the distance in eta-phi space between the photon and muon; c) the
angular separation in phi between the muon and MET; and d) the invariant mass of the Muon+Photon system.
PS or PDF, or JPG,
Additional Plots for Lepton+Photon+MET Search in the Electron channel
Electron+Photon+MET. The distributions for events in in a) the E_T of the photon; b) the E_T of the electron; c) the missing transverse energy; and d) the transverse mass of the
PS or PDF, or JPG,
Electron+Photon+MET. The distributions for events in a) H_T, the sum of the transverse energies of the electron, photon, jets and MET; b) the distance in eta-phi space between the photon and
electron; c) the angular separation in phi between the electron and MET; and d) the invariant mass of the Electron+Photon system.
PS or PDF, or JPG,
Additional Plots for Multi-Lepton + Photon Search
Multi-Lepton + Photon. The distributions for events in a) H_T, the sum of the transverse energies of the lepton, photon, jets and MET; b) the distance in eta-phi space between the photon and each
of the two leptons.
PS or PDF, or JPG,
Additional Plots for Multi-Lepton + Photon Search: Muon Channel
Muon+Photon+MET. The distributions for events in a) the E_T of the photon; b) the P_T of the muon, c) the 2-body mass of the dimuon system, and d) the 3-body invariant mass of Muons + Photon.
PS or PDF, or JPG,
Multi-Muon + Photon. The distributions for events in a) H_T, the sum of the transverse energies of the muon, photon, jets and MET; b) the distance in eta-phi space between the photon and each of
the two muons.
PS or PDF, or JPG,
Additional Plots for Multi-Lepton + Photon Search: Electron Channel
Electron+Photon+MET. The distributions for events in a) the E_T of the photon; b) the E_T of the electron, c) the 2-body mass of the dielectron system, and d) the 3-body invariant mass of Electrons
+ Photon.
PS or PDF, or JPG,
Multi-Electron + Photon. The distributions for events in a) H_T, the sum of the transverse energies of the electron, photon, jets and MET; b) the distance in eta-phi space between the photon and
each of the two electrons.
PS or PDF, or JPG,
Summary slide
Lepton+Photon+X summary slide
PS or PDF, or JPG,
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Anderson-Darling Test
The Anderson-Darling procedure is a general test to compare the fit of an observed cumulative distribution function to an expected cumulative distribution function. This test gives more weight to the
tails than the Kolmogorov-Smirnov test.
The Anderson-Darling statistic (A^2) is defined as
Hypothesis Testing
The null and the alternative hypotheses are:
• H[0]: the data follow the specified distribution;
• H[A]: the data do not follow the specified distribution.
The hypothesis regarding the distributional form is rejected at the chosen significance level (^2, is greater than the critical value obtained from a table. The fixed values of [0]) at various
significance levels. A value of 0.05 is typically used for most applications, however, in some critical industries, a lower
In general, critical values of the Anderson-Darling test statistic depend on the specific distribution being tested. However, tables of critical values for many distributions (except several the most
widely used ones) are not easy to find.
The Anderson-Darling test implemented in EasyFit uses the same critical values for all distributions. These values are calculated using the approximation formula, and depend on the sample size only.
This kind of test (compared to the "original" A-D test) is less likely to reject the good fit, and can be successfully used to compare the goodness of fit of several fitted distributions.
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You, Me, Attack Power and Dupree
Attack Power is one of the more interesting stats for physical DPS classes. At the most basic level, we can all understand one simple truth – more of it is better. And yet, why it is better and the
intricacies of the mechanics it impacts are often misunderstood. I have often noted this to myself when discussions about Attack Power come up in Guild Chat and the ill-informed start chirping up
with bad information. A series of blog entries over at Priestly Endeavors illustrated to me that even people I generally consider pretty knowledgeable can misunderstand Attack Power.
Before I go into detail about Attack Power, I want to provide a quick caveat. I
what I am about to discuss is applicable to Rogues, Warriors and Feral Druids. I am less familiar with the mechanics of Shamans, Paladins and Hunters – so I won’t talk directly about how this impacts
them. Broadly speaking, these are general principles and SHOULD apply to the other classes, but I’m not going to offer prescriptive guidance on a class that I don’t have intimate knowledge about.
Attack Power and White Damage
The term white damage is simply a way to describe our standard auto-attack (or auto-shot) damage that comes from swinging our melee weapon. If we use an ABILITY other than auto-attacks, this is
considered yellow damage – which I’ll talk about in a later section.
When we think about white damage, the easiest thing to figure out is how much damage is caused by our weapon. All you need to do is hover over your weapon and look at the item tooltip to see how much
damage it deals. However, the formula for the actual white damage caused by each weapon strike also includes Attack Power.
All classes convert Attack Power to DPS at a 14:1 ratio (14 AP = 1 DPS). This holds true at level 1 and level 70. For example, 140 Attack Power converts to 10 DPS. Now, different classes derive
Attack Power in different ways, but that 14:1 ratio always holds true. For example, a Warrior gains 2 AP for every 1 STR and a Rogue gains 1 AP per 1 STR. However, both of them convert AP to DPS at
the 14:1 ratio.
So the formula for Damage Dealt by Attack Power looks something like this:
(AP/14) * Weapon Speed = Damage Dealt Per Strike due to Attack Power
Example: (2000 AP / 14) * 2.6 Weapon Speed = 371 Damage Per Strike
One thing to consider is that the amount of DPS remains constant at 14:1 regardless of the actual Weapon Speed. You don’t get more white damage DPS from a slower or faster weapon; it will always be
the same.
Attack Power and Dual Wielding
The notable exception to how Attack Power scales for white damage is Dual Wielding. The reason is because Attack Power is applied at the 14:1 ratio for BOTH weapons. However, because your OFFHAND
weapon suffers from a damage penalty – you don’t get the full benefit.
For your OFFHAND, the formula looks like this:
((AP/14) * Weapon Speed) / 2 = Offhand Damage Dealt Per Strike due to Attack Power
Example: ((2000 AP / 14) * 2.6 Weapon Speed) / 2 = 186 Offhand Damage Per Strike
When you consider that this damage is IN ADDITION TO the Mainhand damage, it becomes pretty clear that the benefit for scaling Attack Power is significantly better for Dual Wielders than Two-Handed
or Single-Handed weapon users. In fact, if you use talents (like Dual Wielding Specialization) to increase your Offhand damage, then you can gain even more incremental benefit for Attack Power.
However – Dual Wielding does come at a cost and it exacts a 19% miss penalty. The base miss penalty on a same level mob is 5%, so anyone who Dual Wields gets an unmodified 24% chance to miss. This is
one reason why +hit stats are very important to Dual Wielders. As that stat increases, more of the benefit from their scaling Attack Power is realized.
White Damage and Haste
As pointed out in the previous two sections, Attack Power is very much a part of White Damage. Therefore, anything that acts as a multiplier on your white damage works as a multiplier on Attack
Power. The primarily multiplier I am talking about is, of course, Haste and it you can get it from items (like Drums of Battle), talents or abilities.
Now, the first thing to remember is that Haste is not going to change the amount of damage dealt in each individual strike. The formulas (above) used to calculate how much damage dealt from the
strike remain unmodified. What DOES change is the frequency of attacks. If we use the first example, our strike will hit for 371 every 2.6 seconds. If we enjoyed a 30% Haste, that same 371 strike
would occur every 2 seconds. The effective DPS from Attack Power increases from 143 DPS to 186.
You’ll notice that this is exactly a 30% increase in DPS for 30% Haste, so we can effectively assume that each 1% of Haste is the equivalent of an extra 1% in WHITE damage. Interestingly, your crit %
has a very similar effect on white damage. The basic theory is that a 1% crit rate would lead to double-damage 1 time in 100 strikes. This is the same damage amount as 101 strikes or 1% more damage
than 100 strikes. Therefore, a 1% crit rate leads to a 1% increase in damage.
Now keep in mind I am strictly talking about the effect on white damage (and I am ignoring the multiplicative effects). Haste vs. Crit vs. Hit can become very convoluted, so I point this out more as
an illustration for how these things impact Attack Power. The purpose is to explain why talents (like Imp Slice and Dice, Flurry) and abilities that provide Haste are so valuable.
Yellow damage and Attack Power
I described yellow damage earlier as damage caused by your non auto-attack abilities. That’s not entirely true since yellow damage can also be sourced from things like Poison procs, but as it relates
to melee classes it is generally true. Yellow damage comes in all sorts of variety because there are lots of different types of abilities. Some of them have nothing to do with Attack Power at all.
Others, like Bloodthirst, are 100% determined by your Attack Power.
You can usually find the “formula” for how an abilities damage is derived by looking it up on
. For simplicity, I am only going to talk about those how Attack Power impacts yellow attacks that are based on Weapon Damage.
By now, you should be able to calculate how much damage is dealt in a single strike for white damage. A slower weapon will deal more damage in a single strike than a fast one.
Let’s compare a 2.6 one-handed sword to a 2.0 one-handed sword:
Example: (2000 AP / 14) * 2.6 Weapon Speed = 371 Damage Per Strike
Example: (2000 AP / 14) * 2.0 Weapon Speed = 286 Damage Per Strike
The 2.6 sword deals 85 damage more per strike at 2000 AP than the 2.0 sword. Well Blizzard, smarty pants that they are, figured out that was a bad idea for attacks that were instant, so they decided
to normalize it based on weapon type in patch 1.08. So, all weapons of a type use the same weapon speed to calculate the Attack Power portion of the instant attack.
1.7 for daggers
2.4 for other one-handed weapons
3.3 for two-handed weapons
2.8 for ranged weapons
So, in our example above – the 2.6 and 2.0 one handed swords would have both been normalized to 342 damage per strike from Attack Power. If we had used a 2.0 Dagger, then the damage per strike would
only have been 243 from the Attack Power. And a two-handed sword would have see even more benefit at 471 damage per strike.
BUT—here’s the kicker—they did NOT normalize the base damage provided in the weapon tooltip. The sword with 189-285 damage and a 2.6 speed and sword with 1.5 speed and 95-178 damage both have
comparable DPS. However, the 2.6 sword has an average damage of 237 for each individual strike and the 1.5 sword has an average damage of 137 per strike. The net effect is that the 2.6 sword will do
roughly 100 damage more on each individual strike.
Why is this important? Because when yellow attacks are INSTANT or faster than your normal weapon speed, then you want each of those attacks to carry as much punch as possible. Many of these attacks
are also use modified weapon damage, like 150% Weapon damage. That means our above 2.6 weapon speed sword would provide 237+342 = 579 Weapon Damage which is modified by 150% to 868 damage. By
contrast, your 1.5 weapon would have done 137+342 = 479 * 150% = 719 in damage. That’s a 149 point difference (or 17% decrease) in the amount of damage dealt by the instant attack of the 1.5 sword.
This is why melee classes are generally advised to put the slowest possible weapon in your Mainhand.
Attack Power compared to other things
One terribly difficult thing to gauge is how important Attack Power is relative to other stats as a contributor to your overall DPS. Quite frankly, it can change pretty dramatically depending on how
the rest of your gear is itemized and even as you level.
I pointed out in the past section that Critical Strike and Haste both effectively provide a 1% increase white damage per % point. At level 20, what is more valuable – a 1% increase or 28 attack
power? Well if my DPS is only 20, then a 1% increase is only .2 DPS while 28 attack power is a 2 DPS increase (10% damage increase). Of course, at level 70, a 2 DPS increase may be almost
insignificant particularly compared to a 1% increase for 800 to 1000 DPS.
Still – Attack Power is incredibly valuable because as I have hopefully pointed out by now, it’s one of the most basic building blocks of the DPS formulas. After all, 1% of zero is still zero. Dual
Wielders, in particular, get even more incremental benefit from Attack Power and talents or abilities that provide Haste.
It also means different things to different classes and specs. One thing I always find pretty comical is the envy that some Rogues get over a Feral Druids Attack Power. The reality is that they are
itemized very differently and for very good reasons. Of course a Rogue is going to have less Attack Power – they Dual Wield and they have abilities (like SnD) that provide significant Haste. Blizzard
would be out of their mind to provide them similar Attack Power in the itemization. Instead, they very much force Rogues to look at +hit gear to take advantage of the Attack Power they are wasting in
Sometimes we are asked things like, what’s better 100 AP or 1.21% to crit? Hmm. Well, that’s not such an easy question to answer now is it?
Edit: It should be pointed out that Kirk at Priestly Endeavors understands Attack Power, he just chose to ignore it in some of the theory crafting he was performing. His latest entry (
) does a much better job of explaining his point and including Attack Power in his discussion.
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Bulletin of the World Health Organization
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Related links
Print version ISSN 0042-9686
Bull World Health Organ vol.79 n.6 Genebra Jan. 2001
On measuring inequalities in health
Michael Wolfson^1 & Geoff Rowe^2
ABSTRACT: In a recent series of papers, Murray et al. have put forward a number of important ideas regarding the measurement of inequalities in health. In this paper we agree with some of these ideas
but draw attention to one key aspect of their approach measuring inequalities on the basis of small area data which is flawed. A numerical example is presented to illustrate the problem. An
alternative approach drawing on longitudinal data is outlined, which preserves and enhances the most desirable aspects of their proposal. These include the use of a life course perspective, and the
consideration of non-fatal health outcomes as well as the more usual information on mortality patterns.
Keywords Health status; Socioeconomic factors; Mortality/trends; Longevity; Outcome assessment (Health care); Analysis of variance; Longitudinal studies; Models, Statistical (source: MeSH).
Mots clés Etat sanitaire; Facteur socioéconomique; Mortalité/orientations; Longévité; Evaluation résultats (Santé); Analyse variance; Etude longitudinale; Modèle statistique (source: INSERM).
Palabras clave Estado de salud; Factores socioeconómicos; Mortalidad/tendencias; Longevidad; Evaluación de resultado (Atención de salud); Análisis de varianza; Estudios longitudinales; Modelos
estadísticos (fuente: BIREME).
In a series of recent papers, Murray et al. (1 3) have put forward a number of important ideas regarding the measurement of inequalities in health. The third paper, by Gakidou et al., gave rise to
some debate (4). Subsequently, WHO published the World health report 2000 (5) which began putting these ideas into practice. The purpose of the current paper is to explore some of them and offer a
number of suggestions.
The core question is what should be meant by inequalities in health. Gakidou et al. define health inequality to be variations in health status across individuals in a population... which allows us
to perform cross-country comparisons and study the determinants of health inequality... WHO is interested in measuring health inequality as a distinct dimension of the performance of health systems
(3). They then go on to focus on the scalar healthy lifespan as the individual-level health measure of interest.
As they observe, healthy lifespan can be estimated by a modified sequence of age-specific survival probabilities (or more precisely, a mortality hazard function). Their key modification of this
standard notion is to combine risks of mortality with risks of disease onset and progression, characterized by an age profile of a summary health index, in order to form a health survivorship
function which in turn underlies estimates of healthy lifespan.
Later in the paper, Gakidou et al. observe that the healthy lifespans of the members of a living population are intrinsically unobservable (since their lives have still to finish unfolding, and we
cannot see the future). But, they claim, ... the distribution of health risks can be reasonably approximated... , and the distribution of these risks determine (subject to elements of chance) the
distribution of ultimately realized healthy lifespans. As a result, any feasible programme of measurement of health inequalities of this sort requires the measurement of health risks.
They appeal in general to methods based on subdividing the population into groups. Inevitably , they observe, this will underestimate the distribution of health expectancy (i.e. healthy
lifespan) in the population even if the groups are perfectly nonoverlapping in terms of their individual health expectancies. The more refined the groupings the more we will approximate the true
underlying distribution of health expectancy. Small area analyses hold out the promise of being one of the most refined methods for revealing the underlying distribution of health expectancy in a
This focus on health risks, and how they can be observed, raises serious measurement issues. Our basic concern is that efforts to describe health inequalities in terms of health survivorship risks,
based on inferences from small area data, are highly problematic, desirable as the approach may be. However, alternative approaches, based on longitudinal data, are much more promising.
Defining inequalities
We begin with two basic points.
First, inequality is a property of a population. It assumes that each member of the population has an attribute, health , that is unidimensional and measurable on a cardinal scale, such as income.
Second, the term health inequalities is used in two quite different ways. Formally, we can think of health as one among a number of relevant attributes of an individual, so that when thinking
about inequality of health in a population, it can be represented by a multivariate or multidimensional joint distribution. These dimensions include health as well as other attributes such as income,
gender, ethnicity, and education. One way to think of health inequality is then in terms of a univariate (or unconditional or marginal) distribution some individuals have high health, others have
low health, and health inequality is intended to indicate the extent of dispersion of health within the population. The second concept is in terms of a bivariate (or conditional) distribution
whether those with high health also have high income, for example.
Gakidou et al. focus on the univariate or marginal approach, much like analyses of income inequality. In this case, the focus is on a single health attribute margin in the multivariate joint
distribution characterizing a population. The main question is how to characterize the shape of the univariate density function describing how health is distributed in the population. In this
case, there is an extensive literature in economics that can inform the choice of inequality or polarization measure (6, 7).
In the other bivariate or conditional case, the focus of attention is on at least two variables for each member of a population: health , and some measure or indicator of socioeconomic status
(SES), such as income. The concerns in this literature are the pervasive observation in many populations of a strong positive correlation of income and health. We agree with Braveman et al. (4) that
the assessment of this kind of bivariate distribution (or the conditional distribution of health given income, or SES more generally) is fundamental to the discourse on health inequalities.
Gakidou & King (8) give the misleading impression that the choice is between invididual-level and group-level approaches to health inequality measurement, with individual-level inequality in their
terms equivalent to what we are calling the univariate approach. Group-level inequality in their terms is the only concept that deals with socioeconomic factors. However, it is entirely feasible to
have both univariate and bivariate notions of health inequality at the individual level in other words based on individual microdata from a representative sample. They also suggest that the
individual-level approach is more of a purely public health perspective , implying that a bivariate or group level approach is not central to public health. This is clearly a contentious claim.
In order to study health inequalities in the bivariate (or multivariate) sense, it is necessary to have a scalar index of individual health analogous to income. Consequently, it is also logically
possible and ultimately practical to study univariate health inequalities as well.
We return to these considerations of univariate and bivariate (or marginal and conditional) health inequalities later. The prior issue raised by Gakidou et al. is how to construct the univariate
index of each individual s health. The simplest approach is to conduct a census or a survey of a representative sample of individuals in a population, and to elicit from each one preferably by
means of a well-structured series of questions and a sophisticated methodology, such as the McMaster Health Utility Index (9) a univariate cardinal index of their health status at the moment.
However, Gakidou et al. want to be more ambitious than this. They want instead to base assessments of health inequality on some notion of inequality in individuals life chances i.e. to include the
entire life cycle, and also to include in their concept of life chances both length of life and health-related quality of life in short, health expectancy.
We certainly agree with them about the great importance of combining both length of life and health-related quality of life during a person s lifetime. However, for this specific discussion, the
inclusion of health status, or non-fatal health outcomes, complicates matters unnecessarily. As a result, we assume that all that matters in assessing health inequality is length of life.
Equivalently, we assume that each individual s health-related quality of life is perfect from birth to the moment of death.
Measuring univariate inequality with small area data
While the notion of individual-level life (health) expectancy is highly appealing as an index of each individual s health, life chances are not directly observable at the individual level. Moreover,
heterogeneity in life chances is not easily observable even at the level of populations. This can be demonstrated by a simple numerical example. Suppose we have two countries, with large populations.
In country A, everyone has an identical mortality risk of 0.5 over a given time span (e.g. a decade), i.e. perfect equality in health as defined by Gakidou et al., bearing in mind that for
simplicity, we are ignoring health status or, equivalently, assuming that it is perfect up to the moment of death. In country B, however, half the population has a mortality risk of 0.25, while the
other half has a mortality risk of 0.75 over the same time span. This is clearly a situation of high univariate health inequality. Nevertheless, by construction, both countries have the same overall
mortality rate. This thought experiment in part recalls the discussion by Vaupel et al. of heterogeneous frailty (10). This should be an ideal case for the use of something like small area data to
reveal the much higher inequality in health in country B as compared to country A, as proposed by Gakidou et al. and illustrated in Fig. 2.3 of the World health report 2000 (5).
Now suppose that we do something even better than the standard combination of the census and death registration statistical systems typically used to estimate small area mortality rates. Instead of
only juxtaposing death counts and the population at risk, we conduct surveys in both of these hypothetical countries, and simply collect individual identification. We then follow all survey
respondents prospectively over a decade to observe whether they die. As a hypothetical numerical experiment, the resulting pattern of deaths is easily simulated, based on the posited equal and
unequal mortality rates in the two countries. We can now apply the suggestion of Gakidou et al. and partition each sample into subgroups, and compare observed mortality rates across these subgroups.
One might expect that the variance in observed mortality rates across the subgroups in country B would be significantly higher than in country A, thereby revealing its much greater inequality in
underlying health (actually mortality) risks. However, Table 1 shows negative results.
Table 1 has eight columns in four pairs (1 through 4) corresponding to four replicates of the sample for each hypothetical country (A and B). Table 1 also has seven rows, corresponding to different
ways of randomly partitioning the samples, in this case 50 000 in each country, into groups of equal size. These range from 2000 groups (implying 25 observations per group) to 25 groups (implying
2000 observations per group). These partitions are analogous to small area observations ranging in size from census tracts to the state or province level in each country. Within each cell of the
table, the standard deviation of the mortality rates across the partitions is shown.
Perhaps surprisingly, there are no observable differences between the two countries for any level of (the analogue of) geographic aggregation, or any of the replicates.^a This kind of numerical
simulation was also run using a sample size of 1 million, as well as with mortality risks of one tenth of the levels in the simulations shown (i.e. 0.05 versus a 50 50 mixture of 0.025 and 0.075).
There was still no difference between A and B.
An intuitive way of accounting for this result is as follows. Let us first consider two urns, A containing a large number of identical gray balls, and B containing a 50 50 mixture of pure black and
pure white balls. If we drew any sample of balls from urn A, the colour of the set of balls would be the same shade of gray. However, in a sequence of random samples of balls from urn B, standing at
a distance so the individual colours merge, some samples would be lighter gray, and others darker gray. So far, it would seem that our urn analogy supports the Gakidou et al. approach. However, the
colours of balls drawn from the urns correspond to the mortality risks, not to observed mortality rates. The reason we do not see an analogous difference in Table 1 between country A and country B is
that the results in Table 1 are not observations of mortality risks per se. Rather they are observations of the outcomes of stochastic processes whose parameters are the mortality risks in question.
The noisiness of these stochastic mortality processes is essentially obscuring the very real (by design) differences between the two (stochastic) mortality processes.
The objective of the approach of Gakidou et al. is to define health inequality in terms of risks. But these risks are inherently unobservable; only their impacts can be seen, and the numerical
example shows that differences in impact, even when the risk distributions are very different, can be invisible.
This result has strong implications. If we are unable to detect even this blatant kind of univariate health inequality in populations where measurement is simple and perfect, what chance is there in
the much more complex realities of imperfect measurement, more detailed indicators of health, and more subtle kinds of heterogeneity?
One reaction to this numerical thought experiment is that it necessarily obscures the underlying heterogeneity (inequality) in mortality risks because the sample has been randomly partitioned to form
the analogue of geographic areas. The alternative would be to partition the samples by some marker of mortality risk. The use of such a marker, whether at the individual or the geographic area
level, however, would breach the objective of revealing heterogeneities in individuals health that are independent of any systematic factors. This independence is required in the Gakidou et al.
approach, since the focus is explicitly on health inequality in the univariate sense, unlike studies such as that of Ross et al., where the focus is on the correlation of health and SES at the level
of subnational geographic regions (11).
Related conceptual and methodological difficulties arise in using mothers rather than geographic areas as a method for grouping individuals (infants in this case) to explore heterogeneities in
mortality risk. A key problem is if the chances of being in the sample (mothers by number of children ever born) and risks are related. An otherwise unobserved mortality risk (e.g., from contaminated
water supplies in certain villages) may affect both the survival of live-born children and the extent to which mothers successfully carry their pregnancies to term. If this happened, larger lower
mortality families would be observed more frequently. This in turn would bias estimates of heterogeneities ( inequalities ) in infant mortality rates derived from samples of mothers from whom
were collected data on the numbers of surviving children compared to the number of births. At the extreme end of risks, an important proportion of women might remain childless. In other words, the
sample would be reduced to zero in just those cases where the greatest mortality risks might have been observed.
From the viewpoint of measuring univariate health inequalities, these are unfortunate conclusions. Measuring health inequalities in terms of heterogeneities in health risks using mother-specific
infant mortality patterns, and small geographic area data as a form of partitioning methods suggested by Gakidou et al. (3), with the former applied as one of the key international rankings in the
World health report 2000 (5) appears not to work.
Cross-sectional inequality
There are several alternative approaches for assessing health inequalities in the univariate sense intended by Gakidou et al. The simplest and most straightforward is what we might call direct or
cross-sectional. Here, we continue with the current more limited but widespread approach where a representative sample of the living population is surveyed and asked structured and standardized
questions about their current health status. The distributions of one or other type of response can then be analysed univariately using standard statistical methods (e.g. means, variances,
quantiles), or the kinds of inequality and polarization measures developed in the income inequality literature.
This approach is most likely to yield rather unsurprising results that some people are sicker than others. And levels of health will probably be highly correlated with age, and therefore not very
informative. However, within age and sex groups, it would be of interest to track trends in the prevalence of excellent or very good health in comparison to fair or poor health, for example, in a
population over time, or across populations in different countries.
In addition, if the surveys also captured socioeconomic characteristics of the respondents (e.g. income, education), this straightforward approach would support bivariate analyses of the kinds
advocated by Braveman et al. (4). One could track the extent to which higher income individuals were also healthier, and whether this association was stronger in one country than another.
However, this approach does not consider individuals over their lifetimes, one of the main objectives of the Gakidou et al. analysis. This necessarily requires longitudinal data, or piecing together
data from different age groups to form a life-cycle or cohort perspective, or both.
Mortality inequality with longitudinal data
We can begin by considering a second broad approach. It draws on longitudinal data analysis, and is more ambitious in terms of its data requirements. In order to illustrate this group of measures,
let us assume the following conditions for a number of countries: a population census is conducted every 10 years which collects SES data, among other things; there is complete and accurate death
registration; and all deaths are linked back to their corresponding census record. As a result, it is possible on a regular basis to estimate a mortality hazard as a function of age, sex, and a range
of baseline SES covariates. Given the institutional and legal possibility of undertaking this kind of data linkage, most developed countries would have more than adequate sample sizes for very rich
mortality hazard estimation.
With these data, a range of health (more simply, mortality) inequality measures is possible. The most direct (multivariate) indication of health inequality, in this case, would simply be the extent
to which the SES covariates were statistically significant. If these covariates were not significant, we could conclude that the health dynamics at work in the country s population, which include not
only health care but a range of other health determinants, were colour blind to SES, so that individuals health risks were independent of socioeconomic status.
However, it is likely that the SES covariates would be statistically significant. One could then construct a further set of indicators. In order to develop the basic idea, we can start with the
easiest, though not the most reasonable measure the dispersion in estimated mortality risks for the population given the actual dispersion in its SES covariates. In this case, we would start with
the estimated hazard function, and then evaluate it for each individual s vector of covariates observed in the population. We could then use the resulting set of predicted hazards (e.g. point
estimates of the five-year mortality risks) as the population distribution of individual-level health risks. In other words, we take the estimated equation plus a sample of individuals SES
characteristics. Then, one at a time, we plug these SES characteristics into the equation and compute each individual s chances of dying over the next five years.
The resulting distribution of these predicted hazards could then be summarized by a statistic such as the variance, an inequality measure such as the Gini coefficient, or a polarization measure. This
is surely the most straightforward indication of the overall inequality in a population s health risks. However, such a measure would confound the extent, for example, of income inequality, with the
strength of the SES association with mortality risk. In other words, this approach mixes together the dispersion in incomes from the observed distribution of SES characteristics in the population,
with the magnitudes of the coefficients on the SES characteristics in the equation for mortality risks. This approach would also understate inequality insofar as the hazard regression did not fit the
data exactly.
As a result, a more informative (conditional) measure could be defined, drawing on an analogue to age standardization. We could estimate separate mortality hazard functions for the same population at
several points in time, or for several different populations, such as different countries. In other words, we have two or more estimated equations relating SES to mortality risk, one for each of the
time periods or populations we wish to compare. We could also define a reference population distribution of SES covariates, analogous to a reference distribution of the population by age and sex
in age standardization.
It is then straightforward to evaluate estimated mortality risks for a large representative sample of points from this reference distribution of SES covariates, for each of the separately estimated
hazard functions over time or across populations. We just have to plug the same standard or reference sample of SES characteristics into the mortality risk equation for country A, then for country B,
and so on. Each resulting distribution of mortality risks is the outcome of interest. For the same posited reference distribution of SES covariates, we can imagine one country having a narrower
distribution of mortality risks than another. In this case, the former country could be said to have less health inequality, as a result of having a weaker link between the standardized distribution
of SES covariates and subsequent mortality.^b
If we are still interested in health inequalities in the univariate or marginal sense preferred by Gakidou et al., rather than in the conditional sense just described, we might imagine instead
focusing on the distribution of residuals to the hazard regressions in each period or each country. In other words, controlling for a vector of SES covariates, we could see whether the hazard
regressions for one country fit the data better than another. In effect, we are seeking to isolate the unobserved heterogeneity in mortality risks, after first removing the systematic part of the
variations associated with an agreed and commonly defined set of SES covariates. This variation in mortality risks that is left over the unobserved heterogeneity is then identified with some sort
of intrinsic health. This seems to be the underlying concept of univariate health being sought by Gakidou et al. If this leftover variation is smaller in country A than in country B (even if
income inequality is higher in country A, or even if the strength of the association between income and mortality risk is higher in country A), we might then claim that (non-systematic) univariate
health inequality is lower in A than in B.
However, this is a weak strategy empirically because residual variation is likely also to be influenced by the omission of important covariates as well. To give a simple example, suppose that the
regressions include age and income as covariates, but not education. If education is also strongly and independently predictive of subsequent mortality, as is typically the case, then the omission of
this variable in the regressions would make it impossible to ascribe any differences in the goodness of fit of the regressions solely to differences in (non-systematic) univariate or intrinsic
health inequality.
Using an estimated synthetic cohort
The largest source of dispersion in mortality risks would be associated with age. This leads to a third alternative set of measures one that combines longitudinal data analysis with life table
concepts. The basic idea is to form a cohort of complete individual health life cycles, and then use this estimated synthetic cohort as the basis for computing a variety of health inequality
In the first and most basic instance, the fact that mortality risks are predominantly associated with age could be accommodated by using only age as a covariate in the hazard regressions indeed,
simply constructing mortality rates and then using these rates to construct a period life table. If the comparison is across populations, the country with the lowest inequality would then be the
one with the lowest variance in age-at-death, or, equivalently, life length, i.e. a rectangularized survival curve. This is essentially the approach used by Le Grand (12).
A considerably more sophisticated life table style or cohort approach would be needed to come close to Gakidou et al. s notion of health expectancy (but still only life expectancy for now, to
simplify the discussion) as the basis for measuring health inequality. Again we can assume no more than a series of decennial population censuses with standardized, internationally comparable SES
questions, and complete mortality follow-up. To do health expectancy, we would also need longitudinal follow-up of health status, not just mortality follow-up. But instead of positing some general
reference distribution of SES covariates, which by its nature will include age, we build up a statistical description of mortality risks over the full life course. We do this by chaining together in
sequence a series of age-specific mortality hazards, say by five-year age groups plus, as is conventional in abridged life tables, a separate infant mortality rate. Additionally, we can assume that
the hazard regressions are fit in a way that also yields an estimated distribution of unobserved heterogeneity.
