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MAGIC: Applications of model theory to algebra and geometry (MAGIC004)
The course begins on Tuesday 17th January, 10 am. See you there!
This will be a 10 hour "version" of what was previously given (Spring 2008, Spring 2010) as a 20 hour course. However, I plan to be somewhat more ambitious, at the expense of leaving many details of
the introductory material to be done in exercises. Among the new aspects of the new version will be the inclusion of stability theory. The first 5 hours will introduce model theory and stability
theory, including stable groups. The last 5 hours will concern connections and applications, largely where groups of one form or the other are involved, and will be taken from among: geometric group
theory, diophantine geometry over number fields and function fields (Manin-Mumford, Mordell-Lang), approximate subgroups,..
Spring 2012 (Monday, January 16 to Friday, March 23)
• Tue 10:05 - 10:55
Anand Pillay
Email pillay@maths.leeds.ac.uk
Phone (0113) 343 5171
Robert Barham
Ahmet Cevik Cong Chen Andrew Furnas
(Leeds) (Leeds) (Leeds)
Ronnie Nagloo Pedro Valencia Dan Dan Yang Rida-e Zenab
(Leeds) (Leeds) (York) (York)
Some familiarity with first order logic would be helpful but not essential.
beginitemize item textbfLectures 1 to 5: BASICS OF MODEL THEORY AND STABILITY THEORY: First order languages, structures and theories, compactness, types, saturation and homogeneity, stability,
stable groups. item textbfLecture 7 to 10: APPLICATIONS: I will cover 4 or 5 topics, explaining why and how material from the earlier lectures can give insights in areas of algebra, geometry, and
number theory. More details will be given closer to the start date. enditemize
Model theory: an introduction Marker
A course in model theory: an introduction to contemporary mathematical logic Poizat and Klein
Lecture notes in Model Theory A. Pillay
Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can
buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for
the first time.)
The assessment will based on doing four assignments. Passing three out of these four assignments is required for a pass overall. The assignments will be connected with filling in details and solving
some problems related to basic material in the first past of the course, and can serve as a means for non-experts to gain some entry into and competence in general model theory.
Assignment 1
Files: Exam paper
Deadline: Friday 23 March 2012 (756.2 days ago)
Assignment 2
Files: Exam paper
Deadline: Friday 23 March 2012 (756.2 days ago)
Assignment 3
Files: Exam paper
Deadline: Friday 23 March 2012 (756.2 days ago)
Assignment 4
Files: Exam paper
Deadline: Friday 23 March 2012 (756.2 days ago)
Files marked L are intended to be displayed on the main screen during lectures.
Week(s) File
magic2012-lecturenotes.pdf L | {"url":"http://maths-magic.ac.uk/course.php?id=201","timestamp":"2014-04-19T04:19:51Z","content_type":null,"content_length":"19648","record_id":"<urn:uuid:c14ce2eb-41a1-462a-8eef-5a5eb154949b>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00538-ip-10-147-4-33.ec2.internal.warc.gz"} |
Multiple Integrals (Volume + Centroid)
November 19th 2007, 04:11 PM
Multiple Integrals (Volume + Centroid)
A) Find the volume bounded by sphere rho=rt(6) and the paraboloid
z=x^2 + y^2
B) Locate the centroid of this region
Any help, tips, work, or similar problems would be greatly appreciated. =)
November 19th 2007, 04:29 PM
I start you off. The upper surface $x^2+y^2+z^2=6^2$ and the lower surface $z=x^2+y^2$. When they intersect the region over we are integrating is the circle $x^2+y^2=6^2$. Now use polar change of
November 20th 2007, 04:31 PM
When you say polar change of variable you mean things like
x = rcos theta
y = rsin theta
z = z
and then i would take the triple integral of that?
I always have trouble setting up the problem. =(
Like setting the integral ranges.
November 20th 2007, 04:44 PM
The upper surface (sphere) is given by $z=\sqrt{36 - x^2-y^2}$.
Thus, the volume is,
$\int_A \sqrt{36-x^2-y^2} - (x^2+y^2) \ dy~dx$ where $A$ is the disk of radius $6$.
Using polar change of variable,
$\int_0^{2\pi} \int_0^6 (\sqrt{36-r^2} - r^2)rdr~d\theta$(Wink)
November 24th 2007, 09:17 AM
When I tried to evaluate the integral, I can do the R part of it, but theres no place for me to put the theta value. Shouldn't there be two different variables in a double integration problem
like this?
November 24th 2007, 09:49 AM
Is that ${\rho}=\sqrt{6}$?.
If so, try:
$\int_{0}^{2\pi}\int_{0}^{\sqrt{2}}\int_{r^{2}}^{\s qrt{6-r^{2}}}rdzdrd{\theta}$
November 24th 2007, 10:00 AM
hmm okay. So when I evaluate it
how do I do the second part of the integral. That rdr complicates everything. | {"url":"http://mathhelpforum.com/calculus/23149-multiple-integrals-volume-centroid-print.html","timestamp":"2014-04-16T06:38:31Z","content_type":null,"content_length":"9799","record_id":"<urn:uuid:01ecadb7-fcd4-4e4c-b645-c03f620b9f3f>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00209-ip-10-147-4-33.ec2.internal.warc.gz"} |
A homology theory for étale groupoids
Results 1 - 10 of 24
, 2000
"... In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. As a first
application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and ..."
Cited by 49 (16 self)
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In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. As a first application
we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu [47]). As a second application we extend van Est’s argument
for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately gives a slight improvement of Hector-Dazord’s integrability criterion [12]. In the third section we describe the
relevant characteristic classes of representations, living in algebroid cohomology, as well as their relation to the van Est map. This extends Evens-Lu-Weinstein’s characteristic class θL [17]
(hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles
- ANN. SCI. ÉCOLE NORM. SUP , 2004
"... In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S 1-gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott
periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framew ..."
Cited by 42 (12 self)
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In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S 1-gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity,
and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological
spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called
“twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C ∗-algebras.
- Advances of Mathematics
"... The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the non-commutative geometry
and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of th ..."
Cited by 17 (6 self)
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The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the non-commutative geometry and
algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation
, 2004
"... We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In
particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantal ..."
Cited by 16 (7 self)
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We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular,
to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales,
which are given a rather simple characterization and are here called inverse quantal
- J. Reine Angew. Math
"... Abstract. We give a superconnection proof of Connes ’ index theorem for proper cocompact actions of étale groupoids. This includes Connes ’ general foliation index theorem for foliations with
Hausdorff holonomy groupoid. 1. ..."
Cited by 13 (2 self)
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Abstract. We give a superconnection proof of Connes ’ index theorem for proper cocompact actions of étale groupoids. This includes Connes ’ general foliation index theorem for foliations with
Hausdorff holonomy groupoid. 1.
"... Abstract. In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via
Chern-Weil theory. For an arbitrary gerbe L, a twisting L Korb(X) of the orbifold K-theory of X is constru ..."
Cited by 9 (2 self)
Add to MetaCart
Abstract. In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via
Chern-Weil theory. For an arbitrary gerbe L, a twisting L Korb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal
[2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold. Contents
, 2001
"... Let G be a Lie groupoid over M such that the target-source map from G to M × M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many
critical points), then the restriction G|U of G to some neighborhood U of O in M is isomorphic to a similar ..."
Cited by 9 (0 self)
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Let G be a Lie groupoid over M such that the target-source map from G to M × M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many
critical points), then the restriction G|U of G to some neighborhood U of O in M is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid G|O on
the normal bundle NO. The proof uses a deformation argument based on a cohomology vanishing theorem, along with a slice theorem which is derived from a new result on submersions with a fibre of
finite type.
"... This paper is concerned with characteristic classes in the cohomology of leaf spaces of foliations. For a manifold M equipped with a foliation F it is well-known that the coarse (naive) leaf
space M/F, obtained from M by identifying each leaf to a point, ..."
Cited by 9 (3 self)
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This paper is concerned with characteristic classes in the cohomology of leaf spaces of foliations. For a manifold M equipped with a foliation F it is well-known that the coarse (naive) leaf space M/
F, obtained from M by identifying each leaf to a point, | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=2697211","timestamp":"2014-04-16T17:32:29Z","content_type":null,"content_length":"32308","record_id":"<urn:uuid:2c30ae4d-f759-4a5b-bca8-9575c5029e7c>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00217-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Logarithms: History and Use
Date: 7/12/96 at 16:50:38
From: Linda Temple
Subject: Logarithms: Why they Work, History, and Name
I have been asked to explain logarithms from a non-numerical sense to
non-math-oriented people. It doesn't seem to be enough for me to show
the equation and how it works, they want to know why. Any thoughts?
Also, do you have short anecdotal history for the development of the
concept of logarithm?
Finally, why is it called a "logarithm"? logos = reason, arithmos =
Date: 7/13/96 at 12:21:19
From: Doctor Anthony
Subject: Re: Logarithms: Why they Work, History, and Name
It is a very great economy of effort if we can reduce multiplication
to the addition of two numbers. The possibility of adding numbers
that can be looked up in tables compiled "forever," as Napier
remarked, instead of carrying out a lengthy process of multiplication,
was suggested in two ways that were quite independent. The first
arose in connection with the preparation of trig. tables for use in
navigation. The second was closely connected with the laborious
calculation involved in reckoning compound interest on investments.
In 1593 two Danish mathematicians suggested the use of trig. tables
for shortening calculations. They used the formula:
sin(A)*cos(B) = (1/2)sin(A+B) + (1/2)sin(A-B)
Thus to multiply 0.17365*0.99027, you look up in tables and find
0.17365 = sin(10), 0.99027 = cos(8)
and the above formula gives sin(10)*cos(8) = (1/2)(sin(18) + sin(2))
From tables sin(18) = 0.30902 sin(2) = 0.03490
sin(18) + sin(2) = 0.34392 and (1/2)(sin(18)+sin(2)) = 0.17196
Giving 0.17365*0.99027 = 0.17196
This device probably suggested to Napier, who is usually called the
inventor of logarithms, a simple method for multiplying by a process
of addition.
Napier had been working on his invention of logarithms for twenty
years before he published his results, and this would place the origin
of his ideas at about 1594. He had been thinking of the sequences
which had been published now and then of successive powers of a given
number. In such sequences it was obvious that sums and differences of
indices of the powers corresponded to products and quotients of the
powers themselves; but a sequence of integral powers of a base, such
as 2, could not be used for computations because the large gaps
between successive terms made interpolation too inaccurate. So to keep
the terms of a geometric progression of INTEGRAL powers of a given
number close together it was necessary to take as the given number
something quite close to 1.
Napier therefore chose to use 1 - 10^(-7) or 0.9999999 as his given
number. To achieve a balance and to avoid decimals, Napier multiplied
each power by 10^7. That is, if N = 10^7[1 - 1/10^7]^L, then L is
Napier's logarithm of the number N. Thus his logarithm of 10^7 is 0.
At first he called his power indices "artificial numbers", but later
he made up the compound of the two Greek words Logos (ratio) and
arithmos (number).
Napier did not think of a base for his system, but nevertheless his
tables were compiled through repeated multiplications, equivalent to
powers of 0.9999999 Obviously the number decreases as the index or
logarithm increases. This is to be expected because he was essentially
using a base which is less than 1. A more striking difference between
his logarithms and ours lies in the fact that his logarithm of a
product or quotient was not equal to the sum or difference of the
logarithms. If L1 = log(N1) and L2 = log(N2), then
N1 = 10^7(1-1/10^7)^L1 and N2 = 10^7(1-1/10^7)^L2, so that
N1*N2/10^7 = 10^7(1-1/10^7)^(L1+L2), so that the sum of Napier's
logarithms will be the logarithm not of N1*N2 but of N1*N2/10^7.
Similar modifications hold, of course, for logarithms of quotients,
powers and roots. These differences are not too significant, for they
merely involve shifting a decimal point.
Napier's work was published in 1614 and was taken up enthusiastically
by Henry Briggs, a professor of Geometry at Oxford. He visited Napier
and discussed improvements and modifications to Napier's method of
logarithms. Briggs proposed that powers of 10 should be used with
log(1) = 0 and log(10) = 1. Napier was nearing the end of his life,
and the task of making up the first table of common logarithms fell to
Briggs. Instead of taking powers of a number close to 1, as had
Napier, Briggs began with log(10) = 1 and then found other logarithms
by taking successive roots. By finding sqrt(10) = 3.162277 for
example, Briggs had log(3.162277) = 0.500000, and from 10^(3/4) =
sqrt(31.62277) = 5.623413 he had log(5.623413) = 0.7500000.
Continuing in this manner, he computed other common logarithms.
Briggs published his tables of logarithms of numbers from 1 to 1000,
each carried out to 14 places of decimals, in 1617. Briggs also
introduced the words "mantissa" for the positive fractional part and
"characteristic" for the integral part (positive or negative).
The first tables of logarithms contained inaccuracies which were
noticed and corrected from time to time. The labor expended in
constructing them was enormous, and it stimulated the search for
better methods of calculating them. This gave a new impetus to the
study of infinite series, for example sqrt(2) = (1 - (1/2))^(-1/2)
which gives rise to an infinite, convergent series when expanded
according to the binomial theorem. This work culminated in the
extremely important exponential series:
where e = Limit {1 + 1/n}^n as n -> infinity. It is easy to show
e^x = Limit {1 + 1/n}^(nx) generates the series shown below:
e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... to infinity, and
e = 1 + 1 + 1/2! + 1/3! + 1/4! + .... = 2.718281828...
e is now used as the base of logarithms in almost all advanced work.
-Doctor Anthony, The Math Forum
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Beyond Higgs: On Supersymmetry (or Lack Thereof) | Guest Blog, Scientific American Blog Network
With the search for the Higgs boson, the last missing piece of the Standard Model of particle physics, apparently reaching its long-anticipated-and-finally-successful conclusion, anticipation of the
next set of discoveries is growing.
Recently the Stanford campus hosted a smallish gathering celebrating the 60^th birthday of Savas Dimopoulos, justly acclaimed by each of the attendees as the (or at least one of the few) most
insightful particle physics model builders of the last 30 years. (And my PhD adviser.) Now you’d think that the leading topic of discussion at such an event would be the details of the ongoing Higgs
search – has it or hasn’t it been discovered? Does the fact that the two relevant experiments at CERN’s Large Hadron Collider (LHC) – ATLAS and CMS – both have a signal indicative of a new particle
with the same mass? And what about the supportive analysis coming from Fermilab’s Tevatron?
Surprisingly (to the outsider) this was all considered old news. Repeatedly, the theorists joked that, with the exception of the actual CERN experimentalists present, all of us know that the Higgs
has now been discovered with a mass of 125 GeV/c^2. (It hasn’t, quite, but the hints are strong.) The message was clear: “We’ve known for decades that the Higgs is going to be found. So break
open the champagne and get the celebrating over with, because what we really want to know is — which is the correct version of Beyond the Standard Model physics?” With a brief nod to large extra
dimensions (a Dimopoulos, and associates A, D and also I, idea) and a fond farewell to Technicolor (another idea that Dimopoulos helped advance), the focus turned again and again to the likely suite
of Supersymmetric (SUSY) particles (yet another stock in which Dimopoulos is heavily invested).
Supersymmetry – a theory that posits that for every known particle there is another (or more than one) yet-to-be-discovered partner particle – is the leading candidate for physics Beyond the Standard
Model. It is central to string theory (a.k.a. super-string theory), required for gauge coupling unification (see below), useful for solving the Higgs Fine Tuning Problem (definitely see below) and
also gives us the leading candidate for dark matter – the Lightest Supersymmetric Particle (LSP).
But I’m getting way ahead of myself, and probably you. Especially since I and my colleagues have come to believe that the principal indictment of the Standard Model, which has been used to argue so
forcefully for Beyond the Standard Model (BSM) physics is, hmmm, dubious. Or as one of those colleagues would say – completely wrong. A main rationale for supersymmetry evaporates on closer
So what is Beyond The Standard Model (BSM) physics, why are people so convinced it is around the corner, and should they be?
At least since the discovery of the W and Z particles at CERN in 1983, physicists have been pretty much convinced that the Standard Model (SM) that emerged from the late 1960s and early 1970s is the
correct model of fundamental physics. At least at energies below the so-called weak-scale – a few hundred GeV – or maybe a few times that. But particle theorists variously hoped/expected/knew that
at higher energies the Standard Model was not the whole story, and a more fundamental theory would need to be found.
There are two types of reasons to doubt the completeness of the Standard Model – aesthetic (philosophical) and mathematical.
Aesthetic problem number one, physicists adore simplicity. Zero and one are our favorite numbers. Two can be suffered. After two comes “too many”, although identical copies (twins, triplets, …)
may receive special dispensation. The Standard Model has too many too-many’s: three fundamental forces (a.k.a. gauge groups); way too many fundamental fermions (particles that make up matter)– three
generations each with at least 5 representations (groups) of them — plus three sets of gauge bosons and the set of particles of which the Higgs boson is a member. It also has far too many (more than
20) independent parameters.
Aesthetic problem number two –– for no apparent reason the weak scale is much (as in about 10^16 times) smaller than what we believe to be the fundamental energy scale of physics – the Planck scale
(about 10^19 GeV), a scale set by the strength of gravity (the one fundamental force not included in the Standard Model). This is known as the (Weak) Hierarchy Problem – and can also be understood
in terms of the absolutely enormous strength of the three Standard Model forces compared to that of gravity between pairs of fundamental particles separated by appropriately microscopic scales.
It is however the technical problem that has carried the most weight in convincing people that there must be physics beyond the Standard Model. It is the story we tell our children — quantum
mechanics makes the Standard Model unstable. Quantum mechanics teaches us that, as a particle such as a Higgs boson travels along, it can emit and reabsorb another particle. This process represents
a “loop contribution” to the mass of the Higgs boson, so-called because a pictorial representation of the process – Feynman diagrams – depicts these processes as loops attached to the traveling Higgs
Unfortunately, when you add up the loop contributions to the mass of the Higgs boson from all possible particles with all possible energies and momenta, they appear to be infinite or at least
proportional to the maximum possible momentum that can be carried. For technical reasons these are called quadratic divergences and are widely derided. For the actual Higgs boson mass to be finite,
there must apparently be subtle and precise cancellation of the loop contributions against the underlying “tree” (loop-free) mass. This Higgs Fine-Tuning Problem, so the lore tells us, must be
BSM physics is the proposed remedy. Supersymmetry cancels the loop of every known particle against the loop of an as-yet-to-be-discovered partner particle. Technicolor eliminates the Higgs boson
– replacing it by a composite of new particles called techni-quarks. If there are large extra dimensions then the largest momentum that can circulate in a loop is actually only a little larger than
the weak scale. Clearly BSM physics is not just desirable but essential.
Recently, however, my colleague Bryan Lynn suggested, and together with Katie Freese and Dmitry Podolsky, he and I explained, how the Standard Model actually comes up with a remedy all on its own.
The Higgs boson is one member of a set of quadruplets in the Standard Model. At energies below the weak scale, its three siblings get eaten – they get incorporated into the W and Z bosons. According
to a famous theorem due to MIT’s Jeffrey Goldstone (hence “Goldstone’s Theorem”), the masses of the three siblings must be exactly zero. In particular, the quadratically divergent contribution to
their masses are zero.
Although this doesn’t force the mass of the Higgs boson to be zero (a good thing, since it seems likely to be about 125 GeV/c^2), it does mean that the quadratic divergences in the Higgs mass that
have worried us for decades are not a problem of the Standard Model after all.
Now, not everybody buys our argument. Some of them prefer to focus on the aesthetic challenge of the Weak Hierarchy Problem, while others argue that we have no choice but to add quantum gravity to
the Standard Model, inevitably resurrecting the Higgs Fine Tuning Problem.
We would counter that the absence of a Higgs Fine Tuning Problem in the Standard Model is such a virtue that, absent any hard evidence for BSM physics, preserving the Standard Model’s Goldstone
miracle should be taken as a requirement of any proposed BSM theories.
The implication is clear. If there is no problem, there may be no need for a solution. Beyond the Standard Model Physics isn’t ruled out by the absence of a Higgs Fine Tuning Problem in the
Standard Model, but it does mean that the Standard Model may well be the whole story, or at least the whole story at the energies that the LHC can command. In short, don’t be surprised if the Higgs
is the last new particle discovered by the LHC. Theorists may hunger for physics beyond the Standard Model, but nature may be quite content without it, thank you very much.
Related at Scientific American: Is Supersymmetry Dead?
1. lumidek 6:42 am 06/21/2012
Exactly one year ago, Dmitry Podolsky, a co-author of yours, had a correspondence – about 10 e-mails – with me. He argued in favor of a simpler “argument” than your paper’s argument why there
were no quadratic divergences to the Higgs mass.
The claim was that they only affected the mass parameters so the quartic coupling wasn’t affected by them and the influence of the quadratic divergences on the Higgs mass is actually linked
to the correction to the tadpole trying to change the vev (location of the minimum of the potential) which has to be zero for stability. So even the former thing is zero.
For hours if not days, I was a bit confused but then I came back to my senses and wrote him an explanation – one that he never replied to again. The explanation is that the correction of the
tadpole is indeed linked to the correction of the Higgs mass and the vev. That’s OK but *none* of these mutually related things is guaranteed to be zero by any principle. In fact, we know in
particular theories it’s not zero. The quadratic divergences reflect the big sensitivity of the Higgs mass/vev on all the detailed parameters of any high-energy theory with start with. We may
choose to say whether it’s the Higgs mass or the Higgs vev that is threatened by these divergent terms but both of them are!
Now, the final paper of yours talks about the pions etc. It is a bizarre treatment. From the viewpoint of the fundamental theory, pions are composite objects and their properties are derived
quantities. We don’t really have special counterterms for pions or even technipions in the Standard Model. They’re not fundamental parameters. So at most, you link the divergences to yet
another quantity that isn’t guaranteed to be zero by any principle. At most, you are supplying a new principle but this principle is equivalent to saying “there shouldn’t be a hierarchy
problem”. You don’t have any independent justification why it’s not there.
It’s only the Goldstone bosons that have a reason to be massless, the Goldstone theorem, but there are only 3 of them, one for each broken generator, and the physical Higgs boson simply isn’t
one of them. That’s why I believe that your paper boldly claiming that the quadratic divergences aren’t really there is wrong.
Link to this
2. jctyler 7:32 am 06/21/2012
“(It hasn’t, quite, but the hints are strong.)”
where have I heard this before? I raise to 400
Link to this
3. christinaak 8:10 am 06/21/2012
I am with jctyler on this one. I am still betting they have not found the Higgs.
Link to this
4. rloldershaw 11:01 am 06/21/2012
The standard model cannot possibly be the final story, as all physicists know.
I do not mean to be disrespectful or excessively negative, but it is important to keep in mind the true status of the Standard Model of particle physics when evaluating our present
understanding of nature at the subatomic level.
1. The Standard Model is primarily a heuristic model with 26-30 fundamental parameters that have to be “put in by hand”.
2. The Standard Model did not and cannot predict the masses of the fundamental particles that make up all of the luminous matter that we can observe.
3. The Standard Model did not and cannot predict the existence of the dark matter that constitutes the overwhelming majority of matter in the cosmos. The Standard Model describes
heuristically the “foam on top of the ocean”.
4. The vacuum energy density crisis clearly suggests a fundamental flaw at the very heart of particle physics. The VED crisis involves the fact that the vacuum energy densities predicted or
measured by particle physicists (microcosm) and cosmologists (macrocosm) differ by up to 120 orders of magnitude (roughly 10^70 to 10^120, depending on how one ‘guess-timates’ the particle
physics VED).
5. The conventional Planck mass is highly unnatural, i.e., it bears no relation to any particle observed in nature, and calls into question the foundations of the quantum chromodynamics
sector of the Standard Model.
6. Many of the key particles of the Standard Model have never been directly observed. Rather, their existence is inferred from secondary, or more likely, tertiary decay products. Quantum
chromodynamics is entirely built on inference, conjecture and speculation. It is too complex for simple definitive predictions and testing.
Robert L. Oldershaw
Discrete Scale Relativity
Fractal Cosmology
Link to this
5. anselm 1:45 am 06/28/2012
“Quantum mechanical fluctuations can produce the the Mirror image of the, physical dark matter and the intervening string bundles caused by Dark energy, cascading towards the SINGULARITY ,
“If you would just, watch the CELESTIAL BODIES Rotation/ORBITALS revolving around the stars,their Geometrical TRIADS in Formation–at and precision FIXED TIME that evolves at extreme low
energy limits — a focused interpretation and the TIME freeze Synchonization in SIMULTANEOUS CUSPS–just twist time and FLIP space the right way, you might physically commence production of the
POSTULATED BEAUTY PHYSICALLY-” THE MAGNIFICIENT HIGGS !!!!!!!!!!
Link to this
6. dpodolsky 4:59 pm 06/29/2012
Dear Lumo,
My memory tells me that it was me who made the last reply in that discussion but if I am wrong, I ask you to forgive me. My answer to your observation was more or less as follows.
1. Stability
Whether renormalized tadpole is zero on not shows one whether Higgs particles can be spontaneously produced from the vacuum or not. If they can be produced, then the vacuum is not stable. The
particles will be produced from vacuum and the physical Higgs’ VEV will change until the renormalized tadpole becomes zero.
A simple observation which goes back to work done by Bryan Lynn long time ago (and which is actually present in the textbook by Peskin and Schroder citing Bryan’s work) is that the
quadratically divergent contribution to the Higgs’ mass coincides identically with the expression for the renormalized tadpole. Hence the first statement of the paper – if renormalized vacuum
is stable w.r.t. spontaneous particle production, quadratic divergences are absent in the Higgs’ mass. This observation is actually well known to CMT physicists. There are plenties of
phenomena in condensed matter physics involving spontaneously broken global symmetries, and none of them feature quadratic (or linear – because they are (2+1)d) divergences in the effective
mass of quasiparticles.
2. Massless modes
I am afraid you cannot simply say that there are several independent generators of global symmetries, so divergences in Goldstone’s masses are unrelated to divergences in Higgs’ mass. Even if
the global symmetry is broken spontaneously, effective potential should respect it in some way or another – that’s why the word “spontaneously” (instead of “explicitly”) is used. Whether
pions are composite or fundamental objects again does not matter – once you have an effective renormalizable lagrangian for low energy degrees of freedom, you can and should study the fate of
divergences appearing in the effective theory.
It so happens that global symmetry relates some contributions to the masses of Goldstones and the Higgs, namely, quadratically divergent contributions. This statement is again known for years
and can be found for example in Peskin and Schroder. So, if you want to see that the Goldstone theorem holds explicitly at a given energy scale (i.e, renormalized mass of Goldstones is zero),
you should conclude that the quadratically divergent contribution to the Higgs’ mass also vanishes.
I share very much your sentiment about the sensitivity of the Higgs’ mass to parameters of HE theories (that’s why we always thought we need SUSY, Technicolor, etc., etc., etc), but that’s
unfortunately not how it works in these theories.
Thanks for the link by the way!
Link to this
Add a Comment
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Harmonic Oscillation
1. The problem statement, all variables and given/known data
1. A man's superelastic suspenders catch on a fence post, he flies back and forth, oscillating with an amplitude A. What distance does he movee in one period ? What is his displacement over 1 period
2. Relevant equations
3. The attempt at a solution
I know that the displacement is 0 since he returns to the original position. I also know that the distance should be 4A but I don't know how to explain it. It just makes sense to me.
How do I prove that distance is 4A ? | {"url":"http://www.physicsforums.com/showthread.php?t=302995","timestamp":"2014-04-17T10:03:25Z","content_type":null,"content_length":"29731","record_id":"<urn:uuid:9a744846-9725-4f3d-a976-cfdb9642db5d>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00516-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Discrete dispersion relation for hp-version finite element approximation at high wave number
Ainsworth, M. (2004) Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM Journal on Numerical Analysis, 42 (2). pp. 553-575. ISSN 0036-1429
Full text not available in this repository. (
Request a copy from the Strathclyde author
The dispersive properties of high order finite element schemes are analyzed in the setting of the Helmholtz equation, and an explicit form of the discrete dispersion relation is obtained for elements
of arbitrary order. It is shown that the numerical dispersion displays three different types of behavior, depending on the size of the order of the method relative to the mesh-size and the wave
number. Quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes. All estimates are fully explicit and are shown to be sharp. Limits
are obtained, where transitions between the different regimes occur, and used to provide guidelines for the selection of the mesh-size and the polynomial order in terms of the wave number so that the
dispersion error is controlled.
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Optimizing repeated modulus within a loop
up vote 6 down vote favorite
I have this statement in my c program and I want to optimize. By optimization I particularly want to refer to bitwise operators (but any other suggestion is also fine).
uint64_t h_one = hash[0];
uint64_t h_two = hash[1];
for ( int i=0; i<k; ++i )
(uint64_t *) k_hash[i] = ( h_one + i * h_two ) % size; //suggest some optimization for this line.
Any suggestion will be of great help.
Edit: As of now size can be any int but it is not a problem and we can round it up to the next prime (but may be not a power of two as for larger values the power of 2 increases rapidly and it will
lead to much wastage of memory)
h_two is a 64 bit int(basically a chuck of 64 bytes).
c optimization bitwise-operators
Do you know anything about size? – Mysticial Jun 15 '12 at 19:33
I have edited the question to make a few things clear as you demanded – Aman Deep Gautam Jun 15 '12 at 19:38
1 There are methods out there that make repeated divisions/modulus over the same number very efficient. But it's not trivial. – Mysticial Jun 15 '12 at 19:40
can you suggest some links for me to read up a bit. – Aman Deep Gautam Jun 15 '12 at 19:41
1 Here it is: gmplib.org/~tege/divcnst-pldi94.pdf But like I said, non-trivial. – Mysticial Jun 15 '12 at 19:44
show 3 more comments
2 Answers
active oldest votes
so essentially you're doing
k_0 = h_1 mod s
k_1 = h_1 + h_2 mod s = k_0 + h_2 mod s
k_2 = h_1 + h_2 + h_2 mod s = k_1 + h_2 mod s
k_n = k_(n-1) + h_2 mod s
Depending on overflow issues (which shouldn't differ from the original if size is less than half of 2**64), this could be faster (less easy to parallelize though):
uint64_t h_one = hash[0];
uint64_t h_two = hash[1];
k_hash[0] = h_one % size;
for ( int i=1; i<k; ++i )
(uint64_t *) k_hash[i] = ( k_hash[i-1] + h_two ) % size;
Note there is a possibility that your compiler already came to this form, depending on which optimization flags you use.
Of course this only eliminated one multiplication. If you want to eliminate or reduce the modulo, I guess that based on h_two%size and h_1%size you can predetermine the steps
where you have to explicitly call %size, something like this:
uint64_t h_one = hash[0]%size;
uint64_t h_two = hash[1]%size;
k_hash[0] = h_one;
up vote 4 down vote step = (size-(h_one))/(h_two)-1;
accepted for ( int i=1; i<k; ++i )
(uint64_t *) k_hash[i] = ( k_hash[i-1] + h_two );
k_hash[i] %= size;
Note I'm not sure of the formula (didn't test it), it's more a general idea. This would greatly depend on how good your branch prediction is (and how big a performance-hit a
misprediction is). ALso it's only likely to help if step is big.
edit: or more simple (and probably with the same performance) -thanks to Mystical:
uint64_t h_one = hash[0]%size;
uint64_t h_two = hash[1]%size;
k_hash[0] = h_one;
for ( int i=1; i<k; ++i )
(uint64_t *) k_hash[i] = ( k_hash[i-1] + h_two );
if(k_hash[i] > size)
k_hash[i] -= size;
+1 It's actually possible to completely remove the modulus in your first approach if you can prove that k_hash[i-1] + h_two will never overflow the integer. But seeing as how
it's a hash, I'm going to assume that the numbers are pretty much random. – Mysticial Jun 15 '12 at 20:34
@Mysticial size was an int though, and the rest are uint_64's, so they shouldn't overflow (h_two can of course be pre-reduced) – harold Jun 15 '12 at 20:37
2 @harold, the it looks like we have a solution. Precompute h_two % size and start with h_one % size. Then at each iteration, add it to an accumulator. Then use an if-statement
to test if it's greater than size and subtract if necessary. – Mysticial Jun 15 '12 at 20:40
@Mysticial: actually my second solution is more or less that, except that I predetermine the step where the modulo should happen. A simple > would probably be more readable and
just as efficient though, I'll add it. – KillianDS Jun 15 '12 at 20:58
@all thank you! – Aman Deep Gautam Jun 15 '12 at 21:33
add comment
If size is a power of two, then applying a bitwise AND to size - 1 optimizes "% size":
up vote 0 down vote (uint64_t *)k_hash[i] = (h_one + i * h_two) & (size - 1)
making size a power of 2 is too much to ask for but we can have it as a prime so can you suggest something in that case – Aman Deep Gautam Jun 15 '12 at 19:40
add comment
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MATH 175(F) Mathematical Politics: Voting, Power, and Conflict (Same as INTR 160) (Q)
Who should have won the 2000 Presidential Election? Do any two senators really have equal power in passing legislation? How can marital assets be divided fairly? While these questions are of interest
to many social scientists, a mathematical perspective can offer a quantitative analysis of issues like these and more. In this course, we will discuss the advantages and disadvantages of various
types of voting systems and show that, in fact, any such system is flawed. We will also examine a quantitative definition of power and the principles behind fair division. Along the way, we will
enhance the critical reasoning skills necessary to tackle any type of problem mathematical or otherwise. Format: lecture/discussion. Evaluation will be based primarily on projects, homework
assignments, and exams. Prerequisites: Mathematics 100 /101/102(or demonstrated proficiency on a diagnostic test) or permission of instructor. No enrollment limit (expected: 25).
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The 7 members of a chess club line up for a picture. Determine the probability that Jordan and Ryan are not beside each other.
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is that 21
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(7! - 2!*6!) /7!
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got it?
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thnx guys
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Others have already mentioned the solution but I thought I would explain how you get it. The way I solved it was by first finding out the probability of them being next to each other. So, we have
7 places. If they're next to each other, we have the following cases: Let A be Jordan and B by Ryan. A B _ _ _ _ _ B A _ _ _ _ _ _ _ A B _ _ _ _ _ B A _ _ _ And so on. In total, we have 12
configurations of this. In each of the blank positions, we do a simple permutation of the remaining people, i.e., 5!. So, we have 12 * 5! ways of arranging them, which equals 2 * 6! ways. There
are 7! ways in total, without the constraint. So, the probability they are next to each other is: 2 * 6! / 7! = 2 / 7 The probability that they are not next to each other is 1 - 2 / 7, then,
which is 5 / 7.
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thnx queelius
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Let's start off with a few simple examples of conduit usage. First, a file copy utility:
>>> :load Data.Conduit.Binary
>>> runResourceT $ sourceFile "input.txt" $$ sinkFile "output.txt"
runResourceT is a function provided by the resourcet package, and ensures that resources are properly cleaned up, even in the presence of exceptions. The type system will enforce that runResourceT is
called as needed. The remainder of this tutorial will not discuss runResourceT; please see the documentation in resourcet for more information.
Looking at the rest of our example, there are three components to understand: sourceFile, sinkFile, and the $$ operator (called "connect"). These represent the most basic building blocks in conduit:
a Source produces a stream of values, a Sink consumes such a stream, and $$ will combine these together.
In the case of file copying, there was no value produced by the Sink. However, often times a Sink will produce some result value. For example:
>>> :load Data.Conduit.List
>>> :module +Prelude
>>> sourceList [1..10] $$ fold (+) 0
sourceList is a convenience function for turning a list into a Source. fold implements a strict left fold for consuming the input stream.
The next major aspect to the conduit library is the Conduit type. This type represents a stream transformer. In order to use a Conduit, we must fuse it with either a Source or Sink. For example:
>>> :load Data.Conduit.List
>>> :module +Prelude
>>> sourceList [1..10] $= Data.Conduit.List.map (+1) $$ consume
Notice the addition of the $=, or left fuse operator. This combines a Source and a Conduit into a new Source, which can then be connected to a Sink (in this case, consume). We can similarly perform
right fusion to combine a Conduit and Sink, or middle fusion to combine two Conduits.
A number of very common functions are provided in the Data.Conduit.List module. Many of these functions correspond very closely to standard Haskell functions.
In addition to connecting and fusing components together, we can also build up more sophisticated components through monadic composition. For example, to create a Sink that ignores the first 3
numbers and returns the sum of the remaining numbers, we can use:
>>> :load Data.Conduit.List
>>> :module +Prelude
>>> sourceList [1..10] $$ Data.Conduit.List.drop 3 >> fold (+) 0
In some cases, we might end up consuming more input than we needed, and want to provide that input to the next component in our monadic chain. We refer to this as leftovers. The simplest example of
this is peek.
>>> :load Data.Conduit.List
>>> :set -XNoMonomorphismRestriction
>>> :module +Prelude
>>> let sink = do { first <- peek; total <- fold (+) 0; return (first, total) }
>>> sourceList [1..10] $$ sink
(Just 1,55)
Notice that, although we "consumed" the first value from the stream via peek, it was still available to fold. This idea becomes even more important when dealing with chunked data such as ByteStrings
or Text.
Final note: Notice in the types below that Source, Sink, and Conduit are just type aliases. This will be explained later. Another important aspect is resource finalization, which will also be covered
Conduit interface
type Source m o = ConduitM () o m ()Source
Provides a stream of output values, without consuming any input or producing a final result.
Since 0.5.0
type Conduit i m o = ConduitM i o m ()Source
Consumes a stream of input values and produces a stream of output values, without producing a final result.
Since 0.5.0
type Sink i m r = ConduitM i Void m rSource
Consumes a stream of input values and produces a final result, without producing any output.
Since 0.5.0
It is important to understand the lifecycle of our components. Notice that we can connect or fuse two components together. When we do this, the component providing output is called upstream, and the
component consuming this input is called downstream. We can have arbitrarily long chains of such fusion, so a single component can simultaneously function as upstream and downstream.
Each component can be in one of four states of operation at any given time:
• It hasn't yet started operating.
• It is providing output downstream.
• It is waiting for input from upstream.
• It has completed processing.
Let's use sourceFile and sinkFile as an example. When we run sourceFile input $$ sinkFile output, both components begin in the "not started" state. Next, we start running sinkFile (note: we always
begin processing on the downstream component). sinkFile will open up the file, and then wait for input from upstream.
Next, we'll start running sourceFile, which will open the file, read some data from it, and provide it as output downstream. This will be fed to sinkFile (which was already waiting). sinkFile will
write the data to a file, then ask for more input. This process will continue until sourceFile reaches the end of the input. It will close the file handle and switch to the completed state. When this
happens, sinkFile is sent a signal that no more input is available. It will then close its file and return a result.
Now let's change things up a bit. Suppose we were instead connecting sourceFile to take 1. We start by running take 1, which will wait for some input. We'll then start sourceFile, which will open the
file, read a chunk, and send it downstream. take 1 will take that single chunk and return it as a result. Once it does this, it has transitioned to the complete state.
We don't want to pull any more data from sourceFile, as we do not need it. So instead, we call sourceFile's finalizer. Each time upstream provides output, it also provides a finalizer to be run if
downstream finishes processing.
One final case: suppose we connect sourceFile to return (). The latter does nothing: it immediately switches to the complete state. In this case, we never even start running sourceFile (it stays in
the "not yet started" state), and so no finalization occurs.
So here are the takeaways from the above discussion:
• When upstream completes before downstream, it cleans up all of its resources and sends some termination signal. We never think about upstream again. This can only occur while downstream is in the
"waiting for input" state, since that is the only time that upstream is called.
• When downstream completes before upstream, we finalize upstream immediately. This can only occur when upstream produces output, because that's the only time when control is passed back to
• If downstream never awaits for input before it terminates, upstream was never started, and therefore it does not need to be finalized.
Note that all of the discussion above applies equally well to chains of components. If you have an upstream, middle, and downstream component, and downstream terminates, then the middle component
will be finalized, which in turn will trigger upstream to be finalized. This setup ensures that we always have prompt resource finalization.
($$) :: Monad m => Source m a -> Sink a m b -> m bSource
The connect operator, which pulls data from a source and pushes to a sink. When either side closes, the other side will immediately be closed as well. If you would like to keep the Source open to be
used for another operations, use the connect-and-resume operator $$+.
Since 0.4.0
($=) :: Monad m => Source m a -> Conduit a m b -> Source m bSource
Left fuse, combining a source and a conduit together into a new source.
Both the Source and Conduit will be closed when the newly-created Source is closed.
Leftover data from the Conduit will be discarded.
Since 0.4.0
(=$) :: Monad m => Conduit a m b -> Sink b m c -> Sink a m cSource
Right fuse, combining a conduit and a sink together into a new sink.
Both the Conduit and Sink will be closed when the newly-created Sink is closed.
Leftover data returned from the Sink will be discarded.
Since 0.4.0
(=$=) :: Monad m => Conduit a m b -> ConduitM b c m r -> ConduitM a c m rSource
Fusion operator, combining two Conduits together into a new Conduit.
Both Conduits will be closed when the newly-created Conduit is closed.
Leftover data returned from the right Conduit will be discarded.
Since 0.4.0
While conduit provides a number of built-in Sources, Sinks, and Conduits, you will almost certainly want to construct some of your own. Previous versions recommended using the constructors directly.
Beginning with 0.5, the recommended approach is to compose existing Pipes into larger ones.
It is certainly possible (and advisable!) to leverage existing Pipes- like those in Data.Conduit.List. However, you will often need to go to a lower level set of Pipes to start your composition. The
following few functions should be sufficient for expressing all constructs besides finalization. Adding in bracketP and addCleanup, you should be able to create any Pipe you need. (In fact, that's
precisely how the remainder of this package is written.)
The three basic operations are awaiting, yielding, and leftovers. Awaiting asks for a new value from upstream, or returns Nothing if upstream is done. For example:
>>> :load Data.Conduit.List
>>> sourceList [1..10] $$ await
Just 1
>>> :load Data.Conduit.List
>>> sourceList [] $$ await
Similarly, we have a yield function, which provides a value to the downstream Pipe. yield features auto-termination: if the downstream Pipe has already completed processing, the upstream Pipe will
stop processing when it tries to yield.
The upshot of this is that you can write code that appears to loop infinitely, and yet will terminate.
>>> :set -XNoMonomorphismRestriction
>>> let infinite = yield () >> infinite
>>> infinite $$ await
Just ()
Or for something a bit more sophisticated:
>>> let enumFrom' i = yield i >> enumFrom' (succ i)
>>> enumFrom' 1 $$ take 5
The final primitive Pipe is leftover. This allows you to return unused input to be used by the next Pipe in the monadic chain. A simple use case would be implementing the peek function:
>>> let peek = await >>= maybe (return Nothing) (\x -> leftover x >> return (Just x))
>>> enumFrom' 1 $$ do { mx <- peek; my <- await; mz <- await; return (mx, my, mz) }
(Just 1,Just 1,Just 2)
Note that you should only return leftovers that were previously yielded from upstream.
:: Monad m
=> o output value
-> ConduitM i o m ()
:: Monad m
=> o
-> m () finalizer
-> ConduitM i o m ()
:: Monad m
=> (Bool -> m ()) True if Pipe ran to completion, False for early termination.
-> ConduitM i o m r
-> ConduitM i o m r
Sometimes, we do not want to force our entire application to live inside the Pipe monad. It can be convenient to keep normal control flow of our program, and incrementally apply data from a Source to
various Sinks. A strong motivating example for this use case is interleaving multiple Sources, such as combining a conduit-powered HTTP server and client into an HTTP proxy.
Normally, when we run a Pipe, we get a result and can never run it again. Connect-and-resume allows us to connect a Source to a Sink until the latter completes, and then return the current state of
the Source to be applied later. To do so, we introduce three new operators. Let' start off by demonstrating them:
>>> :load Data.Conduit.List
>>> (next, x) <- sourceList [1..10] $$+ take 5
>>> Prelude.print x
>>> (next, y) <- next $$++ (isolate 4 =$ fold (Prelude.+) 0)
>>> Prelude.print y
>>> next $$+- consume
data ResumableSource m o Source
A Source which has been started, but has not yet completed.
This type contains both the current state of the Source, and the finalizer to be run to close it.
Since 0.5.0
($$+) :: Monad m => Source m a -> Sink a m b -> m (ResumableSource m a, b)Source
The connect-and-resume operator. This does not close the Source, but instead returns it to be used again. This allows a Source to be used incrementally in a large program, without forcing the entire
program to live in the Sink monad.
Mnemonic: connect + do more.
Since 0.5.0
($$+-) :: Monad m => ResumableSource m a -> Sink a m b -> m bSource
Complete processing of a ResumableSource. This will run the finalizer associated with the ResumableSource. In order to guarantee process resource finalization, you must use this operator after using
$$+ and $$++.
Since 0.5.0
unwrapResumable :: MonadIO m => ResumableSource m o -> m (Source m o, m ())Source
Unwraps a ResumableSource into a Source and a finalizer.
A ResumableSource represents a Source which has already been run, and therefore has a finalizer registered. As a result, if we want to turn it into a regular Source, we need to ensure that the
finalizer will be run appropriately. By appropriately, I mean:
• If a new finalizer is registered, the old one should not be called. * If the old one is called, it should not be called again.
This function returns both a Source and a finalizer which ensures that the above two conditions hold. Once you call that finalizer, the Source is invalidated and cannot be used.
Since 0.5.2
Utility functions
:: Monad m
=> (i1 -> i2) map initial input to new input
-> (i2 -> Maybe i1) map new leftovers to initial leftovers
-> ConduitM i2 o m r
-> ConduitM i1 o m r
Generalized conduit types
It's recommended to keep your type signatures as general as possible to encourage reuse. For example, a theoretical signature for the head function would be:
head :: Sink a m (Maybe a)
However, doing so would prevent usage of head from inside a Conduit, since a Sink sets its output type parameter to Void. The most general type signature would instead be:
head :: Pipe l a o u m (Maybe a)
However, that signature is much more confusing. To bridge this gap, we also provide some generalized conduit types. They follow a simple naming convention:
• They have the same name as their non-generalized types, with a G prepended.
• If they have leftovers, we add an L.
• If they consume the entirety of their input stream and return the upstream result, we add Inf to indicate infinite consumption.
data Flush a Source
Provide for a stream of data that can be flushed.
A number of Conduits (e.g., zlib compression) need the ability to flush the stream at some point. This provides a single wrapper datatype to be used in all such circumstances.
Since 0.3.0
Functor Flush
Eq a => Eq (Flush a)
Ord a => Ord (Flush a)
Show a => Show (Flush a)
Convenience re-exports
data ResourceT m a
The Resource transformer. This transformer keeps track of all registered actions, and calls them upon exit (via runResourceT). Actions may be registered via register, or resources may be allocated
atomically via allocate. allocate corresponds closely to bracket.
Releasing may be performed before exit via the release function. This is a highly recommended optimization, as it will ensure that scarce resources are freed early. Note that calling release will
deregister the action, so that a release action will only ever be called once.
Since 0.3.0
MonadTrans ResourceT
MonadTransControl ResourceT
MonadRWS r w s m => MonadRWS r w s (ResourceT m)
MonadBase b m => MonadBase b (ResourceT m)
MonadBaseControl b m => MonadBaseControl b (ResourceT m)
MonadError e m => MonadError e (ResourceT m)
MonadReader r m => MonadReader r (ResourceT m)
MonadState s m => MonadState s (ResourceT m)
MonadWriter w m => MonadWriter w (ResourceT m)
Monad m => Monad (ResourceT m)
Functor m => Functor (ResourceT m)
Typeable1 m => Typeable1 (ResourceT m)
Applicative m => Applicative (ResourceT m)
MonadIO m => MonadIO (ResourceT m)
MonadCont m => MonadCont (ResourceT m)
(MonadThrow m, MonadUnsafeIO m, MonadIO m, Applicative m) => MonadResource (ResourceT m)
MonadThrow m => MonadThrow (ResourceT m)
(MonadIO m, MonadActive m) => MonadActive (ResourceT m)
class (MonadThrow m, MonadUnsafeIO m, MonadIO m, Applicative m) => MonadResource m
A Monad which allows for safe resource allocation. In theory, any monad transformer stack included a ResourceT can be an instance of MonadResource.
Note: runResourceT has a requirement for a MonadBaseControl IO m monad, which allows control operations to be lifted. A MonadResource does not have this requirement. This means that transformers such
as ContT can be an instance of MonadResource. However, the ContT wrapper will need to be unwrapped before calling runResourceT.
Since 0.3.0
MonadResource m => MonadResource (ListT m)
(MonadThrow m, MonadUnsafeIO m, MonadIO m, Applicative m) => MonadResource (ResourceT m)
MonadResource m => MonadResource (MaybeT m)
MonadResource m => MonadResource (IdentityT m)
MonadResource m => MonadResource (ContT r m)
(Error e, MonadResource m) => MonadResource (ErrorT e m)
MonadResource m => MonadResource (ReaderT r m)
MonadResource m => MonadResource (StateT s m)
MonadResource m => MonadResource (StateT s m)
(Monoid w, MonadResource m) => MonadResource (WriterT w m)
(Monoid w, MonadResource m) => MonadResource (WriterT w m)
MonadResource m => MonadResource (ConduitM i o m)
(Monoid w, MonadResource m) => MonadResource (RWST r w s m)
(Monoid w, MonadResource m) => MonadResource (RWST r w s m)
MonadResource m => MonadResource (Pipe l i o u m)
class Monad m => MonadThrow m where
A Monad which can throw exceptions. Note that this does not work in a vanilla ST or Identity monad. Instead, you should use the ExceptionT transformer in your stack if you are dealing with a non-IO
base monad.