We start with a synthetic cohort of, say, 1 million individuals at birth, as in the radix of a life table. We then expose these newborn individuals, one at a time, to the observed mortality risks in
the 0 to 1 year age group, conditional on a random draw from the posited distribution of SES covariates applicable to this age group (e.g. based on parents SES). The process is then repeated for the
survivors to age 1 year. Each is exposed to the mortality risks estimated for the 1 to 5 year age group, again conditional on a random draw from their distribution of SES covariates. Then the process
is applied to the survivors entering the 5 to 10 year age group, and so on.
We could also construct this synthetic cohort using estimates of the distribution of unobserved heterogeneity, which for convenience we can refer to as frailty or resilience . Each individual
at the start of the simulation used to construct the cohort life table would be tagged with a frailty or resilience index, drawn from the estimated distribution. This index would then also be plugged
into the estimated hazard equation to determine, in each age group, that individual s chance of dying.
Note that standard multi-state life table methods are impractical for constructing synthetic cohorts based on these kinds of complex multivariate transition probability functions. Such life tables
would ultimately need millions of columns to represent the combinatorial explosion of possible states. Fortunately, microsimulation methods such as those developed for Statistics Canada s POpulation
HEalth Model (POHEM) are more than adequate (13).
These processes will generate distributions of life lengths (or in a fuller analysis, health-adjusted life lengths), conditional on the posited age- (and sex-) specific distributions of SES
covariates.^b It will also be possible to construct two versions, one with and the other without an explicit account of the estimated pattern of frailty or resilience (i.e. unobserved heterogeneity).
The latter distribution would be narrower, and the difference can reasonably be identified with the notion of Gakidou et al. of univariate health inequality.
Given the assumed data, the set of hazard regressions, and the microsimulation apparatus for constructing life table measures as just described, at least in thought experiment mode, we now have the
raw material to consider the concept of health inequality both in a full life-cycle framework and in terms of health risks, exactly as suggested by Gakidou et al.
We have just noted what seems to be the most appropriate definition of univariate health inequality in this analytical context, based on the difference in the dispersion in expected life lengths
depending on whether or not unobserved heterogeneity is taken into account. It is also possible to extend this framework to estimate a set of age- (and sex-) specific hazards for country A and for
country B. It would also be possible to estimate the distributions of SES covariates for the two countries, and to posit some sort of reference SES covariate distribution. It would then be possible
to construct various distributions of expected life lengths for the two countries (or time periods in the same country).
More precisely, we now have the following ingredients: four sets of hazard regressions, for populations A and B, and with and without explicit estimates of unobserved heterogeneity namely {H [A]}
and {H [B]} with, and {H[A]} and {H[B]} without unobserved heterogeneity, and two sets of SES covariates (SES[A] and SES[B]) plus a reference set (SES*, say). For any given {H} and SES, the
microsimulation apparatus sketched above generates a distribution of life lengths L based on the set of hazard regressions {H} and SES.
To begin, if the distribution of L ({H [A]}, SES[A]) is more dispersed than L ({H [B]}, SES[B]), this is an overall indication that A has a more unequal distribution of life lengths than B. However,
this seems not to be Gakidou et al. s notion of univariate or marginal health inequality (Chris Murray, personal communication).
Of course, we can compare SES[A] with SES[B] to see which country is more unequal in terms of SES. We can also compare (by inspecting coefficients) {H[A]} and {H[B]} to see which country has the
strongest association between SES and mortality. But this would be rather tedious and complex. A summary approach would compare the two distributions of hypothetical life lengths, L ({H[A]}, SES*)
with L ({H[B]}, SES*) using a standard or reference SES distribution, SES*. The country with the widest distribution of expected life lengths would then be the one with the strongest association
between mortality and SES an indication of health inequality in the bivariate or conditional sense.
Finally, the best way to approach univariate health inequality in the sense apparently desired by Gakidou et al. in this context would be to compare the difference between L ({H [A]}, SES*) and L ({H
[A]}, SES*) with the difference between L ({H [B]}, SES*) and L ({H[B]}, SES*). The country with the greatest difference in the distribution of expected life lengths with and without an explicit
account of unobserved heterogeneity our estimate of frailty or resilience would be the one with the most inequality in health in the univariate sense, over and above any systematic relation with
SES factors.
Extensions and conclusions
We see merits in measures of health inequality in both this univariate or marginal sense, and in the bivariate or conditional sense. However, it is quite possible that the general public and those
concerned with public policy will be most interested in the latter notion of health inequality the social patterning of variations in health. One reason is that the covariates give some clues
as to the causes of inequality, and possible areas of intervention.
Of course, all of the longitudinal or life cycle indicators just sketched are quite ambitious with respect to current data availability. At the same time, they point in a different direction than
Gakidou et al. for health data development. Our approach places primary emphasis on mortality follow-up specifically, and on longitudinal data collections more generally. This is in contrast to
Gakidou et al. s idea of developing small area mortality rate data.
Small area data are certainly of intrinsic interest, and they can generate hypotheses insofar as they show significant patterns such as correlations between life expectancy and unemployment rates (14
) or between mortality rates and income inequality (11). However, small area mortality data also suffer from a number of problems. Particularly with small areas in terms of population, migration may
be a problem, if the circumstances or character of the decedents in a given place are not representative of the population currently living there. The smaller numbers of deaths is also likely to lead
to problems of statistical stability. And the likelihood that place and SES factors are correlated means that any variations are likely to be systematically related to SES, and therefore not
indicative of the kinds of univariate heterogeneities Gakidou et al. are seeking to assess.
In addition, geographic areas for which data are available are usually defined for political or administrative purposes and, as such, they will tend to blur mortality differentials since geographic
areas defined to maximize mortality (and health) homogeneity would probably have different boundaries. Even with significant observed differences in mortality rates across small areas, these
ecological differences will typically underestimate individual-level mortality heterogeneity because various differences between persons are likely to be replicated in each unit of any geographic
partition of the population (for example, there will often be a mixture of smokers and non-smokers in each unit). To be easily compared, geographic areas should be similar in more ways than simply
population size. For example, the population density of areas may make a difference (risks from infectious disease, traffic accidents, etc.). These examples amount to a range of serious problems with
strategies building on small area mortality data.
On the other hand, development of mortality follow-up, and of longitudinal data more generally, seems much more promising. Let us continue with the thought experiment based on a census-mortality
follow-up plus microsimulation-based life table approach as just sketched. The key idea is that once a health inequality indicator is based on a model, it is quite straightforward to pose and
rigorously answer what if questions. A large class of such questions serves to generalize the epidemiological notion of attributable fraction. In effect, we construct two estimates of the
distribution of expected life lengths. The first or base case uses the observed or posited standard distribution of SES covariates. The second snips the connection of one particular covariate
to the estimated morality outcome, and hence the estimated distribution of life lengths.
For example, the coefficients relating income to mortality in the hazard regressions for each age could be set arbitrarily to zero, essentially snipping the (direct) connection between income and
mortality. The resulting differences between the base case and the snipped hypothetical distribution of life lengths then provides an indication of the importance of the snipped SES covariate in
generating health inequalities in the given society e.g. the amount of health (actually life length) inequality attributable to income inequality. A version of this kind of analysis is given at the
end of Wolfson s article (7) where, as a rough estimate, it was concluded that about one-fifth of the SES gradient in life expectancy in Canada could be attributed to differences in smoking
prevalences by SES.
Gakidou et al. raise a significant set of questions. Theirs is a very important objective: to seek a measure of health, for purposes of inequality analysis, that is comprehensive, combining both life
length and health status over the life course, and defining these in terms of risks. The main problem is in the proposed methods for estimating these risks. They suggest the use of small area data.
However, as shown above, this strategy will not work. However other methods, based on longitudinal follow-up data combined with microsimulation-based life table analysis can substantially meet their
ultimate objectives.
We are indebted to serveral anonymous reviewers for helpful comments and suggestions.
Conflicts of interest: none declared.
A propos de la mesure des inégalités de santé
Dans une récente série d articles, Murray et al. ont présenté un certain nombre d idées importantes concernant la mesure des inégalités de santé. Nous partageons ici leur point de vue sur nombre de
questions, mais nous attirons l attention, en illustrant notre démonstration par un exemple numérique, sur les défauts d un des aspects majeurs de leur approche la mesure des inégalités sur la base
de données géographiquement limitées. Nous décrivons une autre approche qui fait appel à des données longitudinales et qui permet de conserver, tout en les améliorant, les aspects les plus séduisants
de leur proposition, à savoir l intégration sur la vie entière et la prise en compte aussi bien des issues non fatales que des données usuelles sur les profils de mortalité.
Sobre la medición de las desigualdades en salud
En una serie de artículos recientes, Murray et al. exponen diversas ideas importantes respecto a la medición de las desigualdades en salud. En este artículo se coincide con muchos de sus
razonamientos, pero se señala a la atención un aspecto fundamental de su método la medición de las desigualdades a partir de datos de áreas pequeñas que presenta fallos. Se da un ejemplo numérico
para ilustrar el problema. También se esboza un método alternativo basado en datos longitudinales, con el que se mantienen y potencian los aspectos más convenientes de su propuesta. Entre ellos
figuran la perspectiva del ciclo vital y la consideración de los problemas de salud no mortales, así como de la información habitual sobre las pautas de mortalidad.
1. Murray CJL. US county patterns of mortality by race: 1965 1994. See: www.hsph.harvard.edu/organizations/bdu/ papers/usbodi/ index.html, 1998. [ Links ]
2. Murray CJL, Gakidou EE, Frenk J. Health inequalities and social group differences: what should we measure? Bulletin of the World Health Organization, 1999, 77: 537 543. [ Links ]
3. Gakidou EE, Murray CJL, Frenk J. Defining and measuring health inequality: an approach based on the distribution of health expectancy. Bulletin of the World Health Organization, 2000, 78: 42 54. [
Links ]
4. Braveman P, Krieger N, Lynch J. Health inequalities and social inequalities in health. Bulletin of the World Health Organization, 2000, 78: 232 235. [ Links ]
5. The world health report 2000 health systems: improving performance. Geneva, World Health Organization, 2000. [ Links ]
6. Cowell FA. Measuring inequality. London, Philip Allen, 1977. [ Links ]
7. Wolfson MC. Divergent inequalities theory and empirical results. Review of Income and Wealth, December 1997. [ Links ]
8. Gakidou E, King G. An individual-level approach to health inequality: child survival in 50 countries. GPE discussion paper series, No. 18. Geneva, World Health Organization, 2000 (unpublished
document). [ Links ]
9. Torrance GW. Utility approach to measuring health-related quality of life. Journal of Chronic Diseases, 1987, 40: 593 600. [ Links ]
10. Vaupel JW, Manton KG, Stallard E. The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 1979, 16: 439 454. [ Links ]
11. Ross N et al. Income inequality and mortality in Canada and the United States. British Medical Journal, 2000, 320: 898 902. [ Links ]
12. Le Grand J. Inequalities in health, some international comparisons. European Economic Review, 1987, 31: 182 191. [ Links ]
13. Wolfson M. POHEM a framework for understanding and modelling the health of human populations. World Health Statistical Quarterly, 1994, 47: 157 176. [ Links ]
14. Statistics Canada. Life expectancy. In: Health reports: how healthy are Canadians? A special issue, 2000, Vol. 11, No. 3, Catalogue No. 82-003-XPB. [ Links ]
^1 Assistant Chief Statistician, Statistics Canada, Analysis and Development Field, 26-K, R.H. Coats Building, Ottawa, K1A 0T6, Canada (email: wolfson@statcan.ca). Correspondence should be addressed
to this author.
^2 Senior Analyst, Socioeconomic Modelling Group, Statistics Canada, Ottawa, Canada.
^a We thank an anonymous reviewer for showing that a similar result can be derived algebraically, assuming a large enough sample. If the mean probability of mortality is the same in both a
homogeneous and a heterogeneous population, then the expected values of the variances of mortality risk will also be identical (within the simplified framework of our simulations, e.g. binomial
mortality risk and no age dependence). This is a more general version of the conclusion we reached through simulation. The simulations are still useful in demonstrating that, even without a large
sample assumption, we should not expect to find excess variability in mortality from thoroughly mixed heterogeneous populations.
^b There is a question of the likely auto-correlation of these SES covariates over the life cycle. The simplest approach in this case would be to posit some standard scenario. But these
auto-correlations too could be measured using longitudinal surveys. Their dynamics, as time-varying covariates, would then have to be simulated as well. This is precisely what POHEM is designed to
Ref. No. 00-0776
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Are S(g) and U(g) isomorphic as g-modules for g Lie algebra over F_p ? Are S(g)^g and U(g)^g isomorphic as com.algs ?
up vote 11 down vote favorite
If $g$ is Lie algebra over field char(k)=0, then the following facts are well-known:
1) S(g) and U(g) are isomorphic as $g$-modules. (Symmetrization map S(g)->U(g) gives isomorphism).
2) S(g)^g and ZU(g)=U(g)^g are isomorphic as commutative algebras. (The Duflo map defines is isomorphism which is combination of symmetrization map with some intricate corrections by terms of smaller
Both facts are based on the symmetrization. If we consider $g$ over field char(k)$\ne$0, there is NO symmetrization map. So I wonder the following:
Question are the facts above true for Lie algebras over char(k)$\ne$0 ?
U(g) - universal enveloping algebra (non-commutative associative algebra defined by relations $[x_i, x_j] = \sum_k c_{ij}^k x_k$, for any linear bases $x_k$ of $g$.
S(g) - symmetric algebra of $g$ (defined as $k[x_1...x_n]$ for any bases $x_k$ of $g$.
S(g)^g, U(g)^g means subspaces of g-invariants (g act by zero).
Symmetrization map is defined as $S(x_1..x_k) = 1/k! \sum_{\sigma} \prod_l {x_{\sigma(l)} $.
Duflo map is not so easy to write so let me just mention some MO questions:
Is the Duflo map for Lie algs. unique ?
Capelli determinant = Duflo ( determinant) - was it known ?
For the case of gl_n there are so-called Capelli generators of the center of U(g), which can be defined over any char. So I think that the center of U(gl) is the same as for char(k)=0 the result is
true. (It is NOT the same as in char=0, as I mistakenly wrote first, since there should be generators corresponding to a^p (which are S(g)^g for any a)).
I think the same is true for other classical semi-simples - there are analogous Capelli like formulas.
rt.representation-theory lie-algebras
add comment
4 Answers
active oldest votes
Question 2 has a negative answer. Indeed, let $L =sl(2,k)$ where $k$ is an algebraically closed field of characteristic $p>3$ and let {$e,h,f$} be the standard basis of $L$. Then it is
well-known (and was first proved by Rudakov-Shafarevich in the late 60s) that the centre $Z(L)$ of $U(L)$ is generated by $E=e^p$, $F=f^p$, $H=h^p-h$ and a Casimir element $C$ subject to
the relation $$4EF+H^2=\prod_{i\in{\mathbb F}_p}(C-i^2)=C^p-2C^{(p+1)/2}+C.$$ Furthermore, the maximal spectrum of $Z(L)$ is a hypersurface in ${\mathbb A}^4$ given by the above equation.
On the other hand, it follows from a result of Brown-Goodearl that the Azumaya locus of $U(L)$ coincides with the smooth locus of ${\rm Specm\ }Z(L)$. So the modular representation theory
of $sl(2,k)$ yields that ${\rm Specm\ }Z(L)$ has exactly $(p-1)/2$ singular points (one can also see this directly by differentiating the above equation).
On the other hand, it is also known (and not difficult to see directly) that $S(L)^L$ is generated by $X=e^p$, $Y=f^p$, $Z=h^p$ and $\Omega=4ef+h^2$ subject to the relation $4XY+Z^2=\
Omega^p$. Moreover, ${\rm Specm\ }S(L)^L$ is a hypersurface in ${\mathbb A}^4$ given by the above equation. This hypersurface has a unique singular point at the origin. Since $(p-1)/2>1$
under our assumptions on $p$, we see that the $k$-algebras $Z(L)$ and $S(L)^L$ cannot be isomorphic.
up vote 11
down vote As far as I know the answer to Question 1 is unknown, in general, although there do exist finite dimensional Lie algebras $L$ for which $L\subset U_1(L)$ is NOT a direct summand of $U(L)$
accepted (if $U(L)$ is isomorphic to $S(L)$ as $ad(L)$-modules, then $U(L)$ must contain an isomorphic copy of $L$ as a direct summand, but that copy doesn't have to lie in the first component
$U_1(L)$ of the canonical filtration of $U(L)$).
If we require the stronger condition that an isomrphism $S(L)\rightarrow U(L)$ sends $S^1(L)$ onto $L\subset U_1(L)$ (which is obviously true for the symmetrisation map), then the answer
to Question 1 is NO. For that condition to hold the Lie algebra $L$ must admit at least one $[p]$-th power map, i.e. must be ${\it restrictable}$ (this observation is due to Michel
Duflo). The majority of finite dimensional simple Lie algebras are non-restrictable, and the smallest example is the $3$-dimensional simple Lie algebra over an algebraically closed field
of characteristic $2$.
@Alexander Premet Thank you very much ! – Alexander Chervov Jun 24 '12 at 17:32
@Alexander Premet are there relations like formula 1 known for sl_n/ gl_n, not just sl_2 ? I mean in gl_n we have n-Casimirs e.g. Tr(E^k), k=1...n. So we may expect that e_{ij}^p are
related to powers of these elements ? May be some nice formulas can be obtained ? – Alexander Chervov Jun 25 '12 at 6:55
There are similar relations for $\mathfrak{gl}_n$ (any $p$ and any $n$) and for $\frak{sl}_n$ with $p\not|n$. Sadly, almost nothing is know about the symmetric invariants of $\mathfrak
1 {sl}_{kp}$. Of course, when $p<n$ the invariants $Tr(E^k)$ will fail to generate $S(\mathfrak{gl}_n)^{GL_n}$ (this is because in characteristic $p$ we have that $Tr(E^p)=(Tr(E))^p$).
But these can be replaced by the basic invarints arising as the coefficients of the characteristic poynomial of a generic matrix. More on this can be found in my joint paper with
Rudolf Tange; see J. Algebra 294 (2005) 177–195. – Alexander Premet Jun 25 '12 at 12:07
@Alexander Premet Thanks again! – Alexander Chervov Jun 26 '12 at 9:41
add comment
Your question may be related to this question about hyperalgebras. In my answer there, I gave a reference to a paper of Friedlander and Parshall (Rational actions associated with the
adjoint representation, Ann. scient. Ec. Norm. Sup., 4e serie, t. 20, 1987, p. 215 a 226), which shows that the answer to your question (1) is yes if $g$ is the Lie algebra of a simple
algebraic group.
up vote 5
down vote For a simple algebraic group $G$ and Lie algebra $g = Lie(G)$, I think that looking at the spaces of $g$-invariants in $S(g)$ and $U(g)$ are probably the same as the spaces of
$G_1$-invariants, where $G_1$ is the first Frobenius kernel of $G$. Then the results in Section 4 of the Friedlander-Parshall paper may be relevant to your question (2).
@Christopher Thank you very much for yours answers and links ! I will look at the paper you suggest. – Alexander Chervov Jun 8 '12 at 5:58
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It seems that in some particular cases it is known, see e.g. this recent result.
I was looking at that a while ago, and at least it became clear to me that one has to be very careful with lifting the $p$th powers from the Poisson centre of $S(\mathfrak{g})$ (which is
up vote 2 where symmetrisation fails most dramatically), e.g. for $\mathfrak{g}=\mathbb{F}_3\{x,y\}$ with $[x,y]=y$ the Poisson central elements $x^3$ and $y^3$ lift to central elements $x^3-x$ and
down vote $y^3$ respectively.
@Vladimir Thank you !!! By the way concerning center of U(n) and U(b) in char =0 there are recent papers by Kostant: arxiv.org/abs/1201.4494 Center of U(n), Cascade of Orthogonal Roots,
and a Construction of Lipsman-Wolf arxiv.org/abs/1205.2362 Coadjoint structure of Borel subgroups and their nilradicals the paper you mentioned discussed more complicated case char=p,
while Kostant char =0. – Alexander Chervov Jun 7 '12 at 9:54
By the way for [x,y]=x are there positive answers ? – Alexander Chervov Jun 7 '12 at 9:55
To be honest, I don't know if it is known, and if yes if it is written somewhere. I think I did convince myself in positive answer for this case, but it was mere curiousity so I did not
dig too deep. – Vladimir Dotsenko Jun 7 '12 at 13:13
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A sort of involved and (seemingly very partial) variant of Duflo isomorphism in characretristic $p$ is claimed in the following paper: N.A. Koreshkov, Central elements and invariants in
modular Lie algebras, Russ. Math. (Izv. VUZ) 46 (2002), N7, 20-24 (Russian original is available, for example, at http://www.ksu.ru/journals/izv_vuz/arch/2002/07/05-7.PDF ). Roughly, it is
up vote proved there that for a finite-dimensional Lie algebra over a finite field $\mathbb Z_p$, a certain modification of $S(\overline{L})^{\overline L}$ is isomorphic to a certain subring of $U(\
1 down overline L)^{\overline L}$, where $\overline L$ is obtained from $L$ by some (infinite) field extension.
Thank you very much! – Alexander Chervov Nov 4 '12 at 20:21
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Not the answer you're looking for? Browse other questions tagged rt.representation-theory lie-algebras or ask your own question.
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Normandy Park, WA Geometry Tutor
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Generalized Majority-Minority Operations are Tractable
Victor Dalmau
In: 20th Annual IEEE Symposium on Logic in Computer Science (LICS 05), 26-29 June 2004, Chicago, USA.
Let $A$ be a finite set and let $\varphi:A^k\rightarrow A$ with $k\geq 3$ be a $k$-ary operation on $A$. We say that $\varphi$ is a generalized majority-minority (GMM) operation if for all $a,b\in A$
we have that \noindent \begin{center} $\varphi(x,y,..,y)=\varphi(y,x,..,y)=\cdots=\varphi(y,y,..,x)=y$ \hspace{5cm} for all $x,y\in\{a,b\}$ or $\varphi(x,y,..,y)=\varphi(y,y,..,x)=x \;\; \text{ for
all } x,y\in\{a,b\}$ \end{center} Near-unanimity and Mal'tsev operations are particular instances of GMM operations. We prove that every CSP instance where all constraint relations are invariant
under a (fixed) GMM operation is solvable in polynomial time. This constitutes one of the largest tractable cases of the CSP.
Postscript - Requires a viewer, such as GhostView
EPrint Type: Conference or Workshop Item (Paper)
Project Keyword: Project Keyword UNSPECIFIED
Subjects: Theory & Algorithms
ID Code: 1827
Deposited By: Victor Dalmau
Deposited On: 29 November 2005
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So you found the Higgs???
Nobel prizes for everyone! Congratulations you managed to see the the entire magnetic field from 1 volt to 300+ Teravolts.Have you solved the Basic Dimensional Analysis problem for mass? Mass is a
one dimentional measure of an object's weight. Gravity is a two dimensional data structure that holds mass and distance. A Matrix cannot ever equal a vector * a vector, or a vector * a scalar. So I
have serious problems with the concept of mass.Let's evaluate a real object and what mathematically occurs when you reduce the natural three dimensional traits to the one dimensional concept of mass.
When reducing the object from three dimensions to two, you lose x,y,z, size of the object. You lose I,j,k position of the object. You lose temperature data. You lose E/M data. You lose spectral
identification data. You have effectively disregarded vital information about the object just to have mass. More importantly you have no manner of retaining that information or using it.I know some
recent reports proved that gold travelling through the particle accellators pass through the double slit test as waves not particles. I believe that was done at the LHC. I could be wrong as to where
the test was done. But that alone kills the particle concept.Now let's again look at that funny concept of gravity. The larger mass will always attract the less massive object at a rate of... So if
an apple hits the ground something more massive then the earth is the only thing that can reverse the apple's direction.Marketing in physics. I know we all need to get paid. Do the math. Don't just
say we found the "GOD" particle. Show me the math that proves you found many particles in the Higg's range and how the attach.Another thing. Particle accellerators are showing what happens after
traumatic collisions of proton beams. This is a decay model. It does nothing to show interactions of bosons prior to the collision. Archemedes disproved gravity by running naked yelling Eureka. It
was not gravity that he discovered it was density!!! Using density you get to keep all your amazing data about an object. So good luck with the god thing. One more thing. Gravity requires
omnisence,omnipresence, and omnipotence. Gravity requires that distance data travel instantanously from object to object. That exceeds the speed of light.The Standard Vibration Model as described
here is an expression model. There is a vital difference in those two concepts.
22 comments:
> A Matrix cannot ever equal a vector * a vector, or a vector * a scalar.
Actually, a matrix CAN equal a vector times a vector, given the right "times" operator. Vectors come equipped with two natural products, the inner and outer product. The former produces a scalar,
and the latter produces a matrix. Cool!
Also, your [E = m c^2] example is terribly flawed. E, m, and c are all scalars. Take c, for example. It is the "speed" of light, not the velocity. c says nothing about which direction light
moves, only how fast. It is a simple 1 dimensional quantity.
Likewise for E. Energy is just a simple number... -5 or 42 or 1.2e9. Yes, you can define an energy field, where you supply a value for energy at every point. But those are two different things.
Energy is a simple number. An energy field is a bunch of energy values spread across space. The E in [E = m c^2] is the former, not the latter.
> When reducing the object from three dimensions to two, you lose x,y,z, size of the object. You lose I,j,k position of the object. You lose temperature data. You lose E/M data. You lose spectral
identification data.
That makes little sense. You claim by eliminating a single dimension, that you lose x, y, z, i, j, k, temp, and whatever form you think e/m and spectral data takes. At the very least, that's 9
values, if not more. How does eliminating 1 dimension result in a loss of at least 9 dimensions?
We need to step back and define a dimension. There are lots of nice rigorous ways to do it, but let's keep it simple. Let's treat dimensions as independently variable quantities.
> The larger mass will always attract the less massive object at a rate of...
Where did this sentence go?
> So if an apple hits the ground something more massive then the earth is the only thing that can reverse the apple's direction.
No. Classical gravity is inversely proportional to distance. An object with less mass than Earth, but denser, could potentially act on the apple from a shorter distance, such that the resulting
ratio exceeds that of Earth's.
> Marketing in physics. I know we all need to get paid. Do the math. Don't just say we found the "GOD" particle. Show me the math that proves you found many particles in the Higg's range and how
the attach.
They are showing the math. Here is a link to the presentations being given at the CERN conference on the Higgs...
That's serious, concrete data. And that's just the tip of the iceberg.
> It does nothing to show interactions of bosons prior to the collision.
What happens prior to the collision impacts what happens after. You can leverage your theory to predict the expected outcome of a set of conditions, set up those conditions, play it forward,
possibly through some unmeasured states, measure the final outcome, and compare to the prediction. That's called science.
> Gravity requires that distance data travel instantanously from object to object.
300 years ago, perhaps. These days, no.
In the case of general relativity, gravity is NOT a force acting at a distance. It's a fundamental curvature of spacetime itself. Changes in mass propagate instigate changes in curvature at the
speed of light.
And in quantum field theory, gravity is modeled as a force-carrying particle. This is obviously not instantaneous action as it must be transmitted by the particle, and this also agrees with
classical gravity in the classical limit of quantization, showing that classical gravity can in fact be explained without instantaneous action.
Didn't like my previous comment?
Thank you for reading my work.
It is not that I did not like your previous comment. I am just dealing with my mother's multiple mylomia. So I am away from my home and work. But most importantly I am holding my mom's hand for
the last times.
I will respond to every point. You bring to my blog an interesting discussion. I will be home on Friday. So by Monday I hope to post a decent response to every part of your envigorating comment.
Every difficult comment provides more thought and improved results to the model.
You should find the basic equations in the 'Motion of a Baryon' and other papers on the non-mobile version of the blog.
Thank you
Good Day and thanks for the comment.
I knew the deminsional analysis had to be the strongest part of this model.
1st dimension. The data.
2nd dimension. The Data structure. This is creating an object Boson. It has data of its own existence and transmitted baryon data.
3rd dimension. The physical structures in stasis. This is a NOW function of Dark Energy to calculate at this point in rotation.
4th Dimension. Time. Data in motion. Functions interacting.
That is how Dimensional Analysis works!
So a bunch of summed vectors cannot equal an energy field?
I will add that to the next comment here.
Thank you
> That is how Dimensional Analysis works!
But that doesn't show how it works. It's just vague words.
The first hurdle in acceptance of your theory is redefining well-known words like "dimension". You may want to find a better word that doesn't conflict with standard usage.
But more so, at least as a first step, I think to relate to people with more mainstream views, you need to calculate actual things using explicit math and show that they're consistent with
empirical evidence.
For example, since you don't like gravity, show the math that predicts the fact that a person experiences an acceleration of ~9.81 m/s^2 when standing on Earth's surface.
Or since you don't like e=mc^2, what's the maximum at-rest energy potential of 1 gram of Unranium-235, using your theory? Is it different than mc^2? Why?
Or what percentage of vertically polarized photons will pass through a polarized filter rotated 83 degrees with respect to it, using your theory? Is it different than cos(83)? Why?
Answering relatively familiar questions such as these will go a long way to convincing the mainstream that you're brand of physics is at least capable of deriving answers consistent with what we
Maybe you can't yet compute these. Is there something you can compute explicitly as evidence? If not, why not? What steps are required to get past it? And lacking these things, what exactly gives
you confidence in your theory at this point in time?
Although a person standing on earth experiences acceleration of -9.81 m/s^2. But a helium balloon does not experience the same acceleration rate. This is due to baryonic density within a media.
This is describe through out the blog.
I will work on the next point
You are right, my model is essentially worthless if it cannot describe and visualize in detail your questions. I am coding these interactions so as not to look like a fool when I bring this
theory into a working visualization model.
This is why I am asking for the funding.
I am going to do this in stages. I will make sure classical e/m rules and this model are statistically close in operation in my current program. Then I will add non-linear and other chaos systems
to improve the program.
> Although a person standing on earth experiences acceleration of -9.81 m/s^2.
Be careful when being precise, because that's not correct either. Acceleration is actually vector, and its magnitude is always positive.
> But a helium balloon does not experience the same acceleration rate.
No, it doesn't. However, it does receive EXACTLY THE SAME contribution from gravity as a person. But to offset this, it also receives a huge mess of forces from the surrounding air, which when
all summed together, counteract the gravitational force to provide net lift.
I know your theory claims to provide an alternate explanation, which is completely acceptable. But you must first understand and accept that the existing theory does in fact accurately explain
the phenomenon.
Because as it stands, if people can't accept that you know the existing theory, they won't accept your criticism of it, nor your alternative to it.
> This is why I am asking for the funding.
Speaking honestly, a blog post won't get you what you need, unless your situation garners enough media attention (which needs some sort of spark) to overcome the rather unconventional method your
Also, I assume you have a MUCH more detailed write up of what exact projects you intend to take on and what each one will produce as an outcome, as you won't get funding without it.
Also, check out kickstarter. It's all the rage right now, and a great way for good ideas to get funding from the masses. But beware: you need clear and achievable goals.
> I will make sure classical e/m rules and this model are statistically close in operation in my current program. Then I will add non-linear and other chaos systems to improve the program.
It's amazing what quantum theory can formalize that we don't have the capacity to calculate yet. Just a warning... you won't get the same level of success trying to build a computer model that
you will get trying to build a mathematical theory.
Thankyou for the pointers about making the project more acceptable to being funded. I am working on all of the documents describing the clear and achievable goals. I hope the blog is a stepping
stone for understanding this model.
I understand the force defined as Gravity is a vector and that the forces of the surrounding air impact the density of the medium and the object. This is why I asked these questions. Through out
the blog I show a fundamental and objective view of other models. I show how e/m functions work as fundamental the interactions involved in this model. I show how particles defined by QCD
interact to form energy, density and motion in this model.