Since 0.3.0
MonadThrow []
MonadThrow IO
MonadThrow Maybe
MonadThrow (Either SomeException)
MonadThrow m => MonadThrow (ListT m)
MonadThrow m => MonadThrow (ResourceT m)
Monad m => MonadThrow (ExceptionT m)
MonadThrow m => MonadThrow (MaybeT m)
MonadThrow m => MonadThrow (IdentityT m)
MonadThrow m => MonadThrow (ContT r m)
(Error e, MonadThrow m) => MonadThrow (ErrorT e m)
MonadThrow m => MonadThrow (ReaderT r m)
MonadThrow m => MonadThrow (StateT s m)
MonadThrow m => MonadThrow (StateT s m)
(Monoid w, MonadThrow m) => MonadThrow (WriterT w m)
(Monoid w, MonadThrow m) => MonadThrow (WriterT w m)
MonadThrow m => MonadThrow (ConduitM i o m)
(Monoid w, MonadThrow m) => MonadThrow (RWST r w s m)
(Monoid w, MonadThrow m) => MonadThrow (RWST r w s m)
MonadThrow m => MonadThrow (Pipe l i o u m)
class Monad m => MonadUnsafeIO m where
A Monad based on some monad which allows running of some IO actions, via unsafe calls. This applies to IO and ST, for instance.
Since 0.3.0
MonadUnsafeIO IO
(MonadTrans t, MonadUnsafeIO m, Monad (t m)) => MonadUnsafeIO (t m)
MonadUnsafeIO (ST s)
MonadUnsafeIO (ST s)
runResourceT :: MonadBaseControl IO m => ResourceT m a -> m a
Unwrap a ResourceT transformer, and call all registered release actions.
Note that there is some reference counting involved due to resourceForkIO. If multiple threads are sharing the same collection of resources, only the last call to runResourceT will deallocate the
Since 0.3.0
newtype ExceptionT m a
The express purpose of this transformer is to allow non-IO-based monad stacks to catch exceptions via the MonadThrow typeclass.
Since 0.3.0
MonadTrans ExceptionT
MonadTransControl ExceptionT
MonadRWS r w s m => MonadRWS r w s (ExceptionT m)
MonadBase b m => MonadBase b (ExceptionT m)
MonadBaseControl b m => MonadBaseControl b (ExceptionT m)
MonadError e m => MonadError e (ExceptionT m)
MonadReader r m => MonadReader r (ExceptionT m)
MonadState s m => MonadState s (ExceptionT m)
MonadWriter w m => MonadWriter w (ExceptionT m)
Monad m => Monad (ExceptionT m)
Monad m => Functor (ExceptionT m)
Monad m => Applicative (ExceptionT m)
MonadCont m => MonadCont (ExceptionT m)
Monad m => MonadThrow (ExceptionT m)
class MonadBase b m => MonadBaseControl b m | m -> b
MonadBaseControl [] []
MonadBaseControl IO IO
MonadBaseControl STM STM
MonadBaseControl Maybe Maybe
MonadBaseControl Identity Identity
MonadBaseControl b m => MonadBaseControl b (MaybeT m)
MonadBaseControl b m => MonadBaseControl b (ListT m)
MonadBaseControl b m => MonadBaseControl b (IdentityT m)
MonadBaseControl b m => MonadBaseControl b (ResourceT m)
MonadBaseControl b m => MonadBaseControl b (ExceptionT m)
(Monoid w, MonadBaseControl b m) => MonadBaseControl b (WriterT w m)
(Monoid w, MonadBaseControl b m) => MonadBaseControl b (WriterT w m)
MonadBaseControl b m => MonadBaseControl b (StateT s m)
MonadBaseControl b m => MonadBaseControl b (StateT s m)
MonadBaseControl b m => MonadBaseControl b (ReaderT r m)
(Error e, MonadBaseControl b m) => MonadBaseControl b (ErrorT e m)
(Monoid w, MonadBaseControl b m) => MonadBaseControl b (RWST r w s m)
(Monoid w, MonadBaseControl b m) => MonadBaseControl b (RWST r w s m)
MonadBaseControl ((->) r) ((->) r)
MonadBaseControl (Either e) (Either e)
MonadBaseControl (ST s) (ST s)
MonadBaseControl (ST s) (ST s) | {"url":"http://hackage.haskell.org/package/conduit-1.0.0.2/docs/Data-Conduit.html","timestamp":"2014-04-19T08:34:08Z","content_type":null,"content_length":"86273","record_id":"<urn:uuid:0886c700-7b63-4e1b-9130-f78cba3c13e3>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00422-ip-10-147-4-33.ec2.internal.warc.gz"} |
trig equation
solve the trig equation sec(x)-2=0 for exact solutions in the interval 0<= x <= 2pi
Quote: Originally Posted by c_323_h sec(x)=2 Now, sec=1/cos Thus cos(x)=1/2. So for what values of x is cos(x)=1/2? -Dan
Quote: Originally Posted by c_323_h That means, $\sec x=2$ because secant is reciprocal function of cosine we have, $\cos x=\frac{1}{2}$ for $0<x<2\pi$ Thus, $x=\frac{\pi}{6},\frac{11\pi}{6}$
Quote: Originally Posted by c_323_h Yup. ThePerfectHacker isn't having a good day with trig. I hope he's okay... :) -Dan
Quote: Originally Posted by topsquark No I am so tired now. Not really realizing what I am doing. If I make another mistake I am leaving this site for today.
Quote: Originally Posted by ThePerfectHacker It's okay. We still love you. Get some sleep for me! -Dan | {"url":"http://mathhelpforum.com/trigonometry/2349-trig-equation-print.html","timestamp":"2014-04-20T22:15:02Z","content_type":null,"content_length":"7446","record_id":"<urn:uuid:22914536-77eb-4d6f-a776-4db796fb142c>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00168-ip-10-147-4-33.ec2.internal.warc.gz"} |
2-Category theory
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Higher category theory
Basic concepts
Basic theorems
Universal constructions
Extra properties and structure
1-categorical presentations
A bicategory is a particular algebraic notion of weak 2-category (in fact, the earliest to be formulated, and still the one in most common use). The idea is that a bicategory is a category weakly
enriched over Cat: the hom-objects of a bicategory are hom-categories, but the associativity and unity laws of enriched categories hold only up to coherent isomorphism.
A bicategory $B$ consists of
• A collection of objects $x,y,z,\dots$, also called $0$-cells;
• For each pair of $0$-cells $x,y$, a category $B(x,y)$, whose objects are called morphisms or $1$-cells and whose morphisms are called 2-morphisms or $2$-cells;
• For each $0$-cell $x$, a distinguished $1$-cell $1_x\in B(x,x)$ called the identity morphism or identity $1$-cell at $x$;
• For each triple of $0$-cells $x,y,z$, a functor ${\circ}\colon B(y,z)\times B(x,y) \to B(x,z)$ called horizontal composition;
• For each pair of $0$-cells $x,y$, natural isomorphisms called unitors: $id_{B(x,y)} \circ const_{1_x} \cong id_{B(x,y)} \cong const_{1_y} \circ id_{B(x,y)}\colon B(x,y) \to B(x,y)$; and
• For each quadruple of $0$-cells $w,x,y,z$, a natural isomorphism called the associator between the two functors from $B_{y,z} \times B_{x,y} \times B_{w,x}$ to $B_{w,z}$ built out of ${\circ}$;
such that
• Such that the pentagon identity is satisfied by the associators.
If there is exactly one $0$-cell, say $*$, then the definition is exactly the same as a monoidal structure on the category $B(*,*)$. This is one of the motivating examples behind the delooping
hypothesis and the general notion of k-tuply monoidal n-category.
Here we spell out the above definition in full detail. Compare to the detailed definition of strict $2$-category, which is written in the same style but is simpler.
A bicategory $B$ consists of
• a collection $Ob B$ or $Ob_B$ of objects or $0$-cells,
• for each object $a$ and object $b$, a collection $B(a,b)$ or $Hom_B(a,b)$ of morphisms or $1$-cells $a \to b$, and
• for each object $a$, object $b$, morphism $f\colon a \to b$, and morphism $g\colon a \to b$, a collection $B(f,g)$ or $2Hom_B(f,g)$ of $2$-morphisms or $2$-cells $f \Rightarrow g$ or $f \
Rightarrow g\colon a \to b$,
equipped with
• for each object $a$, an identity $1_a\colon a \to a$ or $\id_a\colon a \to a$,
• for each $a,b,c$, $f\colon a \to b$, and $g\colon b \to c$, a composite $f ; g\colon a \to c$ or $g \circ f\colon a \to c$,
• for each $f\colon a \to b$, an identity or $2$-identity $1_f\colon f \Rightarrow f$ or $\Id_f\colon f \to f$,
• for each $f,g,h\colon a \to b$, $\eta\colon f \Rightarrow g$, and $\theta\colon g \Rightarrow h$, a vertical composite $\theta \bullet \eta\colon f \Rightarrow h$,
• for each $a,b,c$, $f\colon a \to b$, $g,h\colon b \to c$, and $\eta\colon g \Rightarrow h$, a left whiskering $\eta \triangleleft f\colon g \circ f \Rightarrow h \circ f$,
• for each $a,b,c$, $f,g\colon a \to b$, $h\colon b \to c$, and $\eta\colon f \Rightarrow g$, a right whiskering $h \triangleright \eta \colon h \circ f \Rightarrow h \circ g$,
• for each $f\colon a \to b$, a left unitor $\lambda_f\colon f \circ \id_a \Rightarrow f$ and an inverse left unitor $\bar{\lambda}_f\colon f \Rightarrow f \circ \id_a$,
• for each $f\colon a \to b$, a right unitor $\rho_f\colon \id_b \circ f \Rightarrow f$, and an inverse right unitor $\bar{\rho}_f\colon f \Rightarrow \id_b \circ f$, and
• for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, an associator $\alpha_{f,g,h}\colon h \circ (g \circ f) \Rightarrow (h \circ g) \circ f$ and an inverse associator $\bar{\alpha}_
{f,g,h}\colon (h \circ g) \circ f \Rightarrow h \circ (g \circ f)$,
such that
• for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\eta \bullet \Id_f$ and $\Id_g \bullet \eta$ both equal $\eta$,
• for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b$, the vertical composites $\iota \bullet (\theta \bullet \eta)$ and $(\iota \
bullet \theta) \bullet \eta$ are equal,
• for each $a \overset{f}\to b \overset{g}\to c$, the whiskerings $\Id_g \triangleleft f$ and $g \triangleright \Id_f$ both equal $\Id_{g \circ f }$,
• for each $f\colon a \to b$ and $g \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c$, the vertical composite $(\theta \triangleleft f) \bullet (\eta \triangleleft f)$ equals
the whiskering $(\theta \bullet \eta) \triangleleft f$,
• for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b$ and $i\colon b \to c$, the vertical composite $(i \triangleright \theta) \bullet (i \triangleright \eta)$
equals the whiskering $i \triangleright (\theta \bullet \eta)$,
• for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\lambda_g \bullet (\eta \triangleleft \id_a)$ and $\eta \bullet \lambda_f$ are equal,
• for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\rho_g \bullet (\id_b \triangleright \eta)$ and $\eta \bullet \rho_f$ are equal,
• for each $a \overset{f}\to b \overset{g}\to c$ and $\eta\colon h \Rightarrow i\colon c \to d$, the vertical composites $\alpha_{f,g,i} \bullet (\eta \triangleleft (g \circ f))$ and $((\eta \
triangleleft g) \triangleleft f) \bullet \alpha_{f,g,h}$ are equal,
• for each $f\colon a \to b$, $\eta\colon g \Rightarrow h\colon b \to c$, and $i\colon c \to d$, the vertical composites $\alpha_{f,h,i} \bullet (i \triangleright (\eta \triangleleft f))$ and $((i
\triangleright \eta) \triangleleft f) \bullet \alpha_{f,g,i}$ are equal,
• for each $\eta\colon f \Rightarrow g\colon a \to b$ and $b \overset{h}\to c \overset{i}\to d$, the vertical composites $\alpha_{g,h,i} \bullet (i \triangleright (h \triangleright \eta))$ and $((i
\circ h) \triangleright \eta) \bullet \alpha_{f,h,i}$ are equal,
• for each $\eta\colon f \Rightarrow g\colon a \to b$ and $\theta\colon h \Rightarrow i\colon b \to c$, the vertical composites $(i \triangleright \eta) \bullet (\theta \triangleleft f)$ and $(\
theta \triangleleft g) \bullet (h \triangleright \eta)$ are equal,
• for each $f\colon a \to b$, the vertical composites $\lambda_f \bullet \bar{\lambda}_f\colon f \Rightarrow f$ and $\bar{\lambda}_f \bullet \lambda_f\colon f \circ \id_a \Rightarrow f \circ \id_a$
equal the appropriate identity $2$-morphisms,
• for each $f\colon a \to b$, the vertical composites $\rho_f \bullet \bar{\rho}_f\colon f \Rightarrow f$ and $\bar{\rho}_f \bullet \rho_f\colon \id_b \circ f \Rightarrow \id_b \circ f$ equal the
appropriate identity $2$-morphisms,
• for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, the vertical composites $\alpha_{f,g,h} \bullet \bar{\alpha}_{f,g,h}\colon (h \circ g) \circ f \Rightarrow (h \circ g) \circ f$
and $\bar{\alpha}_{f,g,h} \bullet \alpha_{f,g,h}\colon h \circ (g \circ f) \Rightarrow h \circ (g \circ f)$ equal the appropriate identity $2$-morphisms,
• for each $a \overset{f}\to b \overset{g}\to c$, the vertical composite $(\lambda_g \triangleleft f) \bullet \alpha_{f,\id_b,g}$ equals the whiskering $g \triangleright \rho_f$, and
• for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d \overset{i}\to e$, the vertical composites $((\alpha_{g,h,i} \triangleleft f) \bullet \alpha_{f,h \circ g,i}) \bullet (i \
triangleright \alpha_{f,g,h})$ and $\alpha_{f,g,i \circ h}\bullet \alpha_{g \circ f,h,i}$ are equal.
It is quite possible that there are errors or omissions in this list, although they should be easy to correct. The point is not that one would want to write out the definition in such elementary
terms (although apparently I just did anyway) but rather that one can.
• Any strict 2-category is a bicategory in which the unitors and associator are identities. This includes Cat, MonCat?, the algebras for any strict 2-monad, and so on, at least as classically
• Categories, anafunctors, and natural transformations, which is a more appropriate definition of Cat in the absence of the axiom of choice, form a bicategory that is not a strict 2-category.
Indeed, without the axiom of choice, the proper notion of bicategory is anabicategory.
• Rings, bimodules, and bimodule homomorphisms are the prototype for many similar examples. Notably, we can generalize from rings to enriched categories.
• Objects, spans, and morphisms of spans in any category with pullbacks also form a bicategory.
• The fundamental 2-groupoid? of a space is a bicategory which is not necessarily strict (although it can be made strict fairly easily when the space is Hausdorff by quotienting by thin homotopy,
see path groupoid and fundamental infinity-groupoid). When the space is a CW-complex, there are easier and more computationally amenable equivalent strict 2-categories, such as that arising from
the fundamental crossed complex.
Coherence theorems
One way to state the coherence theorem for bicategories is that every bicategory is equivalent to a strict $2$-category. This “strictification” is not obtained naively by forcing composition to be
associative, but (at least in one construction) by freely adding new composites which are strictly associative. Another way to state the coherence theorem is that every formal diagram of the
constraints (associators and unitors) commutes.
Note that $n=2$ is the greatest value of $n$ for which every weak $n$-category is equivalent to a fully strict one; see semi-strict infinity-category and Gray-category.
The proof of the coherence theorem is basically the same as the proof of the coherence theorem for monoidal categories. An abstract approach can be found in Power’s paper “A general coherence
Classically, “2-category” meant strict 2-category, with “bicategory” used for the weak notion. This led to the more general use of the prefix “2-” for strict (that is, strictly Cat-enriched) notions
and “bi-” for weak ones. For example, classically a “2-adjunction” means a Cat-enriched adjunction, consisting of two strict 2-functors $F,G$ and a strictly Cat-natural isomorphism of categories $D(F
X, Y)\cong C(X, G Y)$, while a “biadjunction” means the weak version, consisting of two weak 2-functors and a pseudo natural equivalence $D(F X, Y)\simeq C(X, G Y)$. Similarly for “2-equivalence” and
“biequivalence,” and “2-limit” and “bilimit.”
We often use “2-category” to mean a strict or weak 2-category without prejudice, although we do still use “bicategory” to refer to the particular classical algebraic notion of weak 2-category. We try
to avoid the more general use of “bi-” meaning “weak,” however. For one thing, is it confusing; a “biproduct” could mean a weak 2-limit, but it could also mean an object which is both a product and a
coproduct (which happens quite frequently in additive categories).
Moreover, in most cases the prefix is unnecessary, since once we know we are working in a bicategory, there is usually no point in considering strict notions at all. Fully weak limits are really the
only sensible ones to ask for in a bicategory, and likewise for fully weak adjunctions and equivalences. Even in a strict 2-category, while we might need to say “strict” sometimes to be clear, we
don't need to say “$2$-”, since we know that we are not working in a mere category. (Max Kelly pushed this point.)
When we do have a strict 2-category, however, other strict notions can be quite technically useful, even if our ultimate interest is in the weak ones. This is somewhat analogous to the use of strict
structures to model weak ones in homotopy theory; see here and here for good introductions to this sort of thing.
Discussion about the use of the term “weak enrichment” above is at weak enrichment.
See also the references at 2-category.
Power, A. J. A general coherence result. J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. doi:10.1016/0022-4049(89)90113-8 MR0985657 | {"url":"http://www.ncatlab.org/nlab/show/bicategory","timestamp":"2014-04-20T18:25:33Z","content_type":null,"content_length":"118461","record_id":"<urn:uuid:ba330d58-1718-41aa-9605-3607af284341>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00211-ip-10-147-4-33.ec2.internal.warc.gz"} |
Diamonds Aren't Forever
Copyright © University of Cambridge. All rights reserved.
'Diamonds Aren't Forever' printed from http://nrich.maths.org/
The ideal gas equation is a simple equation which can be used to model certain gases under certain conditions. In this question, assume that it is always applicable. The equation is as follows:
$$pV = nRT$$
$p$ = pressure
$V$ = volume
$n$ = number of moles
$R$ is 8.314 JK$^{-1}$mol$^{-1}$
$T$ = temperature
I vaporise a diamond using a laser, such that the gas fills 55000 cm$^3$ at a pressure of 900 mmHg and a temperature of 49$^\circ$C.
How many moles of carbon are there in my original sample of diamond? Note that data for this question are at the bottom of the page.
Assuming that the diamond can be modelled as a sphere of density 3.52 $\times$ 10$^{9}$ mg/m$^3$, what would the radius of my diamond have been in cm?
I now cool the vapour to -20$^\circ$C at a constant volume. There is no other heat transfer to or from the gas.
What is the new pressure in Nm$^{-2}$?
If I allow the volume of the gas to double (against a vacuum), what would the new pressure be?
If the expansion had not been against a vacuum, would the new pressure be larger or smaller than that calculated previously? Why?
The gas is now returned to its original temperature and volume, and 5 moles of air are introduced into the container.
What is the new pressure of the gas in container?
Given that air is 0.93% argon (by volume), what is the partial pressure of argon?
If the argon molecules are evenly distributed in the container, what volume does each molecule occupy?
I now try to compress the gas into the smallest volume possible.
Making suitable modelling assumptions, estimate what this smallest volume would be.
1 atm = 760 mmHg
1 bar = 10$^5$ Pa = 0.987 atm
N radius = 75 pm
C radius = 77 pm | {"url":"http://nrich.maths.org/6634/index?nomenu=1","timestamp":"2014-04-21T07:23:00Z","content_type":null,"content_length":"5877","record_id":"<urn:uuid:45c9dcc8-0c8d-4efe-88b5-696435d29ed5>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00636-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Summary: UNIVERSITY OF CALIFORNIA, SANTA BARBARA
BERKELEY · DAVIS · IRVINE · LOS ANGELES · MERCED · RIVERSIDE · SAN DIEGO · SAN FRANCISCO
Geometry, Topology, and Physics Seminar
Exact half-BPS solutions to type IIB supergravity
Eric D'Hoker
Friday, November 2, 2007, 4:00 p.m.
Room 6635 South Hall
Abstract: The complete Type IIB supergravity solutions with 16 supersymmetries
are obtained on the manifold AdS4 × S2
× S2
× with SO(2, 3) × SO(3) × SO(3)
symmetry in terms of two holomorphic functions on a Riemann surface , which
generally has a boundary. This is achieved by reducing the BPS equations using the
above symmetry requirements, proving that all solutions of the BPS equations solve
the full Type IIB supergravity field equations, mapping the BPS equations onto a
new integrable system akin to the Liouville and Sine-Gordon theories, and mapping
this integrable system to a linear equation which can be solved exactly. Amongst the
infinite class of solutions, a non-singular Janus solution is identified which provides | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/426/1383573.html","timestamp":"2014-04-17T22:58:11Z","content_type":null,"content_length":"8218","record_id":"<urn:uuid:6665ae1d-281f-491d-a76b-79ef32e2cd2a>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00171-ip-10-147-4-33.ec2.internal.warc.gz"} |
Lowell, WA Precalculus Tutor
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Applied Mathematics and Mathematical Physics Seminar
Applied Mathematics and Mathematical Physics Seminar 2010/2011 (sponsored by MITACS and PIMS)
This is the web page for the Applied Mathematics and Mathematical Physics Seminar at the Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada. This seminar is
organized and jointly run by W. Abou Salem, J. Brooke, A. Cheviakov, G. Patrick, A. Sowa, J. Szmigielski (Applied Mathematics/ Mathematical Physics) and R. Spiteri (Department of Computer Science).
For inquiries concerning the seminar send email to Jacek Szmigielski, szmigiel@math.usask.ca.
The seminar takes place in McLean Hall rm 242.1 on Wednesdays at 3:00 till 4:00 unless advertised differently.
APPLIED MATHEMATICS/ MATHEMATICAL PHYSICS SEMINAR 2007-2008 (sponsored by MITACS)
APPLIED MATHEMATICS/ MATHEMATICAL PHYSICS SEMINAR 2009-2010 (sponsored by MITACS)
Next meetings: August 11, 2011,
Previous meetings : July 7, 2011, June 30, 2011, May 20, 2011, April 20, 2011, March 25, 2011, March 23, 2011, March 14, 2011, March 3, 2011, February 15, 2011, February 4, 2011, February 2, 2011,
November 24, 2010, November 17, 2010, November 10, 2010, November 4, 2010, November 3, 2010, November 2, 2010, September 23, 2010, September 9, 2010,
Speaker: Dr. Oluwaseun Sharomi Department of Mathematics University of Manitoba
Place: McLean Hall 242.1
Time: 3:00
Title: Mathematical Analysis of dynamics of Chlamydia trachomatis
Chlamydia, caused by the bacterium Chlamydia trachomatis, is one of the most important sexually-transmitted infections globally. In addition to accounting for millions of cases every year, the
disease causes numerous irreversible complications such as chronic pelvic pain, infertility in females and pelvic inflammatory disease. The talk will focus on the use of mathematical models, of the
form of deterministic systems of non-linear differential equations, for gaining qualitative insight into the transmission dynamics and control of Chlamydia within an infected host in vivo) and in a
About The Speaker: Dr. Oluwaseun Sharomi received his Ph.D. in Mathematics from the University of Manitoba in 2010. In 2002/2003, he was awarded the United Bank Africa Prize, the LYNX Club Abeokuta
Prize, the Professor Ishola Adamson Prize, and the University Prize for the Best Graduating Student in the Department of Mathematical Sciences, University of Agriculture, Abeokuta, Ogun State,
Nigeria. His research interests include: mathematical modeling, analysis, and numerical methods for the spread of infectious diseases.
Back to top of page Speaker: Professor George Corliss Department of Electrical and Computer Engineering Marquette University
Place: McLean Hall 242.1
Time: 3:00
Title: Propagating Uncertainties in Modeling Nonlinear Dynamic Systems
In many practical applications of ordinary differential equations modeling physical phenomina, parameters and initial conditions are given with some uncertainties. How can we propagate these
uncertainties rigorously through a solution approximation algorithm? We describe an approach that expands the solution in a Taylor model [Makino & Berz] in uncertain parameters and initial
conditions. We evaluate the Taylor models using p-boxes [Ferson] and gradual numbers representing fuzzy numbers to represent the uncertainties in the state variables of the ODE. We give examples from
reaction process dynamics to demonstrate the potential of this approach for studying the effect of uncertainties with imprecise probability distributions.
Back to top of page Speaker: Dr Giulio Chiribella Perimeter Institute for Theoretical Physics, Waterloo, Ontario b>
Place: Arts 134
Time: 4:00
Title: Optimal Estimation of Quantum Signals in the Presence of Symmetry
Quantum systems can be used as elementary gyroscopes that indicate directions in space or as elementary clocks that indicate moments in time. However, the laws of quantum mechanics impose fundamental
precision limits to the corresponding measurements of orientation and time, limits that cannot be violated no matter how advanced our technology is. Assessing the exact value of these limits is
important for many applications in interferometry, magnetometry, GPS systems, and the study of quantum communication protocols where the communicating parties try to establish a common reference
frame. Finding the most precise estimate for the direction of an ensemble of atomic spins or for the phase of a quantum oscillator are instances of a very general problem: the optimal estimation of
parameters pertaining to the action of symmetry groups. In this talk I will review the framework of quantum estimation theory in the presence of symmetries and highlight the basic group theoretic
structures that underlie the optimal estimation strategies. In particular, I will highlight the role of quantum entanglement between the representation spaces and the multiplicity spaces associated
to the group action of interest.
About the speaker: Dr Chiribella was, just recently, awarded the Hermann Weyl Prize at the 28th International Colloquium on Group Theoretical Methods in Physics, Northumbria University, Newcastle
upon Tyne, UK, July 2010.
Back to top of page Speaker: Dr Giulio Chiribella Perimeter Institute for Theoretical Physics, Waterloo, Ontario b>
Place: McLean Hall 242.1
Time: 3:30
Title:Quantum measurement theory: from quantum states to quantum networks
The main aspect in which quantum theory is richer than classical probability theory is that the quantum framework contains an explicit description of the measurement process, through the concepts of
POVM (positive operator-valued measure) and quantum instrument. POVMs and quantum instruments provide the probability of outcomes and the input-output evolution of quantum states in a measurement,
respectively. However, measuring quantum systems is not the only type of measurement-like operation one can perform: for example, one can try to measure properties of quantum devices, like the gain
of an amplifier or the loss in an optical fibre. In this talk I will review the theoretical description and the physical implementation of these general measurement processes, which has been provided
very recently with the theory of quantum combs and testers [1]. As examples of application of the theory, I will present the optimal architectures for the estimation of an unknown group
transformation, and the optimal estimation-disturbance trade-off in the estimation of an unknown unitary dynamics. [1] G. Chiribella, G. M. D'Ariano, and P. Perinotti, Theoretical framework for
quantum networks, Phys. Rev. A 80, 022339 (2009)
Speaker: Michael Ward University of British Columbia Dept. of Mathematics b>
Place: Arts 134
Time: 4:00
Title: An Asympotic Analysis of Localized Solutions to Some Diffusive and Reaction-Diffusion Systems
A survey of the development of a unified singular perturbation methodology to analyze some linear and nonlinear PDE models of diffusion and reaction-diffusion type with localized solutions is
presented. Specific results from this theory are given for three diverse applications. The first problem is to determine the mean first passage time (MFPT) for free diffusion from within a sphere to
small localized traps on its boundary. In the context of cellular signal transduction, the results predict the time-scale needed for a diffusing molecule to arrive at localized signalling
compartments on the boundary of a biological cell. From a mathematical viewpoint, the problem of optimizing this MFPT is shown to be closely related to the well-known Fekete point problem of finding
the minimum energy configuration of repelling point charges on the surface of a sphere. Secondly, in the context of spatial ecology, a long-standing problem is to determine the persistence threshold
for extinction of a species in a heterogeneous spatial landscape consisting of either favorable or unfavorable local habitats. For a 2-D spatial landscape consisting of such localized patches, and in
the context of the diffusive logistic model, this extinction threshold is calculated asymptotically and the effects of both habitat fragmentation and habitat location on the persistence threshold are
obtained. From a mathematical viewpoint, the persistence threshold represents the principal eigenvalue of an indefinite weight singularly perturbed eigenvalue problem. Finally, the dynamics,
stability, and self-replication behavior of localized spot-type solutions to the well-known Gray-Scott reaction-diffusion model of chemical physics in a two-dimensional domain are discussed. Reduced
ODE systems for the dynamics of spots are given together with phase diagrams in parameter space classifying the different types of spot instabilities. In this lecture I will emphasize the common
mathematical features in the analysis of these three problems, most notably the role of the Neumann Green's function for the Laplacian. Applications of the results to more theoretical questions in
PDE and spectral theory will also be emphasized.
Back to top of page Walid Abou Salem, Department of Mathematics and Statistics
Place: McLean Hall 242.1
Time: 3:00-4:00 pm
Title: Renormalization group approach to singular perturbation theory for nonlinear PDEs.
I discuss the rigorous application of the renormalization group method to (singular) perturbation theory for nonlinear partial differential equations. As a paradigm, I consider the concrete example
of the nonlinear Schrodinger equation with quadratic nonlinearity in three spatial dimensions. I show how to obtain an approximate solution using the RG method together with an estimate of the
difference between the true and approximate solutions. The analysis applies to differential equations where (space-time) resonances are present.
Back to top of page Samuel Butler, Department of Geological Sciences
Place: McLean Hall 242.1
Time: 3:00-4:00 pm
Title: Transport, freezing and melting in mushy layers
Mushy layers consist of a porous medium in which the solid matrix and interstitial liquid are close to thermodynamic equilibrium. Mass can be transferred between the solid and liquid by melting and
freezing. Mushy layers are found in nature in magma chambers and sea ice and have been postulated to occur at the Earth's inner-core outer core, and outer-core mantle boundaries. They are also found
in industrial settings including metal castings. In this talk, I will compare transport processes in mushy layers with those in non-reactive porous media and give expressions for effective transport
velocities and diffusion coefficients and look at the degree of phase change that occurs for given system parameters. Some numerical simulations for a mushy system in which natural convection occurs
will also shown and discussed.
Back to top of page Saeed Torabi Ziaratgahi, Department of Mathematics and Statistics
Place: McLean Hall 242.1
Time: 3:00
Title: Domain decomposition for solving PDEs using RBF collocation methods
The well-known finite difference, finite element, and finite volume methods for solving partial differential equations are based on a mesh discretization that may be a complicated and time-consuming
process, particularly for complex, higher-dimensional geometries. The meshfree or meshless methods try to circumvent the cumbersome issues of mesh generation. One of the most common basic meshfree
approximation methods is the Radial Basis Function (RBF) collocation method. Originally, the motivation for this method came from applications in geodesy, geophysics, mapping, and meteorology. Later,
applications were found in areas such as the numerical solution of ODEs and PDEs, artificial intelligence, learning theory, neural networks, signal processing, sampling theory, statistics, finance,
and optimization. RBF collocation methods are simple to implement because the collocation points need not have any connectivity requirement. They scale well with spatial dimension making them
attractive for modeling high-dimensional problems. They also possess a high rate of convergence. For small to moderate-sized problems, RBF collocation methods outperform traditional methods, but for
large problems, the resultant coefficient matrix is highly ill-conditioned, hindering the applicability of the RBF collocation methods. One of the best remedies to ill-conditioning problem is Domain
Decomposition (DD) method, which splits the original domain into smaller sub-domains and solves the sub-problems in parallel. In this talk, we first describe the interpolation by RBFs, which is used
to construct RBF collocation methods. Then we describe the RBF collocation methods for solving ODEs and PDEs. After that we present different type of DD methods and combine them with RBF collocation
methods. Finally we show the efficiency of the approach by numerical examples.
Back to top of page Ray Spiteri, Department of Computer Science
Place: McLean Hall 242.1
Time: 2:30-4:00 pm
Title: Stiffness analysis of cardiac cell models
The electrophysiology in a cardiac cell can be modelled as a system of ordinary differential equations. The efficient solution of these systems is important because they must be solved many times as
sub-problems of tissue- or organ-level simulations of cardiac electrophysiology. The wide variety of existing cardiac cell models encompasses many different properties, including the complexity of
the model and the degree of stiffness. Accordingly, no single numerical method can be expected to be the most efficient for every model. In this talk, I discuss the stiffness properties of a range of
cardiac cell models and discuss the implications for their numerical solution. This analysis allows us to select or design numerical methods that are highly effective for a given model and hence
outperform commonly used methods.
Back to top of page Professor Walter Craig (McMaster University, Department of Mathematics and Statistics ) PIMS distinguished Applied Mathematics Colloquium
Place: Arts 217
Time: 4:00-5:00 pm
Title: Waves and wave interactions: a paradigm in ocean waves
Wave phenomena occur on an enormous range of scales, from the sub-quantum mechanical to the astrophysical. This talk will discuss the problem of free surface waves in water, which is a classical
problem in mathematical hydrodynamics. We will discuss some of the common scale-independent features of wave prop- agation and wave interaction in this context, including detailed descriptions of
nonlinear wave collisions, and the beginnings of a rigorous kinetic theory for a regime of wave turbulence.
Back to top of page Professor Joel Feldman (University of British Columbia, Department of Mathematics)
Place: Arts 217
Time: 3:30-4:30 pm
Title: Towards the Construction of Bosonic Many-body Models
Bosonic many-body models are an important class of mathematical models that are used to study gases of bosonic particles at very low temperatures. I will introduce
(i) the physical systems,
(ii) one way Physicists formulate the models in question,
(iii) why the models are expected to exhibit some very interesting behaviour,
(iv) why the models can be expected to be very difficult to deal with mathematically rigorously, and
(v) the first steps in a programme to construct the models mathematically rigorously.
This is joint work with Tadeusz Balaban of Rutgers University and Horst Knoerrer and Eugene Trubowitz of the ETH-Zurich.
Back to top of page Professor David Brydges, Canada Research Chair (University of British Columbia, Department of Mathematics)
Place: Arts 263
Time: 3:30-4:30 pm
Title: Branched polymers and Mayer expansions
The Mayer expansion is a power series expansion that has a central place in statistical mechanics. It is also full of combinatorial miracles that relate it to graphs, forests and branched polymers. I
will discuss the background, the results, and some open problems.
Back to top of page PIMS Applied Mathematics Series
Prof. Andrea Bertozzi, University of California, Los Angeles, CA, USA
Place:Arts 217
Time: 3:30 pm
Title: Swarming by nature and by design
The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena have attracted a renewed interest from
the engineering community. This talk will cover a survey of the speaker's research and related work in this area ranging from aggregation models in nonlinear partial differential equations to control
algorithms and robotic testbed experiments. We conclude with a discussion of some interesting problems for the applied mathematics community.
About The Speaker
Andrea Bertozzi is a mathematician who is known for her interdisciplinary work with computer scientists, physicists, and engineers. Much of her work has, in one way or another, examined the behavior
of thin liquid films on hard surfaces. In tandem with physicists and engineers, she has worked at the Argonne National Laboratory and at Duke University constructing mathematical models that explain
this and other physical phenomena. Born in 1965, in Boston, Massachusetts, to William and Norma Bertozzi, Andrea was encouraged by both of her parents to study and attend university. Her father, a
professor of physics at the Massachusetts Institute of Technology, encouraged her to pursue her interest in the sciences. In 1991, she married Bradley Koetje, a management consultant. Bertozzi knew
from an early age that she was interested in mathematics. Even in the first grade, she was captivated by the rudimentary math that was being taught and pushed to learn more. By high school, she had
begun to learn advanced math and was concentrating on theory and abstract concepts, which she found to be the most interesting part of mathematics. After graduating from high school in Lexington,
Massachusetts, in 1983, Bertozzi enrolled in Princeton University to study mathematics. She also studied a considerable amount of physics, although she took no degree in that subject. She earned her
B.A. in math in 1987 and remained at Princeton to complete an M.S. in 1988 and a Ph.D. in 1991. After completing her Ph.D., Bertozzi took a position as L. E. Dickson Instructor of Mathematics at the
University of Chicago. At Chicago, Bertozzi first became interested in the mathematics of thin films. She began working with a group of physicists who were studying mathematical models that described
the behavior of phenomena that were similar to thin films. Gradually, the problem centered specifically on a mathematical description of liquids flowing on a solid surface. This was an area of
mathematics that had not received much attention but had been researched by physicists since the 1960s. Bertozzi remained at the University of Chicago until 1995 when she was offered the position of
associate professor at Duke University in Durham, North Carolina. Then during 1995-96, she worked at the Argonne National Laboratory, located outside of Chicago in Argonne, Illinois. Here, as a MARIA
GOEPPERT MAYER Distinguished Scholar, she continued her work in the field of scientific computing, which she had begun at the University of Chicago. The purpose of scientific computing is to create
computer models that simulate physical processes on the computer. In this way, virtual experiments that can mimic actual physical conditions are created. At Argonne, Bertozzi continued her study of
the mathematical-physical properties of thin liquids on dry surfaces. This problem, which seems relatively simple, is actually complicated. A liquid applied to a dry surface will not spread evenly
but will pool and spread onto the surface in fingerlike rivulets. Bertozzi worked on a set of partial differential equations, also called evolution equations because this kind of math describes an
event occurring over time, that fit a model for film-coating behavior into mathematical terms. This work, although basic research, may someday be helpful for industries such as the
microchip-manufacturing sector, which needs to understand this coating process in making their complicated and delicate product. After her year at the Argonne Lab, Bertozzi returned to her job as
associate professor of mathematics at Duke University in 1996. In 1998, she became associate professor of mathematics and physics, and in 1999, she became a full professor in both disciplines.
Currently, she is director of Duke’s Center for Nonlinear and Complex Systems, an interdisciplinary research center that includes scientists from the disciplines of math, biology, engineering,
medical sciences, and environmental studies. In addition to her studies of thin films on hard surfaces, Bertozzi works in more general problems of fluid dynamics. Bertozzi was recognized for her work
by the Sloan Foundation, which awarded her a research fellowship in 1995. In 1996, she was presented the Presidential Early Career Award for Scientists and Engineers by the U.S. Office of Naval
Research. Cambridge University Press published her book, coauthored with Andrew Majda, Vorticity and Incompressible Flow, in 2000.
Back to top of page
Dr. Feride Tiglay Fields Institute, Toronto, ON
Place:MCLH 242.1
Time:4:00 pm
Title: Integrable evolution equations on spaces of tensor densities and their peakon solutions
In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study a variety of equations of interest in mathematical
physics. I will describe how to extend his formalism using tensor densities and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation
while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bihamiltonian structure, the presence
of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and some
results on blow-up as well as global existence of solutions. Time permitting, I will describe the peakon solutions for these equations.
Back to top of page
Dr. Feride Tiglay Fields Institute, Toronto, ON
Place:MCLH 242.1
Time: 2:30 pm
Title: The Periodic Cauchy Problem for Novikov's Equation
We study the periodic Cauchy problem for an integrable equation with cubic nonlinearities introduced by Novikov. Like the Camassa–Holm and Degasperis–Procesi equations, Novikov’s equation has Lax
pair representations and admits peakon solutions, but it has nonlinear terms that are cubic, rather than quadratic. We show the local well-posedness of the problem in Sobolev spaces and existence and
uniqueness of solutions for all time using orbit invariants. Furthermore, we prove a Cauchy–Kowalevski type theorem for this equation, which establishes the existence and uniqueness of real analytic
Back to top of page
Speaker: Marco Merkli (Memorial University)
Place: McLean 242.2
Time: 3:00 pm
Title: Quantum Scattering Measurement
We consider a quantum system (scatterer) which interacts with a sequence of identical, independent quantum systems (scattering probes). The interaction is sequential, one by one. After leaving the
scatterer, a quantum measurement is perfomed on each probe. The measurement outcomes form a random process. We analyze the asymptotic properties of this process, such as the probability of
convergence. If the process converges, then the scatterer is driven to a final state determined by the measurement outcomes. We also examine large deviations for the average of the measurements. We
illustrate the concepts and results on the (truncated) Jaynes-Cummings model, where both the scatterer and the probes are spins 1/2, representing two degrees of freedom active in the scattering
Back to top of page
Speaker: Calin Atanasiu (EURATOM MEdC Association, Bucharest and Max-Planck Institut für Plasmaphysik, Garching bei Munich, Germany)
Place: McLean 242.1
Time: 2:00 pm
Title: Special aspects of MHD calculations in tokamaks
Magneto-hydrodynamic equilibrium and stability calculations (analytical and numerical) for real diverted tokamak configurations are presented. (a) Two families of exact analytical solutions of the
Grad-Shafranov equation are presented by specifying the highest polynomial dependence of the plasma current density on the solution in such a way that the Grad-Shafranov equation becomes a linear
inhomogeneous differential equation. (b) By introducing a "cast function" in a classical flux coordinate system, in the presence of a separatrix, the solution of the equilibrium equation - the
unknown moments - is determined by the difference between the real flux surface contours and those described by the cast functions only. Thus, the necessary number of moments is small enough to make
computations time-efficient. (c) For instability calculation of tearing and external kink type, the expression of the potential energy has been written in terms of the perturbation of the flux
function, and performing an Euler minimization, a system of ordinary differential equations in that perturbation has been obtained. For a diverted configuration, the usual vanishing boundary
conditions for the perturbed flux function at the magnetic axis and at infinity can no longer be used. An approach to fix "natural" boundary conditions for the perturbed flux function just at the
plasma boundary has been developed; this replaces the vanishing boundary conditions at infinity. Special attention is given to the stabilization of external kink modes in the presence of a conducting
wall - the resistive wall modes, the most dangerous instability for the future tokamak reactors.
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Speaker: Prof. Jean-Francois Ganghoffer, LEMTA, Nancy University, France
Place: McLean 242.1
Time: 11:00 am
Title: Mechanics and thermodynamics of surface growth. Application to bone remodeling
The surface growth of biological tissues is presently analyzed at the continuum scale of tissue elements, adopting the framework of the thermodynamics of surfaces and in line with Eshelbian
mechanics. From a kinematic viewpoint, growth is assumed to occur in a moving referential configuration, considered as an open evolving domain exchanging mass, work, and energy with its environment.
The growing surface is endowed with a superficial excess concentration of moles, which is ruled by an appropriate kinetic equation. The material surface forces for growth are evaluated versus a
surface Eshelby stress, the curvature tensor of the growing surface, the gradient of the chemical energy of nutrients and the applied superficial force field. A system of coupled field equations is
written for the superficial density of minerals, their concentration and the surface velocity of the growing surface. Application of the developed formalism to bone external remodeling highlights the
interplay between transport phenomena and generation of surface mechanical forces. The model is able to describe both bone growth and resorption, according to the respective magnitude of the chemical
and mechanical contributions to the material surface driving force for growth.
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Speaker: Ahmed Kaffel, Postdoctoral Fellow, Department of Computer Science, University of Saskatchewan
Place: McLean 242.1
Time: 2:00 pm
Title: On the Stability of plane viscoelastic shear flows in the limit of infinite Weissenberg and Reynolds numbers
Elastic effects on the hydrodynamic instability of inviscid parallel shear flows are investigated through a linear stability analysis. We focus on the upper convected Maxwell model in the limit of
infinite Weissenberg and Reynolds numbers. Specifically, we study the effects of elasticity on the instability of a few classes of simple parallel flows, specifically plane Poiseuille and Couette
flows, the hyperbolic-tangent shear layer and the Bickley jet. The equation for stability is derived and solved numerically using the Chebyshev collocation spectral method. This algorithm is
computationally efficient and accurate in reproducing the eigenvalues. We consider flows bounded by walls as well as flows bounded by free surfaces. In the inviscid, nonelastic case all the flows we
study are unstable for free surfaces. In the case of wall bounded flow, there are instabilities in the shear layer and Bickley jet flows. In all cases, the effect of elasticity is to reduce and
ultimately suppress the inviscid instability. The numerical solutions are compared with the analysis of the long wave limit and excellent agreement is shown between the analytical and the numerical
solutions. We found flows which are long wave stable, but nevertheless unstable to wave numbers in a certain finite range. While elasticity is ultimately stabilizing, this effect is not monotone;
there are instances where a small amount of elasticity actually destabilizes the flow. The linear stability in the short wave limit of shear flows bounded by two parallel free surfaces is
investigated. Unlike the plane Couette flow which has no short wave instability, we show that plane Poiseuille flow has two unstable eigenmodes localized near the free surfaces which can be combined
into an even and an odd eigenfunctions. The derivation of the asymptotics of these modes shows that our numerical eigenvalues are in agreement with the analytic formula and that the difference
between the two eigenvalues tends to zero exponentially with the wavenumber α.
Back to top of page
Speaker: Professor Abba Gumel, Department of Mathematics, University of Manitoba.
Place: McLean 242.1
Time: 2:00 pm
Title: Dynamically consistent finite-difference methods for differential equations
Standard numerical integrators, such as the Runge-Kutta family of explicit finite-difference methods, are known to exhibit numerous scheme-dependent instabilities and generally fail to preserve some
of the main essential qualitative features (such as positivity, boundedness, asymptotic stability, and bifurcation properties) of the governing continuous system they approximate. The talk will
address the problem of designing appropriate discrete-time models that are dynamically consistent with the corresponding continuous-time model they approximate. Some models arising from modeling
real-life phenomena in the natural and engineering sciences will be discussed.
Biodata: Abba Gumel is a Professor of Mathematics at the University of Manitoba. His research work is based on the design and analysis (qualitative and quantitative) of models for the spread and
control of emerging and re-emerging diseases of public health significance. His homepage is http://home.cc.umanitoba.ca/~gumelab.
Back to top of page | {"url":"http://math.usask.ca/~szmigiel/seminar.html","timestamp":"2014-04-17T02:03:39Z","content_type":null,"content_length":"37192","record_id":"<urn:uuid:03f10f25-5fbf-4fe4-ac16-f6ddbf911dce>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00534-ip-10-147-4-33.ec2.internal.warc.gz"} |
G. Gottlob, G. Greco and F. Scarcello (2005) Pure Nash Equilibria: Hard and Easy Games
G. Gottlob, G. Greco and F. Scarcello (2005) "Pure Nash Equilibria: Hard and Easy Games", Volume 24, pages 357-406 2008 IJCAI-JAIR Best Paper Prize
PDF | PostScript | doi:10.1613/jair.1683
We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is
NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested
in quantitative and qualitative restrictions of the way each player's payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player
depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash
equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding
pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems.
Click here to return to Volume 24 contents list | {"url":"http://www.jair.org/papers/paper1683.html","timestamp":"2014-04-17T06:41:23Z","content_type":null,"content_length":"3680","record_id":"<urn:uuid:b9b369d8-094a-45b6-923b-391f4577be22>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00548-ip-10-147-4-33.ec2.internal.warc.gz"} |
As the first example we consider here the one presented in Kamke's book with number 185
If infolevel is set to a greater integer (possible settings are 1 through 5), more detailed information about the computation method is displayed.
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The relative invariant s3 is: 10/27/x^12*(9*x^2+2)
The first absolute invariant s5^3/s3^5 is: -729/100*(90*x^4-15*x^2-14)^3/(9*x^2+2)^5
The second absolute invariant s3*s7/s5^2 is: 5/3*(9*x^2+2)*(972*x^6-324*x^4-15*x^2+98)/(90*x^4-15*x^2-14)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
inverse of the transformation solving the problem is: {t = x, u(t) = y(x)}
<- Abel successful
The integrals above can be evaluated in terms of hypergeometric functions using value. These implicit results can be tested using odetest
The above means mainly that dsolve can solve the whole class associated to this ODE. For instance, by changing variables
and this ODE is also solvable (actually for arbitrary F(t), P(t) and Q(t)):
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The first invariant is non-rational: -729/100*(90*F(x)^4-15*F(x)^2-14)^3/(9*F(x)^2+2)^5
-> Searching for a convenient change of variables...
<- Unable to rationalize the invariant
The relative invariant s3 is: 10/27*(9*F(x)^2+2)/F(x)^12*diff(F(x),x)^3*P(x)^3
The first absolute invariant s5^3/s3^5 is: -729/100*(90*F(x)^4-15*F(x)^2-14)^3/(9*F(x)^2+2)^5
The second absolute invariant s3*s7/s5^2 is: 5/3*(9*F(x)^2+2)*(972*F(x)^6-324*F(x)^4-15*F(x)^2+98)/(90*F(x)^4-15*F(x)^2-14)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
inverse of the transformation solving the problem is: {t = F(x), u(t) = P(x)*y(x)+Q(x)}
<- Abel successful
The next example is still from Kamke's and appears there with number 257
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The equivalent Abel ODE of 1st kind is: diff(u(x),x) = -2*x*(x^2+1)*(x^2-1)*u(x)^3-2/x^2*u(x)^2-1/x*u(x)
The relative invariant s3 is: -8/27*(27*x^8-9*x^4+2)/x^6
The first absolute invariant s5^3/s3^5 is: 729*(135*x^16-36*x^12+54*x^8-15*x^4+2)^3/(27*x^8-9*x^4+2)^5
The second absolute invariant s3*s7/s5^2 is: 1/3*(27*x^8-9*x^4+2)*(2835*x^24+243*x^20+2349*x^16-927*x^12+495*x^8-105*x^4+10)/(135*x^16-36*x^12+54*x^8-15*x^4+2)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
...checking Abel class AIA (by Halphen)
...checking Abel class AIL (205)
...checking Abel class AIA (147)
...checking Abel class AIL (581)
...checking Abel class AIL (200)
...checking Abel class AIL (257)
inverse of the transformation solving the problem is: {t = 1/x^2, u(t) = x*y(x)}
<- Abel successful
This ODE actually belongs to the Abel class represented by the simpler ODE
Actually, by converting ode[257] from Second Kind to Abel First Kind format
then changing variables in this ODE above (first element in the sequence) using
and renaming the variables x=t, u=y we obtain:
Isolating y' we arrive at the ODE representative for the Abel Class[257]
A much more complicated example is given by the ODE presented in Kamke's book with number 43:
This ODE belongs to class B (by Liouville); that is: it can be obtained from
by changing variables
and setting the parameter C in the Abel ODE representative of Class[B] as
The result above is in fact ode[43]. The process of determining the value of C for which a equivalence between the ODEs Class[B] and ode[43] exists, as well as the explicit form of the equivalence
transformation, followed by using it to build the answer to ode[43] is now available via
The database of solvable classes dsolve includes representatives for 25 classes. To each class there is associated a number. These numbers can be seen via
Classes 1.1 to 1.9 are parameterized classes (parameter C), and each class representative - for instance, for Class 33 - can be seen via
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Although we might wonder how much tinfoil it takes to cover the Egyptian pyramids, finding out how many mummies fit inside it would be much more interesting. Provided they don't form an undead army
and attack you, that is.