I understand that existing models have solutions for these phenomenon, but we all know there are serious problems with current models. The fact that particles of mass like quarks are not affected
by the earth's gravity. This shows that current models are only supported by mathematical models, not observation.
I am working hard on polishing this model. I feel I am at the point where interactions with others will produce better results in the end.
> I understand that existing models have solutions for these phenomenon
Okay, but a quick search on your front page returns:
> 4) Gravity is a failure... Why, Helium balloons.
I'm very confused about where you stand.
I have several postings on helium ballons and how they rise in this model. But basically The helium in the balloon is less dense than then surrounding air. This allows the balloon material to
rise. Excaped helium will rise to meet its medium density.
By using heat, hot air balloons rise and fall in the lower atmosphere. By changing the temperature inside the balloon this makes the baryons less dense causing the balloon to rise.
> But basically The helium in the balloon is less dense than then surrounding air. This allows the balloon material to rise.
There's a really big logical leap between those two sentences, but I'll assume that you were just summarizing, and that the details really are explained elsewhere.
In any case, that wasn't the point. I wasn't concerned with how your theory explains helium balloons, but instead concerned with the fact that...
a) you declared gravity a failure because, I assume, it can't explain them, and yet...
b) seem to claim earlier in this comment thread (July 19, 2012 10:36 AM) to accept that the standard gravitational explanation makes sense.
Which is it?
In this model, gravity is not necessary. That which is normally defined as gravity in other models, in this model, is easily defined as density within a medium.
I also show mathematically and logically as to why one dimensional data cannot exist, in this model, as physical structures in motion.
> In this model, gravity is not necessary.
I was simply asking whether you thought gravity's explanation of how helium balloons rise was valid, because you made a statement implying that you did, which contradicts much of your blog.
> I also show mathematically
I have not yet seen you show anything mathematically.
Compute something. Anything. And show the equations, the work, and the solution.
I am working on the coding part to express the mathematics. I expect its precision to incease over time. I will be using classical Maxwell equations to start then move to more advanced systems of
The more I read Maxwell, Planck, Tesla and others, the more I am impresses with what we already knew. We are discarding this wonderful work for Einstein's rehasing of Two dimensional physics.
Even if you are bending the plane of space to show three dimensions, objects within space can only work on an x,y basis. That produces the science fiction of wormholes, timetravel, blackholes as
attractors, and the Big Bang as the story is told currently. It is currently not relevant to think of objects in space as similar to balls on a fabric cloth. Yet it is the foundation of our
current models.
Why did we move aside these impressive theorists for the simple solution? Media. It is easier to describe 2 dimensions than it is to describe the interactions in 3d.
I will post the math and logic that produces the most accurate result. This will take time since I am working alone.
> We are discarding this wonderful work for Einstein's rehasing of Two dimensional physics.
I would claim you are doing the same injustice to Einstein's work.
> Even if you are bending the plane of space to show three dimensions, objects within space can only work on an x,y basis. ... It is currently not relevant to think of objects in space as similar
to balls on a fabric cloth. Yet it is the foundation of our current models.
That is NOT the foundation of our models. The classic example of balls on a cloth and such is horribly flawed, and is only an analogy used to explain the idea in popular science articles and
books, because laymen have a hard time beyond 2 dimensions.
The actual Theory of General Relativity does NOT deal with curved 2 dimensional spaces, but instead intrinsic curvature of the entirety of 4 dimensional space-time. There is no 2D simplification
present in the real theory.
> Why did we move aside these impressive theorists for the simple solution? Media. It is easier to describe 2 dimensions than it is to describe the interactions in 3d.
Exactly. The version of general relativity you are describing is the popular science version portrayed in the media. It's not the real thing.
For a gentle introduction to the actual theory, I would recommend Susskind's 2008 lectures, available on youtube. (and as a pre-req, check out his lectures on classical physics and special
relativity, and have a solid handle on vector calculus).
Thank you for the pointers. I have watched Susskind's lectures on youtube. I have even linked some of those lectures to this site.
Einstein came along with media based equations that were easier to publish and diagram on newsprint then it was to describe the amazing work coming from Planck's and others. Planck's and others
working on E/M had their work pushed aside for something easier to explain.
Try explaining physics without metaphors or analogies.
1: In this model the descriptions of the dimensional analysis is laid out
2: In this model bosons store information
3: Bosons in stasis interaction (potential)
4: Baryons in motion causes bosons to dynamically interact.
Asymptotes in the equation are where boson interact. The asymptote of the electron occurs at n/0. Three major interactions occur at this electron. It is the junction of the photon, ZBoson, and
the WBoson. The electron functions to convert received external baryon information an initallize transfer to the baryon through a vortex of Zbosons. The electron receives internal baryonic data
and expresses it with a photon.
I have shown that the Zeeman Effect bends and stops the rotation of photons. This has nothing to do with gravitational lensing.
I have explaned information transformation through every boson. The real problem is this. This model is a non-linear equation. There are no constants. The real shapes are not the standard
geometric shapes. All numbers are Real. So that Real problem becomes a series of ratios determining the density, energy, and position of a baryon.
Trigonometry begins at the baryon. Three gluons are required to form a baryon. So every point where the gluon meets another gluon (quark) is the angle and the taut gluon is the line. This is a
fundimental basis of mathematics that shows proof in the model.
In motion a triangle takes on the shape of a cone. This motion of the baryon is the basis proof for conics and conic section. This include Kepler's elipses and conic mathematics.
> Einstein came along with media based equations ...
What does "media based" mean to you?
His field equations were full solutions to gravitation in general relativistic 4D space-time. In no way were they dumbed down or simplified for any particular audience. They weren't for the
media... they were the correct solution to the problem of gravity in the context of the special theory, and was compatible with classical electromagnetism. I'm struggling with how that is "media
> Planck's and others working on E/M had their work pushed aside for something easier to explain.
So you define "media based" as "turned out to be more media friendly than competing work". Okay. But do you honestly think that Einstein's theory won out because the media loved it?
And in what way was prior work pushed aside? Einstein didn't invalidate prior results... he built upon them. Maxwell's equations live on in fullly relativistic, curved glory. Planck's blackbody
results lived on in the quantum, which Einstein himself helped solidify (and which Planck actually resisted). And the Quantum Mechanical ball which Planck got rolling has established itself
fully, despite whatever media attention Relativity stole.
I just don't see where you're coming from.
> In this model ...
I (and other readers) have a hard time accepting any explanations of your model unless you can convince me (and other readers) that you understand the model you are claiming to replace in the
first place.
But claiming that e=mc^2 fails "dimensional analysis", or that gravity can't explain helium balloons, or that gravity fails in the multi-body case, or that Einstein's equations are "media based"
simplifications of prior work, or that light is bent in magnetic fields; make it nearly impossible to take you seriously when it comes to your own work.
For example, gravity DOES explain helium balloons (as commenters have explained on your blog), even if your theory ALSO explains them.
A more fundamental one that I see you mention often: E=mc^2. As defined, it forms a sound statement (independent of truth).
You may be inclined to argue that E, m, and c are not scalars in your theory. Okay. But if you choose to redefine E, m, or c, you can no longer plug them in to E=mc^2 and say anything about it,
because you're now talking about different things. Einstein's statement was in the context of the E, m, and c which HE was using, not whatever alternative definitions you came up with. If you're
going to make statements about E=mc^2, you must do so in the context in which the equation is defined.
> I have shown that the Zeeman Effect bends and stops the rotation of photons. This has nothing to do with gravitational lensing.
Another great example: I know of no support for light being bent in a magnetic field (but feel free to supply proper experimental evidence). And forget about gravitationally lensing... such a
result would invalidate much more foundational pre-20th century principles such as superposition. As a result, little things like Maxwell's equations would be invalidated, interference patterns
wouldn't look as they do, and the world as we know it would be a very different place.
My overall point is that you're not providing proper support for your statements about mainstream physics, which erodes the trust of your reader and serves to deter them from accepting your
I would suggest either:
a) Providing proper, rigorous, explicit proofs of your objections to mainstream physics (which you have not done), independent of whether your theory provides an alternative, or
b) Accepting mainstream physics openly as a valid model, and simply promoting yours as an alternative (which may be more accurate or powerful in some ways you are free to elaborate on).
Thank you for reading my work.
I am not doing well at this time. Ill be back on Monday.
I should add...
Although light can not be bent by magnetism directly, it can indirectly, since a strong enough field stores energy, which is mass, and gives rise to gravitation. But the effect is orders of
magnitude smaller (except possibly in extreme edge cases of highly polarized black holes) than that predicted by computing lensing from gravitational mass directly. Additionally, light is
delected in the direction of the magnetic field, which would likely be inconsistent with the bending of gravitational lensing.
I believe there are some quantum effects at very (very!) high energies that could deflect light as well.
I make these claims without reference, I know, but I just wanted to offer that there ARE situations where light bends to magnetism, but they are extreme edge cases, and would not enable you to
explain away something like gravitational lensing, for instance.
Sorry... just an aside.
This comment has been removed by a blog administrator.
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Physics Forums - View Single Post - 3 dimensional torque from angular velocity
Sorry what i suggested was not physics by linear algebra. Out of curiosity, could you explain about the Euler angle you said. What are X,Y and Z ? I have heard of Euler angles but not Euler vector.
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Creating good looking survival curves – the ‘ggsurv’ functionCreating good looking survival curves - the 'ggsurv' function
Creating good looking survival curves – the ‘ggsurv’ function
This is a guest post by Edwin Thoen
Currently I am doing my master thesis on multi-state models. Survival analysis was my favourite course in the masters program, partly because of the great survival package which is maintained by
Terry Therneau. The only thing I am not so keen on are the default plots created by this package, by using plot.survfit. Although the plots are very easy to produce, they are not that attractive (as
are most R default plots) and legends has to be added manually. I come across them all the time in the literature and wondered whether there was a better way to display survival. Since I was getting
the grips of ggplot2 recently I decided to write my own function, with the same functionality as plot.survfitbut with a result that is much better looking. I stuck to the defaults of plot.survfit as
much as possible, for instance by default plotting confidence intervals for single-stratum survival curves, but not for multi-stratum curves. Below you’ll find the code of the ggsurv function. Just
as plot.survfit it only requires a fitted survival object to produce a default plot. We’ll use the lung data set from the survival package for illustration. First we load in the function to the
console (see at the end of this post).
Once the function is loaded, we can get going, we use the lung data set from the survival package for illustration.
lung.surv <- survfit(Surv(time,status) ~ 1, data = lung)
Censored observations are denoted by red crosses, by default a confidence interval is plotted and the axes are labeled. Everything can be easily adjusted by setting the function parameters. Now lets
look at differences in survival between men and women, creating a multi-stratum survival curve.
lung.surv2 <- survfit(Surv(time,status) ~ sex, data = lung)
(pl2 <- ggsurv(lung.surv2))
The multi-stratum curves are by default of different colors, the standard ggplot colours. You can set them to your favourite color of course. As always with ggplots a legend is created by default.
However we note that levels of the variable sex are called 1 and 2, not very informative. Fortunately the output of ggsurv can still be modified by adding layers after using the function, it is just
an ordinary ggplot object.
(pl2 <- pl2 + guides(linetype = F) +
scale_colour_discrete(name = 'Sex', breaks = c(1,2), labels=c('Male', 'Female')))
That’s better. Note that the function had also created a legend for linetype, that was non-informative in this case because the linetypes are the same. We removed the legend for linetype before
adjusting the one for color.
Finally we can also adjust the plot itself. Maybe the oncologist is very interested in median survival of men and women. Lets help her by showing this on the plot.
med.surv <- data.frame(time = c(270,270, 426,426), quant = c(.5,0,.5,0),
sex = c('M', 'M', 'F', 'F'))
pl2 + geom_line(data = med.surv, aes(time, quant, group = sex),
col = 'darkblue', linetype = 3) +
geom_point(data = med.surv, aes(time, quant, group =sex), col = 'darkblue')
I hope survival researchers will take the effort to produce better looking plots after reading this post, although copy pasting the code won’t be too much of an effort I guess.
ggsurv <- function(s, CI = 'def', plot.cens = T, surv.col = 'gg.def',
cens.col = 'red', lty.est = 1, lty.ci = 2,
cens.shape = 3, back.white = F, xlab = 'Time',
ylab = 'Survival', main = ''){
strata <- ifelse(is.null(s$strata) ==T, 1, length(s$strata))
stopifnot(length(surv.col) == 1 | length(surv.col) == strata)
stopifnot(length(lty.est) == 1 | length(lty.est) == strata)
ggsurv.s <- function(s, CI = 'def', plot.cens = T, surv.col = 'gg.def',
cens.col = 'red', lty.est = 1, lty.ci = 2,
cens.shape = 3, back.white = F, xlab = 'Time',
ylab = 'Survival', main = ''){
dat <- data.frame(time = c(0, s$time),
surv = c(1, s$surv),
up = c(1, s$upper),
low = c(1, s$lower),
cens = c(0, s$n.censor))
dat.cens <- subset(dat, cens != 0)
col <- ifelse(surv.col == 'gg.def', 'black', surv.col)
pl <- ggplot(dat, aes(x = time, y = surv)) +
xlab(xlab) + ylab(ylab) + ggtitle(main) +
geom_step(col = col, lty = lty.est)
pl <- if(CI == T | CI == 'def') {
pl + geom_step(aes(y = up), color = col, lty = lty.ci) +
geom_step(aes(y = low), color = col, lty = lty.ci)
} else (pl)
pl <- if(plot.cens == T & length(dat.cens) > 0){
pl + geom_point(data = dat.cens, aes(y = surv), shape = cens.shape,
col = cens.col)
} else if (plot.cens == T & length(dat.cens) == 0){
stop ('There are no censored observations')
} else(pl)
pl <- if(back.white == T) {pl + theme_bw()
} else (pl)
ggsurv.m <- function(s, CI = 'def', plot.cens = T, surv.col = 'gg.def',
cens.col = 'red', lty.est = 1, lty.ci = 2,
cens.shape = 3, back.white = F, xlab = 'Time',
ylab = 'Survival', main = '') {
n <- s$strata
groups <- factor(unlist(strsplit(names
(s$strata), '='))[seq(2, 2*strata, by = 2)])
gr.name <- unlist(strsplit(names(s$strata), '='))[1]
gr.df <- vector('list', strata)
ind <- vector('list', strata)
n.ind <- c(0,n); n.ind <- cumsum(n.ind)
for(i in 1:strata) ind[[i]] <- (n.ind[i]+1):n.ind[i+1]
for(i in 1:strata){
gr.df[[i]] <- data.frame(
time = c(0, s$time[ ind[[i]] ]),
surv = c(1, s$surv[ ind[[i]] ]),
up = c(1, s$upper[ ind[[i]] ]),
low = c(1, s$lower[ ind[[i]] ]),
cens = c(0, s$n.censor[ ind[[i]] ]),
group = rep(groups[i], n[i] + 1))
dat <- do.call(rbind, gr.df)
dat.cens <- subset(dat, cens != 0)
pl <- ggplot(dat, aes(x = time, y = surv, group = group)) +
xlab(xlab) + ylab(ylab) + ggtitle(main) +
geom_step(aes(col = group, lty = group))
col <- if(length(surv.col == 1)){
scale_colour_manual(name = gr.name, values = rep(surv.col, strata))
} else{
scale_colour_manual(name = gr.name, values = surv.col)
pl <- if(surv.col[1] != 'gg.def'){
pl + col
} else {pl + scale_colour_discrete(name = gr.name)}
line <- if(length(lty.est) == 1){
scale_linetype_manual(name = gr.name, values = rep(lty.est, strata))
} else {scale_linetype_manual(name = gr.name, values = lty.est)}
pl <- pl + line
pl <- if(CI == T) {
if(length(surv.col) > 1 && length(lty.est) > 1){
stop('Either surv.col or lty.est should be of length 1 in order
to plot 95% CI with multiple strata')
}else if((length(surv.col) > 1 | surv.col == 'gg.def')[1]){
pl + geom_step(aes(y = up, color = group), lty = lty.ci) +
geom_step(aes(y = low, color = group), lty = lty.ci)
} else{pl + geom_step(aes(y = up, lty = group), col = surv.col) +
geom_step(aes(y = low,lty = group), col = surv.col)}
} else {pl}
pl <- if(plot.cens == T & length(dat.cens) > 0){
pl + geom_point(data = dat.cens, aes(y = surv), shape = cens.shape,
col = cens.col)
} else if (plot.cens == T & length(dat.cens) == 0){
stop ('There are no censored observations')
} else(pl)
pl <- if(back.white == T) {pl + theme_bw()
} else (pl)
pl <- if(strata == 1) {ggsurv.s(s, CI , plot.cens, surv.col ,
cens.col, lty.est, lty.ci,
cens.shape, back.white, xlab,
ylab, main)
} else {ggsurv.m(s, CI, plot.cens, surv.col ,
cens.col, lty.est, lty.ci,
cens.shape, back.white, xlab,
ylab, main)}
Great post. Is it also possible to use ggsurf if I use glm or glmer to estimate a (multi-level) discrete-time hazard model (see http://www.ats.ucla.edu/stat/r/examples/alda/ch12.htm)?
The function only works if it is used on an object of class survfit. If you want it to work on a different object you should tweak the code a bit. Note that the first part of the function is
creating data frames that are fed to the ggplot code below. If you can turn your fit into a data frame just alike you can readily use the code that produces the plots. Good luck!
See also slide 68 onwards in http://timchurches.github.io/ggplot2er/ which illustrates the use of a similar approach by Ramon Saccilotto of the Basel Institute for Clinical Epidemiology and
Biostatistics (links to Ramon’s work are in the presentation).
Didn’t see that one before, thanks. Decided to write the function because I couldn’t find any function or code. Nice slides by the way, Lung data set is popular!
Either your function or Ramon’s (no idea which one works best) should get submitted to the GGally package, or perhaps even to the autoplot package.
Thanks for the suggestion, I will look into the options.
No worries. I’ll suggest little things if you submit it to GGally, like vectorizing the strata loops or leaving out the xlab and ylab arguments to encourage the use of the labs()
function in ggplot2.
Back with another suggestion: Christopher Gandrud’s simPH package seems very relevant here. It plots everything with ggplot2 and has a ggfitStrata function.
While these are a nice example of using ggplot syntax, you might want to also look at the features associated with survplot() in the rms package by Frank Harrell – all based in base graphics. The
confidence bands are worth their weight in gold.
The survival rate in my data set is 20%. Following your syntax, the range of the y-axis was constrained between 0.75 and 1.00. So what if I want the y-axis ranges from 0 to 1?Thank you!
You can just use the ggplot2 function ylim to adjust the y-axis
ggsurv(my.survfit) + ylim(0, 1)
Although non-R, there is also a win-based tool to draw survival time plots:
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A Study of Malaria and Sickle Cell Anemia
This investigation of the genetics of the Sickle Cell trait via a mathematical model uses probability and teaches properties of quadratic functions and the concept of optimization of a function. The
properties of quadratic functions brought out by this investigation are
-the relationship between the zeros of the function and its factors, -the relationship between the zeros and the location of its vertex, -the symmetry of its graph, and the location of its extreme
point, -factors of quadratics of the form ax^2 + bx
|
{"url":"http://www.curriki.org/xwiki/bin/view/Coll_Group_SanDiegoCommunityCollegesDevelopmentalMathExchange/AStudyofMalariaandSickleCellAnemia?bc=;Coll_Group_SanDiegoCommunityCollegesDevelopmentalMathExchange.IntermediateAlgebra;Coll_Group_SanDiegoCommunityColleg","timestamp":"2014-04-19T15:02:53Z","content_type":null,"content_length":"51787","record_id":"<urn:uuid:116997ca-66a6-446c-bcb1-172efd13af9a>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00611-ip-10-147-4-33.ec2.internal.warc.gz"}
|
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Yoshikazu, Giga and Jurgen, Saal AN APPROACH TO ROTATING BOUNDARY LAYERS BASED ON VECTOR RADON MEASURES. (10 May 2011);
Preprint Series # 976
Fukunaga, Tomonori THE WARPING DEGREE OF A NANOWORD. (25 April 2011); (Submitted)
Preprint Series # 975
IWATA, YUKIKO and OGIHARA, TOMOHIRO RANDOM PERTURBATIONS OF NON-SINGULAR TRANSFORMATIONS ON [0; 1]. (22 February 2011);
Preprint Series # 974
Katsurada, Hidenori ON THE PERIOD OF THE IKEDA LIFT FOR U(m,m). (02 February 2011); (Submitted)
Preprint Series # 973
Yoshikazu, Giga and Qing, Liu and Hiroyoshi, Mitake Large-time Asymptotics for One-dimensional DirichletProblems for Hamilton-Jacobi Equations withNoncoercive Hamiltonians. (20 January 2011);
Preprint Series # 972
G., Ishikawa and Y., Machida and M., Takahashi The asymmetry on singularities of tangent surfaces in contact-cone Legendre-null duality. (12 January 2011);
Preprint Series # 971
GIGA, YOSHIKAZU and LIU, QING and MITAKE, HIROYOSHI SINGULAR NEUMANN PROBLEMS AND LARGE-TIME BEHAVIOR OF SOLUTIONS OF NONCOERCIVE HAMILTON-JACOBI EQUATIONS. (26 October 2010);
Preprint Series # 970
Kawaharada, Akane Estimating topological entropy of multidimensional nonlinear cellular automata. (22 October 2010);
Preprint Series # 969
Takamura, Hiroyuki and Wakasa, Kyouhei The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions. (24 September 2010); (Submitted)
Preprint Series # 968
Yoshikazu, Giga and Noriko, Mizoguchi and Takasi, Senba Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type. (03 September 2010); (Submitted)
Preprint Series # 967
Takamura, Hiroyuki and Uesaka, Hiroshi and Wakasa, Kyouhei Sharp blow-up for semilinear wave equations with non-compactly supported data. (03 September 2010); (Submitted)
Preprint Series # 966
Arai, Asao Spectral Analysis of an Effective Hamiltonian in Nonrelativistic Quantum Electrodynamics. (20 August 2010); (Submitted)
Preprint Series # 965
Giga, Yoshikazu and Saal, Jürgen L1 MAXIMAL REGULARITY FOR THE LAPLACIAN AND APPLICATIONS. (30 July 2010); (Submitted)
Preprint Series # 964
Takahashi, Masatomo On completely integrable implicit ordinary differential equations. (23 July 2010); (Submitted)
Preprint Series # 963
Giga, Yoshikazu and Kohn, Robert V. Scale-Invariant Extinction Time Estimates for Some Singular Diffusion Equations. (20 July 2010); (Submitted)
Preprint Series # 962
Asayama, Mikuri and Izumiya, Shyuichi and Tamaoki, Aiko and Yildirim, Handan Slant Geometry of Spacelike Hypersurfaces in Hyperbolic space and de Sitter space. (11 May 2010);
Preprint Series # 961
Izumiya, Shyuichi and Nagai, Takayuki and Saji, Kentaro Great circular surfaces in the three-sphere. (06 May 2010);
Preprint Series # 960
Giga, Mi-Ho and Giga, Yoshikazu Very Singular Diffusion Equations-Second and Fourth Order Problems. (06 May 2010);
Preprint Series # 959
Ohkitani, Koji and Sakajo, Takashi Oscillatory damping in long-time evolution of the surface quasi-geostrophic equations with generalised viscosity: a numerical study. (May 2010);
Preprint Series # 958
Katsurada, Hidenori CONGRUENCE BETWEEN DUKE-IMAMOGLU-IKEDA LIFTS AND NON-DUKE-IMAMOGLU-IKEDA LIFTS. (20 April 2010);
Preprint Series # 957
Arai, Asao Heisenberg Operators of a Dirac Particle Interacting with the Quantum Radiation Field. (13 April 2010);
Preprint Series # 956
Giga, Yoshikazu and Miura, Hideyuki On vorticity directions near singularities for the Navier-Stokes ows with infinite energy. (12 April 2010);
Preprint Series # 955
Giga, Yoshikazu and Gorka, Przemyslaw and Rybka, Piotr A Comparison Principle for Hamilton-Jacobi equations with discontinuous Hamiltonians. (25 February 2010);
Preprint Series # 954
Katsurada, Hidenori and Kawamura, Hisa-aki IKEDA'S CONJECTURE ON THE PERIOD OF THE DUKE-IMAMOGLU-IKEDA LIFT. (23 February 2010);
Preprint Series # 953
Izumiya, Shyuichi and Handan, Yildirim Slant Geometry of Spacelike Hypersurfaces in the Lightcone. (01 February 2010);
Preprint Series # 952
Giga, Yoshikazu and Mahalov, Alex and Yoneda, Tsuyoshi ON A BOUND FOR AMPLITUDES OF NAVIER-STOKES FLOW WITH ALMOST PERIODIC INITIAL DATA. (08 December 2009);
Preprint Series # 951
Kasedou, Masaki Spacelike submanifolds in de Sitter space. (06 November 2009);
Preprint Series # 950
Dummigan, Neil and Ibukiyama, Tomoyoshi and Katsurada, Hidenori SOME SIEGEL MODULAR STANDARD L-VALUES, AND SHAFAREVICH-TATE GROUPS. (23 October 2009);
Preprint Series # 949
Katsurada, Hidenori and Mizumoto, S. Congruences for Hecke Eigenvalues of Siegel Modular Forms. (20 October 2009);
Preprint Series # 948
Izumiya, Shyuichi and Yildirim, Handan Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space. (15 October 2009);
Preprint Series # 947
Izumiya, Shyuichi and Saji, Kentaro The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat'' spacelike surfaces. (30 September 2009);
Preprint Series # 946
Aikawa, Hiroaki MODULUS OF CONTINUITY OF THE DIRICHLET SOLUTIONS. (30 September 2009);
Preprint Series # 945
Giga, Yoshikazu and Seki, Yukihiro and Umeda, Noriaki On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. (10 September 2009);
Preprint Series # 944
Katsurada, Hidenori and Mizuno, Yoshinori Linear dependence of certain L-values of half-integral weight modular forms. (20 August 2009);
Preprint Series # 943
Fukunaga, Tomonori and Ito, Noboru KHOVANOV HOMOLOGY AND WORDS. (01 July 2009);
Preprint Series # 942
Katsurada, Hidenori A REMARK ON THE NORMALIZATION OF THE STANDARD ZETA VALUES FOR SIEGEL MODULAR FORMS. (01 June 2009);
Preprint Series # 940
Fukunaga, Tomonori HOMOTOPY CLASSIFICATION OF GENERALIZED PHRASES IN TURAEV'S THEORY OF WORDS. (May 2009);
Preprint Series # 941
Katsurada, Hidenori and Kawamura, Hisa-aki On the Andrianov type identity for power series attached to Jacobi forms and its application. (29 April 2009);
Preprint Series # 941
Katsurada, Hidenori and Kawamura, Hisa-aki ON THE ANDRIANOV TYPE IDENTITY FOR POWER SERIES ATTACHED TO JACOBI FORMS AND ITS APPLICATION. (28 April 2009);
Preprint Series # 939
Shibuya, Kazuhiro and Noda, Takahiro On implicit second order PDE of a scalar function on a plane via differential systems. (05 March 2009);
Preprint Series # 938
Giga, Yoshikazu and Seki, Yukihiro and Umeda, Noriaki Mean curvature flow closes open ends of noncompact surfaces of rotation. (20 February 2009);
Preprint Series # 937
Kawano, Shinji On the maximum value of ground states for the scalar field equation with double power nonlinearity. (12 February 2009);
Preprint Series # 936
Kawano, Shinji Uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. (12 February 2009);
Preprint Series # 935
Kawano, Shinji On semilinear elliptic equations with nonlocal nonlinearity. (12 February 2009);
Preprint Series # 934
Kawano, Shinji A remark on the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. (12 February 2009);
Preprint Series # 933
Giga, Yoshikazu and Gorka, Przemyslaw and Rybka, Piotr Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. (02 February 2009);
Preprint Series # 932
Buosi, Marcelo and Izumiya, Shyuichi and Maria Aparecida, Soares Ruas Horo-tight spheres in Hyperbolic space. (14 January 2009); (Submitted)
Preprint Series # 931
KANG, HUNSEOK SPATIO-TEMPORAL CHAOS IN A DISCRETE TURING MODEL. (08 December 2008);
Preprint Series # 930
Izumiya, Shyuichi and Tari , Farid Projections of timelike surfaces in the de Sitter space. (01 December 2008);
Preprint Series # 928
Watanabe, Humihiko On the general transformation of the Wirtinger integral. (23 October 2008);
Preprint Series # 927
Yamaguchi, K. CONTACT GEOMETRY OF SECOND ORDER I. (08 October 2008);
Preprint Series # 929
Abe, Takuro and Terao, Hiroaki A primitive derivation and logarithmic differential forms of Coxeter arrangements. (October 2008);
Preprint Series # 927
Yamaguchi, Keizo Contact Geometry of second order I. (October 2008);
Preprint Series # 926
Shibuya, Kazuhiro On the prolongation of 2-jet space off 2 independent and 1 dependent variables. (September 2008);
Preprint Series # 925
Shibuya, Kazuhiro A set of integral elements of higher order jet spaces. (September 2008);
Preprint Series # 924
Giga, Yoshikazu and Inui, Katsuya and Mahalov, Alex and Saal, Jurgen Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data. (September 2008);
Preprint Series # 923
Chen, Liang and Izumiya, Shyuichi A mandala of Legendrian dualities for pseudo-spheres in semi-Euclidean space. (2008);
Preprint Series # 922
Giga, Yoshikazu and Liu, Qing A Billiard-Based Game Interpretation of the Neumann Problem for the Curve Shortening Equation. (2008);
Preprint Series # 921
Fukunaga, Tomonori HOMOTOPY CLASSIFICATION OF NANOPHRASES WITH LESS THAN OR EQUAL TO FOUR LETTERS. (2008);
Preprint Series # 920
Izumiya, Shyuichi and Tari, Farid Pairs of foliations on timelike surfaces in the de Sitter space S^3_1. (2008);
Preprint Series # 919
Abe, Takuro A generalized logarithmic module and duality of Coxeter multiarrangements. (2008);
Preprint Series # 918
Izumiya, Shyuichi and Romero Fuster, Maria del Carmen and Saji, Kentaro Flat Lightlike Hypersurfaces in Lorentz-Minkowski 4-space. (2008);
Preprint Series # 917
Hattori, Shin On a ramification bound of semi-stable torsion representations over a local field. (2008);
Preprint Series # 916
Giga, Yoshikazu and Umeda, Noriaki On instant blow-up for semilinear heat equations with growing initial data. (2008);
Preprint Series # 915
Abe, Takuro and Terao, Hiroaki and Yoshinaga, Masahiko Totally free arrangements of hyperplanes. (2008);
Preprint Series # 914
Izumiya, Shyuichi and Pei, Donghe and Romero Fuster, Maria del Carmen Spacelike surfaces in Anti de Sitter four-space from a contact viewpoint. (2008);
Preprint Series # 913
., . (deleted). (2008);
Preprint Series # 912
Hiroshima, Fumio and Suzuki, Akito Physical state of non-relativistic quantum electrodynamics. (2008);
Preprint Series # 911
Kasedou, Masaki Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space. (2008);
Preprint Series # 910
Kasai, Kota REPRESENTATION OF SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS VIA TWO-PERSON GAME WITH INTEREST RATE. (2008);
Preprint Series # 909
Arai, Asao and Matsuzawa, Yasumichi Time Operators of a Hamiltonian with Purely Discrete Spectrum. (2008);
Preprint Series # 908
Ashikaga, Tadashi and Ishizaka, Mizuho Another form of the reciprocity law of Dedekind sum. (2008);
Preprint Series # 907
Ashikaga, Tadashi Local signature defect of fibered complex surfaces via monodromy and stable reduction. (2008);
Preprint Series # 906
Chen, Liang and Izumiya, Shyuichi Singularities of Anti de Sitter Torus Gauss maps. (2008);
Preprint Series # 905
Sasaki, Hironobu and Suzuki, Akito Inverse scattering problem for the Klein Gordon equation in quantum field theory. (2008);
Preprint Series # 904
SAKAJO, Takashi Non self-similar, partial and robust collapse of four point vortices on sphere. (2008);
Preprint Series # 903
Kamiya, Hidehiko and Takemura, Akimichi and Terao, Hiroaki Periodicity of non-central integral arrangements modulo positive integers. (2008);
Preprint Series # 902
Shibukawa, Youichi (H,X)-Bialgebroids Associated with Dynamical Yang-Baxter Maps. (2008);
Preprint Series # 901
Giga, Yoshikazu and Liu, Qing A Remark on the Discrete Deterministic Game Approach for Curvature Flow Equations. (2008);
Preprint Series # 900
Garcke, Harald and Kohsaka, Yoshihito and Sevcovic, Daniel Nonlinear stability of stationary solutions for curvature flow with triple junction. (2008);
Preprint Series # 899
Sawada, Okihiro and Usui, Toshiomi The Navier-Stokes equations for linearly growing velocity with nondecaying initial disturbance. (2008);
Preprint Series # 898
OKAMOTO, Hisashi and SAKAJO, Takashi and WUNSCH, Marcus On a generalization of the Constantin-Lax-Majda equation. (2008);
Preprint Series # 897
Fukunaga, Tomonori HOMOTOPY CLASSIFICATION OF NANOPHRASES IN TURAEV’S THEORY OF WORDS. (2008);
Preprint Series # 896
Maekawa, Yasunori Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations. (2008);
Preprint Series # 895
Nakazi, Takahiko Multipliers For A Quotient Banach Space And The Nevanlinna-Pick Theorem. (2008);
Preprint Series # 894
Yamazaki, Noriaki Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator. (2008);
Preprint Series # 893
Abe, Takuro and Nuida, Koji and Numata, Yasuhide Bicolor-eliminable graphs and free multiplicities on the braid arrangement. (2008);
Preprint Series # 892
Chen, Lian Singularities of timelike Anti de Sitter Gauss images. (2008);
Preprint Series # 891
Nakazi, Takahiko Some Fredholm Integration Operators on A Hilbert Space of Holomorphic Functions on The Unit Disc. (2008);
Preprint Series # 890
Inoue, Akihiko and Kasahara, Yukio and Phartyal, Punam Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2. (2008);
Preprint Series # 889
Giga, Yoshikazu and Rybka, Piotr Facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term. (2008);
Preprint Series # 888
Arai, Asao and Matsuzawa, Yasumichi Construction of a Weyl Representation from a Weak Weyl Representation of the Canonical Commutation Relation. (2008);
Preprint Series # 887
Izumiya, Shyuichi and Tari, Farid Projections of surfaces in the hyperbolic space along horocycles. (2008);
Preprint Series # 886
Suzuki, Akito Physical subspace in a model of the quantized electromagnetic field coupled to an external field with an indefinite metric. (2007);
Preprint Series # 885
Yokoyama, Etsuro and Giga, Yoshikazu and Rybka, Piotr A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation.