If we have a pyramid with the same base (B) and height (h) as a prism, we can try and deduce its volume. The empty space in front and back of the pyramid can form another pyramid, and same with the
empty space on either side of the pyramid. That means the prism can holds the volume of more or less three pyramids.
If you have trust issues with us, fill these prisms and pyramids with sand or water or something and see for yourself. We don't lie (that much). You should end up with this formula for the volume of
a pyramid.
Using this formula, we can calculate how much space we have inside the Egyptian pyramids.
Sample Problem
If an Egyptian pyramid has a square base with an edge length of 700 feet and a height of 450 feet, what's the pyramid's volume?
Drumroll, please.
V = 73,500,000 cubic feet
Or seven times as much in cubic dog-feet.
Find the volume of the square pyramid.
Find the volume of the regular (but not right) pentagonal pyramid.
Find the volume of the rectangular pyramid.
Find the volume of the triangular pyramid.
We're vacationing in Egypt and we come across the Pyramid of Giza. Its square base has an edge of 756 feet and it's 453 feet high. If it were filled to the brim with nothing but mummies, how creepy
would that be? Answer: A lot. How many mummies would it hold? Assume a mummy takes up about 12 cubic feet of space. | {"url":"http://www.shmoop.com/surface-area-volume/volume-pyramids-help.html","timestamp":"2014-04-20T11:19:30Z","content_type":null,"content_length":"40903","record_id":"<urn:uuid:6fe4a8f5-152c-40df-94fd-c4081f18eb2d>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00632-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sketch of a Phylogenetic Query Language
Names on Nodes
for two primary purposes:
• Delineating phylogenetic hypotheses (as directed, acyclic graphs).
• Associating identifiers with definitions.
In some ways this works out to be a bit like a query language. You can use it to set up data constructs, and then search them for groups of interest. For example, suppose you wanted a list of all
stem-humans from Kenya. Assuming that your dataset included 1) a taxonomic unit called
Homo sapiens
, 2) a group called
for all extant taxonomic units, and 3) a group called
for all Kenyan taxonomic units, that query might look like this:
<apply xmlns="http://www.w3.org/1998/Math/MathML">
<csymbol definitionURL="http://namesonnodes.org/ns/math/2009#def-Total"/>
<ci>Homo sapiens</ci>
<ci>Homo sapiens</ci>
MathML is great for being flexible and extensible enough to cover concepts like this. But ... it's also really verbose. This is fine for my purposes so far, but it may be cumbersome for other
purposes. So I've been playing around with a more succinct way to write these expressions. Today I tossed up some rough ideas here:
This is a plain-text format loosely inspired by mathematical notation, the C language, etc. Using it, the above query becomes:
"Kenya" & (total("Homo sapiens", "extant") - "Homo sapiens")
...which is quite a bit shorter.
This is still in very early stages, so I thought I'd post it to get some feedback.
Here are a few of the simpler clade definition examples:
"Aves" := clade("Struthio camelus" | "Tetrao major" |
"Vultur gryphus").
"Saurischia" := clade("Megalosaurus bucklandii" <-
"Iguanodon bernissartensis").
"Avialae" := clade("wings used for powered flight" @
"Vultur gryphus").
2 comments:
1. All very impressive and i can se its uses. However, I can't really comment on its efficacy as I am still getting my head around "simple" SQL.
2. This proposal actually has very little in common with SQL. I did write on using SQL to make phylogenetic queries earlier here and here. As you can see, SQL makes things considerably more | {"url":"http://3lbmonkeybrain.blogspot.com/2010/04/sketch-of-phylogenetic-query-language.html?showComment=1270551872110","timestamp":"2014-04-21T04:35:10Z","content_type":null,"content_length":"101039","record_id":"<urn:uuid:f585ae5f-30bd-4c3b-8b20-05b68486c9ff>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00041-ip-10-147-4-33.ec2.internal.warc.gz"} |
Progressive mixture rules are deviation suboptimal
Jean-Yves Audibert
In: NIPS 2007, 3-6 Dec 2007, Vancouver, Canada.
We consider the learning task consisting in predicting as well as the best function in a finite reference set $G$ up to the smallest possible additive term. If $R(g)$ denotes the generalization error
of a prediction function $g$, under reasonable assumptions on the loss function (typically satisfied by the least square loss when the output is bounded), it is known that the progressive mixture
rule satisfies E R(progressive mixture) < min_{g in G} R(g) + Cst ( log |G| ) / n, where $n$ denotes the size of the training set, and $E$ denotes the expectation with respect to the training set
distribution. This work shows that, surprisingly, for appropriate reference sets $\G$, the deviation convergence rate of the progressive mixture rule is no better than $Cst/sqrt n$: it fails to
achieve the expected $Cst/n$. We also provide an algorithm which does not suffer from this drawback,and which is optimal in both deviation and expectation convergence rates. | {"url":"http://eprints.pascal-network.org/archive/00003153/","timestamp":"2014-04-16T19:14:40Z","content_type":null,"content_length":"7173","record_id":"<urn:uuid:9d63fffa-caed-4a0b-8f62-cb1159e8f967>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
How do scientists measure or calculate the weight of a planet?
Barry Lienert, a geophysicist at the University of Hawaii, provides the following explanation.
We start by determining the mass of the Earth. Issac Newton's Law of Universal Gravitation tells us that the force of attraction between two objects is proportional the product of their masses
divided by the square of the distance between their centers of mass. To obtain a reasonable approximation, we assume their geographical centers are their centers of mass.
Because we know the radius of the Earth, we can use the Law of Universal Gravitation to calculate the mass of the Earth in terms of the gravitational force on an object (its weight) at the Earth's
surface, using the radius of the Earth as the distance. We also need the Constant of Proportionality in the Law of Universal Gravitation, G. This value was experimentally determined by Henry
Cavendish in the 18th century to be the extemely small force of 6.67 x 10^-11 Newtons between two objects weighing one kilogram each and separated by one meter. Cavendish determined this constant by
accurately measuring the horizontal force between metal spheres in an experiment sometimes referred to as "weighing the earth."
Knowing the mass and radius of the Earth and the distance of the Earth from the sun, we can calculate the mass of the sun (right), again by using the law of universal gravitation. The gravitational
attraction between the Earth and the sun is G times the sun's mass times the Earth's mass, divided by the distance between the Earth and the sun squared. This attraction must be equal to the
centripetal force needed to keep the earth in its (almost circular) orbit around the sun. The centripetal force is the Earth's mass times the square of its speed divided by its distance from the sun.
By astronomically determining the distance to the sun, we can calculate the earth's speed around the sun and hence the sun's mass.
Once we have the sun's mass, we can similarly determine the mass of any planet by astronomically determining the planet's orbital radius and period, calculating the required centripetal force and
equating this force to the force predicted by the law of universal gravitation using the sun's mass.
Additional details are provided by Gregory A. Lyzenga, a physicist at Harvey Mudd College in Claremont, Calif.
The weight (or the mass) of a planet is determined by its gravitational effect on other bodies. Newton's Law of Gravitation states that every bit of matter in the universe attracts every other with a
gravitational force that is proportional to its mass. For objects of the size we encounter in everyday life, this force is so minuscule that we don't notice it. However for objects the size of
planets or stars, it is of great importance.
In order to use gravity to find the mass of a planet, we must somehow measure the strength of its "tug" on another object. If the planet in question has a moon (a natural satellite), then nature has
already done the work for us. By observing the time it takes for the satellite to orbit its primary planet, we can utilize Newton's equations to infer what the mass of the planet must be.
For planets without observable natural satellites, we must be more clever. Although Mercury and Venus (for example) do not have moons, they do exert a small pull on one another, and on the other
planets of the solar system. As a result, the planets follow paths that are subtly different than they would be without this perturbing effect. Although the mathematics is a bit more difficult, and
the uncertainties are greater, astronomers can use these small deviations to determine how massive the moonless planets are.
Finally, what about those objects such as asteroids, whose masses are so small that they do not measurably perturb the orbits of the other planets? Until recent years, the masses of such objects were
simply estimates, based upon the apparent diameters and assumptions about the possible mineral makeup of those bodies.
Now, however, several asteroids have been (or soon will be) visited by spacecraft. Just like a natural moon, a spacecraft flying by an asteroid has its path bent by an amount controlled by the mass
of the asteroid. This "bending" is measured by careful tracking and Doppler radio measurement from Earth. Recently, the NEAR spacecraft flew by the asteroid Mathilde, determining for the first time
its actual mass. It turned out to be considerably lighter and more "frothy" in structure than had been expected, a fact that is challenging planetary scientists for an explanation.
Originally published on March 16, 1998. | {"url":"http://www.scientificamerican.com/article/how-do-scientists-measure/","timestamp":"2014-04-17T03:14:02Z","content_type":null,"content_length":"61261","record_id":"<urn:uuid:1e6f7641-b46c-4c14-96ff-c81063aa4ed5>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00498-ip-10-147-4-33.ec2.internal.warc.gz"} |
How do scientists measure or calculate the weight of a planet?
Barry Lienert, a geophysicist at the University of Hawaii, provides the following explanation.
We start by determining the mass of the Earth. Issac Newton's Law of Universal Gravitation tells us that the force of attraction between two objects is proportional the product of their masses
divided by the square of the distance between their centers of mass. To obtain a reasonable approximation, we assume their geographical centers are their centers of mass.
Because we know the radius of the Earth, we can use the Law of Universal Gravitation to calculate the mass of the Earth in terms of the gravitational force on an object (its weight) at the Earth's
surface, using the radius of the Earth as the distance. We also need the Constant of Proportionality in the Law of Universal Gravitation, G. This value was experimentally determined by Henry
Cavendish in the 18th century to be the extemely small force of 6.67 x 10^-11 Newtons between two objects weighing one kilogram each and separated by one meter. Cavendish determined this constant by
accurately measuring the horizontal force between metal spheres in an experiment sometimes referred to as "weighing the earth."
Knowing the mass and radius of the Earth and the distance of the Earth from the sun, we can calculate the mass of the sun (right), again by using the law of universal gravitation. The gravitational
attraction between the Earth and the sun is G times the sun's mass times the Earth's mass, divided by the distance between the Earth and the sun squared. This attraction must be equal to the
centripetal force needed to keep the earth in its (almost circular) orbit around the sun. The centripetal force is the Earth's mass times the square of its speed divided by its distance from the sun.
By astronomically determining the distance to the sun, we can calculate the earth's speed around the sun and hence the sun's mass.
Once we have the sun's mass, we can similarly determine the mass of any planet by astronomically determining the planet's orbital radius and period, calculating the required centripetal force and
equating this force to the force predicted by the law of universal gravitation using the sun's mass.
Additional details are provided by Gregory A. Lyzenga, a physicist at Harvey Mudd College in Claremont, Calif.
The weight (or the mass) of a planet is determined by its gravitational effect on other bodies. Newton's Law of Gravitation states that every bit of matter in the universe attracts every other with a
gravitational force that is proportional to its mass. For objects of the size we encounter in everyday life, this force is so minuscule that we don't notice it. However for objects the size of
planets or stars, it is of great importance.
In order to use gravity to find the mass of a planet, we must somehow measure the strength of its "tug" on another object. If the planet in question has a moon (a natural satellite), then nature has
already done the work for us. By observing the time it takes for the satellite to orbit its primary planet, we can utilize Newton's equations to infer what the mass of the planet must be.
For planets without observable natural satellites, we must be more clever. Although Mercury and Venus (for example) do not have moons, they do exert a small pull on one another, and on the other
planets of the solar system. As a result, the planets follow paths that are subtly different than they would be without this perturbing effect. Although the mathematics is a bit more difficult, and
the uncertainties are greater, astronomers can use these small deviations to determine how massive the moonless planets are.
Finally, what about those objects such as asteroids, whose masses are so small that they do not measurably perturb the orbits of the other planets? Until recent years, the masses of such objects were
simply estimates, based upon the apparent diameters and assumptions about the possible mineral makeup of those bodies.
Now, however, several asteroids have been (or soon will be) visited by spacecraft. Just like a natural moon, a spacecraft flying by an asteroid has its path bent by an amount controlled by the mass
of the asteroid. This "bending" is measured by careful tracking and Doppler radio measurement from Earth. Recently, the NEAR spacecraft flew by the asteroid Mathilde, determining for the first time
its actual mass. It turned out to be considerably lighter and more "frothy" in structure than had been expected, a fact that is challenging planetary scientists for an explanation.
Originally published on March 16, 1998. | {"url":"http://www.scientificamerican.com/article/how-do-scientists-measure/","timestamp":"2014-04-17T03:14:02Z","content_type":null,"content_length":"61261","record_id":"<urn:uuid:1e6f7641-b46c-4c14-96ff-c81063aa4ed5>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00498-ip-10-147-4-33.ec2.internal.warc.gz"} |
Static Program Slicing Algorithms are Minimal for Free Liberal Program Schemas
, 2004
"... A program schema defines a class of programs, all of which have identical statement structure, but whose expressions may differ. We define a class of syntactic similarity binary relations
between linear structured schemas and show that these relations characterise schema equivalence for structured s ..."
Cited by 6 (6 self)
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A program schema defines a class of programs, all of which have identical statement structure, but whose expressions may differ. We define a class of syntactic similarity binary relations between
linear structured schemas and show that these relations characterise schema equivalence for structured schemas which are linear, free and liberal. In this paper we prove that similarity implies
equivalence for linear schemas; the proof of a near-converse for schemas that are linear, free and liberal (LFL), which is much longer, is given in a Technical Report, which also contains the results
of this paper. Our main result considerably extends the class of program schemas for which equivalence is known to be decidable, and suggests that linearity is a constraint worthy of further
investigation. Key words: structured program schemas, conservative schemas, liberal schemas, free schemas, linear schemas, schema equivalence, static analysis, program slicing Preprint submitted to
Elsevier Science 2 August 2006 1
"... Aprogramschemadefinesaclassofprograms,allofwhichhaveidenticalstatement structure, but whose functions and predicates may differ. A schema thus defines an entire class of programs according to
how its symbols are interpreted. A slice of a schema is obtained from a schema by deleting some of its state ..."
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Aprogramschemadefinesaclassofprograms,allofwhichhaveidenticalstatement structure, but whose functions and predicates may differ. A schema thus defines an entire class of programs according to how its
symbols are interpreted. A slice of a schema is obtained from a schema by deleting some of its statements. We prove that given a schema S which is predicate-linear, free and liberal, such that the
true and false parts of every if predicate satisfy a simple additional condition, and a slicing Preprint submitted to Elsevier Sciencecriterion defined by the final value of a given variable after
execution of any program defined by S, theminimalsliceofSwhich respects this slicing criterion contains all the symbols ‘needed ’ by the variable according to the data dependence and control
dependence relations used in program slicing, which is the symbol set given by Weiser’s static slicing algorithm. Thus this algorithm gives predicate-minimal slices for classes of programs
represented by schemas satisfying our set of conditions. We also give an example to show that the corresponding result with respect to the slicing criterion defined by termination behaviour is
incorrect. This strengthens a recent result in which S was required to be linear, free and liberal, and termination behaviour as a slicing criterion was not considered.
"... program schemas ..."
, 2010
"... Given a program, a quotient can be obtained from it by deleting zero or more statements. The field of program slicing is concerned with computing a quotient of a program which preserves part of
the behaviour of the original program. All program slicing algorithms take account of the structural prope ..."
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Given a program, a quotient can be obtained from it by deleting zero or more statements. The field of program slicing is concerned with computing a quotient of a program which preserves part of the
behaviour of the original program. All program slicing algorithms take account of the structural properties of a program such as control dependence and data dependence rather than the semantics of
its functions and predicates, and thus work, in effect, with program schemas. The dynamic slicing criterion of Korel and Laski requires only that program behaviour is preserved in cases where the
original program follows a particular path, and that the slice/quotient follows this path. In this paper we formalise Korel and Laski’s definition of a dynamic slice as applied to linear schemas, and
also formulate a less restrictive definition in which the path through the original program need not be preserved by the slice. The less restrictive definition has the benefit of leading to smaller
slices. For both definitions, we compute complexity bounds for the problems of establishing whether a given slice of a linear schema is a dynamic slice and whether a linear schema has a non-trivial
dynamic slice and prove that the latter problem is NP-hard in both cases. We also give an example to prove that minimal dynamic slices (whether or not they preserve the original path) need not be
unique. 1.
"... Abstract—A highly efficient lightweight forward static slicing method is introduced. The method is implemented as a tool on top of srcML, an XML representation of source code. The approach does
not compute the program dependence graph but instead dependency information is computed as needed while co ..."
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Abstract—A highly efficient lightweight forward static slicing method is introduced. The method is implemented as a tool on top of srcML, an XML representation of source code. The approach does not
compute the program dependence graph but instead dependency information is computed as needed while computing the slice on a variable. The result is a list of line numbers, dependent variables,
aliases, and function calls that are part of the slice for a given variable. The tool produces the slice in this manner for all variables in a given system. The approach is highly scalable and can
generate the slices for all variables of the Linux kernel in less than 13 minutes. Benchmark results are compared with the CodeSurfer slicing tool and the approach compares well with regards to
accuracy of slices. Keywords- program slicing; software maintenance; impact analysis; minimal slice I. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=417702","timestamp":"2014-04-20T13:43:22Z","content_type":null,"content_length":"23823","record_id":"<urn:uuid:20d55d1f-7596-4f66-adc4-fd736c9e3f88>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00238-ip-10-147-4-33.ec2.internal.warc.gz"} |
Int -> [a] -> [[a]]
drop n xs returns the suffix of xs after the first n elements, or [] if n > length xs: > drop 6 "Hello World!" == "World!" > drop 3 [1,2,3,4,5] == [4,5] > drop 3 [1,2] == [] > drop 3 [] == [] > drop
(-1) [1,2] == [1,2] > drop 0 [1,2] == [1,2] It is an instance of the more general Data.List.genericDrop, in which n may be of any integral type.
take n, applied to a list xs, returns the prefix of xs of length n, or xs itself if n > length xs: > take 5 "Hello World!" == "Hello" > take 3 [1,2,3,4,5] == [1,2,3] > take 3 [1,2] == [1,2] > take 3
[] == [] > take (-1) [1,2] == [] > take 0 [1,2] == [] It is an instance of the more general Data.List.genericTake, in which n may be of any integral type.
replicateM n act performs the action n times, gathering the results.
The intersperse function takes an element and a list and `intersperses' that element between the elements of the list. For example, > intersperse ',' "abcde" == "a,b,c,d,e"
replicate n x is a list of length n with x the value of every element. It is an instance of the more general Data.List.genericReplicate, in which n may be of any integral type.
The non-overloaded version of insert.
The deleteBy function behaves like delete, but takes a user-supplied equality predicate.
The genericDrop function is an overloaded version of drop, which accepts any Integral value as the number of elements to drop.
The genericTake function is an overloaded version of take, which accepts any Integral value as the number of elements to take.
Append two lists, i.e., > [x1, ..., xm] ++ [y1, ..., yn] == [x1, ..., xm, y1, ..., yn] > [x1, ..., xm] ++ [y1, ...] == [x1, ..., xm, y1, ...] If the first list is not finite, the result is the first
O(n) Splits a Text into components of length k. The last element may be shorter than the other chunks, depending on the length of the input. Examples: > chunksOf 3 "foobarbaz" == ["foo","bar","baz"]
> chunksOf 4 "haskell.org" == ["hask","ell.","org"]
The deleteFirstsBy function takes a predicate and two lists and returns the first list with the first occurrence of each element of the second list removed.
The unionBy function is the non-overloaded version of union.
scanl is similar to foldl, but returns a list of successive reduced values from the left: > scanl f z [x1, x2, ...] == [z, z `f` x1, (z `f` x1) `f` x2, ...] Note that > last (scanl f z xs) == foldl f
z xs.
The insert function takes an element and a list and inserts the element into the list at the last position or equal to the next element. In particular, if the list is sorted before the call, the
result will also be sorted. It is a special case of insertBy, which allows the programmer to supply their own comparison function.
delete x removes the first occurrence of x from its list argument. For example, > delete 'a' "banana" == "bnana" It is a special case of deleteBy, which allows the programmer to supply their own
equality test.
scanr is the right-to-left dual of scanl. Note that > head (scanr f z xs) == foldr f z xs.
intercalate xs xss is equivalent to (concat (intersperse xs xss)). It inserts the list xs in between the lists in xss and concatenates the result.
Replaces all instances of a value in a list by another value.
Outputs indented XHTML. Because space matters in HTML, the output is quite messy.
zipWith generalises zip by zipping with the function given as the first argument, instead of a tupling function. For example, zipWith (+) is applied to two lists to produce the list of corresponding
The \\ function is list difference ((non-associative). In the result of xs \\ ys, the first occurrence of each element of ys in turn (if any) has been removed from xs. Thus > (xs ++ ys) \\ xs == ys.
It is a special case of deleteFirstsBy, which allows the programmer to supply their own equality test.
The intersect function takes the list intersection of two lists. For example, > [1,2,3,4] `intersect` [2,4,6,8] == [2,4] If the first list contains duplicates, so will the result. > [1,2,2,3,4]
`intersect` [6,4,4,2] == [2,2,4] It is a special case of intersectBy, which allows the programmer to supply their own equality test.
The union function returns the list union of the two lists. For example, > "dog" `union` "cow" == "dogcw" Duplicates, and elements of the first list, are removed from the the second list, but if the
first list contains duplicates, so will the result. It is a special case of unionBy, which allows the programmer to supply their own equality test.
Flexible type extension
Replace all locations in the input with the same value. The default definition is fmap . const, but this may be overridden with a more efficient version.
The elemIndices function extends elemIndex, by returning the indices of all elements equal to the query element, in ascending order.
Substitute various time-related information for each %-code in the string, as per formatCharacter. For all types (note these three are done here, not by formatCharacter): * %% % * %t tab * %n newline
glibc-style modifiers can be used before the letter (here marked as z): * %-z no padding * %_z pad with spaces * %0z pad with zeros * %^z convert to upper case * %#z convert to lower case
(consistently, unlike glibc) For TimeZone (and ZonedTime and UTCTime): * %z timezone offset in the format -HHMM. * %Z timezone name For LocalTime (and ZonedTime and UTCTime): * %c as dateTimeFmt
locale (e.g. %a %b %e %H:%M:%S %Z %Y) For TimeOfDay (and LocalTime and ZonedTime and UTCTime): * %R same as %H:%M * %T same as %H:%M:%S * %X as timeFmt locale (e.g. %H:%M:%S) * %r as time12Fmt locale
(e.g. %I:%M:%S %p) * %P day-half of day from (amPm locale), converted to lowercase, am, pm * %p day-half of day from (amPm locale), AM, PM * %H hour of day (24-hour), 0-padded to two chars, 00 - 23 *
%k hour of day (24-hour), space-padded to two chars, 0 - 23 * %I hour of day-half (12-hour), 0-padded to two chars, 01 - 12 * %l hour of day-half (12-hour), space-padded to two chars, 1 - 12 * %M
minute of hour, 0-padded to two chars, 00 - 59 * %S second of minute (without decimal part), 0-padded to two chars, 00 - 60 * %q picosecond of second, 0-padded to twelve chars, 000000000000 -
999999999999. * %Q decimal point and fraction of second, up to 12 second decimals, without trailing zeros. For a whole number of seconds, %Q produces the empty string. For UTCTime and ZonedTime: * %s
number of whole seconds since the Unix epoch. For times before the Unix epoch, this is a negative number. Note that in %s.%q and %s%Q the decimals are positive, not negative. For example, 0.9 seconds
before the Unix epoch is formatted as -1.1 with %s%Q. For Day (and LocalTime and ZonedTime and UTCTime): * %D same as %m/%d/%y * %F same as %Y-%m-%d * %x as dateFmt locale (e.g. %m/%d/%y) * %Y year,
no padding. Note %0y and %_y pad to four chars * %y year of century, 0-padded to two chars, 00 - 99 * %C century, no padding. Note %0C and %_C pad to two chars * %B month name, long form (fst from
months locale), January - December * %b, %h month name, short form (snd from months locale), Jan - Dec * %m month of year, 0-padded to two chars, 01 - 12 * %d day of month, 0-padded to two chars, 01
- 31 * %e day of month, space-padded to two chars, 1 - 31 * %j day of year, 0-padded to three chars, 001 - 366 * %G year for Week Date format, no padding. Note %0G and %_G pad to four chars * %g year
of century for Week Date format, 0-padded to two chars, 00 - 99 * %f century for Week Date format, no padding. Note %0f and %_f pad to two chars * %V week of year for Week Date format, 0-padded to
two chars, 01 - 53 * %u day of week for Week Date format, 1 - 7 * %a day of week, short form (snd from wDays locale), Sun - Sat * %A day of week, long form (fst from wDays locale), Sunday - Saturday
* %U week of year sundayStartWeek), 0-padded to two chars, 00 - 53 * %w day of week number, 0 (= Sunday) - 6 (= Saturday) * %W week of year mondayStartWeek), 0-padded to two chars, 00 - 53
The zipWith3 function takes a function which combines three elements, as well as three lists and returns a list of their point-wise combination, analogous to zipWith.
The foldM function is analogous to foldl, except that its result is encapsulated in a monad. Note that foldM works from left-to-right over the list arguments. This could be an issue commutative. >
foldM f a1 [x1, x2, ..., xm] == > do > a2 <- f a1 x1 > a3 <- f a2 x2 > ... > f am xm If right-to-left evaluation is required, the input list should be reversed.
Lift a binary function to actions.
Like foldM, but discards the result.
Extend a generic reader
Lift a ternary function to actions.
Monadic fold over the elements of a structure, associating to the left, i.e. from left to right.
Monadic fold over the elements of a structure, associating to the right, i.e. from right to left.
Promote a function to a monad, scanning the monadic arguments from left to right. For example, > liftM2 (+) [0,1] [0,2] = [0,2,1,3] > liftM2 (+) (Just 1) Nothing = Nothing
applies lookup to an interval
Show more results | {"url":"http://www.haskell.org/hoogle/?hoogle=Int+-%3E+%5Ba%5D+-%3E+%5B%5Ba%5D%5D&start=41","timestamp":"2014-04-24T11:10:09Z","content_type":null,"content_length":"65318","record_id":"<urn:uuid:74d083a1-d41e-4822-9b19-8fdb71031de2>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00627-ip-10-147-4-33.ec2.internal.warc.gz"} |
Magnitudes of Vectors Don't Add Up
Date: 11/14/2010 at 02:15:12
From: David
Subject: Why does the vector law of addition work?
I don't understand how adding vectors results in a triangle in which the
third side is equivalent to the sum of the original two vectors. In
particular, I don't understand how the sum of the two added vectors can
have the same magnitude as the vector sum.
A vector is defined as something with magnitude and direction, so vectors
are equal if and only if they have the same magnitude and direction. The
addition of vectors means combining two vectors, so the result of vector
addition should give a vector with the same direction and magnitude as
that of the combination of the added vectors, right?
I can see how the sum of vectors A and B, if combined, would have the same
direction as the third side of a triangle. What I don't understand here is
how the magnitude of the third side can be equal to the magnitude of the
other two sides. That would mean two sides of a triangle sum to the third
side, wouldn't it?
Date: 11/14/2010 at 10:01:22
From: Doctor Ian
Subject: Re: Why does the vector law of addition work?
Hi David,
Suppose you are standing on a giant grid. You are given two numbers (a,b).
You move a units to the east, and b units to the north. Now you are given
two more numbers (c,d). You move c units to the east, and d units to the
What is the total distance you've moved to the east? It's a + c, right?
And what is the total distance you've moved to the north? It's b + d,
right? So you could have got to the same final point by being given the
numbers (a + c, b + d) along with the same instructions.
Does this make sense? Do you see how it illustrates the rule for vector
> A vector is defined as something with magnitude and direction, so
> vectors are equal if and only if they have the same magnitude and
> direction. The addition of vectors means combining two vectors, so the
> result of vector addition should give a vector with the same direction
> and magnitude as that of the combination of the added vectors, right?
Right. And they are combined by adding their components, as illustrated in
the example above.
> I can see how the sum of vectors A and B, if combined, would have the
> same direction as the third side of a triangle. What I don't understand
> here is how the magnitude of the third side can be equal to the
> magnitude of the other two sides. That would mean two sides of a
> triangle sum to the third side, wouldn't it?
The magnitudes don't add directly. If you add two vectors, the magnitude
of the resulting vector will be somewhere between zero and the sum of the
individual magnitudes.
The latter occurs when they have the same direction, e.g.,
(3,0) + (4,0) = (7,0)
The vectors on the left have magnitudes of 3 and 4, and the sum has a
magnitude of 7.
The former occurs when they have opposite directions, but the same
magnitude, e.g.,
(3,0) + (-3,0) = (0,0)
The vectors on the left have magnitude 3, but they cancel each other out,
leaving a null vector, with no direction or magnitude.
In between, we might have something like
(3,0) + (0,4) = (3,4)
Here, the vectors on the left have magnitudes 3 and 4, but the sum has a
magnitude of 5. That would correspond to a situation like
Two guys are pushing on a box. One pushes to the east
with a force of 3 lbs, while the other pushes to the
north with a force of 4 lbs. What is the resultant force
on the box?
We can add the vectors to get (3,4). The magnitude of that is
sqrt(3^2 + 4^2) = sqrt(25) = 5
The direction of that is
tan^-1(4/3) = about 53 degrees
So we could replace the two guys with one guy, pushing with a force of
5 lbs, at an angle of 53 degrees from the x-axis, and the box would move
in the same way as when the two guys push it.
Now, why doesn't the combined force have a magnitude of 7 lbs? Well, think
of it this way: the box is moving at an angle to the force being applied
by the guy pushing to the east. So only SOME of his force is going to
moving the box. And the same is true for the guy pushing to the east. So
we should expect the resultant force to be less than either of the
individual forces.
Now try thinking about those other kinds of cases. In one, the two guys
are both pushing east, and their forces add up -- so the box moves to the
east, under a force of 3 + 4 = 7 lbs. In the other, one guy is pushing
east while the other pushes west, and since they apply the same magnitude
of force, the box doesn't move at all. That is, it's like a force of
3 + -3 = 0 lbs is being applied to it.
In terms of triangles, you can think of it this way. The hands of a clock
always form two sides of a triangle, right? And the third side of that is
the line connecting the hands. Would you expect the length of that third
side to always be the sum of the lengths of the individual hands? Or does
the angle between them have something to do with it?
Does this help?
- Doctor Ian, The Math Forum
Date: 11/15/2010 at 09:49:43
From: David
Subject: Thank you (Why does the vector law of addition work?)
Yes that clarified it, thank you!!! | {"url":"http://mathforum.org/library/drmath/view/76184.html","timestamp":"2014-04-19T08:09:30Z","content_type":null,"content_length":"10480","record_id":"<urn:uuid:1bcd1a90-0555-4d7b-9c4a-2bdab7d5c907>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00137-ip-10-147-4-33.ec2.internal.warc.gz"} |
All problems © Copyright 2003 Stan Wagon. Reproduced with permission.
Please don't ask us for answers! Solutions for almost every problem are posted to the mailing list, but at Professor Wagon's request we do not reproduce them here. If you can't sleep without
knowing the answer to one of these problems, please contact Professor Wagon at wagon@macalester.edu. | {"url":"http://mathforum.org/wagon/fall03/","timestamp":"2014-04-24T16:35:20Z","content_type":null,"content_length":"3988","record_id":"<urn:uuid:9af052ab-03f8-4fcb-9a68-ab772462a04e>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00109-ip-10-147-4-33.ec2.internal.warc.gz"} |
How do scientists measure or calculate the weight of a planet?
Barry Lienert, a geophysicist at the University of Hawaii, provides the following explanation.
We start by determining the mass of the Earth. Issac Newton's Law of Universal Gravitation tells us that the force of attraction between two objects is proportional the product of their masses
divided by the square of the distance between their centers of mass. To obtain a reasonable approximation, we assume their geographical centers are their centers of mass.
Because we know the radius of the Earth, we can use the Law of Universal Gravitation to calculate the mass of the Earth in terms of the gravitational force on an object (its weight) at the Earth's
surface, using the radius of the Earth as the distance. We also need the Constant of Proportionality in the Law of Universal Gravitation, G. This value was experimentally determined by Henry
Cavendish in the 18th century to be the extemely small force of 6.67 x 10^-11 Newtons between two objects weighing one kilogram each and separated by one meter. Cavendish determined this constant by
accurately measuring the horizontal force between metal spheres in an experiment sometimes referred to as "weighing the earth."
Knowing the mass and radius of the Earth and the distance of the Earth from the sun, we can calculate the mass of the sun (right), again by using the law of universal gravitation. The gravitational
attraction between the Earth and the sun is G times the sun's mass times the Earth's mass, divided by the distance between the Earth and the sun squared. This attraction must be equal to the
centripetal force needed to keep the earth in its (almost circular) orbit around the sun. The centripetal force is the Earth's mass times the square of its speed divided by its distance from the sun.
By astronomically determining the distance to the sun, we can calculate the earth's speed around the sun and hence the sun's mass.
Once we have the sun's mass, we can similarly determine the mass of any planet by astronomically determining the planet's orbital radius and period, calculating the required centripetal force and
equating this force to the force predicted by the law of universal gravitation using the sun's mass.
Additional details are provided by Gregory A. Lyzenga, a physicist at Harvey Mudd College in Claremont, Calif.
The weight (or the mass) of a planet is determined by its gravitational effect on other bodies. Newton's Law of Gravitation states that every bit of matter in the universe attracts every other with a
gravitational force that is proportional to its mass. For objects of the size we encounter in everyday life, this force is so minuscule that we don't notice it. However for objects the size of
planets or stars, it is of great importance.
In order to use gravity to find the mass of a planet, we must somehow measure the strength of its "tug" on another object. If the planet in question has a moon (a natural satellite), then nature has
already done the work for us. By observing the time it takes for the satellite to orbit its primary planet, we can utilize Newton's equations to infer what the mass of the planet must be.
For planets without observable natural satellites, we must be more clever. Although Mercury and Venus (for example) do not have moons, they do exert a small pull on one another, and on the other
planets of the solar system. As a result, the planets follow paths that are subtly different than they would be without this perturbing effect. Although the mathematics is a bit more difficult, and
the uncertainties are greater, astronomers can use these small deviations to determine how massive the moonless planets are.
Finally, what about those objects such as asteroids, whose masses are so small that they do not measurably perturb the orbits of the other planets? Until recent years, the masses of such objects were
simply estimates, based upon the apparent diameters and assumptions about the possible mineral makeup of those bodies.
Now, however, several asteroids have been (or soon will be) visited by spacecraft. Just like a natural moon, a spacecraft flying by an asteroid has its path bent by an amount controlled by the mass
of the asteroid. This "bending" is measured by careful tracking and Doppler radio measurement from Earth. Recently, the NEAR spacecraft flew by the asteroid Mathilde, determining for the first time
its actual mass. It turned out to be considerably lighter and more "frothy" in structure than had been expected, a fact that is challenging planetary scientists for an explanation.
Originally published on March 16, 1998. | {"url":"http://www.scientificamerican.com/article/how-do-scientists-measure/","timestamp":"2014-04-17T03:14:02Z","content_type":null,"content_length":"61261","record_id":"<urn:uuid:1e6f7641-b46c-4c14-96ff-c81063aa4ed5>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00498-ip-10-147-4-33.ec2.internal.warc.gz"} |
Matches for:
Contemporary Mathematics
1988; 730 pp; softcover
Volume: 78
Reprint/Revision History:
third printing 1998
ISBN-10: 0-8218-5088-1
ISBN-13: 978-0-8218-5088-6
List Price: US$88
Member Price: US$70.40
Order Code: CONM/78
Artin introduced braid groups into mathematical literature in 1925. In the years since, and particularly in the last five to ten years, braid groups have played diverse and unexpected roles in widely
different areas of mathematics, including knot theory, homotopy theory, singularity theory, and dynamical systems. Most recently, the area of operator algebras has brought striking new applications
to knots and links.
This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Artin's Braid Group, held at the University of California, Santa Cruz, in July 1986. This
interdisciplinary conference brought together leading specialists in diverse areas of mathematics to discuss their discoveries and to exchange ideas and problems concerning this important and
fundamental group. Because the proceedings present a mix of expository articles and new research, this volume will be of interest to graduate students and researchers who wish to learn more about
braids, as well as more experienced workers in this area. The required background includes the basics of knot theory, group theory, and low-dimensional topology.
• K. Aomoto -- A construction of integrable differential system associated with braid groups
• J. S. Birman -- Mapping class groups of surfaces
• E. Brieskorn -- Automorphic sets and braids and singularities
• A. L. Carey and D. E. Evans -- The operator algebras of the two-dimensional Ising model
• F. R. Cohen -- Artin's braid groups, classical homotopy theory, and sundry other curiosities
• W. D. Dunbar -- Classification of solvorbifolds in dimension three, I
• M. Falk and R. Randell -- Pure braid groups and products of free groups
• L. V.Hansen -- Polynomial covering maps
• Y. Ihara -- Arithmetic analogues of braid groups and Galois representations
• B. Jiang -- Application of braids to fixed points of surface maps
• L. H. Kauffman -- Statistical mechanics and the Jones polynomial
• P. Kluitmann -- Hurwitz action and finite quotients of braid groups
• T. Kobayashi -- Heights of simple loops and pseudo-Anosov homeomorphisms
• T. Kohno -- Linear representations of braid groups and classical Yang-Baxter equations
• G. I. Lehrer -- A survey of Hecke algebras and the Artin braid groups
• A. Libgober -- On divisibility properties of braids associated with algebraic curves
• W. B. R. Lickorish -- The panorama of polynomials for knots, links, and skeins
• R. J. Milgram and P. Löffler -- The structure of deleted symmetric products
• B. Moishezon and M. Teicher -- Braid group technique in complex geometry, I: Line arrangements in \(CP^2\)
• H. R. Morton -- Problems
• H. R. Morton -- Polynomials from braids
• H. R. Morton and P. Traczyk -- The Jones polynomial of satellite links around mutants
• M. Oka -- On the deformation of certain type of algebraic varieties
• P. Orlik and L. Solomon -- Braids and discriminants
• J. H. Przytycki -- \(t_k\) moves on links
• L. Rudolph -- Mutually braided open books and new invariants of fibered links
• M. Salvetti -- Generalized braid groups and self-energy Feynman integrals
• B. Wajnryb -- Markov classes in certain finite symplectic representations of braid groups
• R. F. Williams -- The braid index of an algebraic link
• D. N. Yetter -- Markov algebras | {"url":"http://ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-78","timestamp":"2014-04-19T10:29:12Z","content_type":null,"content_length":"17158","record_id":"<urn:uuid:ba33a96d-5b96-4279-a4e7-9cd63c451ce0>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00507-ip-10-147-4-33.ec2.internal.warc.gz"} |
Optimizing a Function of One Variable
This article describes an algorithm for optimizing a function of one variable without requiring a function for the derivative. Methods that require derivatives are often fast but unstable. Methods
that do not require derivatives are often stable but slow. The method implemented here, however, is stable and efficient. It was first published by Richard Brent in his book "Algorithms for
Minimization Without Derivatives."
Given a function f(x) and an interval [a, b], the method finds the minimum of f(x) over [a, b]. The method can be used to maximize functions as well. To find the maximum of a function, pass in the
negative of the original function. That is, the maximum of f(x) occurs at the minimum of -f(x).
Brent's method is more robust than the Newton's more familiar method. When Newton's method works, it is very fast. But when it does not work, it can go wildly wrong, producing worse answers with each
iteration. Newton's method is like the girl in the nursery rhyme:
When she was good, she was very, very good.
But when she was bad she was horrid.
Brent's method is also easier to use than Newton's method because it does not require the user to supply a derivative function. The method sacrifices some efficiency for the sake of robustness, but
the it is more efficient than other robust methods such as the golden section method.
Using the Code
The code given here is a single C++ function and a demo project for using it. To use the function in your own project, simply #include the header file Brent.h.
The main input to the minimization function is a templated argument, a function object implementing the objective function to minimize. The objective function must implement a public method with the
signature double operator()(double x). For example, here is a class for a function object to compute the function f(x) = -x exp(-x).
class foo
double operator()(double x) {return -x*exp(-x);}
The primary reason our code requires a function object rather than simply a function is that functions that need to be optimized in applications often depend on parameters in addition to function
arguments. A function object may have a dozen parameters that are fixed before finding the minimum of the resulting function of one variable.
The other arguments are the end points of the interval over which the function is minimized, the tolerance for stopping, and an output parameter for returning the location of the minimum. The return
value of the minimization function is the value of the objective function at its minimum.
The signature for our minimization function is as follows.
template <class TFunction>
double Minimize
TFunction& f, // [in] objective function to minimize
double leftEnd, // [in] smaller value of bracketing interval
double rightEnd, // [in] larger value of bracketing interval
double epsilon, // [in] stopping tolerance
double& minLoc // [out] location of minimum, where the minimum occurs
Suppose we want to find the minimum of the function -x exp(x) represented by the class foo above. Below is a plot of this function.
We find the minimum of this function twice. First we find the minimum over the interval [-1, 2] to illustrate finding a minimum in the middle of an interval. Then we find the minimum over the
interval [2, 3] to illustrate the case case of the minimum occuring at one of the ends of the interval.
foo f;
double minloc; // location of the minimum
double minval; // value at the minimum
std::cout << std::setprecision(7); // display seven decimal places in output
// First minimize over [-1, 2]
minval = Minimize<foo>(f, -1, 2, 1e-6, minloc);
std::cout << "Computed minimum location: " << minloc << "\n"
<< "Exact value: " << 1.0 << "\n"
<< "Computed minimum value: " << minval << "\n"
<< "Exact minimum value: " << f(1.0) << "\n\n";
// Now minimize over [2, 3]
minval = Minimize<foo>(f, 2, 3, 1e-6, minloc);
std::cout << "Computed minimum location: " << minloc << "\n"
<< "Exact value: " << 2.0 << "\n"
<< "Computed minimum value: " << minval << "\n"
<< "Exact minimum value: " << f(2.0) << "\n\n";
In the first case, minimizing over [-1, 2], the exact minimum occurs at 1. This is the global minimum of the function. In the second case, minimizing over [2, 3], the function is increasing on this
interval and so the minimum occurs at the left end point. The output shows that the results were correct to the specified tolerance of 10^-6.
Computed minimum location: 1
Exact value: 1
Computed minimum value: -0.3678794
Exact minimum value: -0.3678794
Computed minimum location: 2.000001
Exact value: 2
Computed minimum value: -0.2706704
Exact minimum value: -0.2706706
Brent's method is an overlooked treasure from the past. It was developed decades ago (1972), and not too many people know about it now. However, the method is ideal for many applications since it is
both robust and efficient. People often use the method without realizing it because it is used internally by other software packages, such as Mathematica. | {"url":"http://www.codeproject.com/Articles/30201/Optimizing-a-Function-of-One-Variable?fid=1528885&df=90&mpp=10&noise=1&prof=True&sort=Position&view=Expanded&spc=None&fr=11","timestamp":"2014-04-16T21:24:46Z","content_type":null,"content_length":"75126","record_id":"<urn:uuid:1b4a466e-714d-4fba-9808-78112ae33966>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00566-ip-10-147-4-33.ec2.internal.warc.gz"} |
Forest Grove, PA Math Tutor
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The PH-KH-CO2 equation completely wrong? UPDATE on 22nd post
THIS POST IS LONG. BUT PLEASE, ALL CHEM BUFFS, STICK TO IT.
Four days ago I sat down to figure out the equation that governs all pH-KH-CO2 tables. I feel pretty comfortable with chemistry, so it seemed like a relatively simple task. But thats one nothing
started to make sense. Since then, I've been searching, reading, searching...and more reading...and think I have come to a conclusion. Except, my conclusion is so outrageous I believe I must have
messed up somewhere! To all other chem buffs, tell me if I'm right or wrong.
First I'm going to leave out all the chemical equations, gas and solubility laws, equilibrium constants/equations...everything that explains why we can find CO2 from KH and pH. If you understand what
I'm going to go through then you you know about all that. I'll begin with the current origins of the equation we use to calculate CO2, the Henderson-Hasselbach equation.
pH = pKa + log([A-]/[HA])
Knowing the pKa of a certain weak acid, and the molar concentrations of the weak acid and its conjugate base (a buffer exists), we can find pH. In our case, A- is HCO3 and HA is H2CO3. H2CO3 by
convention can be replaced by CO2. the pKa of H2CO3 is 6.37. So equation becomes (first line) and can be rearranged to find [CO2]:
pH = 6.37 + log([HCO3]/[CO2])
pH = 6.37 + log[HCO3] - log[CO2]
log[CO2] = log[HCO3] + 6.37 - pH
[CO2] = [HCO3]*10^(6.37-pH)
The last equation to calculate [CO2] is 100% correct. We could use this for our purposes, but it's impractical for the average aquarist to use foreign molar concentration written in scientific
notation. It's better to just plug in our well known unit of dKH and get our answer. Thats what
this link
written by George Booth on APD provides for us, and is the current equation that we are using now.
To make the equation aquarist-friendly, we need to first find a conversion factor that interchanges dKH to molarity HCO3. According to the website 1 dKH = 2.92E-4M HCO3*. The author did as so:
dKH*17.8 = 'HCO3'mg/L
'HCO3'mg/L * (mol/61020mg) =
2.92E-4M HCO3
This is NOT correct. The author uses the common 17.8 factor to change dKH into mg/L, but doesn't realize that he has mg/L
. He assumes the units are mg/L HCO3 so proceeds in converting the mg into mol, by the molar mass of HCO3. Obviously as you know, using the molar mass of HCO3 is not going to convert mg/L CaCO3 into
molarity HCO3. First we need to change mg/L CaCO3 into mg/L HCO3, then proceed into molarity. Before making the correction, I will continue the author's process:
[CO2] = [HCO3]*10^(6.37-pH)
[CO2] = 2.92E-4*dKH*10^(6.37-pH)
Right now the answer is [CO2]. We need mg/L CO2.
[CO2]*44010(**) = 2.92E-4*dKH*10^(6.37-pH)
CO2 mg/L = 12.838*dKH*10^(6.37-pH)
This is the current equation that we use for our CO2 charts. However, there was an error in finding it. So it must be wrong. THIS IS WHERE I NEED SOMEONE TO TELL ME THERE WAS NO ERROR AND I WAS
WRONG. . I'll go ahead and show how to find the 'assumed' correct equation.
dKH*17.8 = CaCO3 mg/L
CaCO3 mg/L * (2*61.02/100.09) = HCO3 mg/L
HCO3 mg/L * (mol/61020) =
3.56E-4M HCO3
Therefore the only difference between my equation and the current one: 1dKH = 3.56E-4M HCO3 instead of 1dKH = 2.92E-4M HCO3. Completing the new equation...
[CO2] = [HCO3]*10^(6.37-pH)
[CO2] = 3.56E-4*dKH*10^(6.37-pH)
[CO2]*44010 = 3.56E-4*dKH*10^(6.37-pH)
CO2 mg/L = 15.65*dKH*10^(6.37-pH)
Comparing and testing (KH=5, pH=6.8 ) both equations side-by-side:
Original - CO2 mg/L = 12.838*dKH*10^(6.37-pH)
Correct? - CO2 mg/L = 15.65*dKH*10^(6.37-pH)
Original - CO2 = 23.85 mg/L
Correct? - CO2 = 29.10 mg/L
Just this example shows an 18% error. So now comes the scary part, am I wrong????. I believe I'm going to find out very fast I am because saying the current, long used CO2 charts are completely wrong
is outrageous. At the same time, the author of the current equation made a very obvious error, so I do not know which equation to trust!
-Chris Rolson
*Corrected. The author should have used 61020 mg/mol, not 61. This isn't the source of error however.
**Corrected. The author should have used 44010 mg/mol, not 44. This isn't the source of error however. | {"url":"http://www.plantedtank.net/forums/showthread.php?t=9839","timestamp":"2014-04-20T19:40:38Z","content_type":null,"content_length":"153249","record_id":"<urn:uuid:663ee1dc-0c5e-4995-bdcb-1bc3af952140>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00003-ip-10-147-4-33.ec2.internal.warc.gz"} |
1007 -- DNA Sorting
DNA Sorting
Time Limit: 1000MS Memory Limit: 10000K
Total Submissions: 79843 Accepted: 32095
One measure of ``unsortedness'' in a sequence is the number of pairs of entries that are out of order with respect to each other. For instance, in the letter sequence ``DAABEC'', this measure is 5,
since D is greater than four letters to its right and E is greater than one letter to its right. This measure is called the number of inversions in the sequence. The sequence ``AACEDGG'' has only one
inversion (E and D)---it is nearly sorted---while the sequence ``ZWQM'' has 6 inversions (it is as unsorted as can be---exactly the reverse of sorted).