Preprint Series # 884
Agemi, Rentaro and Takamura, Hiroyuki The energy decay of divergence-free displacements for elastic waves with Neumann boundary condition. (2007);
Preprint Series # 883
Tsutaya, Kimitoshi and Machihara, Shuji Scattering Theory for the Dirac Equation with a Nonlocal Term. (2007);
Preprint Series # 882
Fukuda, Kei and Inoue, Akihiko and Nakano, Yumiharu Optimal intertemporal risk allocation applied to insurance pricing. (2007);
Preprint Series # 881
Kashima, Yohei On the double critical-state model for type-II superconductivity in 3D. (2007);
Preprint Series # 880
Buosi, Marcelo and Izumiya, Shyuichi and Soares Ruas, Maria Aparecida Total absolute horospherical curvature of submanifolds in hyperbolic space. (2007);
Preprint Series # 879
Oura, Manabu Eisenstein polynomials associated to binary codes. (2007);
Preprint Series # 878
Izumiya, Shyuichi and Nuno Ballesteros, Juan Jose and Romero Fuster, Maria del Carmen Global properties on spacelike submanifolds of codimension two in Minkowski space. (2007);
Preprint Series # 877
NEWTON, Paul and SAKAJO, Takashi Statistical properties of point vortex equilibria on the sphere. (2007);
Preprint Series # 876
Cho, Yonggeun and Ozawa, Tohru and Shim, Yong-Sun ELLIPTIC ESTIMATES INDEPENDENT OF DOMAIN EXPANSION. (2007);
Preprint Series # 875
Ohtsuka, Takeshi and Shirakawa, Ken and Yamazaki, Noriaki OPTIMAL CONTROL OF A SINGULAR DIFFUSION EQUATION WITH CONSTRAINT. (2007);
Preprint Series # 874
Giga, Yoshikazu EVOLUTION EQUATIONS WITH ALMOST PERIODIC INITIAL DATA. (2007);
Preprint Series # 873
Abe, Takuro and Yoshinaga, Masahiko Coxeter multiarrangements with quasi-constant multiplicities. (2007);
Preprint Series # 872
GOTO, Shun'ichi and NAKAGAWA, Maki and OHTSUKA, Takeshi Uniqueness and existence of generalized motion for spiral crystal growth. (2007);
Preprint Series # 871
Kasahara, Yukio and Pourahmadi, Mohsen and Inoue, Akihiko Applications of a finite-dimensional duality principle to some prediction problems. (2007);
Preprint Series # 870
Inoue, Akihiko and Anh, V.V. Fractional Processes with Long-range Dependence. (2007);
Preprint Series # 869
Konishi, Yukiko and Minabe, Satoshi On solutions to Walcher's extended holomorphic anomaly equation. (2007);
Preprint Series # 868
Arai, Asao Representations of the Quantum Plane and the Quantum Algebra $U_{q}({\rm sl}_2)$ on $L^2(\R^d)$. (2007);
Preprint Series # 867
Izuki, Mitsuo Wavelets and modular inequalities in variable L^p spaces. (2007);
Preprint Series # 866
Abe, Takuro and Numata, Yasuhide Exponents of $2$-multiarrangements and multiplicity lattice. (2007);
Preprint Series # 865
Kamiya, Hidehiko and Takemura, Akimichi and Terao, Hiroaki The characteristic quasi-polynomials of the arrangements of root systems. (2007);
Preprint Series # 864
Garcke, Harald and Ito, Kazuo and Kohsaka, Yoshihito Nonlinear stability of stationary solutions for surface diffusion with boundary conditions. (2007);
Preprint Series # 863
Machide, Tomoya Elliptic Dedekind-Rademacher Sums and Transformation Formulae of Certain Infinite Series. (2007);
Preprint Series # 862
Sato, Norifumi A simple proof of the Allen-Cahn equation to Brakke's motion. (2007);
Preprint Series # 861
Ozawa, Tohru and Sasaki, Hironobu Inequalities associated with dilations. (2007);
Preprint Series # 860
Giga, Yoshikazu and Jo, Hideaki and Mahalov, Alex and Yoneda, Tsuyoshi On time analyticity of the Navier-Stokes equations in a rotating frame with spatially almost periodic data. (2007);
Preprint Series # 859
Cho, Yonggeun and Ozawa, Tohru Sobolev inequalities with symmetry. (2007);
Preprint Series # 858
SAKAJO, Takashi and YAGASAKI, Kazuyuki Chaotic motion of the N-vortex problem on a sphere I: Saddle-centers in two-degree-of-freedom. (2007);
Preprint Series # 857
Le Coz, Stefan and Fukuizumi, Reika and Fibich, Gadi and Ksherim, Baruch and Sivan, Yonatan Instability of bound states of a nonlinear Schrodinger equation with a Dirac potential. (2007);
Preprint Series # 856
Giga, Yoshikazu and Seki, Yukihiro and Umeda, Noriaki Blow-up at space infinity for nonlinear heat equations. (2007);
Preprint Series # 855
Carles, Remi and Ozawa, Tohru A NONLINEAR POISSON FORMULA FOR THE SCHR\"ODINGER OPERATOR. (2007);
Preprint Series # 854
Izuki, Mitsuo and Sawano, Yoshihiro and Tachizawa, Kazuya Wavelet bases in the weighted Besov and Triebel-Lizorkin spaces with A^loc_p-weights. (2007);
Preprint Series # 853
Izumiya, Shyuichi and Takahashi, Masatomo and Tari, Farid Folding maps on spacelike and timelike surfaces and duality. (2007);
Preprint Series # 852
Furuhata, Hitoshi Hypersurfaces in Statistical Manifolds. (2007);
Preprint Series # 851
Watanabe, Humihiko Linear differential relations satisfied by Wirtinger integrals. (2007);
Preprint Series # 850
Douglas, Ronald G. and Nakazi, Takahiko and Seto, Michio Shift Operators on the ${\Bbb C}^2$-valued Hardy Space. (2007);
Preprint Series # 849
Sasaki, Itaru and Suzuki, Akito Fermionic renormalization group method based on the smooth Feshbach map. (2007);
Preprint Series # 848
Nakazi, Takahiko Invariant Subspaces In The Bidisc And Wandering Subspaces. (2007);
Preprint Series # 847
SAKAJO, Takashi A fast tree-code algorithm for the vortex method on a sphere. (2007);
Preprint Series # 846
Maekawa, Yasunori On the existence of Burgers vortices for high Reynolds numbers. (2007);
Preprint Series # 845
Hiroshima, Fumio and Sasaki, Itaru Enhanced binding for N-particle system interacting with a scalar bose field I. (2007);
Preprint Series # 844
Sasaki, Itaru Schr\"odinger operators with rapidly oscillating potentials. (2007);
Preprint Series # 843
Hattori, Shin Tame characters and ramification of finite flat group schemes. (2007);
Preprint Series # 842
De Bouard, Anne and Fukuizumi, Reika Stochastic fluctuations in the Gross-Pitaevskii equation. (2007);
Preprint Series # 841
Maekawa, Yasunori On a linearized operator of the equation for Burgers vortices. (2007);
Preprint Series # 840
ROGER, MATTHIAS and TONEGAWA, YOSHIHIRO CONVERGENCE OF PHASE–FIELD APPROXIMATIONS TO THE GIBBS–THOMSON LAW. (2007);
Preprint Series # 839
Kamiya, Hidehiko and Takemura, Akimichi and Terao, Hiroaki Periodicity of hyperplane arrangements with integral coefficients modulo positive integers. (2007);
Preprint Series # 838
Izumiya, Shyuichi and Saji, Kentaro and Takahashi, Masatomo Horospherical flat surfaces in Hyperbolic 3-space. (2007);
Preprint Series # 837
Geissert, Matthias and Giga, Yoshikazu On the Stokes resolvent equations in locally uniform L^p spaces in exterior domains. (2007);
Preprint Series # 836
Takahashi, Masatomo Uniqueness for geometric solutions of implicit second order ordinary differential equations. (2007);
Preprint Series # 835
Sabau, Sorin V and Shimada, Hideo Riemann-Finsler surfaces. (2007);
Preprint Series # 834
Aoyagi, Yutaka and Tsutaya, Kimitoshi and Yamauchi, Yusuke GLOBAL EXISTENCE OF SOLUTIONS FOR A REACTION-DIFFUSION SYSTEM. (2007);
Preprint Series # 833
Izumiya, Shyuichi and Tari, Farid Projections of surfaces in the hyperbolic space to hyperhorospheres and hyperplanes. (2007);
Preprint Series # 832
Izuki, Mitsuo and Sawano, Yoshihiro Atomic decomposition for the weighted Besov / Triebel-Lizorkin spaces with A^loc_p weights. (2007);
Preprint Series # 831
Sasaki, Hironobu Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. (2007);
Preprint Series # 830
Carles, Remi and Ozawa, Tohru On the wave operators for the critical nonlinear Schr\"odinger equation. (2007);
Preprint Series # 829
Inoue, Akihiko AR and MA representation of partial autocorrelation functions, with applications. (2007);
Preprint Series # 828
Inoue, Akihiko and Nakano, Yumiharu Remark on optimal investment in a market with memory. (2007);
Preprint Series # 827
Cho, Yonggeun and Ozawa, Tohru and Sasaki, Hironobu and Shim, Yong-sun Remarks on the relativistic Hartree equations. (2007);
Preprint Series # 826
Abe, Takuro Free and non-free multiplicity on the arrangement of type$A_3-1$. (2007);
Preprint Series # 825
Izuki, Mitsuo and Tachizawa, Kazuya Wavelet characterizations of weighted Herz spaces. (2007);
Preprint Series # 824
Ishikawa, Goo Global classification of curves on the symplectic plane. (2007);
Preprint Series # 823
Abe, Takuro and Terao, Hiroaki and Wakefield, Max The e-multiplicity and addition-deletion theorems for multiarrangements. (2006);
Preprint Series # 822
Ozawa, Tohru and Yamauchi, Kazuyuki Remarks on analytic smoothing effect for the Schr\"odinger equation. (2006);
Preprint Series # 821
Reznikoff, Maria G and Tonegawa, Yoshihiro Higher Multiplicity in the One-Dimensional Allen-Cahn Action Functional. (2006);
Preprint Series # 820
Morita, Hideaki Garsia-Haiman modules for hook partitions and Green polynomials with two variables. (2006);
Preprint Series # 819
Sato, Yuzuru and Ay, Nihat Adaptive Dynamics for Interacting Markovian Processes. (2006);
Preprint Series # 818
ABE, TAKURO and TERAO, HIROAKI and WAKEFIELD, MAX THE CHARACTERISTIC POLYNOMIAL OF A MULTIARRANGEMENT. (2006);
Preprint Series # 817
Ozawa, Tohru and Zhai, Jian Global Existence of Small Classical Solutions to Nonlinear Schr\"odinger Equations. (2006);
Preprint Series # 816
Kobayashi, Mahito On the cusped fan in a planar portrait of a manifold. (2006);
Preprint Series # 815
Nakazi, Takahiko Invariant Subspaces Of Toeplitz Operators And Uniform Algebras. (2006);
Preprint Series # 814
Hirata, Kentaro The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations. (2006);
Preprint Series # 813
Cho, Yonggeun and Ozawa, Tohru Global solutions to semirelativistic Hartree equations. (2006);
Preprint Series # 812
Okabe, Shinya THE MOTION OF ELASTIC PLANAR CLOSED CURVES UNDER THE AREA-PRESERVING CONDITION. (2006);
Preprint Series # 811
Izuki, Mitsuo The characterizations of weighted Sobolev spaces by wavelets and scaling functions. (2006);
Preprint Series # 810
Hirata, Kentaro Boundary behavior of solutions of the Helmholtz equation. (2006);
Preprint Series # 809
Oura, Manabu and Poor, Cris and Yuen, David S. Toward the Siegel ring in genus four. (2006);
Preprint Series # 808
Oura, Manabu On the integral ring spanned by genus two weight enumerators. (2006);
Preprint Series # 807
Watanabe, Humihiko Twisted homology and cohomology groups associated to the Wirtinger integral. (2006);
Preprint Series # 806
Katsurada, Hidenori and Kawamura, Hisa-aki A certain Dirichlet series of Rankin-Selberg type associated with the Ikeda lifting. (2006);
Preprint Series # 805
Abe, Takuro Faces of arrangements of hyperplanes and Arrow's impossibility theorem. (2006);
Preprint Series # 804
Machide, Tomoya Sums of Products of Kronecker's Double Series. (2006);
Preprint Series # 803
Nagase, Yuko and Tonegawa, Yoshihiro A singular perturbation problem with integral curvature bound. (2006);
Preprint Series # 802
Nakazi, Takahiko Isometric Composition Operators Between Two Weighted Hardy Spaces. (2006);
Preprint Series # 801
Maekawa, Yasunori On a free boundary problem of viscous incompressible flows. (2006);
Preprint Series # 800
Mikami, Toshio Stochastic control with fixed marginal distributions. (2006);
Preprint Series # 799
Terao, Hiroaki Chambers of Arrangements of Hyperplanes and Arrow's Impossibility Theorem. (2006);
Preprint Series # 798
ADACHI, Jiro and ISHIKAWA, Go-o Classification of phase singularities for complex scalar waves. (2006);
Preprint Series # 797
NEWTON, Paul, K and SAKAJO, Takashi The N-vortex problem on a rotating sphere: IV. Ring configurations coupled to a background field. (2006);
Preprint Series # 796
Aikawa, Hiroaki and Hirata, Kentaro DOUBLING CONDITIONS FOR HARMONIC MEASURE IN JOHN DOMAINS. (2006);
Preprint Series # 795
Giga, Yoshikazu and Inui, Katsuya and Mahalov, Alex and Saal, Jürgen Global solvabiliy of the Navier-Stokes equations in spaces based on sum-closed frequency sets. (2006);
Preprint Series # 794
Giga, Yoshikazu and Zhai, Jian Uniqueness of Constant Weakly Anisotropic Mean Curvature Immersion of Sphere $S^2$ In $\Bbb R^3$. (2006);
Preprint Series # 793
Nakazi, Takahiko and Yamamoto, Takanori Finite dimensional semisimple Q-algebras. (2006);
Preprint Series # 792
Cho, Yonggeun and Ozawa, Tohru On radial solutions of semi-relativistic Hartree equations. (2006);
Preprint Series # 791
Yamauchi, Yusuke BLOW-UP RESULTS FOR A REACTION-DIFFUSION SYSTEM. (2006);
Preprint Series # 790
Abe, Takuro The stability of the family of $B_2$-type arrangements. (2006);
Preprint Series # 789
Nakazi, Takahiko Integral Operators on a Subspace of Holomorphic Functions on the Disc. (2006);
Preprint Series # 788
Maekawa, Yasunori A lower bound for fundamental solutions of the heat convection equations. (2006);
Preprint Series # 787
Nakazi, Takahiko and Seto, Michio Double commuting compressed shifts and generalized interpolation in the Hardy space over the bidisk. (2006);
Preprint Series # 786
Sasaki, Hironobu Convergence of scattering operators for the Klein-Gordon equation with a nonlocal nonlinearity. (2006);
Preprint Series # 785
Machide, Tomoya An Elliptic Analogue of the Generalized Dedekind-Rademacher Sums. (2006);
Preprint Series # 784
FUSHO, Takesi and IZUMIYA, Shyuichi Lightlike surfaces of spacelike curves in de Sitter 3-space. (2006);
Preprint Series # 783
Nakazi, Takahiko Hyponormal Toeplitz Operators And Zeros Of Polynomials. (2006);
Preprint Series # 782
Suzuki, Akito Scaling limit for a generalization of the Nelson model and its application to nuclear physics. (2006);
Preprint Series # 781
CHINO, Sachiko and IZUMIYA, Shyuichi Lightlike developables in Minkowski 3-space. (2006);
Preprint Series # 780
Karageorgis, Paschalis and Tsutaya, Kimitoshi Small-data scattering for nonlinear waves of critical decay in two space dimensions. (2006);
Preprint Series # 779
SAKAJO, Takashi Integrable four-vortex motion on sphere with zero moment of vorticity. (2006);
Preprint Series # 778
Giga, Yoshikazu and Kuroda, Hirotoshi and Yamazaki, Noriaki Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation. (2006);
Preprint Series # 777
Izumiya, Shyuichi and Romero Fuster, Maria del Carmen The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space. (2006);
Preprint Series # 776
Cho, Yonggeun High regularity of solutions of compressible Navier-Stokes equations. (2006);
Preprint Series # 775
Hoffmann, Karl-Heinz and Kubo, Masahiro and Yamazaki, Noriaki optimal control problems for elliptic-parabolic variational inequalities with time-dependent constraints. (2006);
Preprint Series # 774
Morita, Hideaki and Wachi, Akihito and Watanabe, Junzo Zero-dimensional Gorenstein algebras with the action of the symmetric group S^k. (2006);
Preprint Series # 773
Cho, Yonggeun and Ozawa, Tohru On the semi-relativistic Hartree type equation. (2006);
Preprint Series # 772
Sasaki, Hironobu The inverse scattering problem for Schr\"odinger and Klein-Gordon equations with a nonlocal nonlinearity. (2006);
Preprint Series # 771
Cho, Muneo and Nakazi, Takahiko Spectral Area Estimates For Norms Of Commutators. (2006);
Preprint Series # 770
Karageorgis, Paschalis and Tsutaya, Kimitoshi On the asymptotic behavior of nonlinear waves in the presence of a short-range potential. (2006);
Preprint Series # 769
Shibukawa, Youichi Construction of dynamical Yang-Baxter maps and their characterization. (2006);
Preprint Series # 768
Fukuizumi, Reika and Ohta, Masahito and Ozawa, Tohru Nonlinear Schr\"odinger equation with a point defect. (2006);
Preprint Series # 767
Izumiya, Shyuichi and Takahashi, Masatomo Spacelike Parallels and Evolutes in Minkowski pseudo-spheres. (2006);
Preprint Series # 766
Izuki, Mitsuo The Haar wavelets and the Haar scaling function in weighted $L^p$ spaces with $A_p^{\dy ,m}$ weights. (2006);
Preprint Series # 765
Machide, Tomoya Functional Equations and the Harmonic Relations for Multiple Zeta Values. (2006);
Preprint Series # 764
Cho, Yonggeun and Ozawa, Tohru On small amplitude solutions to the generalized Boussinesq equations. (2006);
Preprint Series # 763
Giga, Yoshikazu and Rybka, Piotr Facet bending in the driven crystalline curvature flow in the plane. (2005);
Preprint Series # 762
Tadokoro, Satoru and Yamaguchi, Yutaka and Tsuda, Ichiro and Fujii, Hiroshi In-Out Intermittency in Gap Junction-Coupled Class I^* Neurons. (2005);
Preprint Series # 761
Giga, Y and Inui, K and Mahalov, A and Matsui, S and Saal, J Rotating Navier-Stokes Equations in ${\mathbb R}^{3}_{+}$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem.
Preprint Series # 760
Giga, Yoshikazu and Umeda, Noriaki Blow-up directions at space infinity for solutions of semilinear heat equations. (2005);
Preprint Series # 759
Aikawa, Hiroaki Equivalence between the boundary Harnack principle and the Carleson estimate. (2005);
Preprint Series # 758
SAKAJO, Takashi Invariant dynamical systems embedded in the N-vortex problem on a sphere with pole vortices. (2005);
Preprint Series # 757
Pourahmadi, Mohsen and Inoue, Akihiko and Kasahara, Yukio A Prediction Problem in L^2(w). (2005);
Preprint Series # 756
Yamaguchi, Keizo and Yatsui, Tomoaki PARABOLIC GEOMETRIES ASSOCIATED WITH DIFFERENTIAL EQUATIONS OF FINITE TYPE. (2005);
Preprint Series # 755
Saji, Kentaro and Takahashi, Masatomo Singularities of smooth mappings with patterns. (2005);
Preprint Series # 754
Nakazi, Takahiko and Yamamoto, Takanori Generalized Riesz Projections and Toeplitz Operators. (2005);
Preprint Series # 753
Giga, Yoshikazu and Rybka, Piotr FACETED CRYSTALS GROWN FROM SOLUTION - A STEFAN TYPE PROBLEM WITH A SINGULAR INTERFACIAL ENERGY. (2005);
Preprint Series # 752
Lin, Ching-Lung and Nakamura, Gen and Sini, Mourad Unique continuation for the transversally isotropic dynamical systems and its application. (2005);
Preprint Series # 751
Nakamura, Gen and Potthast, Roland and Sini, Mourad The no-response approach and its relation to non-iterative methods for the inverse scattering. (2005);
Preprint Series # 750
Choi, Cheok and Nakamura, Gen and Shirota, Kenji Variational approach for identifying the coefficient of wave equation. (2005);
Preprint Series # 749
Daido, Yuki and Kang, Hyeonbae and Nakamura, Gen Reconstruction of Inclusions for the Inverse Boundary Value Problem of Non-stationary Heat Equation. (2005);
Preprint Series # 748
Tsujii, Masato Decay of correlations in suspension semi-flows of angle-multiplying maps. (2005);
Preprint Series # 747
Izumiya, Shyuichi Differential Geometry from the viewpoint of Lagrangian or Legendrian singularity theory. (2005);
Preprint Series # 746
Suzuki, Akito A scaling limit for a general class of quantum field models and its application to nuclear physics and condensed matter physics. (2005);
Preprint Series # 745
Sasaki, Hironobu Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. (2005);
Preprint Series # 744
Cho, Yonggeun and Ozawa, Tohru Global existence on nonlinear Schr\"{o}dinger-IMBq equations. (2005);
Preprint Series # 743
Machide, Tomoya Elliptic Bernoulli Functions And Their Identities. (2005);
Preprint Series # 742
Cho, Yonggeun and Kim, Hyunseok Existence result for heat-conducting viscous incompressible fluids with vacuum. (2005);
Preprint Series # 741
Maekawa, Yasunori and Terasawa, Yutaka The Navier-Stokes equations with initial data in uniformly local $ L^{p} $ spaces. (2005);
Preprint Series # 740
Ozawa, Tohru and Tsutaya, Kimitoshi On the Cauchy Problem for Schro"dinger-improved Boussinesq equations. (2005);
Preprint Series # 739
Morita, Hideaki Green polynomials at roots of unity and its application. (2005);
Preprint Series # 738
Giga, Yoshikazu and Ohtsuka, Takeshi and Schaetzle, Reiner On a uniform approximation of motion by anisotropic curvature by the Allen-Cahn equations. (2005);
Preprint Series # 737
Sasaki, Itaru Ground State Energy of the Polaron in the Relativistic Quantum Electrodynamics. (2005);
Preprint Series # 736
Giga, Yoshikazu and Inui, Katsuya and Mahalov, Alex and Matsui, Shin'ya Uniform local solvability for the Navier-Stokes equations with the Coriolis force. (2005);
Preprint Series # 735
Hirata, Kentaro Sharp estimates for the Green function, 3G inequalities, and nonlinear Schr\"{o}dinger problems in uniform cones. (2005);
Preprint Series # 734
Giga, Yoshikazu and Kuroda, Hirotoshi and Yamazaki, Noriaki An existence result for a discretized constrained gradient system of total variation flow in color image processing. (2005);
Preprint Series # 733
Mikami, Toshio A generalization of a curvature flow of graphs on R. (2005);
Preprint Series # 732
Inoue, Akihiko and Nakano, Yumiharu Optimal long term investment model with memory. (2005);
Preprint Series # 731
Shibuya, Kazuhiro and Yamaguchi, Keizo Drapeau theorem for differential systems. (2005);
Preprint Series # 730
Izumiya, Shyuichi and Saji, Kentaro and Takeuchi, Nobuko Circular surfaces. (2005);
Preprint Series # 729
Morita, Hideaki and Nakajima, Tatsuhiro A formula of Lascoux-Leclerc-Thibon and representations of symmetric groups. (2005);
Preprint Series # 728
Cheng, Jin and Liu, Jijun and Nakamura, Gen Recovery of Boundaries and Types for Multiple Obstacles from the Far-field Pattern. (2005);
Preprint Series # 727
Hirata, Kentaro Martin boundary points of cones generated by spherical John regions. (2005);
Preprint Series # 726
Hirata, Kentaro Comparison estimates for the Green function and the Martin kernel. (2005);
Preprint Series # 725
Hirata, Kentaro Boundary behavior of quotients of Martin kernels. (2005);
Preprint Series # 724
Mikami, Toshio Semimartingales from the Fokker-Planck equation. (2005);
Preprint Series # 723
Cho, Yonggeun and Ozawa, Tohru Remarks on modified improved Boussinesq equations in one space dimension. (2005);
Preprint Series # 722
Kubo, Masahiro and Yamazaki, Noriaki Periodic solutions of elliptic-parabolic variational inequalities with time-dependent constraints. (2005);
Preprint Series # 721
Terasawa, Yutaka A remark on weak type (1, 1) estimates of Hardy-Littlewood maximal operators on metric spaces acting on dirac measures. (2005);
Preprint Series # 720
Morita, Hideaki and Watanabe, Junzo Zero dimensional Gorenstein algebras with the action of the symmetric group $S_k$. (2005);
Preprint Series # 719
Giga, Yoshikazu and Ohnuma, Masaki On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations. (2005);
Preprint Series # 718
Katsurada, Hidenori Congruence of Siegel modular forms and special values of their standard zeta functions. (2005);
Preprint Series # 717
Shibukawa, Youichi Dynamical Yang-Baxter Maps. (2005);
Preprint Series # 716
maekawa, yasunori Large time behavior of derivatives of the vorticity for the two dimensional Navier-Stokes flow. (2005);
Preprint Series # 715
Arai, Asao Generalized Weak Weyl Relation and Decay of Quantum Dynamics. (2005);
Preprint Series # 714
Tonegawa, Yoshihiro Diffused interface with the chemical potential in the Sobolev space. (2005);
Preprint Series # 713
Izumiya, Shyuichi and Kikuchi, Makoto and Takahashi, Masatomo Global properties of spacelike curves in Minkowski 3-space. (2005);
Preprint Series # 712
Forbes, Brian and Jinzenji, Masao Extending the Picard-Fuchs system of local mirror symmetry. (2005);
Preprint Series # 711
Buosi, Marcelo and Izumiya, Shyuichi and Soares Ruas, Mria Aoarecida HORO-TIGHT IMMERSIONS OF S^1. (2005);
Preprint Series # 710
Jinzenji, Masao and Nakamura, Iku and Suzuki, Yasuki Conics on a generic hypersurface. (2005);
Preprint Series # 709
Kohsaka, Yoshihito and Nagasawa, Takeyuki On the existence for the Helfrich flow and its center manifold near spheres. (2005);
Preprint Series # 708
Kato, Jun and Nakamura, Makoto and Ozawa, Tohru A generalization of the weighted Strichartz estimates for wave equations large and an application to self-similar solutions. (2005);
Preprint Series # 707
Sakajo, Takashi High-dimensional heteroclinic and homoclinic manifolds in odd point-vortex ring on sphere with pole vortices. (2005);
Preprint Series # 706
Ozawa, Tohru Remarks on proofs of conservation laws for nonlinear Schr\"odinger equations. (2005);
Preprint Series # 705
Kohn, Robert V. and Maria G, Reznikoff and Tonegawa, Yoshihiro The sharp-interface limit of the action functional for Allen-Cahn in one space dimension. (2005);
Preprint Series # 704
Cho, Yonggeun and Shim, Yongsun Global estimates of maximal operators generated by dispersive equations. (2005);
Preprint Series # 703
van Manen, Martijn On dangerous self-tangencies in families of conflict sets. (2005);
Preprint Series # 702
Tachizawa, Kazuya Weighted L^p Sobolev-Lieb-Thirring inequalities. (2005);
Preprint Series # 697
Izumiya, Shyuichi Timelike hypersurfaces in de Sitter space and Legendrian singularities. (2005);
Preprint Series # 696
Yamazaki, Noriaki A class of nonlinear evolution equations governed by time-dependent operators of subdifferential type. (2005);
Preprint Series # 695
Morita, Hideaki Decomposition of Green polynomials of type $A$ and DeConcini-Procesi-Tanisaki algebras of certain types. (2005);
Preprint Series # 694
Ishikawa, Go-o and Machida, Yoshinori Singularities of improper affine spheres and surfaces of constant Gaussian curvature. (2005);
Preprint Series # 693
Chen, Yunmei and Rao, Murali and Tonegawa, Yoshihiro and Wunderli, T Partial regularity for a selective smoothing functional for image restoration in BV space. (2005);
Preprint Series # 691
Hishida, Toshiaki $L^q$ estimates of weak solutions to the stationary Stokes equations around a rotating body. (2005);
Preprint Series # 690
Mikami, Toshio and Thieullen, Michele Optimal Transportation Problem by Stochastic Optimal Control. (2005);
Preprint Series # 689
Takamura, Hiroyuki Mixed amplitude solutions of semilinear systems of 3-dimensional wave equations. (2005);
Preprint Series # 688
Katayama, Soichiro and Yokoyama, Kazuyoshi Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds. (2005);
Preprint Series # 687
Nakamura, Gen and Wang, Shengzhang and Wang, Yanbo Numerical differentiation of the second order derivative of functions with several variables. (2005);
Preprint Series # 686
Nakazi, Takahiko and Osawa, Tomoko Invariant subspaces and Hankel type operators on a Bergman space. (2005);
Preprint Series # 685
Cho, Yonggeun and Lee, Sanghyuk and Shim, Yongsun A maximal inequality associated to Schr\{o}dinger type equation. (2005);
Preprint Series # 684
Cho, Yonggeun and Kim, Youngcheol and Lee, Sanghyuk and Shim, Yongsun $L^p-L^q$ estimates for convolutions with distribution kernels having singularities on the light cone. (2005);
Preprint Series # 618
Takahashi, Masatomo Holonomic systems of general Clairaut type. (30 November 2004);
Preprint Series # 613
Watanabe, Humihiko Sur la d'eg'en'erescence de quelques formules de connexion pour les fonctions hyperg'eom'etriques de Gauss. (21 July 2004);
Preprint Series # 701
FUKUIZUMI, Reika and OZAWA, Tohru On a decay property of solutions to the Haraux-Weissler equation. (2004);
Preprint Series # 700
DE BOUARD, Anne and FUKUIZUMI, Reika Stability of standing waves for nonlinear Schr\"odinger equations with inhomogeneous nonlinearities. (2004);
Preprint Series # 699
FUKUIZUMI, Reika and OZAWA, Tohru Exponential decay of solutions to nonlinear elliptic equations with potentials. (2004);
Preprint Series # 698
FUKUIZUMI, Reika and OHTA, Masahito Instability of standing waves for nonlinear Schr\"odinger equations with inhomogeneous nonlinearities. (2004);
Preprint Series # 692
Terasawa, Yutaka Outer measures and weak type (1,1) estimates of Hardy-Littlewood maximal operators. (2004);
Preprint Series # 683
Giga, Yoshikazu and Mahalov, Alex and Nicolaenko, Basil The Cauchy problem for the Navier-Stokes equations with spatially almost periodic initial data. (2004);
Preprint Series # 682
Forbes, Brian Computations on B model geometric transitions. (2004);
Preprint Series # 680
Cho, Yonggeun and Jin, Bum Ja Blow-up of the viscous heat-conducting compressible flow. (2004);
Preprint Series # 679
Giga, Yoshikazu and Rybka, Piotr Stability of facets of crystals growing from vapor. (2004);
Preprint Series # 678
Arai, Asao and Hayashi, Kunimitsu and Sasaki, Itaru Spectral Properties of a Dirac Operator in the Chiral Quark Soliton Model. (2004);
Preprint Series # 677
Mikami, Toshio A Simple Proof of Duality Theorem for Monge-Kantorovich Problem. (2004);
Preprint Series # 676
Cho, Yonggeun and Kim, Hyunseok On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. (2004);
Preprint Series # 675
Cho, Yonggeun and Kim, Hyunseok Existence results for viscous polytropic fluids with vacuum. (2004);
Preprint Series # 674
Nakamura, Gen and Potthast, Roland and Sini, Mourad The convergence proof of the no-response test for localizing an inclusion. (2004);
Preprint Series # 673
Izumiya, Shyuichi Legendrian dualities and spacelike hypersurfaces in the lightcone. (2004);
Preprint Series # 672
Watanabe, Humihiko Transformation relations of matrix functions associated to the hypergeometric function of Gauss under modular transformations. (2004);
Preprint Series # 671
Choie, YoungJu and Oura, Manabu Remark on the weight enumerators and Siegel modular forms. (2004);
Preprint Series # 670
Giga, Yoshikazu and Umeda, Noriaki On Blow up at Space Infinity for Semilinear Heat Equations. (2004);
Preprint Series # 669
Kin, Eiko and Sakajo, Takashi Efficient topological chaos embedded in the blinking vortex system. (2004);
Preprint Series # 668
Saal, Juergen Maximal regularity for the Stokes system on noncylindrical space-time domains. (2004);
Preprint Series # 667
Yamazaki, Noriaki Doubly nonlinear evolution equation associated with elliptic-parabolic free boundary problems. (2004);
Preprint Series # 666
Giga, Yoshikazu and Kubo, Motohiko and Tonegawa, Yoshihiro Magnetic clusters and fold energies. (2004);
Preprint Series # 665
FURUHATA, Hitoshi and VRANCKEN, Luc THE CENTER MAP OF AN AFFINE IMMERSION. (2004);
Preprint Series # 664
Giga, Yoshikazu and Inui, Katsuya and Mahalov, Alex and Matsui, Shin'ya Navier-Stokes Equations in a Rotating Frame in ${\mathbb R}^3$ with Initial Data Nondecreasing at Infinity. (2004);
Preprint Series # 663
Miyao, Tadahiro and Sasaki, Itaru Stability of Discrete Ground State. (2004);
Preprint Series # 662
Sato, Yuzuru and Akiyama, Eizo and Crutchfield, James P. Stability and Diversity in Collective Adaptation. (2004);
Preprint Series # 661
Inoue, Akihiko and Nakano, Yumiharu and Anh, Vo Binary market models with memory. (2004);
Preprint Series # 660
Hidano, Kunio and Yokoyama, Kazuyoshi A new proof of the global existence theorem of Klainerman for quasi-linear wave equations. (2004);
Preprint Series # 659
Hidano, Kunio and Yokoyama, Kazuyoshi A remark on the almost global existence theorems of Keel, Smith and Sogge. (2004);
Preprint Series # 658
Sasaki, Hironobu and Watanabe, Michiyuki Uniqueness on identification of cubic convolution nonlinearity. (2004);
Preprint Series # 657
Inoue, Akihiko and Nakano, Yumiharu and Anh, Vo Linear filtering of systems with memory. (2004);
Preprint Series # 656
Arai, Asao and Hayashi, Kunimitsu Spectral analysis of a Dirac operator with a meromorphic potential. (2004);
Preprint Series # 655
Hidano, Kunio and Yokoyama, Kazuyoshi Space-time L^2-estimates and life-span of the Klainerman-Machedon radial solutions to some semi-linear wave equations. (2004);
Preprint Series # 654
Kubo, Masahiro and Yamazaki, Noriaki Quasilinear parabolic variational inequalities with time-dependent constraints. (2004);
Preprint Series # 653
Inoue, Jyunji and Nakazi, Takahiko Nonnegative functions in weighted hardy spaces. (2004);
Preprint Series # 652
Mikami, Toshio and Thieullen, Mich¥`ele Duality Theorem for Stochastic Optimal Control Problem. (2004);
Preprint Series # 651
Watanabe, Michiyuki Inverse problem for the nonselfadjoint Schr"odinger Operator with energy dependent potential in Two dimensions. (2004);
Preprint Series # 650
Izuchi, Keiji and Nakazi, Takahiko and Seto, Michio Backward shift invariant subspaces in the bidisc III. (2004);
Preprint Series # 649
Giga, Yoshikazu and Giga, Mi-Ho and Hontani, Hidekata SELFSIMILAR EXPANDING SOLUTIONS IN A SECTOR FOR A CRYSTALLINE FLOW. (2004);
Preprint Series # 648
Morassi, Antonino and Nakamura, Gen and Sini, Mourad An inverse dynamical problem for connected beams. (2004);
Preprint Series # 647
Nagase, Yuko and Tonegawa, Yoshihiro Interior gradient estimate for 1-D anisotropic curvature flow. (2004);
Preprint Series # 646
Inoue, Akihiko and Kasahara, Yukio Explicit representation of finite predictor coefficients and its applications. (2004);
Preprint Series # 645
Yamazaki, Noriaki Attractors of asymptotically periodic multivalued dynamical systems governed by time-dependent subdifferentials. (2004);
Preprint Series # 644
Nakazi, Takahiko Rouch\'{e} type theorems and a theorem of Adamyan, Arov and Krein. (2004);
Preprint Series # 643
Itoh, Shigeharu and Tanaka, Naoto and Tani, Atusi Steady solution and its stability to Navier-Stokes equations with general Navier slip boundary condition. (2004);
Preprint Series # 642
SAKAJO, Takashi Transition of global dynamics of a polygonal vortex ring on a sphere with pole vortices. (2004);
Preprint Series # 641
Ozawa, Tohru and Yamauchi, Kazuyuki and Yamazaki, Yasuko Analytic Smoothing Effect for Solutions to Schr\"odinger Equations with Nonlinearity of Integral. (2004);
Preprint Series # 640
Izumiya, Shyuichi and Pei, Donghe and Romero Fuster, M. C. and Takahashi, Masatomo The horospherical geomoetry of submanifolds in hyperbolic space. (2004);
Preprint Series # 639
Giga, Yoshikazu and Kuroda, Hirotoshi On breakdown of solutions of a constrained gradient system of total variation. (2004);
Preprint Series # 638
Saal, Juergen The Stokes operator with Robin boundary conditions in solenoidal subspaces of L^1({\mathbb R}^n_+) and L^\infty({\mathbb R}^n_+). (2004);
Preprint Series # 637
Nakazi, Takahiko Brown-Halmos type theorems of weighted Toeplitz operators II. (2004);
Preprint Series # 636
Giga, Yoshikazu and Rybka, Piotr Stability of facets of self-similar motion of a crystal. (2004);
Preprint Series # 635
Nakajima, Fumio Predator-prey system model of singular equations; back to D'Ancona's question. (2004);
Preprint Series # 634
Nakajima, Fumio Uniformly ultimate boundedness of solutions for some 3-dimensional systems. (2004);
Preprint Series # 633
Saal, Juergen Stokes and Navier-Stokes equations with Robin boundary conditions in a half-space. (2004);
Preprint Series # 632
Katsurada, Hidenori Special values of the standard zeta functions for elliptic modular forms. (2004);
Preprint Series # 631
Ibukiyama, Tomoyoshi and Katsurada, Hidenori An explicit formula for the Koecher-Maa\ss Dirichlet series for the Ikeda lifting. (2004);
Preprint Series # 630
Kubo, Masahiro and Yamazaki, Noriaki Elliptic-parabolic variational inequalities with time-dependent constraints. (2004);
Preprint Series # 629
Yamazaki, Noriaki Stability for asymptotically periodic multivalued dynamical systems generated by double obstacle problems. (2004);
Preprint Series # 628
Seto, Michio Submodules of L^2(R^2). (2004);
Preprint Series # 627
Nakazi, Takahiko Factorizations Of Functions In H^p(T^n ). (2004);
Preprint Series # 620
Umeda, Noriaki Existence, Nonexistence of Global Solution and Large Time Behavior of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations. (02 December 2003);
Preprint Series # 619
Jinzenji, Masao Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation. (01 December 2003);
Preprint Series # 617
Nakazi, Takahiko Interpolation Of Weighted l^q Sequences By H^p Functions. (29 November 2003);
Preprint Series # 616
Yamada, Kazuyuki On visicous conservation laws with growing initial data. (25 November 2003);
Preprint Series # 615
Ishikawa, Go-o Infinitesimal deformations and stabilities of singular Lugendre submanifolds. (23 November 2003);
Preprint Series # 614
Sakajo, Takashi Motion of a vortex sheet on a sphere with pole vortices. (22 November 2003);
Preprint Series # 612
Kita, Naoyasu and Ozawa, Tohru Sharp asymptotic behavior of solutions to nonlinear schr\"odinger equations with repulsive interactions. (19 November 2003);
Preprint Series # 610
Hatakeyama, Motohiko and Tsuda, Ichiro Internal logic viewed from observation space: Theory and a case study. (20 October 2003);
Preprint Series # 609
GIGA, YOSHIKAZU and KASHIMA, YOHEI and YAMAZAKI, NORIAKI LOCAL SOLVABILITY OF A CONSTRAINED GRADIENT SYSTEM OF TOTAL VARIATION. (18 October 2003);
Preprint Series # 608
Tachizawa, Kazuya Weighted Sobolev-Lieb-Thirring inequalities. (15 October 2003);
Preprint Series # 606
Giga, Yoshikazu and Rybka, Piotr Existence of self-similar evolution of crystals grown from supersaturated vapor. (10 October 2003);
Preprint Series # 681
Forbes, Brian Open string mirror maps from Picard-Fuchs equations on relative cohomology. (2003);
Preprint Series # 626
Hontani, Hidetaka and Giga, Miho and Giga, Yoshikazu and Deguchi, Koichiro Expanding selfsimilar solutions of a crystalline flow with applications to contour figure analysis. (2003);
Preprint Series # 625
Sini, Mourad On the one dimensional Gelfand and Borg-Levinson spectral problems for discontinuous coefficients. (2003);
Preprint Series # 624
Nakamura, Gen and Sini, Mourad Remarks on the Inverse Scattering Problem for Ocean Acoustics. (2003);
Preprint Series # 623
Yamamoto, Takahiro Classification of singular fibres of stable maps from 4-manifolds to 3-manifolds and its applications. (2003);
Preprint Series # 622
Oura, Manabu OBSERVATION ON THE WEIGHT ENUMERATORS FROM CLASSICAL INVARIANT THEORY. (2003);
Preprint Series # 621
Umeda, Noriaki Large Time Behavior and Uniqueness of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations. (2003);
Preprint Series # 611
Ozawa, Tohru and Yamauchi, Kazuyuki Structure of Dirac matrices and invariants for nonlinear Dirac equations. (2003);
Preprint Series # 607
Machihara,, Shuji and Nakamura,, Makoto and Nakanishi,, Kenji and Ozawa, Tohru Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. (2003);
Preprint Series # 605
Bardi, Martino and Giga, Yoshikazu Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. (2003);
Preprint Series # 604
Giga, Yoshikazu Singular diffusivity facets, shocks and more. (2003);
Preprint Series # 603
Kato, Jun and Ozawa, Tohru Weighted Strichartz estimates for the wave equation in even space dimensions. (2003);
Preprint Series # 602
Kato, Jun and Ozawa, Tohru Weighted Strichartz estimates and existence of self-similar solutions for semilinear wave equations. (2003);
Preprint Series # 601
Kato, Jun and Ozawa, Tohru On solutions of the wave equation with homogeneous Cauchy data. (2003);
Preprint Series # 600
Ozawa, Tohru and Yamazaki, Yasuko SMOOTHING EFFECT AND LARGE TIME BEHAVIOR OF SOLUTIONS TO SCHR\"ODINGER EQUATIONS WITH NONLINEARITY OF INTEGRAL TYPE. (2003);
Preprint Series # 599
Ozawa, T and Yamazaki, Y Life-span of smooth solutions to the complex Ginzburg-Landau type equation on torus. (2003);
Preprint Series # 598
Kashima, Y A subdifferential formulation of fourth order singular diffusion equations. (2003);
Preprint Series # 597
Tonegawa, Y On stable critical points for a singular perturbation problem. (2003);
Preprint Series # 596
Mikami, T Monge's problem with a quadratic cost by the zero-noise limit h-pass processes. (2003);
Preprint Series # 595
Nakazi, T and Osawa, T Invariant subspaces of finite codimension and uniform algebras. (2003);
Preprint Series # 594
Giga, Y and Matsui, S and Sasayama, S On blow up rate for sign-changing solutions in a convex domain. (2003);
Preprint Series # 593
Jimbo, S Singular perturbation of domains and Semilinear elliptic equations III. (2003);
Preprint Series # 592
Izuchi, K and Nakazi, T and Seto, M Backward shift invariant subspaces in the bidisc II. (2003);
Preprint Series # 591
Tsai, Y and Giga, Y A numerical study of anisotropic crystal growth with bunching under very singular vertical diffusion. (2003);
Preprint Series # 590
Nakazi, T Exposed points and extremal problems in H1 on a bidisc. (2003);
Preprint Series # 589
Sawada, O On analyticity rate estimates of the solutions to the Navier-Stokes equations in Bessel-potential spaces. (2003);
Preprint Series # 588
Giga, Y and Kobayashi, R On constrained equations with singular diffusivity. (2003);
Preprint Series # 587
Miyao, T Momentum operators with a winding gauge potential. (2003);
Preprint Series # 586
Yamauchi, Kazuyuki On an underlying structure for the consistency of viscosity solutions. (2003);
Preprint Series # 585
Davydov, A. A and Ishikawa, G and Izumiya, S and Sun, W.-Z Generic singularities of implicit systems of first order differential equations on the plane. (2003);
Preprint Series # 584
Giga, M.-H and Giga, Y A PDE approach for motion of phase-boundaries by a singular interfacial energy. (2003);
Preprint Series # 583
Daido, Y and Nakamura, G Reconstruction of Inclusions for the Inverse Boundary Value Problem with Mixed Type Boundary Condition and Source Term. (2003);
Preprint Series # 582
Daido, Y and Ikehata, M and Nakamura, G Reconstruction of Inclusions for the Inverse Boundary Value Problem with Mixed Type Boundary Condition. (2003);
Preprint Series # 581
Nakazi, T and Osawa, T Spectra of Toeplitz Operators and Uniform Algebras. (2003);
Preprint Series # 580
Giga, Y and Yamada, K On viscous Burgers-like equations with linearly growing initial data. (2003);
Preprint Series # 579
Tsujii, M Physical measures for partially hyperbolic surface endomorphisms. (2003);
Preprint Series # 578
Giga, Y and Matsui, S and Sasayama, S Blow up rate for semilinear heat equation with subcritical nonlinearity. (2002);
Preprint Series # 577
Izumiya, S and Pei, D and Takahashi, M Evolutes of hypersurfaces in Hyperbolic space. (2002);
Preprint Series # 576
Izumiya, S and Kossowski, M and Pei, D and Romero-Fuster, M. C Singularities of C$B!g!>(Blightlike hypersurfaces in Minkowski 4-space. (2002);
Preprint Series # 575
Izumiya, S and Janeczko, S A symplectic framework for multiplane gravitational lensing. (2002);
Preprint Series # 574
Izumiya, S and Pei, D and Romero-Fuster, M. C The hyperbolic Gauss-Bonnet type theorem. (2002);
Preprint Series # 573
Izumiya, S and Pei, D and Romero-Fuster, M. C The horospherical geometry of surfaces in Hyperbolic 4-space. (2002);
Preprint Series # 572
Izuchi, K and Nakazi, T Backward shift invariant subspaces in the bidisc. (2002);
Preprint Series # 571
Nakano, Y Minimization of shortfall risk in a jump-diffusion model. (2002);
Preprint Series # 570
Ishii, H and Mikami, T Convexified Gauss curvature flow of bounded open sets in an anisotropic external field a stochastic apporoximation and PDE. (2002);
Preprint Series # 569
Hwang, J. M and Yamaguchi, K Characterization of Hermitian symmetric spaces by fundamental forms. (2002);
Preprint Series # 568
Miyao, T Strongry supercommuting serf-adjoint operators. (2002);
Preprint Series # 567
Giga, Y and Sawada, O regularizing-decay rate estmates for solutions to the Navier-Stokes initial value problem. (2002);
Preprint Series # 566
Ishikawa, G Perturbations of Caustics and fronts. (2002);
Preprint Series # 565
Ishikawa, G Classifying singular Legendre curves by contactomorphisms. (2002);
Preprint Series # 564
Takahashi, M Bifurcations of ordinary differential equations of Clairaut type. (2002);
Preprint Series # 563
Harris, A and Tonegawa, Y A $B"_"_!>(Bpoincar lemma for forms near an isolated complex singularity. (2002);
Preprint Series # 562
Nakazi, T and Yamamoto, T Norms of some singular integral operators on weighted L2 spaces. (2002);
Preprint Series # 561
Sasaki, M and Sasaki, T An Approach to N = 4 ADE gauge Theory on K3. (2002);
Preprint Series # 560
Kato, J On the uniqueness of nondecaying solutions for the Navier-Stokes equations. (2002);
Preprint Series # 559
Izumiya, S and Pei, D and Takahashi, M Curves and surfaces in Hyperbolic space. (2002);
Preprint Series # 558
Izumiya, S and Pei, S and Fuster, M. C. Romero Umbilicity of spacelike submanifolds of Minkowski space. (2002);
Preprint Series # 557
Izumiya, S. and Pei, D. and Fuster, M. C. Romero The lightcone Gauss map of a spacelike surface in Minkowski 4-space. (2002);
Preprint Series # 556
Izumiya, S and Saji, K and Takeuchi, N Singularities of line congruences. (2002);
Preprint Series # 555
Izumiya, S and Takeuchi, N New special curves and developable surfaces. (2002);
Preprint Series # 554
Tonegawa, Y Domain dependent monotonicity formula for a singular perturbation problem. (2002);
Preprint Series # 553
Giga, Y and Rybka, P Berg's effect. (2002);
Preprint Series # 552
Giga, M.-H and Giga, Y Minimal vertical singular diffusion preventing overturning for the Burgers equation. (2002);
Preprint Series # 551
Ohtsuka, T A level set method for spiral crystal growth. (2002);
Preprint Series # 550
Nakazi, T and Watanabe, T Properties of a Rubin's orthogonal function which is a linear combination of two inner functions. (2002);
Preprint Series # 549
Nakazi, T Absolute values and real parts for functions in the Smirnov class. (2002);
Preprint Series # 548
Tachizawa, K A generalization of the Lieb-Thirring inequalities in low dimensions. (2002);
Preprint Series # 547
Nakano, Y Minimizing coherent risk measures of shortfall in discrete-time models with cone constraints. (2002);
Preprint Series # 546
Elliott, C. M and Giga, Y and Goto, S Dynamic boundary conditions for Hamilton-Jacobi equations. (2001);
Preprint Series # 545
Sawada, O On time-local solvability of the Navier-Stokes equations in Besov spaces. (2001);
Preprint Series # 544
Arai, A Non-relativistic limit of a Dirac-Maxwell operator in relativistic quantum electrodynamics. (2001);
Preprint Series # 543
Sugano, K On H-separable extensions of QF-3 rings. (2001);
Preprint Series # 542
Nakazi, T The Nevanlinna counting functions for Rudin's orthogonal functions. (2001);
Preprint Series # 541
Jinzenji, M and Sasaki, T N = 4 supersymmetric Yang-Mills theory on orbifold-T4/Z2 higher rank case, 17 pages. (2001);
Preprint Series # 540
Ishii, H and Mikami, T Motion of a graph by R-curvature. (2001);
Preprint Series # 539
Jinzenji, M Gauss-Manin system and the virtual structure constants. (2001);
Preprint Series # 538
Giga, Y On the two-dimensional nonstationary vorticity equations. (2001);
Preprint Series # 537
Nakazi, T and Yamamoto, T Norm of a linear combination of two operators of a Hilbert space. (2001);
Preprint Series # 536
Inoue, A What does the partial autocorrelation function look like for large lags. (2001);
Preprint Series # 535
Yamagami, A On Gouva's conjecture in the unobstructed case. (2001);
Preprint Series # 534
Tsai, Y.-H. R and Giga, Y and Osher, S A level set approach for computing discontinuous solutions of a class of Hamilton-Jacobi equations. (2001);
Preprint Series # 533
Escher, Y. Giga and K. Ito, J and Giga, Y and Ito, K On a limiting motion and self-intersections for the intermediate surface diffusion flow. (2001);
Preprint Series # 532
Suwa, T Characteristic classes of singular varieties. (2001);
Preprint Series # 531
Tonegawa, Y Remarks on convergence of the Allen-Cahn equation. (2001);
Preprint Series # 530
Giga, Y and Rybka, P Quasi-static evolution of 3-D crystals grown from supersaturated vapor. (2001);
Preprint Series # 529
Giga, Y and Ishimura, N and Kohsaka, Y Spiral solutions for a weakly anisotropic curvature flow equation. (2001);
Preprint Series # 528
Nakazi, T and Yamamoto, t Two dimensional commutative Banach algebras and von Neumann inequality. (2001);
Preprint Series # 527
Izawa, T and Suwa, T Multiplicity of functions on singular varieties. (2001);
Preprint Series # 526
Anh, V and Inoue, A Dynamic models of asset prices with long memory. (2001);
Preprint Series # 526
Anh, V and Inoue, A Dynamic models of asset prices with long memory. (2001);
Preprint Series # 525
Suwa, T Residues of Chern classes. (2001);
Preprint Series # 524
Yamaguchi, K and Yatsui, T Geometry of higher order differential equations of finite type associated with symmetric spaces. (2001);
Preprint Series # 523
Mikami, T Covariance kernel and the central limit theorem in the total variation distance. (2001);
Preprint Series # 522
Nakazi, T Toeplitz operators and weighted norm inequalities on the bidisc. (2001);
Preprint Series # 521
Nakano, Yumiharu Efficient hedging with coherent risk measure. (2001);
Preprint Series # 520
Inoue, A On the worst conditional expectation. (2001);
Preprint Series # 519
Giga, Y Viscosity solutions with shocks. (2001);
Preprint Series # 518
Jinzenji, M and Sasaki, T N=4 supersymmetric Yang-Mills theory on orbifold-T4/Z2. (2001);
Preprint Series # 517
Yoneda, R The composition operators on weighted bloch space. (2001);
Preprint Series # 516
Izumiya, S and Pei, D-H and Sano, T Horospherical surfaces of curves in hyperbolic space. (2001);
Preprint Series # 515
Izumiya, S and Takeuchi, N Generic special curves. (2001);
Preprint Series # 514
Izumiya, S and Pei, D-H and Sano, T Singularities of hyperbolic Gauss maps. (2001);
Preprint Series # 513
Izumiya, S and Maruyama, K Transversal topology and singularities of Haefliger foliations. (2001);
Preprint Series # 512
Izumiya, S Generating families of developable surfaces in R3. (2001);
Preprint Series # 511
Izumiya, S and Takeuchi, N Special curves and raled surfaces. (2001);
Preprint Series # 510
Giga, Y Shocks and very strong vertical diffusion. (2000);
Preprint Series # 509
Tonegawa, Y Phase field model with a variable chemical potential. (2000);
Preprint Series # 508
Arai, A Supersymmetric methods for constructing soliton-type solutions to multi-component nonlinear Schrdinger and Klein-Gordon equations. (2000);
Preprint Series # 507
Arai, A Instability in the spectral and the Fredholm properties of an infinite dimensional Dirac operator on the abstract Boson-Fermion Fock space. (2000);
Preprint Series # 506
Kobayashi, R and Giga, Y On anisotropy and curvature effects for growing crystals. (2000);
Preprint Series # 505
Nakazi, T Interpolation problem for l1 and a uniform algebra. (2000);
Preprint Series # 504
Inoue, A and Kasahara, Y Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing. (2000);
Preprint Series # 503
Giga, Y and Matsui, S and Sawada, O Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity. (2000);
Preprint Series # 502
Arai, A Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation. (2000);
Preprint Series # 501
Tsujii, M Fat solenoidal attractors. (2000);
Preprint Series # 500
Tsuda, I and Kuroda, S Cantor coding in the hippocampus. (2000);
Preprint Series # 499
Escher, J and Giga, Y and Ito, K On a limiting motion and self-interactions of curves moved by the intermediate surface diffusion flow. (2000);
Preprint Series # 498
Giga, Y and Paolini, M and Rybka, P On the motion by singular interfacial energy. (2000);
Preprint Series # 497
Giga, Y and Inui, K and Kato, J and Matsui, S Remarks on the uniqueness of bounded solutions of the NavierStokes equations. (2000);
Preprint Series # 496
Nakazi, T Backward shift invariant subspaces in the bidisc. (2000);
Preprint Series # 495
Arisawa, M and Giga, Y Anisotropic curvature flow in a very thin domain. (2000);
Preprint Series # 494
Mikami, T Optimal control for absolutely continuous stochastic processes and the mass transportation problem. (2000);
Preprint Series # 493
Tsuda, I Towards an interpretation of dynamic neural activity in terms of chaotic dynamical systems. (2000);
Preprint Series # 492
Tsuda, I and Hatakeyama, M Making sense of internal logic Theory and a case study. (2000);
Preprint Series # 491
Yamagami, A On Gouvs conjecture on controlling the conductor. (2000);
Preprint Series # 490
Nakazi, T and Yamamoto, T The real part of an outer function and a Helson-Szeg weight. (2000);
Preprint Series # 489
Ohtani, S Construction of unramified Galois extensions over maximal abelian extensions of algebraic number fields. (2000);
Preprint Series # 488
Inoue, A Asymptotic behaviour for partial autocorrelation functions of fractional ARIMA processes. (2000);
Preprint Series # 487
Shibukawa, Y Classification of the R-operator. (2000);
Preprint Series # 486
Nakazi, T Functions in N+ with the positive real parts on the boundary. (2000);
Preprint Series # 485
Arai, A and Kawano, H A class of deformations of the Schrdinger representation of the Heisenberg commutation relation and exact solution to a Heisenberg equation and a Schrdinger equation. (2000);
Preprint Series # 484
Bogaevski, I. A and Ishikawa, G Lagrange mappings of the first open Whitney umbrella. (2000);
Preprint Series # 483
Bingham, N. H and Inoue, A Abelian, Tauberian and Mercerian theorems for arithmetic sums. (2000);
Preprint Series # 482
Bingham, N. H. and Inoue, A Tauberian and Mercerian theorems for systems of kernels. (2000);
Preprint Series # 481
Nakazi, T Two dimensional Q-algebras. (2000);
Preprint Series # 480
Arai, A and Hirokawa, M Stability of ground states in sectors and its application to the Wigner-Weisskopf model. (2000);
Preprint Series # 479
Giga, M.-H and Giga, Y Crystalline and level set flow - Convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane. (2000);
Preprint Series # 478
Giga, M.-H and Giga, Y Generalized motion by nonlocal curvature in the plane. (2000);
Preprint Series # 477
Nakazi, T On an invariant subspace whose common zero set is the zeros of some function. (1999);
Preprint Series # 476
Sano, T On affine parallels of generic plane curves. (1999);
Preprint Series # 475
Inoue, A and Kasahara, Y Asymptotics for prediction errors of stationary processes with reflection positivity. (1999);
Preprint Series # 474
Nakazi, T Norm inequalities for some singular integral operators. (1999);
Preprint Series # 473
Yoneda, R Characterization of Bloch space and Besov spaces by oscillations. (1999);
Preprint Series # 472
Giga, Y and Sato, M.-H A level set approach to semicontinuous viscosity solutions for Cauchy problems. (1999);
Preprint Series # 471
Ito, K and Kohsaka, Y Stability of a stationary solution for evolving boundaries of symmetric three-phases driven by surface diffusion. (1999);
Preprint Series # 470
Nakamura, N and Ozawa, T Small solutions to nonlinear wave equations in the Sobolev spaces. (1999);
Preprint Series # 469
Nakamura, M and Ozawa, T Small solutions to nonlinear Schrdinger equations in the Sobolev spaces. (1999);
Preprint Series # 468
Higuchi, A and Matsue, K and Tsujishita, T Deductive hyperdigraphs - a method of describing diversity of coherences. (1999);
Preprint Series # 467
Kubota, K and Yokoyama, Y Global existence of classical solutions to systems of nonlinear wave equations with different speed of propagation. (1999);
Preprint Series # 466
Izumiya, S and Kossioris, G and Makrakis, G Multivalued solutions to the eikonal equation in stratified media. (1999);
Preprint Series # 465
Kawazumi, N and Shibukawa, Y The meromorphic solutions of the Bruschi-Calogero equation. (1999);
Preprint Series # 464
Mikami, T Dynamical systems in the variational formulation of the Fokker-Plank equation by the Wasserstein metric. (1999);
Preprint Series # 463
Agemi, R Global existence of nonlinear elastic waves. (1999);
Preprint Series # 462
Nakazi, T Essential norms of some singular integral operators. (1999);
Preprint Series # 461
Giga, M.-H and Giga, Y and Kobayashi, R Very singular diffusion equations. (1999);
Preprint Series # 460
Furuhata, H and Kurose, T Self-dual centroaffine surfaces of codimension two with constant affine mean curvature. (1999);
Preprint Series # 459
Giga, Y and Sato, M.-H On semicontinuous solutions for general Hamilton-Jacobi equations. (1999);
Preprint Series # 458
Izumiya, S and Takeuchi, N Singularities of ruled surfaces in R3. (1999);
Preprint Series # 457
Arai, A and Hirokawa, H Grounds states of a general class of quantum field Hamiltonians. (1999);
Preprint Series # 456
Pei, D and Sano, T and Torii, E Generic differential geometry of hyperbolic plane curves. (1999);
Preprint Series # 455
Oda, F and Yoshida, T Crossed burnsiope rings [I] The fundamental theorem. (1999);
Preprint Series # 454
Kubo, H and Kubota, K Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions. (1999);
Preprint Series # 453
Adachi, J Generic knots in contact 3$B!>(Bmanifolds. (1999);
Preprint Series # 452
Inoue, A Asymptotics for the partial autocorrelation function of a stationary process. (1999);
Preprint Series # 451
Nakazi, T Riesz's functions and Carleson inequalities. (1999);
Preprint Series # 450
Arai, A Infinite dimensional analysis and analytic number theory. (1999);
Preprint Series # 449
Ishikawa, G Developable hypersurfaces and homogeneous spaces in a real projective space. (1999);
Preprint Series # 448
Ishikawa, G Singularities of Developable Surfaces. (1999);
Preprint Series # 447
Arai, A Fundamental properties of the Hamiltonian of a Dirac particle coupled to the quantized radiation field. (1999);
Preprint Series # 446
Nakazi, T and Yoneda, R Compact Toeplitz operators with continuous symbols on weighted Bergman spaces. (1999);
Preprint Series # 445
Arai, A Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians. (1999);
Preprint Series # 444
Ishii, H and Mikami, T A two dimensional random crystalline algorithm for Gauss curvature flow. (1999);
Preprint Series # 443
Nakazi, T Some special bounded homomorphisms of a uniform algebra. (1999);
Preprint Series # 442
Tsujii, M Absolutely continuous invariant measures for expanding piecewise linear maps. (1999);
Preprint Series # 441
Kirillov, A. N and Schilling, A and Shimozono, M A bijection between Littlewood-Richardson tableaux and rigged configurations. (1999);
Preprint Series # 440
Arai, A and Hirokawa, M and Hiroshima, F On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff. (1998);
Preprint Series # 439
Inoue, A and Kasahara, Y On the asymptotic behavior of the prediction error of a stationary process. (1998);
Preprint Series # 438
Yoneda, R Compact Toeplitz operators on Bergman spaces. (1998);
Preprint Series # 437
Nakazi, T and Osawa, T Finite rank intermediate Hankel operators on the Bergman space. (1998);
Preprint Series # 436
Inoue, A and Kikuchi, H Abel-Tauber theorems for Hankel and Fourier transforms and a problem of Boas. (1998);
Preprint Series # 435
Uemura, T Morita-Mumford classes on finite cyclic subgroups of the mapping class group of closed surfaces. (1998);
Preprint Series # 434
Shimizu, Y L$B!g!>(Bestimate of first-order space derivatives of Stokes flow in a half space. (1998);
Preprint Series # 433
Ito, K Loss of convexity of compact hypersurfaces moved by surface diffusion. (1998);
Preprint Series # 432
Yoshida, T Categorical aspects of generating functions(II) Operations on categories and functors. (1998);
Preprint Series # 431
Ishikawa, G Determinacy, transversality and Lagrange stability. (1998);
Preprint Series # 430
Diejen, J. F. Van and Kirillov, A. N Formulas for q-spherical functions using inverse scattering theory of reflectionless Jacobi operators. (1998);
Preprint Series # 429
Hiroshima, F Uniqueness of the ground state of a model in quantum electrodynamics A functional integral approach. (1998);
Preprint Series # 428
Hiroshima, F Ground states of a model in quantum electrodynamics. (1998);
Preprint Series # 427
Sano, T Bifurcations of affine invariants for one parameter family of generic convex plane curves. (1998);
Preprint Series # 426
Arai, A On the essential spectra of quantum field Hamiltonians. (1998);
Preprint Series # 425
Nakamura, M Strichartz estimates for wave equations in the homogeneous Besov space. (1998);
Preprint Series # 424
Kirillov, A. N and Shimozono, M A generalization of the Kostka-Foulkes polynomials. (1998);
Preprint Series # 423
Tsukada, T Reticular Legendrian Singularities. (1998);
Preprint Series # 422
Kobayashi, R and Warren, J. A and Carter, W. C Modeling grain boundaries using a phase field technique. (1998);
Preprint Series # 421
Pei, D and Sano, T The focal developable and the binormal indicatrix of a nonlightlike curve in Minkowski 3-space. (1998);
Preprint Series # 420
Kobayashi, R and Giga, Y Equations with singular diffusivity. (1998);
Preprint Series # 419
Izumiya, S and Katsumi, H and Yamasaki, T The rectifying developable and the spherical Darboux image of a space curve. (1998);
Preprint Series # 418
Ishikawa, M and Matsui, S Existence of a forward self-similar stagnation flow of the Navier-Stokes equations. (1998);
Preprint Series # 417
Yamaguchi, K G2-Geometry of overdetermined systems of second order. (1998);
Preprint Series # 416
Yoshida, T Categorical aspects of generating functions (I) Exponential formulas and Krull-Schmidt categories. (1998);
Preprint Series # 415
Tsujii, M Piecewise expanding maps on the plane with singular ergodic properties. (1998);
Preprint Series # 414
Suwa, T Dual class of a subvariety. (1998);
Preprint Series # 413
Brasselet, J.-P and Lehmann, D and Seade, J and Suwa, T Milnor classes of local complete intersections. (1998);
Preprint Series # 412
Tsujii, M Absolutely continuous invarient measures for piecewise real-analytic expanding maps on the plane. (1998);
Preprint Series # 411
Izumiya, S and Pei, D and Sano, T The lightcone gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space. (1998);
Preprint Series # 410
Giga, Y and Inui, K and Matsui, S On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data. (1998);
Preprint Series # 409
Giga, Y and Ito, K Loss of convexity of simple closed curves moved by surface diffusion. (1998);
Preprint Series # 408
Nakazi, T and Okubo, K Generalized Numerical Radius And Unitary p-Dilation. (1998);
Preprint Series # 407
Kawazumi, N and Uemura, T Riemann-Hurwitz formula for Morita-Mumford classes and surface symmetries. (1998);
Preprint Series # 406
Hiroshima, F Ground states and spectrum of quantum electrodynamics of non-relativistic particles. (1998);
Preprint Series # 405
Yokoyama, K Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions. (1998);
Preprint Series # 404
Kohsaka, Y Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary. (1998);
Preprint Series # 403
Nakazi, T and Okubo, K p-contraction and 2$B!_(B2 matrix. (1998);
Preprint Series # 402
Lehmann, D and Suwa, T Generalization of variations and Baum-Bott residues for holomorphic foliations on singular varieties. (1998);
Preprint Series # 401
Nakamura, M and Ozawa, T Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth. (1998);
Preprint Series # 400
Tsujii, M A simple proof for monotonicity of entropy in the quadratic family. (1997);
Preprint Series # 399
Mikami, T Markov marginal problems and their applications to Markov optimal control. (1997);
Preprint Series # 398
Pei, D Singularities of P2-valued Gauss maps of surfaces in Minkowski. (1997);
Preprint Series # 397
Hirata, K and Sugano, K On semisimple extensions of serial rings. (1997);
Preprint Series # 396
Nishiura, Y and Ueyama, D A skeleton structure of self-replicating dynamics. (1997);
Preprint Series # 395
Bingham, N. H and Inoue, A Ratio Mercerian theorems with applications to Hankel and Fourier transforms. (1997);
Preprint Series # 394
Bingham, N. H and Inoue, A Extension of the Drasin-Shea-Jordan theorem. (1997);
Preprint Series # 393
Arase, T A remark on the quantale structure of multisets. (1997);
Preprint Series # 392
Tsujishita, T and Watanabe, H Monoidal closedness of the category of simulations. (1997);
Preprint Series # 391
Giga, Y and Ohnuma, M and Sato, M.-H On strong maximum principle and large time behaviour of generalized mean curvature flow with the Neumann boundary condition. (1997);
Preprint Series # 390
Nakamura, M and Ozawa, T The Cauchy problem for nonlinear wave equations in the homogeneous Sobolev space. (1997);
Preprint Series # 389
Tsukada, T Reticular Lagrangian singularities. (1997);
Preprint Series # 388
Giga, M.-H and Giga, Y Remarks on convergence of evolving graphs by nonlocal curvature. (1997);
Preprint Series # 387
Nakazi, T and Yamamoto, T Norms of some singular integral operators and their inverse operators. (1997);
Preprint Series # 386
Aviles, P and Giga, Y On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg-Landau type energy for gradient fields. (1997);
Preprint Series # 385
Arai, A Representation-theoretic aspects of two-dimensional quantum systems in singular vector potentials canonical commutation relations, quantum algebras, and reduction to lattice quantum systems.