You are responsible for cataloguing a sequence of DNA strings (sequences containing only the four letters A, C, G, and T). However, you want to catalog them, not in alphabetical order, but rather in
order of ``sortedness'', from ``most sorted'' to ``least sorted''. All the strings are of the same length.
The first line contains two integers: a positive integer n (0 < n <= 50) giving the length of the strings; and a positive integer m (0 < m <= 100) giving the number of strings. These are followed by
m lines, each containing a string of length n.
Output the list of input strings, arranged from ``most sorted'' to ``least sorted''. Since two strings can be equally sorted, then output them according to the orginal order.
Sample Input
Sample Output | {"url":"http://poj.org/problem?id=1007","timestamp":"2014-04-16T21:51:38Z","content_type":null,"content_length":"6911","record_id":"<urn:uuid:a5e42348-8ffd-471d-ba38-546d3f0a3b7c>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00433-ip-10-147-4-33.ec2.internal.warc.gz"} |
Middle School Math Worksheets
On this page you will find: a complete list of all of our math worksheets, lessons, math homework, and quizzes. All for the middle levels of Grade 6, Grade 7, and Grade 8. These worksheets are geared
for students between the ages of eleven and fifteen. | {"url":"http://www.mathworksheetscenter.com/grades/68/","timestamp":"2014-04-17T18:34:08Z","content_type":null,"content_length":"34651","record_id":"<urn:uuid:625884f2-3c8f-4f4d-9a15-155b91af7fa5>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00140-ip-10-147-4-33.ec2.internal.warc.gz"} |
Halting Problem
One of the most interesting theoretical parts of
is the study of what can (and cannot) be computed. For instance, take the question, "does this program complete?" I.e., will it not go into an infinite loop. How would you answer this question, given
an arbitrary piece of code? You could try running it. But what if it takes a long time? How long are you willing to wait? The
asks the question: "Given a program and its input, determine whether the program will complete or run forever."
proved in 1936 that there cannot exist a general algorithm for answering this question for
arbitrary program and input. Turing introduced the
in this proof. The first link below relates this result to
for mathematical formal systems. Links:
The halting problem is no longer an interesting question. No one actually using computers asks whether a program is going to halt, because nowadays they have the Ctrl-C and other measures to stop the
machine if there seems to be a problem. Otherwise, I challenge you to come up with anything other than a ToyProblem where there is a real issue. Please, don't delete my comment again, until you can
cite a real issue in the field.
The halting problem isn't a question, even though it's phrased that way above. It actually leads to an important proof that certain types of algorithm cannot be implemented. These are not just
s, but real issues. For example, when your boss asks you to write a program to determine if anyone in the development team has written code that will get caught in an endless loop, the proof allows
you to provide a reason why it can't be done. As for pressing Ctrl-C, sure, you can do that if you're at a console and have made a decision that a long-running program should be stopped. However,
what if it's a background process? How does a monitoring program know whether the process is simply busy and will finish eventually, or actually caught in a loop? Based on the source code and input
data, it can't know. In short, the
tells us that a certain category of what otherwise might be very useful, pragmatic monitoring programs or compile-time code checkers -- those that look at source code and input and determine whether
they'll eventually produce output or not -- can't exist.
The proof does not tell you that the existence of an endless loop cannot be determined; rather, it tells you that there is no
Turing machine which can determine the existence of the endless loop
for any arbitrary program.
In fact, for any arbitrary program, there
a Turing machine which can determine whether it halts; such a machine need only be large enough to contain a directed graph of all possible states (a finite set, assuming your computer is a Turing
machine, which I imagine it is, and your input is finite (or at least, it gets its input from one or more Turing machines)) of the system and check that graph for cycles. Granted, such a machine
would be impossibly large for all but the simplest of programs, but the point is that for any
program, the halting question is answerable given sufficient resources. At any rate, the semantics of whatever programming language you use typically allow you to take a
[No, its not always possible to construct a specific machine for another specific machine to determine if its halts. If it was possible, your proof program could be constructed automatically just by
brute force. Such a brute force prover could verify your program as you write it. What if I write a program to compute pi and search for a certain string of digits somewhere within. Can I write a
program to tell me if the first will halt? What if I have a program that accepts a stream of 'a' and halts when it receives a 'b' (failing on any other input). What if my pi search program generates
an 'a' every time it computes a new digit and outputs a 'b' when it finds the sub string. It's output is fed into my a-or-b program. Can I write a program that tells me if my a-or-b program halts? ]
• Thank you, and furthermore, to the original commentor, your example is ridiculous, no boss ever asks "write a program to determine if code will get caught in an endless loop", they ask instead
"write code without any bugs". See, we programmers, monkeys that we are, actually learned to write code so we can see progress. This is how programming culture evolved: to refine the ability to
prevent the need to abort drastically or dump core.
Sorry your comment was deleted.
spoofed your
cookie and thereby merged his edits with yours, so the
reverted the lot. Related: | {"url":"http://c2.com/cgi/wiki?HaltingProblem","timestamp":"2014-04-18T00:53:09Z","content_type":null,"content_length":"7550","record_id":"<urn:uuid:f532e715-344e-4972-aef6-aa8d5c8ae35c>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00615-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Theoretical Minimum, Released today, January 29th 2013
Is this intended to be the first book in a series covering all branches of physics, as Landau's?
Suskind gives Landau credit for the title of the book,
theoretical minimum
And from the following quote he does say that his plan is to do more books, sort of:
"One of the main inquiries [I get] is whether I will ever convert the [online] lectures into books? the Theoretical Minimum is the answer."
So he does sort of say he's going to make more books because he uses plural when referring to
. But since he mentions it in such a subtle way it kind of makes me wonder how many he actually wants to do. | {"url":"http://www.physicsforums.com/showthread.php?p=4254989","timestamp":"2014-04-19T17:33:32Z","content_type":null,"content_length":"70076","record_id":"<urn:uuid:98cd6f36-bbaa-40a5-8016-c004182d958c>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00293-ip-10-147-4-33.ec2.internal.warc.gz"} |
Southeastern Math Tutor
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
This is the testimonial you wrote.
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Bell, J. and Casteels, K. and Launois, S. (2012) Enumeration of H-strata in quantum matrices with respect to dimension. Journal of Combinatorial Theory, Series A, 119 (1). pp. 83-98. ISSN 0097-3165.
(The full text of this publication is not available from this repository)
Bell, J. and Casteels, K. and Launois, S. (2012) Primitive ideals in quantum Schubert cells: Dimension of the strata. Forum Mathematicum . ISSN 0933-7741 (print); 1435-5337 (online). (The full text
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Book section
Bell, J. and Casteels, K. and Launois, S. (2012) Enumeration of torus-invariant strata with respect to dimension in the big cell of the quantum minuscule Grassmannian of type B_n. In: Ara, P. and
Brown, K.A. and Lenagan, T.H. and Stafford, J.T. and Zhang, J.J., eds. New Trends in Noncommutative Algebra. Contemporary Mathematics, 562 . American Mathematical Society, pp. 27-40. ISBN
9780821852972. (Full text available) | {"url":"http://kar.kent.ac.uk/view/people/Casteels=3AK=2E=3A=3A.html","timestamp":"2014-04-18T20:49:47Z","content_type":null,"content_length":"12345","record_id":"<urn:uuid:c8409a14-3853-462d-b725-790b9a0677b6>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00453-ip-10-147-4-33.ec2.internal.warc.gz"} |
[duxuser] math
• From: "simensen" <simensen@xxxxxxxxx>
• To: <duxuser@xxxxxxxxxxxxx>
• Date: Tue, 6 Jun 2006 20:17:00 -0400
Thank you Neal! We now have our first two sheets of Spatial Math. Your steps and helps brought it down to a level even I could understand.
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MzScheme 360
Note: This is MzScheme v360, which is what's packaged with Ubuntu Gutsy. I haven't had a chance to try PLT Scheme 4 yet, which by all accounts departs from the version 3 series considerably.
From the old PLT Scheme web page:
PLT Scheme is an umbrella name for a family of implementations of the Scheme programming language.
MzScheme is the lightweight, embeddable, scripting-friendly PLT Scheme implementation.
To reproduce my results on Ubuntu Gutsy Gibbon, install MzScheme with
sudo apt-get install mzscheme
and run the interpreter with
mzscheme --case-insens 2>&1
MzScheme makes the following choices: | {"url":"http://web.mit.edu/~axch/www/scheme/choices/mzscheme.html","timestamp":"2014-04-16T13:24:06Z","content_type":null,"content_length":"4109","record_id":"<urn:uuid:099093b5-adf4-4371-99c0-ea5e5f93078e>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00559-ip-10-147-4-33.ec2.internal.warc.gz"} |
High quality gap analysis PDF Ebooks are listed below.
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gap analysis 10 out of 10 based on 14 ratings.
The San Francisco Bay Trail Project
PDF Ebook PDF Book Pages: -
In addition to this Gap Analysis Study, another important aspect of this project has been the meticulous. cataloguing of each unfinished segment of Bay Trail into a geographic ...
Santa Clara
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ATIS Next Generation Network (NGN) Framework Part III:
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The purpose of this Gap Analysis is to identify standards activities and assess the ... Unique to this Gap. Analysis is also the participation of the elected leaders of the ATIS ...
Information Needed for Consensus on Policies and Programs to
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supplementation in regions where malaria transmission is year. round, the risk to iron sufficient children (discussed below) is ... anemia, supplementation trials in populations at risk for iron ...
SEO GAP Analysis
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Before the implementation of the Iowa Aquatic Gap Analysis, project coordinators had no sense ... diversity might be assisted by the Gap Analysis data provided by our project. ...
GAP Program Reports
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The Mission of the Gap Analysis Program (GAP) http://gapanalysis.nbii.gov > is to promote ... imperative for GAP to be better poised to participate in. other national efforts, such as ...
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confer additional advantages for gap analysis because their high species richness, mix of habitat ... effort is intended to show how the Gap Analysis method of identifying gaps in biodiversity ...
Report on the University of Michigan-Flint Student
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A Critique of Info-Gap Robustness Model
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Missouri GAP
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Business Continuity professionals put their minds together to create Generally Accepted Business ... The DRJ Editorial Advisory Board Generally Accepted BC Practices Committee in concert with DRI.
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When the National Gap Analysis Program began in the 1980s, land cover mapping was a ... From the onset, the National Gap Analysis Program became a lead agency in the move to ...
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This year will see 36 states with Gap Analysis projects, with perhaps ten ... Gap Analysis is a scientific method for identifying the degree to which native ...
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DMCA | ebook free download | {"url":"http://asaha.com/ebooks/gap-analysis.html","timestamp":"2014-04-21T07:05:01Z","content_type":null,"content_length":"25539","record_id":"<urn:uuid:0d321dff-5806-435b-892b-7965276a04aa>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00129-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ron Schuler's Parlour Tricks
Maxwell's Equations
"The most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics."
- Richard Feynman.
A shy, somewhat dull child who earned the nickname "Dafty" while at Edinburgh Academy, James Clerk Maxwell -- born on this day in 1831 in Edinburgh -- had an intense curiosity about the mechanics of
everyday objects, and later, much ahead of his schoolmates, developed an appreciation for the power of mathematical models. At 14 he wrote a paper on a method of drawing elipses using pins and thread
which was published by the Edinburgh Royal Society (a not entirely new idea -- the mathematical basis for it was proposed by
-- but it was a remarkable adaptation for a 14 year-old).
He read
at the University of Edinburgh and in 1850 went to Cambridge, where he met the cream of Britain's young scientists; they found him eccentric and a bit difficult to follow as he jumped excitedly from
topic to topic, but nonetheless they seemed to recognize his intellectual gifts. After graduation in 1854, he went to teach at Marischal College in Aberdeen, where he studied the rings of Saturn and
described them as being composed of numerous small solid particles (that being the best mathematical explanation for their stability), a description which was verified by NASA's Voyager probe in
1980. He married the daughter of the principal at Marischal in 1859, but that did not save his job as a junior professor when Marischal merged with King's College Aberdeen the following year. He
managed to obtain the chair of natural philosophy at King's College London shortly thereafter, where he did his most important work.
Before Maxwell's work on electromagnetism, although predecessors such as Michael Faraday did develop a sophisticated understanding of the circumstances in which electrical induction could exist and
how to produce it, they did not have a mature or very clear picture of the shape of electricity and its movements. Maxwell replaced the old "machine-like" models of electricity (put it in here and it
comes out here) with a mathematical model which would predict electrical phenomena. Before a mystified Royal Society meeting in 1864, Maxwell read his "Dynamical Theory of the Electromagnetic Field,"
unveiling the equations which comprise the basic laws of electromagnetism, and showing that an electric charge sends waves through space at some frequency.
From his work, Maxwell could predict the existence of the whole invisible spectrum of electromagnetic frequencies, including radio waves, microwaves, infrared and ultraviolet waves, X-rays and gamma
rays, as well as pinpointing the speed of electricity at about 300,000 kilometers per second - so close to the speed of light as to suggest that light itself was an electromagnetic disturbance.
Maxwell's equations, which finally appeared in their most developed form in
Electricity and Magnetism
(1874), were puzzling to scientists of the time; but after Maxwell's death, Heinrich Hertz (at the urging of Heinrich von Helmholtz) confirmed Maxwell's theory of electromagnetism in a series of
experiments which measured electromagnetic "waves" and showed how electricity behaved like light.
The verification of Maxwell's equations, then, became an important first step in the development of 20th century atomic physics; on the practical side of things, Maxwell's equations are used today in
the design of everything from integrated circuits to cellular phones to predict and reduce levels of electromagnetic interference. Applying a similar theoretical basis to the study of gases, Maxwell
also shares credit for the "Maxwell-Boltzmann" kinetic theory of gases: working independently of Ludwig Boltzmann, Maxwell showed that temperatures and heat were manifestations of molecular movement,
and Maxwell attempted to describe such movement mathematically.
At the time of his death on November 5, 1879 of abdominal cancer (at age 49, the same age his mother died of the same disease), most scientists were pretty sure Maxwell was onto something, but only
time would show the true extent of his brilliance and influence.
Categories: Physicists
1 Comments:
Note TYPO, at the beginning of last paragraph. Otherwise, the blog is well-written and very informative. Thanks -
(Typo : "At the time of his death on died of abdominal cancer on November 5, 1879 at age 49"...)
Subscribe to Post Comments [Atom]
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Tangent: Introduction to the Tangent Function (subsection Tan/05)
Students usually learn the following basic table of tangent function values for special points of the circle:
For real values of argument , the values of are real.
In the points , the values of are algebraic. In several cases they can be integers , 0, or 1:
The values of can be expressed using only square roots if and is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.
The function is an analytical function of that is defined over the whole complex ‐plane and does not have branch cuts and branch points. It has an infinite set of singular points:
(a) are the simple poles with residues –1. (b) is an essential singular point.
It is a periodic function with the real period :
The function is an odd function with mirror symmetry:
The first derivative of has simple representations using either the function or the function:
The derivative of has much more complicated representations than symbolic derivatives for and :
where is the Kronecker delta symbol: and .
The function satisfies the following first-order nonlinear differential equation:
The function has a simple series expansion at the origin that converges for all finite values with :
where are the Bernoulli numbers.
The function has a well-known integral representation through the following definite integral along the positive part of the real axis:
The function has the following simple continued fraction representations:
Indefinite integrals of expressions involving the tangent function can sometimes be expressed using elementary functions. However, special functions are frequently needed to express the results even
when the integrands have a simple form (if they can be evaluated in closed form). Here are some examples:
Definite integrals that contain the tangent function are sometimes simple. For example, the famous Catalan constant can be defined as the value of the following integral:
This constant also appears in the following integral:
Some special functions can be used to evaluate more complicated definite integrals. For example, the generalized hypergeometric and polygamma functions are needed to express the following integral:
The following finite sums that contain the tangent function can be expressed using cotangent functions:
Other finite sums that contain the tangent function can be expressed using polynomial functions:
The evaluation limit of the first formula from the previous subsubsection for gives the following value for the corresponding infinite sum from the tangent:
Other infinite sums that contain the tangent can also be expressed using elementary functions:
The following finite product from the tangent has a very simple value:
The tangent of a sum can be represented by the rule: "the tangent of a sum is equal to the sum of tangents divided by one minus the product of tangents." A similar rule is valid for the tangent of
the difference:
In the case of multiple arguments , , , …, the function can be represented as the ratio of the finite sums including powers of tangents:
The tangent of the half‐angle can be represented using two trigonometric functions by the following simple formulas:
The sine function in the last formula can be replaced by the cosine function. But it leads to a more complicated representation that is valid in some vertical strips:
To make this formula correct for all complex , a complicated prefactor is needed:
where contains the unit step, real part, imaginary part, the floor, and the round functions.
The sum of two tangent functions can be described by the rule: "the sum of tangents is equal to the sine of the sum multiplied by the secants." A similar rule is valid for the difference of two
The product of two tangent functions and the product of the tangent and cotangent have the following representations:
The most famous inequality for the tangent function is the following:
There are simple relations between the function and its inverse function :
The second formula is valid at least in the vertical strip . Outside of this strip a much more complicated relation (that contain the unit step, real part, and the floor functions) holds:
Tangent and cotangent functions are connected by a very simple formula that contains the linear function in the argument:
The tangent function can also be represented using other trigonometric functions by the following formulas:
The tangent function has representations using the hyperbolic functions:
The tangent function is used throughout mathematics, the exact sciences, and engineering. | {"url":"http://functions.wolfram.com/ElementaryFunctions/Tan/introductions/Tan/05/","timestamp":"2014-04-19T19:54:51Z","content_type":null,"content_length":"64711","record_id":"<urn:uuid:422b7c9e-1dca-45b1-b2cb-ab666db0c12c>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00276-ip-10-147-4-33.ec2.internal.warc.gz"} |
Convert inverse (square angstrom) to inverse (square centimeters) - Conversion of Measurement Units
›› Convert inverse (square angstrom) to inverse (square centimeters)
›› More information from the unit converter
How many inverse (square angstrom) in 1 inverse (square centimeters)? The answer is 1.0E-16.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between inverse (square angstrom) and inverse (square centimeters).
Type in your own numbers in the form to convert the units!
›› Metric conversions and more
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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Math Forum: Math 7 - Alejandre
Student Page
Teacher Lesson Plan
You have read and followed the directions to the activity called Hop, Skip, Jump on page 5 of Glencoe's Interactive Mathematics text.
Here's the problem, which is also sometimes called Traffic Jam:
There are seven stepping stones and six people. On the three lefthand stones, facing the center, stand three of the people. The other three people stand on the three righthand stones, also facing
the center. The center stone is not occupied.
The challenge: exchanging places
Everyone must move so that the people originally standing on the righthand stepping stones are on the lefthand stones, and those originally standing on the lefthand stepping stones are on the
righthand stones, with the center stone again unoccupied.
The rules:
1. After each move, each person must be standing on a stepping stone.
2. If you start on the left, you may only move to the right. If you start on the right, you may only move to the left.
3. You may "jump" another person if there is an empty stone on the other side. You may not "jump" more than one person.
4. Only one person can move at a time.
You have tried this activity using two manipulatives (your bodies and the small plastic people). Now try it using the computer. Go to:
Traffic Jam Game - Java applet
Be sure to manipulate the various options that Mike Morton has made available, including:
1. background color
2. foreground color
3. level of difficulty
4. show history and redraw history
Look for a pattern. What do you see? Are there any rules to completing this activity sucessfully? What are they?
1. What if there are only 2 people and 3 spaces?
How many moves does it take for the two people to exchange positions?
2. What if there are 4 people and 5 spaces?
How many moves does it take for 4 people to exchange positions?
3. What about 6?
4. What about 8?
5. What about 10?
6. Can you find a pattern for any number of people?
There's another version of Traffic Jam at the Math Forum. | {"url":"http://mathforum.org/alejandre/frisbie/student.jam.html","timestamp":"2014-04-17T02:06:57Z","content_type":null,"content_length":"5981","record_id":"<urn:uuid:5760fa8d-a0df-4e73-bfcc-4c0b76c9560f>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00107-ip-10-147-4-33.ec2.internal.warc.gz"} |
Quality Control
Main Entry: quality control Function: noun : an aggregate of activities (as design analysis and inspection for defects) designed to ensure adequate quality especially in manufactured products
Quality control measure must be instituted for every sample matrix to confirm preparatory method performance. Although standard operating procedures governing ASC method development and verification
require investigation into preparatory and analytical interferences, it is not realistic to identify all interferences involved in real world samples. The Standard quality assurance for ASC projects
requires that each batch of 20 samples or every different matrix type be accompanied by one set of quality control parameters. A set of quality control parameters includes: four preparation blanks,
one blank spike, one certified reference material (similar in matrix to the associated samples), one matrix duplicate, and one matrix spike and matrix spike duplicate. Most analyses are also
accompanied by analytical duplicates and analytical spikes for additional information to account for analytical, not preparatory, interferences. Analytical duplicates and spikes should never be
confused with matrix duplicates and spikes as they are a focused quality control measure for the analytical method only. ASC may also include a method of standard addition curve to more specifically
identify biases associated with specific matrices.
All quality control measures not complying with the associated standard operating procedure must comply with corrective action measures. ASC does not believe that a singular quality control measure
that does not meet the quality control objective necessitates reanalysis if it does not negatively impact the validity of the results. All clients are informed of any variances encountered in the
preparatory or analytical procedures and the reasoning for the actions taken.
Preparation blanks are reagent water samples generated by the laboratory that are included in all sample preparatory procedures. Preparation blanks identify the background concentration for the
preparatory method as well as the inherent variability at low levels. Preparation blanks also assist in the identification of the limit of detection for any given method. As variability increases,
limits of detection will naturally increase as the confidence for reporting below the background concentration is compromised. Method detection limits are often represented by three times the
standard deviation of the preparation blanks.
Applied Speciation and Consulting, LLC recognized both performance based method detection limits and 40CFR based reporting limits. Performance based detection limits conform to equations 1-3:
(1) Variance = ( (PB01 -PBavg )2 + (PB02 -PBavg )2 + (PB03 -PBavg )2 + ··· + (PBn -PBavg )2 )/(n-1)
(3) Estimated Method Detection Limit (eMDL) = 3 * s
Method 40CFR allows more latitude when generating method detection limits; however, it should be noted that method detection limits generated under 40CFR are not necessarily representative of
instrument performance at the time of sample analysis. The method detection limit, in accordance with 40CFR is generated by 7 replicate analysis of either a known spiked standard or reagent water.
The MDL is generated by multiplying the student's “t” value at a 99% confidence level with the standard deviation of the replicate results. In accordance with 40CFR, the method detection limit is
only generated for analysis of different matrix types or if instrument performance varies significantly (check samples identify that the instrument can no longer detect within 10 times the method
detection limit at any confidence level). It should also be noted that 40CFR specifies blank correction be applied to known spiked standard recoveries which may not comply with specific contracts.
Performance based detection limits can supply a better representation of instrument performance at the time of sample analysis. Generation of method detection limits using the 40CFR method can
produce generalized accounts of the instrument's ability with given matrices but may not represent fluctuations in instrument performance over time.
Blank Spikes
Blank spikes (BS), or laboratory control samples (LCS), identify the performance of the preparation method on a clean matrix void of interferences. If a preparation method is not capable of
extracting the analytes of interest from reagent water and solubilizing them for analysis the probability of acceptable efficiency on real world samples is minimal.
The recovery for blank spikes is determined by the following equation:
Recovery (%) = 100 * (measured concentration/certified concentration)
Go to Top
Blank spikes (BS), or laboratory control samples (LCS), identify the performance of the preparation method on a clean matrix void of interferences. If a preparation method is not capable of
extracting the analytes of interest from reagent water and solubilizing them for analysis the probability of acceptable efficiency on real world samples is minimal.
The recovery for blank spikes is determined by the following equation:
Certified Reference Materials
Certified reference materials (CRMs) are standards rigorously tested by an outside source that represent a specific matrix type. Each batch of samples, or every different matrix type, must be
accompanied by a certified reference material (if available) of similar constitution. For instance, the use of a soil CRM is not appropriate for determination of trace metals in tissues. The CRM
sample accompanies the batch of samples through the preparatory and analytical procedure.
The importance of CRMs cannot be understated as their recoveries represent the performance of the methods on real world samples. Low recoveries of CRMs often identify poor efficiency of the
preparatory method to extract and/or solubilize the analytes of interest. It should be noted that CRMs are not available for all analytes and matrices, especially for trace metals speciation.
Applied Speciation and Consulting, LLC always participates in intercomparison studies that generate these CRMs.
Go to Top
Certified reference materials (CRMs) are standards rigorously tested by an outside source that represent a specific matrix type. Each batch of samples, or every different matrix type, must be
accompanied by a certified reference material (if available) of similar constitution. For instance, the use of a soil CRM is not appropriate for determination of trace metals in tissues. The CRM
sample accompanies the batch of samples through the preparatory and analytical procedure.