Preprint Series # 384
Ito, K Asymptotic stability of planar rarefaction wave for scalar viscous conservation law. (1997);
Preprint Series # 383
Nakamura, M and Ozawa, T The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. (1997);
Preprint Series # 382
Arai, A A new estimate for the ground state energy of Schrdinger operators. (1997);
Preprint Series # 381
Giga, Y and Matsui, S and Shimizu, Y On estimates in Hardy spaces for the Stokes flow in a half space. (1997);
Preprint Series # 380
Hiroshima, F Weak coupling limit removing an ultraviolet cut-off for a Hamiltonian of particles interacting with a scalar field. (1997);
Preprint Series # 379
Giga, Y and Ito, K On pinching of curves moved by surface diffusion. (1997);
Preprint Series # 378
Inoue, J and Nakazi, T On the zeroes of solutions of an extremal problem in H1. (1997);
Preprint Series # 377
Ishikawa, G A relative transversality theorem and its applications. (1997);
Preprint Series # 376
Ishikawa, G and Morimoto, T Solution surfaces of Monge-Ampre equations. (1997);
Preprint Series # 375
Solomon, L and Terao, H The double coxeter arrangement. (1997);
Preprint Series # 374
Inoue, J and Takahashi, S.-E On characterizations of the image of Gelfand transform of commutative Banach algebras. (1997);
Preprint Series # 373
Nakazi, T Brown-Halmos type theorems of weighted Toeplitz operators. (1997);
Preprint Series # 372
Giga, M.-H and Giga, Y Stability for evolving graphs by nonlocal weighted curvature. (1997);
Preprint Series # 371
Nakazi, T and Sawada, H The commutator ideal in Toeplitz algebras for uniform algebras and the analytic structure. (1997);
Preprint Series # 370
Bingham, N. H and Inoue, A An Abel-Tauber theorem for Hankel transforms. (1997);
Preprint Series # 369
Nakamura, M and Ozawa, T Nonlinear Schrdinger equations in the Sobolev space of critical order. (1997);
Preprint Series # 368
Ito, K BV-solutions of a hyperbolic$B!>(Belliptic system for a radiating gas. (1997);
Preprint Series # 367
Arai, A A class of representations of the *$B!>(Balgebra of the canonical commutation relations over a Hilbert space and instability of embedded eigenvalues in quantum field models. (1996);
Preprint Series # 366
Arai, A and Hirokawa, M On the existence and uniqueness of ground states of a generalized spin$B!>(Bboson model. (1996);
Preprint Series # 365
Tsukada, T Stability of reticular optical caustics. (1996);
Preprint Series # 364
Izumiya, S and Sano, T Generic affine differential geometry of space curves. (1996);
Preprint Series # 363
Mikami, T Equivalent conditions on the centrallimit theorem for a sequence of probability measures on R. (1996);
Preprint Series # 362
Nakamura, I Hilbert schemes and simple singularities E6, E7 and E8. (1996);
Preprint Series # 361
Nakamura, M and Ozawa, T Low energy scattering for nonlinear Schrdinger equations in fractional order Sobolev spaces. (1996);
Preprint Series # 360
Tsuda, I and Yamaguchi, A Singular$B!>(Bcontinuous nowhere$B!>(Bdifferentiable attractors in neural systems. (1996);
Preprint Series # 359
Tsuda, I and Tadaki, K A logic$B!>(Bbased dynamical theory for a genesis of biological threshold. (1996);
Preprint Series # 358
Hayashi, N and Naumkin, P.I and Ozawa, T Scattering theory for the Hartree equation. (1996);
Preprint Series # 357
Izumiya, S Singularities of solutions for first order partial differential equations. (1996);
Preprint Series # 356
Bingham, N. H and Inoue, A The Drasin$B!>(BShea$B!>(BJordan theorem for Hankel transforms of arbitrarily large order. (1996);
Preprint Series # 355
Ishikawa, G Topology of plane trigonometric curves and the strangeness of plane curves derived from real pseudo$B!>(Bline arrangements. (1996);
Preprint Series # 354
Gyoja, A and Yamashita, H Associated variety, Kostant$B!>(BSekiguchi correspondence, and locally free U(n)$B!>(Baction on Harish$B!>(BChandra modules. (1996);
Preprint Series # 353
Chen, Y.$B!>(BG and Giga, Y and Sato, K On instant extinction for very fast diffusion equations. (1996);
Preprint Series # 352
Nishiura, Y and Suzuki, H Nonexistence of stable turing patterns with smooth limiting interfacial configurations in higher dimensional spaces. (1996);
Preprint Series # 351
Nakazi, T and Watatani,, Y Invariant subspace theorems for subdiagonal algebras. (1996);
Preprint Series # 350
Hiroshima, F Weak coupling limit with a removal of an ultraviolet cut$B!>(Boff for a Hamiltonian of particles interacting with a massive scalar field. (1996);
Preprint Series # 349
Agemi, R and Yokoyama, K The null condition and global existence of solutions to systems of wave equations with different speeds. (1996);
Preprint Series # 348
Ito, Y and Nakamura, I Hilbert schemes and simple singularities An and Dn. (1996);
Preprint Series # 347
Nakazi, T and Yamamoto, T Weighted Norm Inequalities For Some Singular Integral Operators. (1996);
Preprint Series # 346
Kubo, H and Kubota, K Asymptotic behaviors of radially symmetric solutions of $B""(Bu = |u|P for super critical values p in even space dimensions. (1996);
Preprint Series # 345
Douai, A and Terao, H The determinant of a hypergeometric period matrix. (1996);
Preprint Series # 344
Giga, M.$B!>(BH and Giga, Y A subdifferential interpretation of crystalline motion under nonuniform driving force. (1996);
Preprint Series # 343
Kawazumi, N and S. Morita, S The primary approximation to the cohomology of the moduli space of curves and cocycles for the stable characteristic classes. (1996);
Preprint Series # 342
Shimizu, Y A remark on estimates of bilinear forms of gradients in Hardy space. (1996);
Preprint Series # 341
Ishikawa, G Topological classification of the tangent developables of space curves. (1996);
Preprint Series # 340
Inoue, A Abel$B!>(BTauber theorems for Fourier$B!>(BStieltjes coefficients. (1996);
Preprint Series # 339
Tominaga, N Analysis of a family of strongly commuting self$B!>(Badjoint operators with applications to perturbed Dirac operators. (1996);
Preprint Series # 338
Hiroshima, F A scaling limit of a Hamiltonian of many nonrelativistic particles interacting with a quantized radiation field. (1996);
Preprint Series # 337
Tsuda, I A new type of self$B!>(Borganization associated with chaotic dynamics in neural networks. (1996);
Preprint Series # 336
Giga, Y and Gurtin, M.E and Matias, J On the dynamics of crystalline motions. (1996);
Preprint Series # 335
Seade, J and Suwa, T Residues and topological invariants of singular holomorphic foliations1. (1996);
Preprint Series # 334
Khanedani, B and Suwa, T First variation of horomorphic forms and some applications. (1996);
Preprint Series # 333
Nakazi, T The spectra of Toeplitz operators with unimodular symbols. (1996);
Preprint Series # 332
Ohnuma, M and Sato, K Singular degenerate parabolic equations with applications to the p$B!>(Blaplace diffusion equation. (1996);
Preprint Series # 331
Giga, M.$B!>(BH and Giga, Y Evolving graphs by singular weighted curvature. (1996);
Preprint Series # 330
Kurokawa, Y Singularities for projections of contour lines of surfaces onto planes. (1996);
Preprint Series # 329
Ogurisu, O Anticommutativity and spin 1/2 Schrdinger operators with magnetic fields. (1996);
Preprint Series # 328
Ozawa, T and Tsutsumi, Y Space$B!>(Btime estimates for null gauge forms and nonlinear Schrdinger equations. (1996);
Preprint Series # 327
Tsutaya, K Local regularity of non$B!>(Bresonant nonlinear wave equations. (1996);
Preprint Series # 326
Lehmann, D and Suwa, T Residues of holomorphic vector fields on singular varieties. (1995);
Preprint Series # 325
Sugano, K On strongly separable Frobenius extensions. (1995);
Preprint Series # 324
Arai, A Factorization of self$B!>(Badjoint operators by abstract Dirac operators and its application to second quantizations on Boson Fermion Fock spaces. (1995);
Preprint Series # 323
Kawazmi, N An infinitesimal approach to the stable cohomology of the moduli of Riemann surfaces. (1995);
Preprint Series # 322
Nakazi, T An outer function and several important functions in two variables. (1995);
Preprint Series # 321
Ohnuma, M Axisymmetric solutions and singular parabolic equations in the theory of viscosity solutions. (1995);
Preprint Series # 320
Teruya, T Normal intermediate subfactors. (1995);
Preprint Series # 319
Izumiya, S and Kossioris, G. T Bifurcations of shock waves for viscosity solutions of Hamilton$B!>(BJacobi equations of one space variable. (1995);
Preprint Series # 318
Sun, Wei$B!>(BZhi Shadows of moving surfaces. (1995);
Preprint Series # 317
Giga, M.-H and Giga, Y Consistency in evolutions by crystalline curvature. (1995);
Preprint Series # 316
Shibukawa, Y Vertex$B!>(Bface correspondence in elliptic solutions of the Yang$B!>(BBaxter equation. (1995);
Preprint Series # 315
Arai, A Canonical commutation relations, the Weierstrass Zetafunction, and infinite dimensional Hilbert space representations of the quantum group Uq (sl2). (1995);
Preprint Series # 314
Mikami, T Weak convergence on the first exit time of randomly perturbed dynamical systems with a repulsive equilibrium point. (1995);
Preprint Series # 313
Nakazi, T Slice maps and multipliers of invariant subspaces. (1995);
Preprint Series # 312
Ito, H. M and Mikami, T Poissonian asymptotics of a randomly perturbed dynamical system Flip$B!>(Bflop of the Stochastic Disk Dynamo. (1995);
Preprint Series # 311
Kawazumi, N On the stable cohomology algebra of extended mapping class groups for surfaces. (1995);
Preprint Series # 310
Izumiya, S and Sano, T Generic affine differential geometry of plane curves. (1995);
Preprint Series # 309
Aarai, A and Hirokawa, M On the existence and uniqueness of ground states of the spin$B!>(Bboson Hamiltonian. (1995);
Preprint Series # 308
Arai, A Representation of canonical commutation relations in a gauge theory, the Aharonov$B!>(BBohm effect, and Dirac Weyl operator. (1995);
Preprint Series # 307
Yoshizaki, J On the structure of the singular set of a complex analytic foliation. (1995);
Preprint Series # 306
Honda, T and Suwa, T Residue formulas for singular foliations defined by meromorphic functions on surfaces. (1995);
Preprint Series # 305
Bingham, N.H and Inoue, A Jordan's theorem for fourier and hankel transforms. (1995);
Preprint Series # 304
Ozawa, T On the nonlinear Schrdinger equations of derivative type. (1995);
Preprint Series # 303
Jimbo, S and Zhai, J Ginzburg‐Landau equation with magnetic effect non‐simply‐connected domains. (1995);
Preprint Series # 302
Giga, M.‐H and Giga, Y Geometric evolution by nonsmooth interfacial energy. (1995);
Preprint Series # 301
Zhai, J Harmonic maps and Ginzburg‐Landau type elliptic syste m. (1995);
Preprint Series # 300
Lehmann, D. and Soares, M. and Suwa, T. On the index of a holomorphic vector field tangent to a singular variety. (1995);
Preprint Series # 299
Izumiya, S. and Sun, W-Z. Singularities of solution surfaces for quasilinear 1st order partial differential equations. (1995);
Preprint Series # 298
Higuchi, A. Lattices of closure operators. (1995);
Preprint Series # 297
F., Hiroshima Diamagnetic Inequalities for Systems of Nonrelativistic Particles with a Quantized Field. (1995);
Preprint Series # 296
A., Kishimoto A Rohlin property for one-parameter automorphism groups. (1995);
Preprint Series # 295
S., Izumiya Local classifications of multi-valued solutions of quasilinear first order partial differential equations. (1995);
Preprint Series # 294
S., Izumiya and A., Takiyama A time-like surface in Minkowski 3-space which contains pseudocodes. (1995);
Preprint Series # 293
P., Aviles and Y., Giga The distance function and defect energy. (1995);
Preprint Series # 292
N., Kawazumi A Generalization of the Morita-Mumford Classes to Extended Mapping Class Groups for Surfaces. (1995);
Preprint Series # 291
F., Hiroshima Functional Integral Representation of a Model in QED. (1995);
Preprint Series # 290
Y., Giga Interior derivative blow-up for quasilinear parabolic equations. (1995);
Preprint Series # 289
K., Goto and A., Yamaguchi and I., Tsuda Nine-bit states cellular automata are capable of simulating the pattern dynamics of coupled map lattice. (1995);
Preprint Series # 288
A., Kishimoto The Rohlin property for automorphisms of UHF algebras. (1995);
Preprint Series # 287
T., Nakazi Factorizations of outer functions and extremal problems. (1995);
Preprint Series # 286
A., Arai Strong coupling limit of the zero-energy-state density of the Dirac-Weyl operator with a singular vector potential. (1995);
Preprint Series # 285
K., Hidano and K., Tsutaya Scattering theory for nonlinear wave equations in the invariant Sobolev space. (1995);
Preprint Series # 284
T., Nakazi and M., Yamada Riesz's Functions In Weighted Hardy And Bergman Spaces. (1995);
Preprint Series # 283
K., Iwata and J., Schfer Markov property and cokernels of local operators. (1995);
Preprint Series # 282
T., Mikami Asymptotic behavior of the first exit time of randomly perturbed dynamical systems with a repulsive equilibrium point. (1995);
Preprint Series # 281
A., Arai and N., Tominaga Analysis of a family of strongly commuting self-adjoint operators with applications to perturbed d'Alembertians and the external field problem in quantum field theory.
Preprint Series # 280
N., Terai and T., Hibi Monomial ideals and minimal non-faces of Cohen-Macaulay complexes. (1995);
Preprint Series # 279
N., Terai and T., Hibi Finite free resolutions and 1-skeletons of simplicial(d - l)-spheres. (1995);
Preprint Series # 278
N., Terai and T., Hibi Computation of Betti numbers of monomial ideals associated with stacked polytopes. (1995);
Preprint Series # 277
N., Terai and T., Hibi Computation of Betti numbers of monomial ideals associated with cyclic polytopes. (1995);
Preprint Series # 276
N., Terai and T., Hibi Stanley-Reisner rings whose Betti numbers are independent of the base field. (1995);
Preprint Series # 275
N., Terai and T., Hibi Alexander duality theorem and second Betti numbers of Stanley-Reisner rings. (1995);
Preprint Series # 274
H., Kubo On the critical decay and power for semilinear wave equations in odd space dimensions. (1994);
Preprint Series # 273
K., Tsutaya Global existence of small amplitude solutions for the Klein-Gordon-Zakharov equations. (1994);
Preprint Series # 272
S., Izumiya and A., Takiyama A time-like surface in Minkowski 3-space which contains light-like lines. (1994);
Preprint Series # 271
S., Jimbo and Y., Morita and J., Zhai Ginzburg landau equation and stable steady state solutions in a non-trivial domain. (1994);
Preprint Series # 270
A., Arai Gauge theory on a non-simply-connected domain and representations of canonical commutation relations. (1994);
Preprint Series # 269
A., Arai Operator-theoretical analysis of representation of a supersymmetry algebra in Hilbert space. (1994);
Preprint Series # 268
Y., Giga Evolving curves with boundary conditions. (1994);
Preprint Series # 267
T., Mikami A sufficient condition for the uniqueness of solutions to a class of integro-differential equations. (1994);
Preprint Series # 266
T., Ozawa On the resonance equations of long and short waves. (1994);
Preprint Series # 265
N., Hayashi and T., Ozawa Schrdinger Equations with nonlinearity of integral type. (1994);
Preprint Series # 264
J., Wierzbicki On Commutativity of Diagrams of Type II1 Factors. (1994);
Preprint Series # 263
Y., Giga and M.E., Gurtin A comparison theorem for crystalline evolution in the plane. (1994);
Preprint Series # 262
N., Kawazumi Homology of hyperelliptic mapping class groups for surfaces. (1994);
Preprint Series # 261
A., Inoue An Abel-Tauber theorem for Fourier sine transforms. (1994);
Preprint Series # 260
T., Mikami Copula fields and its applications. (1994);
Preprint Series # 259
K., Iwata and J., Schfer Markov property and cokernels of local operators. (1994);
Preprint Series # 258
Y-G., Chen and Y., Giga and T., Hitaka and M., Honma A Stable Difference Scheme for Computing Motion of Level Surfaces by the Mean Curvature. (1994);
Preprint Series # 257
A., Arai Trace Formulas, a Golden-Thompson Inequality and Classicallimit in Boson Fock Space. (1994);
Preprint Series # 256
T., Tsujishita Construction of Universal Modal World based on Hyperset Theory. (1994);
Preprint Series # 255
T., Tsujishita On Triple Mutual Information. (1994);
Preprint Series # 254
T., Suwa Residues of complex analytic foliations relative to singular invariant subvarieties. (1994);
Preprint Series # 253
T., Nishimori Some remarks in a qualitative theory of similarity pseudogroups. (1994);
Preprint Series # 252
T., Mikami Large Deviations and Centrallimit Theorems for Eyraud-Farlie-Gumbel-Morgenstern Processes. (1994);
Preprint Series # 251
H., Kubo and K., Kubota Asymptotic behaviors of radially symmetric solutions of \Box u = |u|P for super critical values p in odd space dimensions. (1994);
Preprint Series # 250
S., Izumiya and G., T. Singularities for viscosity solutions of Hamilton-Jacobi equations. (1994);
Preprint Series # 249
A., Inoue Tauberian theorems for Fourier cosine transforms. (1994);
Preprint Series # 248
H., Kikuchi Sheaf cohomology theory for measurable spaces. (1994);
Preprint Series # 247
N., Hayashi and K., Kato and T., Ozawa Dilation Method and smoothing Effect of Solutions to the Benjamin-ono Equation. (1994);
Preprint Series # 246
T., Nakazi and M., Yamada (A_2)-Conditions and Carleson Inequalities. (1994);
Preprint Series # 245
H., Kubo Slowly decaying solutions for semilinear wave equations in odd space dimensions. (1994);
Preprint Series # 244
D., Lehmann and T., Suwa Residues of holomorphic vector fields relative to singular invariant subvarieties. (1994);
Preprint Series # 243
N., Hayashi and K., Kato and T., Ozawa Dilation Method and Smoothing Effect of the Schrdinger Evolution Group. (1994);
Preprint Series # 242
J., Zhai Some Estimates For The Blowing up Solutions of Semilinear Heat Equations. (1994);
Preprint Series # 241
K., Sugano Note on H-separable Frobenius extensions. (1994);
Preprint Series # 240
C., Dohmen Existence of Fast Decaying Solutions to a Haraux-Weissler Equation With a Prescribed Number of Zeroes. (1994);
Preprint Series # 239
Y., Giga and N., Mizoguchi On time periodic solutions of the Dirichlet problem for degenerate parabolic equations of nondivergence type. (1994);
Preprint Series # 238
T., Hibi Buchsbaum complexes with linear resolutions. (1994);
Preprint Series # 237
T., Ozawa Characterization of Trudinger's inequality. (1994);
Preprint Series # 236
T., Mikami Large deviations for the first exit time on small random perturbations of dynamical systems. (1994);
Preprint Series # 235
C., Dohmen A Pohozaev-type inequality for a quasilinear Haraux-Weissler equation. (1994);
Preprint Series # 234
T., Nakazi and M., Yamada Invertible Toeplitz operators and uniform algebras. (1994);
Preprint Series # 233
C., Dohmen and Y., Giga and N., Mizoguchi Existence of selfsimilar shrinking curves for anisotropic curvature flow equations. (1994);
Preprint Series # 232
Y., Giga and N., Mizoguchi Existence of periodically evolving convex curves moved by anisotropic curvature. (1994);
Preprint Series # 231
A., Hoshiga The asymptotic behaviour of radial solutions near the blow-up point to quasi-linear wave equations in two space dimensions. (1994);
Preprint Series # 230
H., Okuda and I., Tsuda A coupled chaotic system with different time scales: Toward the implication of observation with dynamical systems. (1994);
Preprint Series # 229
K., Tsutaya Lower bounds for the life span of solutions of semilinear wave equations with data of non compact support. (1994);
Preprint Series # 228
S., Izumiya Characteristic vector fields for first order partial differential equations. (1994);
Preprint Series # 227
N., Hayashi and T., Ozawa Global, small radially symmetric solutions to nonlinear Schrdinger equations and a gauge transformation. (1994);
Preprint Series # 226
T., Nakazi and K., Takahashi Two dimensional representations of uniform algebras. (1994);
Preprint Series # 225
H., Kubo and K., Kubota Asymptotic behaviors of radial solutions to semilinear wave equations in odd space dimensions. (1994);
Preprint Series # 177
Sugano, Kozo On bicommutators of modules over H‐separable extension rings III. (19 November 1993);
Preprint Series # 224
I., Tsuda Can stochastic renewal of maps be a model for cerebral cortex? (1993);
Preprint Series # 223
I., Nakai Notes on versal deformation offirst order PDE and web structure. (1993);
Preprint Series # 222
T., Miyake and Y., Maeda On a property of Fourier coefficients of cusp forms of half-integral weight. (1993);
Preprint Series # 221
S., Jimbo and Y., Morita Ginzburg Landau equation and stable solutions in a rotational domain. (1993);
Preprint Series # 220
W., Bruns and T., Hibi Cohen-Macaulay partially ordered sets with pure resolutions. (1993);
Preprint Series # 219
Y., Giga and K., Yama-uchi On instability of evolving hypersurfaces. (1993);
Preprint Series # 218
S., Izumiya and G., T. Realization theorems of geometric singularities for Hamilton-Jacobi equations. (1993);
Preprint Series # 217
T., Suwa Indices holomorphic vector fields relative to invariant curves. (1993);
Preprint Series # 216
Y., Giga and N., Mizoguchi Existence of periodic solutions for equations of evolving curves. (1993);
Preprint Series # 215
A., Arai Seeding limit of anticommuting self-adjoint operators and nonrelativistic limit of Dirac operators. (1993);
Preprint Series # 214
A., Hoshiga Blow-up of the radial solitions to the equations of vibrating membrane. (1993);
Preprint Series # 213
K., Sugano Note on non-commutative local field. (1993);
Preprint Series # 212
A., Arai On self-adjointness of Dirac operators in Boson-Fermion Fock spaces. (1993);
Preprint Series # 211
S., Izumiya and G.-T., Kossioris Geometric singularities for solutions of single conservation laws. (1993);
Preprint Series # 210
T., Hibi Star-shaped complexes and Ehrhart polynomials. (1993);
Preprint Series # 209
T., Hibi Hochster's formula on Betti numbers and Buchsbaum complexes. (1993);
Preprint Series # 208
S., Izumiya and B., Li Overdetermined systems of first order partial differential equations with singular solution. (1993);
Preprint Series # 207
A., Inoue Regularly varying correlations. (1993);
Preprint Series # 206
T., Hibi Betti number sequences of simplicial complexes, Cohen-Macaulay types and Mbius functions of partially ordered sets, and related topics. (1993);
Preprint Series # 205
N., Honda Vanishing theorem for the tempered distributions. (1993);
Preprint Series # 204
J., Wierzbicki An estimation of the depth from an intermediate subfactor. (1993);
Preprint Series # 203
A., Arai Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications. (1993);
Preprint Series # 202
K., Iwata On Markov properties of Gaussian generalized random fields. (1993);
Preprint Series # 201
K.-S., Saito and Y., Watatani Subdiagonal algebras for subfactors. (1993);
Preprint Series # 200
S., Izumiya and Y., Kurokawa Holonomic systems of Clairaut type. (1993);
Preprint Series # 199
A., Arai and N., Tominaga Quantization of angle - variables. (1993);
Preprint Series # 198
T., Ozawa Local decay estimates for Schrdinger operators with long range potentials. (1993);
Preprint Series # 197
Y., Giga Motion of a graph by convexified energy. (1993);
Preprint Series # 196
H., Kubo Asymptotic behaviors of solutions to semilinear wave equations with initial data of slow decay. (1993);
Preprint Series # 195
T., Ozawa and Y., Tsutsumi Global existence and asymptotic behavior of solutions for the Zakharov equations in three space dimensions. (1993);
Preprint Series # 194
F., Hiroshima Scaling limit of a model of quantum electrodynamics. (1993);
Preprint Series # 193
H., Kubo Blow-up of solutions to semilinear wave equations with initial data of slow decay in low space dimensions. (1993);
Preprint Series # 192
J., Seade and T., Suwa A residue formula for the index of a holomorphic flow. (1993);
Preprint Series # 191
N., Hayashi and T., Ozawa Finite energy solutions of nonlinear Schrdinger equations of derivative type. (1993);
Preprint Series # 190
S., Izumiya and G.T., Kossioris Semi-local classification of geometric singularities for Hamilton-Jacobi equations. (1993);
Preprint Series # 189
T., Nakazi Multipliers of invariant subspaces in the bidisc. (1993);
Preprint Series # 188
A., Jensen and T., Ozawa Existence and non-existence results for wave operators for perturbations of the laplacian. (1993);
Preprint Series # 187
O., Ogurisu Unitary equivalence between a spin 1/2 charged particle in a two-dimensional magnetic field and a spin 1/2 neutral particle with an anomalous magnetic moment in a two-dimensional electric
field,. (1993);
Preprint Series # 186
Y., Kurokawa On functional moduli for first order ordinary differential equations. (1993);
Preprint Series # 185
Y., Giga and K., Yamauchi On a lower bound for the extinction time of surfaces moved by mean curvature. (1993);
Preprint Series # 184
T., Hibi Canonical modules and Cohen-Macaulay types of partially ordered sets. (1993);
Preprint Series # 183
G., Ishikawa and T., Ozawa The genus of a connected compact real algebraic surface in the affine three space. (1993);
Preprint Series # 182
M., Yamada Distance formulas of asymptotic Toeplitz and Hankel operators. (1993);
Preprint Series # 181
K., Sugano Note on H-separable Galois extension. (1993);
Preprint Series # 180
O., Ogurisu Ground state of a spin 1/2 charged particle in an even dimensional magnetic field. (1993);
Preprint Series # 179
Ogurisu, O. Existence and structure of infinitely degenerate zero‐energy ground states of a Wess‐Zumino type model in supersymmetric quantum mechanics. (1993);
Preprint Series # 178
Nakazi, T. Toeplitz operators and weighted norm inequalities. (1993);
Preprint Series # 176
Hoshiga, Akira The initial value problems for quasi‐linear wave equations in two space dimensions with small data. (1992);
Preprint Series # 175
Izumiya, Shyuichi Systems of Clairaut type. (1992);
Preprint Series # 174
Ozawa, Tohru Wave propagation in even dimensional spaces. (1992);
Preprint Series # 173
Suwa, Tatsuo Unfoldings of codimension one complex analytic foliation singularities. (1992);
Preprint Series # 172
Izumiya, Shyuichi A characterization of complete integrability for partial differential equations of first order. (1992);
Preprint Series # 171
Inoue, Junji and Nakazi, Takahiko Finite dimensional solution sets of extremal problems in H1. (1992);
Preprint Series # 170
Fukui, Toshihide and Giga, Yoshikazu Motion of a graph by nonsmooth weighted curvature. (1992);
Preprint Series # 169
Ozawa, Tohru Remarks on quadratic nonlinear Schrodinger equations. (1992);
Preprint Series # 168
Inoue, Akihiko On the equations of stationary precesses with divergent diffusion coefficients. (1992);
Preprint Series # 167
Okabe, Yasunori and Mano, Hajime and Itoh, Yoshiaki Random collision model for interacting populations of two species and its strong law of large numbers. (1992);
Preprint Series # 166
Okabe, Yasunori A new algorithm driven from the view‐point of the fluctuation‐dissipation theorem in the theory of KM2O‐Langevin equations. (1992);
Preprint Series # 165
Wierzbicki, Jerzy and Watatani, Yasuo Commuting squares and relative entropy for two subfactors. (1992);
Preprint Series # 164
Giga, Yoshikazu and Sato, Motohiko Neumann problem for singular degenerate parabolic equations. (1992);
Preprint Series # 163
Albeverio, S. and Iwata, K. and Kolsrud, T. Moments of random fields over a family of elliptic curves, and modular forms. (1992);
Preprint Series # 162
Albeverio, S. and Iwata, K. and Kolsrud, T. Random parallel transport on surfaces of finite type, and relations to homotopy. (1992);
Preprint Series # 161
Arai, Asao Dirac operators in Boson‐Fermion Fock spaces and supersymmetrie quantum field theory. (1992);
Preprint Series # 160
Arai, Asao Properties of the Dirac‐Weyl operator with a strongly singular gauge potential. (1992);
Preprint Series # 159
Hibi, Takayuki Cohen‐Macaulay types of Cohen‐Macaulay complexes. (1992);
Preprint Series # 158
Izumiya, S. and Marar, W.L. The Euler number of a topologically stable singular surface in a 3‐manifold. (1992);
Preprint Series # 157
Arai, Asao Momentum operators with gauge potentials, local quantization of magnetic flux, and representation of canonical commutation relations. (1992);
Preprint Series # 156
Izumiya, S. Perestroikas of optical wave fronts and graphlike Legendrian unfoldings. (1992);
Preprint Series # 155
Ohnuma, Masaki and Sato, Moto-hiko Singular degenerate parabolic equations with applications to geometric evolutions. (1992);
Preprint Series # 154
Ozawa, Tohru On critical cases of Sobolev inequalities. (1992);
Preprint Series # 153
Watatani, Yasuo Lattices of intermediate subfactors. (1992);
Preprint Series # 152
Kubo, Hideo Global existence of solutions of semilinear wave equations with data of non compact support in odd space dimensions. (1992);
Preprint Series # 151
Ishikawa, Goo Developable of a curve and determinancy relative to osculation‐type. (1992);
Preprint Series # 150
Ishikawa, Goo Determinacy of envelope of the osculating hyperplanes to a curve. (1992);
Preprint Series # 149
Giga, Yoshikazu and Takahashi, Shuji On global weak solutions of the nonstationary two‐phase Stokes flow. (1992);
Preprint Series # 148
Okabe, Yasunori Langevin equations and causal analysis. (1992);
Preprint Series # 147
Sato, Moto-Hiko Interface evolution with Neumann boundary condition. (1992);
Preprint Series # 146
Hayashi, Nakao and Ozawa, Tohru Remarks on nonlinear Schrdinger equations in one space dimension. (1992);
Preprint Series # 145
Nakazi, Takahiko and Takahashi, Katsutoshi Hyponormal Toeplitz operators and extremal problems of Hardy spaces. (1992);
Preprint Series # 144
Kubota, K. and Mochizuki, K. On small data scattering for 2‐dimensional semilinear wave equations. (1992);
Preprint Series # 143
Ishikawa, Goo and Ohmoto, Toru Local invariants of singular surfaces in an almost complex four‐manifold. (1992);
Preprint Series # 142
Morimoto, Tohru Geometric structures on filtered manifolds. (1992);
Preprint Series # 141
Takamura, Hiroyuki Weighted deformation theorem for normal currents. (1992);
Preprint Series # 140
Izumiya, Shyuichi What is the Clairaut equation ? (1992);
Preprint Series # 139
Giga, Yoshikazu and Yoshida, Zensho A bound for the pressure integral in a plasma equilibrium. (1992);
Preprint Series # 138
Izumiya, S. and Marar, W.L. The Euler characteristic of the image of a stable mapping from a closed n‐manifold to a (2n ‐ l)‐manifold. (1992);
Preprint Series # 137
Izumiya, S. and Marar, W.L. The Euler characteristic of a generic wave front in a 3‐manifold. (1992);
Preprint Series # 136
Izumiya, Shyuichi Singular solutions of first order differential equations. (1992);
Preprint Series # 135
Izumiya, Shyuichi The Clairaut type equation. (1992);
Preprint Series # 134
Aviles, Ptricio and Giga, Yoshikazu and Komuro, Naoto Duality formulas and variational integrals. (1992);
Preprint Series # 133
Okabe, Yasunori Applications of the theory of KM2O‐Langevin equations to the linear prediction problem for the multi‐dimensional weakly stationary time series. (1992);
Preprint Series # 132
Okabe, Yasunori and Ootsuka, Takashi Applications of the theory of KM2O‐Langevin equations to the non‐linear prediction problem for the one‐dimensional strictly stationary time series. (1992);
Preprint Series # 99
Agemi, R. and Kubota, K. and Takamura, H. On certain integral equations related to nonlinear wave equations. (1991);
Preprint Series # 131
Giga, Mariko and Giga, Yoshikazu and Sohr, Hermann L^p estimates for the Stokes system. (1991);
Preprint Series # 130
Altschuler, Steven and Angenent, Sigurd and Giga, Yoshikazu Mean curvature flow through singularities for surfaces of rotation. (1991);
Preprint Series # 129
Kubota, K. Existence of a global solution to a semi‐linear wave equation with initial data of non‐compact support in low space dimensions. (1991);
Preprint Series # 128
Arai, Asao and Mitoma, Itaru Comparison and nuclearity of spaces of differential forms on topological vector spaces. (1991);
Preprint Series # 127
Ishikawa, G. and Izumiya, S. and Watanabe, K. Vector fields near a generic submanifold. (1991);
Preprint Series # 126
Izumiya, Shyuichi Completely integrable holonomic systems of first order differential equations. (1991);
Preprint Series # 125
Hibi, Takayuki Face number inequalities for matroid complexes and Cohen‐Macaulay types of Stanley‐Reisner rings of distributive lattices. (1991);
Preprint Series # 124
Izumiya, Shyuichi Completely integrable holonomic systems of first order differential equations. (1991);
Preprint Series # 123
Arai, Asao Fock‐space representations of the relativistic supersymmetry algebra in the two‐dimensional spacetime. (1991);
Preprint Series # 122
Chen, Yun-Gang Blow‐up solutions to a finite difference analogue of u_1 = Δu + u^{1+α} in N‐dimensional balls. (1991);
Preprint Series # 121
Arai, Asao Commutation properties of the partial isometries associated with anticommuting self‐adjoint operators. (1991);
Preprint Series # 120
Nakazi, Takahiko Invariant subspaces in the bidisc and commutators. (1991);
Preprint Series # 119
Altschuler, Stephen and Angenent, Sigurd and Giga, Yoshikazu Generalized motion by mean curvature for surfaces of rotation. (1991);
Preprint Series # 118
Agemi, Rentaro and Takamura, Hiroyuki The lifespan of classical solutions to nonlinear wave equations in two space dimensions. (1991);
Preprint Series # 117
Hibi, Takayuki A lower bound theorem for Ehrhart polynomials of convex polytopes. (1991);
Preprint Series # 116
Nishimori, Toshiyuki A note on the classification of non‐singular flows with transverse similarity structures. (1991);
Preprint Series # 115
Arai, Asao De Rham operators, Laplacians, and Dirac operators on topological vectorspaces. (1991);
Preprint Series # 114
Nakazi, Takahiko Sum of two inner functions and exposed points in H1. (1991);
Preprint Series # 113
Takahashi, Shuji On a regularity criterion uo to the boundary for weak solutions of the Navier‐Stokes equations. (1991);
Preprint Series # 112
Matsuda, Kouichi An analogy of the theorem of Hector and Duminy. (1991);
Preprint Series # 111
Nishimori, Toshiyuki A qualitative theory of similarity pseudogroups and an analogy of Sacksteder's theorem. (1991);
Preprint Series # 110
Hibi, Takayuki and Wakayama, Masato A q‐analogue of Capelli's identity for GL(2). (1991);
Preprint Series # 109
Chen, Yung-Gang and Giga, Yoshikazu and Goto, Shun'ichi Analysis toward snow crystal growth. (1991);
Preprint Series # 108
Arai, Asao An abstract sum formula and its applications to special functions. (1991);
Preprint Series # 107
Nakazi, Takahiko p‐dilations and hypo‐Dirichlet algebras. (1991);
Preprint Series # 106
Nakazi, Takahiko Extremal problems in HP. (1991);
Preprint Series # 105
Agemi, Rentaro Blow‐up of solutions to nonlinear wave equations in two space dimensions. (1991);
Preprint Series # 104
Takamura, Hiroyuki Global existence of classical solutions to nonlinear wave equations with spherical symmetry for small data with noncompact support in three space dimensions. (1991);
Preprint Series # 103
Arai, Asao and Ogurisu, Osamu Meromorphic N = 2 Wess‐Zumino supersymmetric quantum mechanics. (1991);
Preprint Series # 102
Munemasa, Akihiro and Watatani, Yasuo Orthogonal pairs of ‐subalgebras and association schemes. (1991);
Preprint Series # 101
Izumiya, S. Legendrian singularities and first order differential equations. (1991);
Preprint Series # 100
Izumiya, S. Geometric singularities for Hamilton‐Jacobi equation. (1991);
Preprint Series # 98
Lawrynowicz, Julian and Koshi, Shozo and Suzuki, Osamu Dualities generated by the generalised Hurwitz problem and variation of the Yang‐Mills field. (1990);
Preprint Series # 97
Okabe, Yasunori and Inoue, Akihiko The theory of KN2O‐Langevin equations and its applications to data analysis (II): Causal analysis (1),. (1990);
Preprint Series # 96
Ninomiya, Syoiti The Fourier‐Sato transformation of pure sheaves. (1990);
Preprint Series # 95
Sano, Takashi and Watatani, Yasuo Angles between two sub factors. (1990);
Preprint Series # 94
Okabe, Yasunori and Inoue, Akihiko On the exponential decay of the correlation functions for KMO‐Langevin equations. (1990);
Preprint Series # 93
Arai, Asao A theorem on essential self‐adjointness with application to Hamiltonians in non‐relativistic quan‐tum field theory. (1990);
Preprint Series # 92
Nakazi, Takahiko and Takahashi, Katsutoshi Homogeneous polynomials and invariant subspaces in the polydiscs II. (1990);
Preprint Series # 91
Nakazi, Takahiko Homogeneous polynomials and invariant subspaces in the polydiscs. (1990);
Preprint Series # 90
Sugano, Kozo On bicommutators of modules over H‐separable extension rings II. (1990);
Preprint Series # 89
Arai, Asao Improper Bogoliubov transformations and instability of embedded eigenvalues. (1990);
Preprint Series # 88
Giga, Yoshikazu and Goto, Shun'ichi and Ishii, Hitoshi and Sato, Moto-hiko Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains.
Preprint Series # 87
Inoue, Akihiko The Alder‐Wainwright effect for stationary processes with reflection positivity (II). (1990);
Preprint Series # 86
Arai, Asao Diffusive behavior of an electron interacting with a quantized radiation field. (1990);
Preprint Series # 85
Morimoto, Tohru Theoreme de Cartan‐Kahler dans une classe de fonctions formelles Gevrey. (1990);
Preprint Series # 84
Morimoto, Tohru Generalized Spencer cohomology groups and quasi‐regular bases. (1990);
Preprint Series # 83
Nakazi, Takahiko and Yamamoto, Takanori Weighted norm inequalities and uniform algebras. (1990);
Preprint Series # 82
Nakazi, Takahiko Absolute values of Toeplitz operators and Hankel operators. (1990);
Preprint Series # 81
Hayakawa, A. and Ishikawa, G. and Izumiya, S. and Yamaguchi, K. Classification of generic integral diagrams and first order ordinary differential equations. (1990);
Preprint Series # 80
Miyake, Toshitsune and Maeda, Yoshitaka On elliptic cyclopean forms. (1990);
Preprint Series # 79
Ishikawa, Goo Maslov class of an isotropic map‐germ arising from one dimensional symplectic reduction. (1990);
Preprint Series # 78
Ishikawa, Goo Topologically extremal real algebraic surfaces in P2×P1 and P1×P1×P1. (1990);
Preprint Series # 77
Takahashi, Shuji On interior regularity criteria for weak solutions of the Navier‐Stokes equations. (1990);
Preprint Series # 76
Arai, Asao De Rham‐Hodge‐Kodaira decomposition in ∞‐dimensions. (1990);
Preprint Series # 75
Kishimoto, Akitaka Actions of finite groups on certain inductive limit C*‐algebras. (1990);
Preprint Series # 74
Kishimoto, Akitaka A weak approximate innerness for abelian actions on C*‐algebras. (1990);
Preprint Series # 73
Sugano, Kozo On bicommutators of modules over H‐separable extension rings. (1990);
Preprint Series # 72
Arai, Asao An asymptotic analysis and its application to the nonrelativistic limit of the Pauli‐Fierz and a spin‐bosonmodel. (1990);
Preprint Series # 71
Kiyohara, Kazuyoshi Compact Liouville surfaces. (1990);
Preprint Series # 70
Giga, Yoshikazu and Goto, Shun'ichi Motion of hypersurfaces and geometric equations. (1990);
Preprint Series # 69
Aviles, Patricio and Giga, Yoshikazu Variational integrals on mappings of bounded variation and their lower semicontinuity. (1990);
Preprint Series # 68
Nakazi, Takahiko Szeg's theorem on a bidisc. (1990);
Preprint Series # 67
Cerveau, Dominique and Suwa, Tatsuo Determinacy of complex analytic foliation germs without integrating factors. (1990);
Preprint Series # 66
Inoue, Akihiko The Alder‐Wainwright effect for stationary processes with reflection positivity (I). (1990);
Preprint Series # 65
Chen, Yung-Gang Blow‐up solutions of a semilinear parabolic equation with the Neumann and Robin boundary conditions. (1989);
Preprint Series # 64
Saito, Masa-Hiko and Zucker, S. Classification of non‐rigid families of K3 surfaces and a finiteness theorem of Arakelov type. (1989);
Preprint Series # 63
Agemi, Rentaro Global existence of classical solutions to nonlinear wave equations in n space dimension. (1989);
Preprint Series # 62
Arai, Asao Exactly solvable supersymmetric quantum mechanics. (1989);
Preprint Series # 61
Arai, Asao general class of infinite dimensional Dirac operators and path integral representation of their index. (1989);
Preprint Series # 60
Giga, Yoshikazu and Sohr, Hermann Abstract LP estimates for the Cauchy problem with applications to the Navier‐Stokes equations in exterior domains. (1989);
Preprint Series # 59
Arai, Asao A general class of (essentially) iso‐spectral perturbations. (1989);
Preprint Series # 58
Nakazi, Takahiko Existence of solutions of extremal problems in H1. (1989);
Preprint Series # 57
Chen, Yung-Gang and Giga, Yoshikazu and Goto, Shun'ichi Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. (1989);
Preprint Series # 56
Sannami, A. An example of a regular Cantor set whose difference set is a Cantor set with positive measure. (1989);
Preprint Series # 55
Arai, Asao A general class of (essentially) iso‐spectral perturbations. (1989);
Preprint Series # 54
Cielak, Waldemar and Koshi, Shozo and Zajac, Jozef On integral formulas for convex domains. (1989);
Preprint Series # 53
Nakazi, Takahiko Norms of Hankel operators on a bidisc. (1989);
Preprint Series # 52
Aviles, Patricio and Giga, Yoshikazu Variational integrals on mappings of bounded variation. (1989);
Preprint Series # 51
Okabe, Yasunori and Nakano, Yuji On the theory of KM2O‐Langevin equations with application to data analysis(I). (1989);
Preprint Series # 50
Arai, Asao On the degeneracy in the ground state of the N = 2 Wess‐Zumino supersymmetric quantum mechanics. (1989);
Preprint Series # 49
Hirokawa, Masao Mori's Langevin equation for a quantum harmonic oscillator coupled to infinitely many scalar Bosons. (1989);
Preprint Series # 48
Yoshida, Zensho and Giga, Yoshikazu Remarks on spectra of operator rot. (1989);
Preprint Series # 47
Giga, Yoshikazu and Yoshida, Zensho A dinamic free‐boundary problem in plasma physics. (1988);
Preprint Series # 46
Giga, Yoshikazu and Kohn, Robert V. Nondegeneracy of blowup for semilinear heat equations. (1988);
Preprint Series # 45
Nakazi, Takahiko Bounded Hankel forms with weighted norms and lifting theorems. (1988);
Preprint Series # 44
Arai, Asao Existence of infinitely many zero‐energy states in a model of supersymmetric quantum mechanics. (1988);
Preprint Series # 43
Arai, Asao Long‐time behavior of two‐point functions of a quantum harmonic oscillator interacting with bosons. (1988);
Preprint Series # 42
Arai, Asao A criterion for the boundedness from below with a class of symmetric operators and its applications. (1988);
Preprint Series # 41
Sugano, Kozo On H‐separable extensions of primitive rings II. (1988);
Preprint Series # 40
Izumiya, S. First order partial differential equations and singularities. (1988);
Preprint Series # 39
Izumiya, S. Topological properties of legendrian singularities. (1988);
Preprint Series # 38
Nakazi, Takahiko Certain invariant subspaces of H2 and L2 on a bidisc. (1988);
Preprint Series # 37
Nakazi, Takahiko Commuting dilations and uniform algebras. (1988);
Preprint Series # 36
Suwa, Tatsuo D‐modules associated to complex analytic singular foliations. (1988);
Preprint Series # 35
Morimoto, Tohru Un critre pour l'existence d'une connexion de Cartan. (1988);
Preprint Series # 34
Morimoto, Tohru Structures gomtriques sur des varits filtres. (1988);
Preprint Series # 33
Suwa, Tatsuo Structure of the singular of a complex analytic foliation. (1988);
Preprint Series # 32
Giga, Y. and Miyakawa, T. Navier‐Stokes flow in R3 with measures as initial vorticity and Morrey spaces. (1988);
Preprint Series # 31
Nakazi, Takahiko and Yamamoto, Takanori Some Singular Integral Operators And Helson‐Szeg Measures. (1988);
Preprint Series # 30
Kishi, Kazuo and Nakazi, Takahiko Some Typical Ideal In a Uniform Algebra. (1988);
Preprint Series # 29
Giga, Y. and Sohr, H. On the Stokes operator in exterior domains. (1988);
Preprint Series # 28
Giga, Yoshikazu A local characterization of blowup points of semilinear heat equations. (1988);
Preprint Series # 27
Sugano, Kozo Note on separable extensions of noncomutative rings. (1988);
Preprint Series # 26
Arai, Asao Supersymmetric embedding of the Hamiltonian for the RWA osci1lator. (1988);
Preprint Series # 25
Matsuoka, Sachiko An algebraic criterion for right‐left equivalence of holomorphic functions on analytic varieties. (1988);
Preprint Series # 24
Matsuoka, Sachiko Nonsingular Algebraic Curves in RP1×RP1. (1988);
Preprint Series # 23
Nakazi, Takahiko Complete Spectral Area Estimates and Selfcommutators. (1988);
Preprint Series # 22
Nakazi, Takahiko A Lifting Theorem And Analytic Operator Algebras. (1988);
Preprint Series # 21
Arai, Asao Perturbation of Embedded Eigenvalues in Fock Spaces A General Class of Exactly Soluble Models. (1988);
Preprint Series # 20
Aviles, Patricio and Giga, Yoshikazu Singularities and rank one properties of Hessian measures. (1988);
Preprint Series # 19
Sannami, Atsuro A topological classification of the periodic orbits of the Henon family. (1988);
Preprint Series # 18
Okabe, Yasunori On the theory of discrete KMO‐Lanegvin equation with reflection positivity (III). (1988);
Preprint Series # 9
Nakazi, Takahiko Weighted norm inequalities and uniform algebras. (1987);
Preprint Series # 8
Suwa, Tatsuo A factorization theorem for unfoldings of analytic functions. (1987);
Preprint Series # 7
Hida, Haruzo A p‐adic measure attached to the zeta functions associated with two elliptic modular forms II. (1987);
Preprint Series # 6
Nakazi, Takahiko A Spectral Dilation of Some Non‐Dirichlet Algebra. (1987);
Preprint Series # 5
Nakazi, Takahiko Notes on Interpolation by Bounded Analytic Functions. (1987);
Preprint Series # 4
Nakamura, Iku Threefolds Homeomorphic to a Hyperquadric in P4. (1987);
Preprint Series # 3
Arai, A. Path Integral Representation of the Index of Kahler-Dirac Operators on an Infinite Dimensional Manifold. (1987);
Preprint Series # 2
Giga, Y. and Kanbe, T. Large time behavior of the vorticity of two‐dimensional flow and its application to vortex formation. (1987);
Preprint Series # 17
Miyake, Toshitsune On Qab‐rationality of Eisenstein series of weight 3/2. (1987);
Preprint Series # 16
Arai, Asao Spectral Analysis of a Quantum Harmonic Oscillator Coupled to Infinitely Many Scalar Bosons. (1987);
Preprint Series # 15
Okabe, Yasunori On the theory of discrete KMO‐Langevin equations with reflection positivity (II). (1987);
Preprint Series # 14
Ishikawa, Goo Parametrization of a Singular Lagrangian Variety. (1987);
Preprint Series # 13
Okabe, Yasunori On a stochastic difference equation for the multidimensional weakly stationary process with discrete time. (1987);
Preprint Series # 12
Kubota, K. Microlocal parametrices and propagation of singularities near gliding points for hyperbolic mixed problems II. (1987);
Preprint Series # 11
Nitta, Kazuo Note on the double centralizers in an H‐separable extension. (1987);
Preprint Series # 10
Miyake, Toshitsune On the spaces of Eisenstein series of Hilbert modular groups. (1987);
Preprint Series # 1
Okabe, Y. On the theory of discrete KMO‐Langevin equations with reflection positivity (I). (1987);
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Topic: 0 divided by 0
Replies: 4 Last Post: Jun 2, 2011 8:48 AM
Messages: [ Previous | Next ]
Re: 0 divided by 0
Posted: Jun 2, 2011 8:48 AM
0÷0 = ? in my opinion.
I think about it this way: the reason any number divided by zero is undefined is because zero can't be multiplied by anything to get another number, making it unable to go into any other number.
However, zero divided by zero is different, because zero mutiplied by ANY number is zero, meaning that zero can go into zero an infinite number of times.
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Hello everyone,
I have a 'small' question regarding the combination of legends. Suppose I
have a model, and the binned version of it in two vectors, say, model, with
the corresponding values for x, x_model and binned_model with the
corresponding values of x, x_binned_model.
What I want to do is to plot this using pyx, but, in the legend, combine
the symbol I use for the binned version of the model with the model itself,
which I plot as a line. So I have something like the following code:
from pyx import *
from numpy import np
# Create a model
x = np.array([0,1,2,3,4,5,6])
model = x**2
# Bin two values; 0.5, binned between 0 and 1, and 4.5, binned between 4
and 5:
x_binned_model = np.array([0.5,4.5])
x_binned_model = np.array([0.33,20.33])
c = canvas.canvas()
g = c.insert(graph.graphxy(height=5,width=10, key=graph.key.key(pos='bl'),\
x = graph.axis.linear(title = 'x axis'),\
y = graph.axis.linear(title = 'y axis')))
# Plot 'full version' of the model:
g.plot(graph.data.values(x=x_model,y=model, title = 'The model',\
styles = [graph.style.line([color.cmyk.CornflowerBlue,\
# ...binned version of the model:
g.plot(graph.data.values(x=x_binned_model, y=binned_model, title = 'The
styles =
[graph.style.symbol(graph.style.symbol.square,symbolattrs =
deco.stroked([color.cmyk.CornflowerBlue])],size = 0.1)])
So when you plot it, you have two legends: one with a line for the model
and one with squares for the bined model: is there a way to combine the two
and just have one legend, with the line and the square overplotted on it?
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Great stuff! As I was doing an Internet search on effective math software, I came across the Algebrator Web site. I decided to purchase the software after seeing the online demo and never regretted
it. Thanks!
Lakeysha Smith, OH
My former algebra tutor got impatient whenever I couldn't figure out an equation. I eventually got tired of her so I decided to try the software. I'm so impressed with it! I can't stress enough how
great it is!
Trish Cooper, CO
Super piece of software! I'm finally acing all of my algebra tests! Thanks a lot!
Stephanie Cummings, AZ
You can now forget about being grounded for bad grades in Algebra. With the Algebrator it takes only a few minutes to fully understand and do your homework.
Barbara, LA
Super piece of software! I'm finally acing all of my algebra tests! Thanks a lot!
Robert Davis, CA
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Using R for Introductory Statistics, Chapter 3.4
...a continuing journey through Using R for Introductory Statistics, by John Verzani.
Simple linear regression
Linear regression is a kooky term for fitting a line to some data. This odd bit of terminology can be blamed on Sir Francis Galton, a prolific victorian scientist and traveler who saw it as related
to his concept of regression toward the mean. Calling it a linear model is a little more straight-forward, and linear modeling through the lm function is bread-and-butter to R.
For example, let's look at the data set diamonds to see if there's a linear relationship between weight and cost of diamonds.
f = price ~ carat
plot(f, data=diamond, pch=5,
main="Price of diamonds predicted by weight")
res = lm(f, data=diamond)
abline(res, col='blue')
We start by creating the formula f using the strange looking tilde operator. That tells the R interpreter that we're defining a symbolic formula, rather than an expression to be evaluated
immediately. So, our definition of formula f says, "price is a function of carat". In the plot statement, the formula is evaluated in the context given by data=diamond, so that the variables in our
formula have values. That gives us the scatter plot. Now let's fit a line using lm, context again given by data=diamond, and render the resulting object as a line using abline. Looks spiffy, but what
just happened?
The equation of a line that we learned in high school is:
Minimizing squared error over our sample gives us estimates of the slope and intercept. The book presents this without derivation, which is a shame.
Maybe later, I'll get brave an try to insert a derivation here.
There's a popular linear model that applies to dating, which goes like this: It's OK for a man to date a younger woman if her age is at least half the man's age plus seven. In other words, this:
Apparently, I should be dating a 27 year old. Let me go ask my wife if that's OK. In the meantime, let's see how our rule compares to results of a survey asking the proper cutoff for dating for
various ages.
plot(jitter(too.young$Male), jitter(too.young$Female),
main="Appropriate ages for dating",
xlab="Male age", ylab="Female age")
abline(7,1/2, col='red')
res <- lm(Female ~ Male, data=too.young)
abline(res, col='blue', lty=2)
legend(15,45, legend=c("half plus 7 rule",
"Estimated from survey data"),
col=c('red', 'blue'), lty=c(1,2))
That's a nice correspondence. On second thought, this is statistical proof that my daughter is not allowed to leave the house 'til she's 30.
Somehow related to that is the data set Animals, comparing weights of body and brain for several animals. The basic scatterplot not revealing much, we put the data on a log scale and find that it
looks much better. As near as I can tell, the I or AsIs function does something like the opposite of the tilde operator. It tells the interpreter to go ahead and evaluate the enclosed expression. The
general gist is to transform our data to log scale then apply linear modeling.
f = I(log(brain)) ~ I(log(body))
plot(f, data=Animals,
main="Animals: brains vs. bodies",
xlab="log body weight", ylab="log brain weight")
res = lm(f, data=Animals)
abline(res, col='brown')
Now the problem is, the line doesn't seem to fit very well. Those three outliers on the right edge have high body weights but less than expected going on upstairs. That seems to unduly influence the
linear model away from the main trend. R contains some alternative algorithms for fitting a line to data. The function lqs is more resistant to outliers, like the large but pea-brained creatures in
this example.
res.lqs = lqs(f, data=Animals)
abline(res.lqs, col='green', lty=2)
That's better. Finally, you might use identify to solve the mystery of the knuckleheaded beasts.
with(Animals, identify(log(body), log(brain), n=3, labels=rownames(Animals)))
Problem 3.31 is about replicate measurements, which might be a good idea where measurement error, noisy data, or other random variation is present. We follow the by now familiar procedure of defining
our formula, doing a scatterplot, building our linear model, and finally plotting it over the scatterplot.
We are then asked to look at the variance of measurements at each particular voltage. To do that, we'll first split our data.frame up by voltage. The result is a list of vectors, one per voltage
breakdown.by.voltage = split(breakdown$time, breakdown$voltage)
List of 7
$ 26: num [1:3] 5.8 1580 2323
$ 28: num [1:5] 69 108 110 426 1067
$ 30: num [1:11] 7.7 17 20 21 22 43 47 139 144 175 ...
$ 32: num [1:15] 0.27 0.4 0.69 0.79 2.75 3.9 9.8 14 16 27 ...
$ 34: num [1:19] 0.19 0.78 0.96 1.31 2.78 3.16 4.15 4.67 4.85 6.5 ...
$ 36: num [1:15] 0.35 0.59 0.96 0.99 1.69 1.97 2.07 2.58 2.71 2.9 ...
$ 38: num [1:7] 0.09 0.39 0.47 0.73 1.13 1.4 2.38
Next, let's compute the variance for each component of the above list and build a data.frame out of it.
var.by.voltage = data.frame(voltage=names(breakdown.by.voltage),
This split-apply-combine pattern looks familiar. It's basically a SQL group by in R. It's also the basis for Hadley Wickham's plyr library. Plyr's ddply function takes breakdown, a data.frame, and
splits it on values of the voltage column. For each part, it computes the variance in the time column, then assembles the results back into a data.frame.
ddply(breakdown, .(voltage), .fun=function(df) {var(df$time)})
While that's not directly related to linear modeling, this kind of exploratory data manipulation is what R is made for.
More fun
Previous episode of Using R for Introductory Statistics
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Tiling troubles solution
September 2008
Tiling troubles
Perpendicula, the magnificent princess of the rectangle, is admiring her magnificent new bathroom. It's rectangular of course; 10m wide and 20m long. It still needs tiling, though, and Perpendicula
has ordered tiles of pure gold to mirror the dimensions of the bathroom: they are 1m wide and 2m long.
There is a problem though: water, milk and honey pipes need to be fitted in two diagonally opposed corners of the bathroom. This means that in each of these two corners a square area of 1m×1m has to
remain untiled. Can the rest of the floor area be tiled without cutting any of the 2m×1m tiles?
The solution
Think of the pricess's bathroom as a rectangular chess board consisting of 1m×1m tiles which are alternately black and white. Now remove two diagonally opposed squares. The problem is whether we can
tile the remaining area by our 2m×1m tiles.
The clue lies in the colours: say that we have removed the top left and bottom right square, and assume that the top left square is black. Since the number of squares in a row is even (there are ten
1m squares) we know that the top right square is white. The number of squares in a column is also even (twenty 1m squares), so the bottom left square is black. We have therefore removed two black
squares. Originally, our board had the same number of black and white squares (100 of each), so the remaining areas has two more white squares than black squares. But each of our 2m×1m tiles covers
one black and one white square, so we cannot possibly cover an area with less black squares than white ones — the princess will have to cut the tiles!
This, incidentally, is quite a well-known puzzle which usually goes by the name of the mutilated chess board problem. It's of great interest to computer scientists because it's a classic example of a
problem whose solution becomes simple once you've discovered a trick — in this case, the trick is to think about colours. At the moment only humans have the ability to come up with such tricks.
Computers can only follow a set of mechanical instructions. The mutilated chess board problem has therefore been a focus of artificial intelligence research — is it possible to program a computer to
think creatively?
Back to main puzzle page
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Statistics And Probability Archive | August 08, 2008 | Chegg.com
Statistics And Probability Archive: Questions from August 08, 2008
• Anonymous asked
In the following regression, X=total assets ($billions),Y=total revenue ($billions), and N=64 large... Show more
In the following regression, X=total assets ($billions),Y=total revenue ($billions), and N=64 large banks.
(a) Write the fitted regression equation
(b) State the degrees of freedom for a two-tailed test forzero slope, and use appendix D to find the critical valueA=.05
(c) What is your conclusion about the slope
(d) Interpret the 955 confidence limits for the slope
(e) Verify that F=+ 2 square
(f) In your own words, describe the fit of thisregression
R 2 square .0519, Standard error 6.977, N=64
ANOVA TABLE
Source SS DF MS F P-Value
Regression 3,260.0981 1 3,260.0981 66.97 1.90E-11
Residual 3,018.3339 62 48.6828
Total 6,278.4320 63
Regression output
Variables coefficients std.err t(df=62) Value95% Value95%
lower uper
Intercept 6.5763 1.9254 3.416 .0011 2.7275
X-1 0.0452 0.0055 8.183 1.90E-11 0.0342
95% upper 10.4252
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To help schedule staffing and equipment needs, a large hospitaluses a multipl... Show more
Please answer, Thanks!
To help schedule staffing and equipment needs, a large hospitaluses a multiple regression model to predict its 'bed census' Y, the number of beds occupied at the end ofeach day. Using hospital
records from the mostrecent 16 days, a total of 3independent variables are used to find the estimated regressionmodel. Let B1, B2, and B3 denote thecoefficients of the 3 variables in this model.
Acomputer printout indicates that the total sum of squares (SST)associated with the model is 580.34 and thecorresponding regression sum of squares (SSR) is371.40. Using a significance level of
0.05, can you conclude that at least one of theindependent variables in the model provides useful (i.e.,statistically significant) information for predicting daily bedcensus?
Perform a one-tailed test.
State the Null Hypothesis
State the Alternate Hypothsis
Type of test statistic is F, please provide degree of freedomfor dfn and dfd.
Provide the p-value
Round to at least three decimal places.
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[Numpy-discussion] Overlap arrays with "transparency"
Cristi Constantin darkgl0w@yahoo....
Mon May 18 07:37:09 CDT 2009
Good day.
I am working on this algorithm for a few weeks now, so i tried almost everything...
I want to overlap / overwrite 2 matrices, but completely ignore some values (in this case ignore 0)
Let me explain:
a = [
[1, 2, 3, 4, 5],
[5,5,5] ]
b = [
[2,2,2,2] ]
Then, we have:
a over b = [
[5,5,5,2] ]
b over a = [
5,5,5] ]
That means, completely overwrite one list of arrays over the other, not matter what values one has, not matter the size, just ignore 0 values on overwriting.
I checked the documentation, i just need some tips.
TempA = [[]]
One For Cicle in here to get the Element data...
Data = vElem.data # This is a list of numpy ndarrays.
for nr_row in range( len(Data) ): # For each numpy ndarray (row) in Data.
NData = Data[nr_row] # New data, to be written over old data.
OData = TempA[nr_row:nr_row+1] or [[]] # This is old data. Can be numpy ndarray, or empty list.
OData = OData[0]
# NData must completely eliminate transparent pixels... here comes the algorithm... No algorithm yet.
if len(NData) >= len(OData):
# If new data is longer than old data, old data will be completely overwritten.
TempA[nr_row:nr_row+1] = [NData]
else: # Old data is longer than new data ; old data cannot be null.
TempB = np.copy(OData)
TempB.put( range(len(NData)), NData )
#TempB[0:len(NData)-1] = NData # This returns "ValueError: shape mismatch: objects cannot be broadcast to a single shape"
TempA[nr_row:nr_row+1] = [TempB]
del TempB
The result is stored inside TempA as list of numpy arrays.
I would use 2D arrays, but they are slower than Python Lists containing Numpy arrays. I need to do this overwrite in a very big loop and every delay is very important.
I tried to create a masked array where all "zero" values are ignored on overlap, but it doesn't work. Masked or not, the "transparent" values are still overwritten.
Please, any suggestion is useful.
Thank you.
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Cosmological horizons and reconstruction of quantum field theories
Dappiaggi, Claudio and Moretti, Valter and Pinamonti, Nicola (2007) Cosmological horizons and reconstruction of quantum field theories. UNSPECIFIED. (Unpublished)
Download (523Kb) | Preview
As a starting point for this manuscript, we remark how the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds shares some non trivial geometric properties with null
infinity in an asymptotically flat spacetime. Such a feature is generalized to a larger class of expanding spacetimes M admitting a geodesically complete cosmological horizon $\Im^-$ common to all
co-moving observers. This property is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat
at null infinity. Under suitable hypotheses on M the algebra of observables for a Klein-Gordon field in M is mapped into a subalgebra of the algebra of observables W($\Im^-$) constructed on the
cosmological horizon. There is exactly one pure quasifree state λ on W($\Im^-$) which fulfills a suitable energy-positivity condition with respect to a generator related with the cosmological time
translations. Furthermore λ induces a preferred physically meaningful quantum state λM for the quantum theory in M. If M admits a timelike Killing generator preserving $\Im^-$, then the associated
self-adjoint generator in the GNS representation of λM has positive spectrum (i.e. energy). λM turns out to be invariant under every symmetry of M which preserve the cosmological horizon. In the case
of an expanding de Sitter spacetime, λM coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case.
Item Type: Departmental Technical Report
Department or Research center: Mathematics
Subjects: UNSPECIFIED
Report Number: UTM 718, December 2007
Repository staff approval on: 07 Dec 2007
Actions (login required)
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Slope of Parallel and Perpendicular LinesAlgebraLAB: Lessons
In this lesson, the
of a
line segment
connecting two points will be compared to the
of segments parallel and perpendicular. A general formula for finding the
of a
perpendicular line segment
will be developed and used.
Parallel Lines:
The line segment shown below connects the points (1, 2) and (3, –2). As we look at this diagram from left to right, the segment slopes downward. We will define a negative number to describe the
downward slope and how steeply this line segment slopes. To do this, we notice that the vertical distance (rise) between these points is 4 and the horizontal distance (run) is 2. The slope is a
ratio of vertical to horizontal distances: Any line parallel to this segment will also have the same slope of –2.
Perpendicular Lines:
Shown below is another segment of the same length perpendicular at (1, 2). The other endpoint is (– 3, 0). The rise between endpoints is 2 and the run is 4, the opposite of the rise and run for
the segment connecting (1, 2) and (3, – 2). Also this new segment is sloping upward. The slope will be positive. We have slope = slope of the perpendicular which is – 2. This is the reciprocal,
and differs in sign.
1. If two lines are perpendicular, their slopes are negative reciprocals.
2. Parallel lines have the same slope.
Let's Practice:
i. If a line has a slope of 5, what is the slope of a line parallel and a line perpendicular?
The parallel line will have the same slope which is 5.
The perpendicular line will have a slope of
To assist you in seeing this relationship, look at 5 as the fraction reciprocal would then be
ii. If the equation of a line is given by slope of a line perpendicular?
Since the slope of the given line is reciprocal is
An extension of this example would be to calculate the equation of the line that is perpendicular to our original line, equation as y = y-axis and be oriented at 90º to each other. Note that
the dimensions of the graphing calculator's screen unfortunately do not give the correct "impression of perpendicularity."
iii. If the equation of a line is given by equation of a line parallel and containing the point (2, 3)?
The equation of a line is given by equation given by
iv. What is an equation of a line perpendicular to the line given by the equation point
(-1, -1)?
We find the slope of the line by solving for y: we have slope is slope of a perpendicular is line with this slope and containing the point (-1, -1) will have an equation given by
Important facts:
1. If two line segments are parallel, their slopes are the same.
2. If two line segments are perpendicular, their slopes are negative reciprocals.
What is slope of a line parallel to the segment connecting the points (–2, 7) and (4, 6)?
What is your answer?
What is the slope of a line perpendicular to the segment connecting the points (–2, 7) and (4, 6)?
What is your answer?
What is an equation of a line parallel to the line given by point (6, –2)?
What is your answer?
What is an equation of a line perpendicular to the line given by point (–10, 12)?
What is your answer?
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Math Forum Discussions
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Search All of the Math Forum:
Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
Topic: GUI to exe problem
Replies: 6 Last Post: Mar 21, 2013 6:33 PM
Messages: [ Previous | Next ]
Jan Re: GUI to exe problem
Posted: Mar 21, 2013 6:33 PM
Posts: 5
Registered: 11/5/12 "Steven_Lord" <slord@mathworks.com> wrote in message <kifsq9$rcr$1@newscl01ah.mathworks.com>...
> "Jan " <urbanecjan@gmail.com> wrote in message
> news:kifqop$kq1$1@newscl01ah.mathworks.com...
> > "Steven_Lord" <slord@mathworks.com> wrote in message
> > <kif1ob$mo6$1@newscl01ah.mathworks.com>...
> *snip*
> > Thanks, I am a bit closer to what I need. Problem still is, that I need to
> > choose one parameter in first radio button group and in second radio
> > button group I need to use those parameters for various simulations
> > (depending what you choose in 2nd group) . Procedure you described works
> > only if I load data under one of the 2nd button group and process them in
> > the same function.
> So you need to share data between the callback functions in your GUI? See
> the two Data Management sections in the Concepts portion of this
> documentation page:
> http://www.mathworks.com/help/matlab/code-to-run-the-gui.html
> --
> Steve Lord
> slord@mathworks.com
> To contact Technical Support use the Contact Us link on
> http://www.mathworks.com
Well, thanks. Finally I got it.
all I need is to use handles.
So my scrtip should looks like
handles.data = load('wtr.mat');
if handles.data.x < 5
% do something
thanks again
regards Jan
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st: RE: Weights vs subscripts
[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
st: RE: Weights vs subscripts
From "Nick Cox" <n.j.cox@durham.ac.uk>
To <statalist@hsphsun2.harvard.edu>
Subject st: RE: Weights vs subscripts
Date Wed, 4 Oct 2006 10:35:23 +0100
You pretty well answered your own question.
Stata permits weights in some commands, although not in
Or rather Stata permits subscripts in -replace-,
by virtue of their being part of an expression,
appearing to the RHS of an = sign. In an example
. gen d = c
. replace d[1] = 1
weights not allowed
Stata's reasoning is, or is equivalent to,
(a) "replace d" I understand.
(b) but what is that "[1]"?
(c) it is to the left of the = sign, so it
is _not_ part of the expression
(d) so the user must intend it as some specification
of weights
(e) but weights are not allowed in -replace-.
There remains the question of what you are trying
to do. I think Alex and Michael misread your question,
as your code is looping over a varlist.
foreach i of varlist x {
replace y=y[`i'-1]+ z[`i']
As the varlist contains a single variable, the
loop is redundant here, so this boils down to
replace y = y[x - 1] + z[x]
The implication is that x contains
observation numbers, in effect pointers. Is that right?
It is a way of getting some subtle effects, or you
might be confused.
Michael Blasnik
Actually, the better (faster) way to do this is
replace y=y[`i'-1] +z[`i'] in `i'
Alex Ogan
Somebody more knowledgeable will probably explain exactly why you can't
use subscripts on the left side like that.
But I can suggest a workaround:
replace y=y[`i'-1] +z[`i'] if _n==`i'
Shihe Fan
Could any one explain to me why it is OK to write the code in the
following way
gen y=0
foreach i of varlist x {
replace y=y[`i'-1]+ z[`i']
but not OK in this way
replace y[`i']=y[`i'-1] +z[`i']
the program always treat the [`i'] on the left side as
weights, instead of subscripts
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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Encyclopaedia IndexBack to start of article
6.5 The multi-fluid model for chemically-reacting flows
6.5.1 The main ideas
The fundamental ideas of MFM, which have been decribed in section 5.3, remain the same whether or not chemical reaction is present.
It needs merely to be emphasised that each fluid of the population, as well as having the attribute which distinguishes it from other members of the population, can have its own temperature and a
full range of chemical-species concentrations.
Often, two-dimensional populations are appropriate. In combustion applications, these will usually be:
1. the fuel/air ratio; and
2. some measure of the reactedness.
6.5.2 Application to the well-stirred reactor
(a) The well-stirred reactor, with a single entering stream
An important idealisation, partly realised experimentally by way of the "Longwell bomb", is the steady-flow chemical reactor that is so well stirred by mechanical means that variations of the
time-average temperature and concentration from place to place are absent.
Turbulent fluctuations remain in it, however; and the multi-fluid turbulence model can simulate their influence on reactor performance.
Some 25-fluid results from one such study will be shown.
It will be seen that the shape of the population distribution depends on the dimensionless micro-mixing and chemical-rate parameters.
The influence of the "population grid", ie the number of fluids, will also be demonstrated.
(b) The 25-fluid results
• The micro-mixing constant is varied over the range: 500 > chsoA > 6 ,
for fixed reaction-rate constant, chsoB = 5 .
• It will be seen that, as the micro-mixing constant diminishes: - the average reactedness diminishes, - the breadth of the population distribution increases, - the proportions of both unburned and
fully-burned gas increase.
The results are, for chsoA = : 500 ; 200; 100; 50; 20; 10; 7; 6
• Reduction of chsoA to 5.0 causes the reaction to be totally extinguished, ie the population consisted entirely of fluid 1, the unburned gas.
(c) Summary of population-grid-dependence results
Similar calculations have been performed for other numbers of fluids, with the following results.
no of fluids | chsoA = 500 200 100 50 20 10 7 6
100 | .980 .964 .940 .897 .780 .591 .416 .313
25 | .980 .964 .941 .898 .782 .593 .417 .302
10 | .965 .965 .942 .901 .789 .604 .431 .315
5 | .976 .964 .944 .908 .805 .638 .492 .409
4 | .970 .960 .943 .910 .815 .668 .548 .484
3 | .948 .941 .931 .910 .847 .742 .652 .601
Conclusion: If the 100-fluid results are taken as correct, the 10-fluid solution may be regarded as good enough for many practical purposes. So the use of MFM need not be very expensive.
(d) The well-stirred reactor with 2 streams of differing compositon
In order to illustrate the use of MFM with a two-dimensional fluid population, two further pictures will be shown.
They come from a study of a reactor into which enter steadily:
1. a cold fuel-lean gas stream, and
2. a hot fuel-rich gas stream.
The two-dimensional histogram has fuel/air ratio as the horizontal axis and reactedness as the vertical axis.
The extent of filling of the grid of boxes indicates how much of each fluid is present.
Case for which the mixing constant is halved
6.5.3 Application to the paddle-stirred reactor
(a) The problem
Industrial reactors, unlike the ideal well-stirred one, are far from having uniform time-mean concentrations and temperatures.
They are also three-dimensional; and unsteady analysis may be needed in order to represent properly the effect of the stirring paddle.
PHOENICS may be used for simulating such reactors; and the use of the multi-fluid model reveals the importance of being able to simulate the micro-mixing process.
These points will be illustrated by the following extract from a recent study.
(b) The geometry
The geometry and computational domain are shown below.
The impeller speed is 500 rpm, the dynamic laminar viscosity is 1.0cP and the water density is 1000 kg/m
The grid is divided into two parts, namely an inner part which rotates at the same speed as the impeller, and an outer part which is at rest.
The total number of cells was 31365 (45 vertical, 41 radial and 17 circumferential).
A view of the 3-dimensional body-fitted grid
(c) The starting condition
The sketch below illustrates the apparatus and the initial state of the two liquids.
They are both at rest, and are separated by a horizontal interface
The paddle is supposed to be suddenly set in motion.
The computational task is to predict both the macro-mixing, represented by the subsequent distributions of velocity, pressure and time-average concentration, but also the extent to which the two
liquids are mixed together at any point.
| ||| |
| upper |.| |
| liquid ||| |
| |.| acid |
| lower |.| alkali |
| liquid ||| |
| --------- |
|paddle ///////// |
| . |
.< axis of
The stirred mixing tank
An 11-fluid version has been employed to simulate the mixing of an acid upper liquid and an alkaline lower liquid in the tank.
(d) The distributions after 10 paddle rotations
The salt-concentration distribution is calculated on two different assumptions, namely:
1. that the multi-fluid model is valid, so that each fluid reacts at its own rate, according to its own acid-base ratio; and
2. that fluctuations can be neglected, so that the salt-production rate depends only on the total-population acid-base ratio.
Assumption (2) is the conventional, ie single-fluid, model.
The salt concentrations predicted by the multi-fluid model.
The salt concentrations predicted by the single-fluid model.
The single-fluid model predicts appreciably higher salt yields than the multi-fluid model.
They are larger than the multi-fluid values, because micro-mixing is presumed (wrongly) to be perfect.
Some fluid-population histograms are now shown, for points located on a radius near the top of the tank. Their shapes would be hard to "presume" correctly.
Radius index =: 1; 2; 3; 4; 5; 6
(e) Conclusions
• It is possible, and not very expensive, to compute the probability-density functions describing the extent of micro- mixing of initally-separated materials in a stirred tank.
• The 11 fluids used in the calculations were probably too few; but any number can be used (100 is common) so that (fluid-) grid- independence can be tested.
• When a chemical reaction can take place between the materials, the yield can be computed accurately only by way of a multi-fluid model.
• Even without further refinement, MFM is probably more reliable than any presumed-pdf method, because it PREDICTS instead of PRESUMING.
6.5.4 Application to a 3D combustor
(a) The problem of predicting smoke production
Smoke is produced in combustors in regions of high temperature and excess fuel. Its rate may be computed, given:
(1) suitable chemical-kinetic rate formula; and (2) the distributions of temperature and fuel concentration.
Of course, it is not the time-mean temperature and concentration that are relevant, because of the turbulent fluctuations.
This will be demonstrated by attachment of a smoke model to PHOENICS library case 492, together with an 11-fluid MFM, with fuel-air ratio as the distinguishing attribute.
The conclusions are similar to those for the paddle-stirred reactor: only when the multi-fluid nature of turbulence is accounted for can chemical-reaction-rate predictions be regarded as credible.
(b) The thermodynamics, stoichiometry and smoke kinetics
* The equilibrium SCRS scheme is | * <-temperature
used, which entails that all | * *
fluid properties are functions | * *
of the mixture fraction, as 1.0 - * + * +<-fuel
indicated. |+ * + + *+
| + * + + + *
* Here "fuel" signifies the | +* + + +
fuel-rich air fuel mixture | *+ + + +
which is injected into this | * + +
particular combustion chamber. |* + +<-oxid + +
| + + + +
* The micro-mixing rate is taken | + + + +<-product
as 10 * epsilon/k * the |+ + + +
product of the mass fractions. 0.0 |_________+_________| 1.0
mixture fraction ->
* Smoke-production rate is taken as: const * (f - f_stoich) * T**5
(c) Computed contours of some all-fluid-average quantities
In the following contour diagrams, the flow is from right to left.
Only one sector of the combustion chamber is shown, because the pattern of injection ports is repeated 6 times around the circumference.
Longitudinal velocity, w1, contours
These and other contoured values are the averages over all fluids
Unburned-fuel contours. It can be seen that a small amount of unburned fuel escapes from the chamber near the combustor wall.
(d) Computed fluid-population distributions, with mixture fraction as population-distinguishing attribute
Now follow a set of fluid-population distributions for 10 points located on a radius in the middle of the exit plane.
Radius index = 1; 2; 3; 4; 5; 6; 7; 8; 9; 10
(e) Corresponding smoke concentration contours
Each of the fluids produces smoke at the rate which corresponds to its own fuel-air ratio and temperature. This leads to an all-fluid-average smoke concentration as follows.
Smoke concentration according to the multi-fluid model.
Next will be shown computations based on the neglect of the mixture-fraction fluctuations.
Smoke concentration according to the single-fluid model
The maximum value is 25 % greater; and the distribution is different.
(f) Conclusions
• The difference between the single-fluid and multi-fluid smoke- production rates is, as expected, significant.
• Neglect of the influence of fluctuations, or basing calculations on unlikely-to-be-correct pdf presumptions, is therefore dangerous for combustor designers seeking to reduce smoke.
• A similar demonstration could easily be made for NOX or other pollutant.
• The sufficiency of 11 fluids can be tested by "fluid-grid- refinement" studies; but these are easy to do, as was shown for the well-stired reactor.
• Greater realism can of course be attained by the use of a two- dimensional population, with fuel-air ratio as one dimension and reactedness (or an equivalent) as another.
(d) Future developments
• Two-dimensional-population calculations have already been made for turbulent diffusion flames (Spalding, 1995). They appear to present no problems of convergence, no matter how many fluids are
• Of course, the greater the number of fluids, the larger is the computer time. Therefore, for large 3D (and especially time-dependent) calculations, the number of fluids will be made no larger
than necessary.
• There is no need for the number to be the same over the whole field; or to remain constant throughout the time span. Population- grid-adaptation strategies have already been devised.
• Computational expense will therefore present no serious barrier to the use of MFM for combustor simulation.
• The main need is for acceptance and use of the model, folllowed by experimental validation.
6.5.5 Implementation in PHOENICS
Examples of chemically-reacting flows, simulated by way of the MFM, are to be found in the MFM-option Library of PHOENICS.
In all the examples provided so far, the number of fluids is uniform and constant, the use of computer-time-economising devices being still a matter for research and development.
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CBM #2 Vocabulary
28 terms · Vocabulary for the CBM #2
The sum of all the items, divided by the number of items in the set (sometimes called an average)
The middle value when the data are in numerical order or the mean of the two middle numbers if there is an even number of items
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Discrete probability distribution problem
October 1st 2013, 03:39 AM #1
Nov 2011
Discrete probability distribution problem
The problem:
Cecile is a nurse and has a 0,25 of working a night shift. In the latest time period there have been 10 cases of death at night time by the nursing home,and a remarkably large number of these
have happened while Cecilie has been on duty. 7 of the cases of death have been on her duty. Therefore, an investigation has been started to find out if she has anything to do with these deaths.
Assume in the following that all the deaths are natural and that they have happened at
dierent nights.
Let X be a stochastic variable that describes the number out of n = 10 natural deaths that
happen on Cecilie's night shifts.
What is the probability of 7 or more deaths, out of the 10 deaths at night time, happening while Cecilie is on duty?
The right answer should be: 0,004
I am not sure how to calculate this, I tried finding the chance of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 deaths happening while she is on duty, but I am not sure how to find the chance of it being 7
or more.
I would really appreciate any feedback, thank you!
Re: Discrete probability distribution problem
The problem:
Cecile is a nurse and has a 0,25 of working a night shift. In the latest time period there have been 10 cases of death at night time by the nursing home,and a remarkably large number of these
have happened while Cecilie has been on duty. 7 of the cases of death have been on her duty. Therefore, an investigation has been started to find out if she has anything to do with these deaths.
Assume in the following that all the deaths are natural and that they have happened at
dierent nights.
The right answer should be: 0,004
You can see the calculation here.
Re: Discrete probability distribution problem
Wow, thank you so much! I really appreciate it! I hope you have a good day.
Re: Discrete probability distribution problem
I was wondering if you could help me with a futher part of the question:
Lets assume that about 300 other nurses have the same sort of position as Cecilie, in
nursing homes around Norway. What is the probability of at least one of these 300
nurses having 7 or more out of 10 natural deaths happening while they are on duty?
Does the answers in this subsection give any reason to strengthen the suspicion towards
Cecilie? Justify your answer.
October 1st 2013, 04:22 AM #2
October 1st 2013, 04:34 AM #3
Nov 2011
October 2nd 2013, 12:06 AM #4
Nov 2011
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Why does 0! = 1
The text book says because it's convenient. Is there a better explanation or proof. Thankyou.
Hopefully this helps do not be alarmed by this. a lot of time mathematicians make definitions because they work, and they don't contradict anything else. but they allow for things to work nicely.
this is fine
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[FOM] Bi-interpretability vs mutual interpretability - and Woodin
Ali Enayat ali.enayat at gmail.com
Sun Jan 24 21:16:28 EST 2010
Thomas Forster's recent posting asks:
>Can you say something about the nature of the interpretability between the
>theory of HC and the theory of P(N)? It's not entirely clear from Woodin's
>article what language these theories are to be expressed in, and that
>clearly matters!
My response: The particular language for P(N) here is none other than
the language of "analysis", which is often formulated in a two-sorted
FIRST ORDER LANGUAGE, a sort for "natural numbers", a sort for "sets
of natural numbers" (aka "real numbers" sometimes). There is also a
relation symbol for expressing the membership relation between natural
numbers and sets of natural numbers, as well as arithmetical operation
symbols {plus and times}, sometimes augmented by other arithmetical
operation and relation symbols.
Nowadays the above language is often called the language of "second
order arithmetic", which can be misleading, since the theory is
handled by first order, not second order logic.
The language for HC, on the other hand, is the usual (one-sorted)
language of set theory {epsilon}.
There is a "natural" interpretation of second order arithmetic plus
the choice scheme in the theory ZF\{Power set} plus "every set if at
most countable". "Numbers" are interpreted as finite von Neumann
ordinals, and "sets of numbers" are interpreted in the obvious way.
As it turns out, the interpretation I can be inverted by an
interpretation J of ZF\{Power set} plus "every set is at most
countable" in second order arithmetic plus the choice scheme. Note
that the choice scheme of second order arithmetic translates to the
replacement scheme of set theory via J.
J is usually defined in terms of trees on natural numbers, but one can
also define J by using *pointed* well-founded extensional relations R
on natural numbers, where "pointed" means that R has a "top" element.
This device is quite versatile: it can be used to "simulate" a
Zermelian structure within Quine's NF (as well as Quine-Jensen's NFU),
as noted by Hinnion, and later but independently by Holmes. More
recently, the same device was used by Koepke to interpret ZFC in the
second order theory of ordinals.
Finally, regarding the last question posed by Forster:
>And how important is it for W's programme that the theories should be mutually interpretable (in whatever sense is in play)?
Ever since Descartes' discovery of the two-way street between Geometry
and Algebra, mathematicians have added the following meta-strategy to
their toolbox: when faced with a difficult problem P pertaining to
some domain D, look for an "equivalent" domain D' and try solving the
corresponding problem P' instead of P; and if successful, translate
the solution S' of P' back to obtain the solution S of P.
In our discussion, the "equivalence" is the bi-interpretaion
relationship between theories, which, as emphasized in my previous
posting, is much stronger than mutual interpretability, since it
provides a *canonical correspondence* between models of a theory T and
models of another theory T' bi-interpretable with T.
Best regards,
More information about the FOM mailing list
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What is AMPS TO KILOWATTS CALCULATOR 3 PHASE?
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