The importance of CRMs cannot be understated as their recoveries represent the performance of the methods on real world samples. Low recoveries of CRMs often identify poor efficiency of the
preparatory method to extract and/or solubilize the analytes of interest. It should be noted that CRMs are not available for all analytes and matrices, especially for trace metals speciation.
Applied Speciation and Consulting, LLC always participates in intercomparison studies that generate these CRMs.
Matrix Duplicates
Matrix duplicates (MD) identify the precision of the preparatory and analytical procedure. Precision is defined as the closeness of agreement between independent test results obtained under
stipulated conditions (typically represented in the form of relative standard deviation). Matrix duplicates are sub-samples of a homogenous sample. Each matrix duplicate should accompany the initial
sample throughout the preparatory and analytical process. Results for matrix duplicates should be identical to the initial sample. Poor precision for a matrix duplicate may denote heterogeneity or
preparatory issues. Field duplicates are not representative of laboratory precision; rather, they represent field homogeneity and sample collection performance. ASC cannot guarantee comparability of
field duplicates as they are a field quality objective.
Precision for matrix duplicates conform to the equation stipulated below:
Relative Percent Difference (RPD) = |(M-MD)/avg| * 100
where M and MD are the concentrations of the respective duplicates and avg is the average of M and MD results
Larger projects may necessitate the analysis of multiple sets of matrix duplicates for a better representation of precision. For this instance, statistical analysis can be performed on the group of
data in accordance with the preceding equation as defined below:
s = [Σ x'(x-x')^2 / Σ x']^1/2
x = initial sample concentration
x' = duplicate sample concentration
Go to Top
Matrix duplicates (MD) identify the precision of the preparatory and analytical procedure. Precision is defined as the closeness of agreement between independent test results obtained under
stipulated conditions (typically represented in the form of relative standard deviation). Matrix duplicates are sub-samples of a homogenous sample. Each matrix duplicate should accompany the initial
sample throughout the preparatory and analytical process. Results for matrix duplicates should be identical to the initial sample. Poor precision for a matrix duplicate may denote heterogeneity or
preparatory issues. Field duplicates are not representative of laboratory precision; rather, they represent field homogeneity and sample collection performance. ASC cannot guarantee comparability of
field duplicates as they are a field quality objective.
Precision for matrix duplicates conform to the equation stipulated below:
where M and MD are the concentrations of the respective duplicates and avg is the average of M and MD results
Larger projects may necessitate the analysis of multiple sets of matrix duplicates for a better representation of precision. For this instance, statistical analysis can be performed on the group of
data in accordance with the preceding equation as defined below:
Matrix Spikes
The purpose of matrix spikes (MS) and matrix spike duplicates (MSD) is to identify method performance and precision. Matrix spikes are generated by the addition of a known amount of target analyte to
a sub-sample. Unless the added target analyte is infused within a similar matrix, the ability of the matrix spike to represent method performance is limited; rather, matrix spikes often assist in the
identification on chemical interferences inherent in the matrix. The efficiency of any method to dissolute an aqueous standard solution will always be significantly greater than a real world sample.
With this in mind, ASC uses certified reference materials as an indication of method performance more than matrix spikes. Matrix spike duplicates serve the same purpose as matrix duplicates with the
added complexity of an increase in concentration. Matrix spike and matrix spike duplicate accuracy and precision values are calculated by the following equations.
Recovery = 100 * (measured concentration/certified concentration) (%)
Relative Percent Difference (RPD) = |(MS-MSD)/avg| * 100
where MS and MSD are the concentrations of the respective duplicates and avg is the average of MS and MSD results
Go to Top
The purpose of matrix spikes (MS) and matrix spike duplicates (MSD) is to identify method performance and precision. Matrix spikes are generated by the addition of a known amount of target analyte to
a sub-sample. Unless the added target analyte is infused within a similar matrix, the ability of the matrix spike to represent method performance is limited; rather, matrix spikes often assist in the
identification on chemical interferences inherent in the matrix. The efficiency of any method to dissolute an aqueous standard solution will always be significantly greater than a real world sample.
With this in mind, ASC uses certified reference materials as an indication of method performance more than matrix spikes. Matrix spike duplicates serve the same purpose as matrix duplicates with the
added complexity of an increase in concentration. Matrix spike and matrix spike duplicate accuracy and precision values are calculated by the following equations.
where MS and MSD are the concentrations of the respective duplicates and avg is the average of MS and MSD results
Analytical Duplicates
The purpose of matrix spikes (MS) and matrix spike duplicates (MSD) is to identify method performance and precision. Matrix spikes are generated by the addition of a known amount of target analyte to
a sub-sample. Unless the added target analyte is infused within a similar matrix, the ability of the matrix spike to represent method performance is limited; rather, matrix spikes often assist in the
identification on chemical interferences inherent in the matrix. The efficiency of any method to dissolute an aqueous standard solution will always be significantly greater than a real world sample.
With this in mind, ASC uses certified reference materials as an indication of method performance more than matrix spikes. Matrix spike duplicates serve the same purpose as matrix duplicates with the
added complexity of an increase in concentration. Matrix spike and matrix spike duplicate accuracy and precision values are calculated by the following equations.
Recovery = 100 * (measured concentration/certified concentration) (%)
Relative Percent Difference (RPD) = |(A-AD)/avg| * 100
where A and AD are the concentrations of the respective duplicates and avg is the average of A and AD results
Go to Top
where A and AD are the concentrations of the respective duplicates and avg is the average of A and AD results
Analytical Spike / Analytical Spike Duplicate
The purpose of analytical spikes (AS) and analytical spike duplicates (ASD) is to identify analytical interferences associated with the sample matrix. Analytical spikes are generated by the addition
of a known amount of target analyte to a sample after digestion. Analytical spike duplicates serve the same purpose as analytical duplicates with the added complexity of an increase in concentration.
Low recoveries for analytical spikes may denote signal suppression which should necessitate further investigation. Analytical spike and analytical spike duplicate accuracy and precision values are
calculated by the following equations.
Recovery = 100 * (measured concentration/certified concentration) (%)
Relative Percent Difference (RPD) = |(AS-ASD)/avg| * 100
where AS and ASD are the concentrations of the respective duplicates and avg is the average of AS and ASD results
Go to Top
The purpose of analytical spikes (AS) and analytical spike duplicates (ASD) is to identify analytical interferences associated with the sample matrix. Analytical spikes are generated by the addition
of a known amount of target analyte to a sample after digestion. Analytical spike duplicates serve the same purpose as analytical duplicates with the added complexity of an increase in concentration.
Low recoveries for analytical spikes may denote signal suppression which should necessitate further investigation. Analytical spike and analytical spike duplicate accuracy and precision values are
calculated by the following equations.
where AS and ASD are the concentrations of the respective duplicates and avg is the average of AS and ASD results
If you have any questions regarding services or would like a quotation, please feel free to email us at info@appliedspeciation.com or call (425) 483-3300. Feel free to visit our website on a regular
basis as we will be providing scientific discussions and useful links to save you time and money. | {"url":"http://www.appliedspeciation.com/Quality-Control.html","timestamp":"2014-04-21T09:36:54Z","content_type":null,"content_length":"32730","record_id":"<urn:uuid:445509a7-cd63-4511-8ca6-855d1d261ed7>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00498-ip-10-147-4-33.ec2.internal.warc.gz"} |
Increasing interval and decreasing interval
February 24th 2006, 06:22 AM #1
Jul 2005
Increasing interval and decreasing interval
I have got a complex function y=(x^2-4*x-1)/(2*x+1) and I have to show the increasing and decreasing intervals.
How to show it when y'=(2*x^2+2*x-2)/(2*x+1)^2 but I am not able to find zero points and therefore it is impossible to find the increasing and decreasing intervals.
Any advice ?
Last edited by totalnewbie; February 24th 2006 at 06:28 AM.
I have got a complex function y=(x^2-4*x-1)/(2*x+1) and I have to show the increasing and decreasing intervals.
How to show it when y'=(2*x^2+2*x-2)/(2*x+1)^2 but I am not able to find zero points and therefore it is impossible to find the increasing and decreasing intervals.
Any advice ?
Sketch the curve:
This is not precise answer.
The function,
$y=\frac{x^2-4x-1}{2x+1},xot =-1/2$ then,
$y'=\frac{2x^2+2x-2}{(2x+1)^2},xot =-1/2$.
By Fermat's Principle the necessary conditions is when the derivative is zero or does not exits. Notice it does not exist at $x=-1/2$ but the function itself does not posses that domain. Thus,
$y'=0$. That happens when $2x^2+2x-2=0$,
Now you can use derivative test to determine whether they are maximum or minimum or neither.
This is not precise answer.
No its not, its a hint.
You have two points at which the derivative is zero and one point at
which it goes to infinity which is between the zeros of the derivative.
It is increasing from -infinity to the first turning point decreasing
from there to the singularity, then decreasing from the singularity
to the next turning point and increasing from there out to +infinity.
February 24th 2006, 07:17 AM #2
Grand Panjandrum
Nov 2005
February 24th 2006, 07:31 AM #3
Jul 2005
February 24th 2006, 08:44 AM #4
Global Moderator
Nov 2005
New York City
February 24th 2006, 08:49 AM #5
Grand Panjandrum
Nov 2005 | {"url":"http://mathhelpforum.com/calculus/1993-increasing-interval-decreasing-interval.html","timestamp":"2014-04-16T07:55:52Z","content_type":null,"content_length":"45545","record_id":"<urn:uuid:4dbd6887-ed6b-4e83-b204-275e762fcd72>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00583-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Problem Set 9
For this entire problem set, R = k[x1, x2, . . . , xn] with k a eld.
Problem 1. Dene a map : A1
by t (t, t2
, t3
, t4
). This induces a map
: k[A, B, C, D] k[t].
a) Find ker().
b) Compute J = ker() k[A, B, C].
c) How does this compare with the ideal of the twisted cubic?
d) Compute I = ker() k[B, C].
e) How does this compare with the ideal in Problem 3 on Set 8?
f) Is I = J k[B, C]?
Problem 2. Let I = (x2
- y2
, xy - 1) be an ideal in k[x, y].
a) Compute a Gröbner basis for I with respect to the lex order.
b) Find a reduced Gröbner basis for I with respect to the lex order. | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/038/2275934.html","timestamp":"2014-04-16T20:00:19Z","content_type":null,"content_length":"7701","record_id":"<urn:uuid:add5bdfb-f69b-444e-b125-ea0a48be48e7>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00190-ip-10-147-4-33.ec2.internal.warc.gz"} |
Comparative statics and dynamics of optimal choice models in Hilbert spaces
Chichilnisky, Graciela and Kalman, P.J. (1979): Comparative statics and dynamics of optimal choice models in Hilbert spaces. Published in: Journal of Mathematical Analysis and Applications , Vol. 70,
No. No. 2 (August 1979): pp. 490-504.
Download (1487Kb) | Preview
We study properties of the solutions to a parametrized constrained optimization problem in Hilbert spaces. A special operator is studied which is of importance in economic theory; sufficient
conditions are given for its existence, symmetry, and negative semidefiniteness. The techniques used are calculus and non linear functional analysis on Hilbert spaces.
Item Type: MPRA Paper
Original Comparative statics and dynamics of optimal choice models in Hilbert spaces
Language: English
Keywords: Hilbert spaces; optimization; operator; parametrized constrained maximization; comparative statics; Slutky; Hicks; Samuelson; matrix; Hilbert; Euclidean spaces; optimal growth; dynamic
models; growth; manifold; constrained optimization
C - Mathematical and Quantitative Methods > C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C61 - Optimization Techniques; Programming Models;
Dynamic Analysis
Item ID: 8001
Depositing Graciela Chichilnisky
Date 31. Mar 2008 05:33
Last 18. Feb 2013 03:19
K. J. Arrow and F. H. Hahn, "General Competitive Analysis," Holdern-Day, San Francisco, 1971.
K.J. ARROW, E. W. BARANKIN, AND D. BLACKWELL., Admissible points in convex sets, in "Contributions to the Theory of Games" (H. W. Kuhn and A.W. Tucker, Eds.), Vol. II, pp. 87-92,
Princeton Univ. Press, Princeton, N.J., 1953.
M. S. BERGER, Generalized differentiation and utility functionas for commodity spaces of arbitrary dimensions, in "Preferences, Utility and Demand" (J. Chipman, L. Hurqicz, M. Richeter,
and H. Sonnenschein, Eds.), Harcourt, Brace, Jovanovich, New York, 1971.
G. CHICHILNISKY, Nonlinear functional analysis and optimal economic growth, J. Math. Anal. 61 (1977), 490-503.
G. CHICHILNISKY AND P. J. KALMAN, Properties of critical points and operators in economics, J. Anal. Appl. 57 (1977) 241-297.
L. COURT, Entrepreneurial and consumer demand theories for commodity spectra, Parts I, II, Econometrica 9 (April, July, Oct. 1941), 241-297.
N. DUNFORD AND J. SCHWARTZ, "Linear Operators," Interscience, New York, 1958.
P. J. Kalman and M. Intriligator, Generalized comparative statics with applications to consumer theory and producer theory, International Economics Review 14 (1973).
P. KALMAN, Theory of consumer behavior when prices enter the utility function, Econometrica (Oct. 1968).
L. V. KANTOROIVICH AND G. P. AKILOV, "Functional Analysis in Normed Spaces," Pergamon Press and Macmillian Co., New York, 1964.
S. LANG, "Differential Manifolds," Series in Mathematics, Addison-Wesley, Reading, Mass., 972.
D. G. LUENBERGER, "Optimization by vector space methods," Wiley, New York, 1969.
F. RIESZ AND B. SZ-NAGY, "Functional Analysis," Unger, New York, 1955.
P. A. Samuelson, "The Foundations of Economic Analysis," Harvard University Press, Cambridge, Mass., 1947.
S. SMALE, An infinite dimensional version of Sard's theorem, Amer. J. Math. 87 (1965), 861-866.
URI: http://mpra.ub.uni-muenchen.de/id/eprint/8001 | {"url":"http://mpra.ub.uni-muenchen.de/8001/","timestamp":"2014-04-20T00:41:19Z","content_type":null,"content_length":"21374","record_id":"<urn:uuid:25c112b7-359c-4eaa-8f51-2707f695426b>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00396-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math (Calculus AB)
Posted by Jake on Monday, January 24, 2011 at 11:13pm.
An equation to the graph of y=x^3+3x^2+2 at its point of inflection is:
a) y=-3x+1
Please help. I am in urgent need.
• Math (Calculus AB) - MathMate, Tuesday, January 25, 2011 at 12:27am
I suppose the question is:
"A line tangent to the graph of y=x^3+3x^2+2 at its point of inflection is"
First find the derivatives:
y' = 3x²+6x
y" = 6x+6
At the point of inflexion, y"=0, or x=-1.
Then the slope at x=-1 is
y'(-1) = 3-6=-3
There are only two choices of lines that have a slope of -3.
From these, we calculate
y(-1) = (-1)³+3(-1)²+2
Which of these two lines (with slope = -3) gives y=4 at x=-1?
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Representations of Finite Subgroups on Homology
up vote 3 down vote favorite
Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting
representation of $H$ on $H_{*}(G;\mathbb{Z})$, the integral homology of $G$. Is anything in general known about this $H$-representation? For instance, are there nice generators of the homology
groups of $G$ on which I might try to describe the action of $H$? Also, I would be grateful for any and all references that might prove relevant.
add comment
1 Answer
active oldest votes
Since $H\leq G$ and $G$ is connected, each $h\in H$ is connected to the identity, hence its action on $G$ is homologous to the identity.
up vote 6 down vote You're not going to get anything interesting here unless you either look at $N_G(H)/H$ acting on $H\backslash G$, or (equivalently) $N_G(H)$ acting on the $H$-equivariant
accepted cohomology of $G$, or something like that.
What about dropping the connected assumption? There are papers by R.L. Taylor in the 1950s on these structures. – Ronnie Brown Feb 16 '13 at 11:43
Hi Allen, thanks for the answer and interesting suggestions! Hi Ronnie, it might prove interesting to remove the connectedness assumption. Thanks for the references! – Peter
Crooks Feb 16 '13 at 16:38
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Not the answer you're looking for? Browse other questions tagged at.algebraic-topology rt.representation-theory lie-groups homotopy-theory group-actions or ask your own question. | {"url":"http://mathoverflow.net/questions/121958/representations-of-finite-subgroups-on-homology","timestamp":"2014-04-19T22:47:28Z","content_type":null,"content_length":"53615","record_id":"<urn:uuid:2c753e99-909f-42d8-9a08-d39ef5a5e48f>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00592-ip-10-147-4-33.ec2.internal.warc.gz"} |
[racket] strange loops
From: Marijn (hkBst at gentoo.org)
Date: Tue Mar 6 08:23:33 EST 2012
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Hash: SHA1
On 06-03-12 12:03, Pierpaolo Bernardi wrote:
> Hello,
> I was expecting the procedure 'fa' below to run in constant
> memory, as, in my understanding, It doesn't use any non-tail
> recursive loops, it does not build any data structure, and only
> performs arithmetic operations on small integers.
> But some of my assumptions must be wrong, as the calls to next-z
> accumulate and eventually fill the available memory.
> Can someone explain me where's my mistake?
> (A secondary note: the debugger shows the value of the squares
> vector as a 100 element vector, with no hint that the value is
> abbreviated. Pretty confusing, IMHO).
> Thanks.
> Pierpaolo
> ================ #lang racket
> (define (perfect-square? n) (= n (sqr (integer-sqrt n))))
> (define (fa) (let ((squares (for/vector ((i (in-range 1 10000))) (*
> i i)))) (let/ec return (let next-sum ((sum 3)) (let ((limit-z
> (quotient sum 3))) (let next-z ((z 1)) (if (= z limit-z) (next-sum
> (add1 sum)) (for ((y+z (in-vector squares)))
This loop is a fiction as all paths in the body go to either next-sum
or next-z and leave the rest of the iteration for whenever. Thus you
never iterate over your vector of squares, but you do build up
stack-frames that promise to do that in a future which never happens.
> (let ((y (- y+z z))) (if (> (+ y z) sum) (next-sum (add1 sum)) (let
> ((x (- sum z y))) (if (and (perfect-square? (+ x y))
> (perfect-square? (- x y)) (perfect-square? (+ x z))
> (perfect-square? (- x z))) (return (list x y z sum)) (next-z (add1
> z))))))))))))))
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Posted on the users mailing list. | {"url":"http://lists.racket-lang.org/users/archive/2012-March/050862.html","timestamp":"2014-04-20T08:15:45Z","content_type":null,"content_length":"7266","record_id":"<urn:uuid:64a80a17-7cc1-46b2-b997-b928673505fe>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00568-ip-10-147-4-33.ec2.internal.warc.gz"} |
BEE4223 Power Electronics
BEE4223 Power Electronics
To supply a dc source from an ac source. The diode will remain off as long as the voltage of ac source is less than dc voltage. ... – PowerPoint PPT presentation
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Transcript and Presenter's Notes | {"url":"http://www.powershow.com/view/1b945-MjgyO/BEE4223_Power_Electronics_powerpoint_ppt_presentation","timestamp":"2014-04-19T15:00:36Z","content_type":null,"content_length":"115290","record_id":"<urn:uuid:f6218b8b-4e30-4ce4-bb14-602d9fe30813>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00406-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics with Calculus/Appendix 2/Examples of Derivatives
From Wikibooks, open books for an open world
For x(t), position as a function of time
Velocity: The rate of change of position with respect to time
$\begin{matrix} \mathbf{v}(t) = f'(t) = {dx \over dt} \end{matrix}$
Acceleration: The rate of change of velocity with respect to time
$\begin{matrix} \mathbf{a}(t) = \mathbf{v}'(t) = f''(t) = {d^2x \over dt^2} \end{matrix}$
Jerk: The rate of change of acceleration with respect to time
$\begin{matrix} \mathbf{j}(t) = \mathbf{a}'(t) = f'''(t) = {d^3x \over dt^3} \end{matrix}$
Jerk is not commonly used in first year motion. Its main application is in dealing with travel of large objects that change their weight as they move due to a change in mass. One example might be a
rocket travelling up from rest. As it burns fuel, its centre of gravity changes and as such, its acceleration is not constant (violation of Newton's Second Law).
Mechanics (Statics)[edit]
Given the details of the loading of a beam, we can represent it on a diagram of the beam with arrows indicating forces, curved arrows indicating moments (resistance to torque) and shaded regions
representing universally varying or distributed loads. We can use this diagram (commonly known as a free body diagram) and the information contained within it to draw a diagram representing the shear
forces (V in the beam, and can also derive an equation that represents these. The equation may not be as simple as a polynomial, and is quite often a series of continuous functions with endpoints at
the points on the beam where the forces occur.
We can perform an indefinite integral on each of these segments of the beam to get more information on it. The indefinite integrals combine to form a diagram of the bending moments in the beam.
Bending moments are a special type of moment, as the beam is most likely to fail where the bending moments are at a relative extrema. By definition, any indefinite integral will contain a constant, C
. In the case of the bending moment diagram, our C is merely the endpoint of the previous segment. The only exception being when we have a moment, we add or subtract its value (depending on
Hence, differentiating the bending moment model will give us the shear force.
Differentiating the shear force model brings us back to our loading diagram, and differentiating that will give us the shape of the deflection of the beam under the loading.
For any bending moment model b, as a function of distance from the end of the beam, x,
$\begin{matrix} b'(x) &=& V(x) \\ b''(x) &=& V'(x) &=& \operatorname{FBD}(x) \\ b'''(x) &=& \operatorname{FBD}'(x) &=& f(x) \end{matrix}$
Where f(x) is a function describing the deflection of the beam. | {"url":"http://en.wikibooks.org/wiki/Physics_with_Calculus/Appendix_2/Examples_of_Derivatives","timestamp":"2014-04-16T19:52:47Z","content_type":null,"content_length":"28442","record_id":"<urn:uuid:0ba1edf5-112b-4340-a219-45e632f56d0d>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00047-ip-10-147-4-33.ec2.internal.warc.gz"} |
Gauge condition
Actually the constraint
\frac{1}{c^2}\frac{\partial\phi}{\partial t} + \nabla \cdot \mathbf A = 0
is due to Lorenz (Lorenz and Lorentz are easily confused):
This equation is sometimes used because it leads to simple and symmetric wave equations for the scalar and vector potential, which are then easily solved for known charge and current distribution and
initial conditions on the field.
The potentials are auxiliary functions without direct physical meaning. The meaning of the constraint is really just simplification of the relativistic equations so they become nice and simple. | {"url":"http://www.physicsforums.com/showthread.php?s=0062868ca59b17772da8b114874e57b6&p=4343088","timestamp":"2014-04-20T14:09:14Z","content_type":null,"content_length":"28542","record_id":"<urn:uuid:afbc3e1f-514a-4fc9-b482-4781e87d091b>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00077-ip-10-147-4-33.ec2.internal.warc.gz"} |
Use Excel Formulas to Highlight Duplicate Rows
If you have a spreadsheet containing several fields of data (as in the example on the right), you can highlight duplicate rows in your spreadsheet by using Excel Formulas.
To highlight Excel duplicates in the example spreadsheet on the right, first collate the data from columns A - C into column D. This can be done using a formula that is made up of the & operator and
the Text function. In cell D2, the formula will be:
=A2 & " " & B2 & " " & TEXT(C2,"dd/mm/yyyy")
Once columns A - C are collated into column D, we need to highlight duplicates in the contents of column D. This can be done using the Countif function. The function to be entered into cell E2 is:
=COUNTIF(D$2:D2, D2)
Note that the above function uses a combination of Absolute and Relative Cell References. The first reference, to cell D2, is an absolute cell reference (shown by the $ sign), while the remaining
cell references are relative cell references.
Therefore, when the formula is copied down to the rows below, the initial reference to cell D$2 remains fixed, while the remaining references are adjusted to refer to cells D3, D4, etc. This causes
the Countif function to only count cells up to and including those in the current row and therefore causes the formula to only highlight duplicate values in column E.
The functions are shown in cells D2-E11 of the spreadsheet below:
The results of the formulas are shown in the spreadsheet below. It is seen that the duplicate entry in row 10 has the value "2" in cell E10, showing that this is a duplicate.
As an additional feature, in the spreadsheet below, Conditional Formatting has been used to highlight rows in which the value in column E is greater than 1. | {"url":"http://www.excelfunctions.net/Excel-Duplicates.html","timestamp":"2014-04-20T15:57:20Z","content_type":null,"content_length":"13962","record_id":"<urn:uuid:feec5340-c058-4b09-8493-d267ee1431df>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00207-ip-10-147-4-33.ec2.internal.warc.gz"} |
Results 1 - 10 of 29
, 2001
"... We propose a fully functional identity-based encryption scheme (IBE). The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational
Diffie-Hellman problem. Our system is based on bilinear maps between groups. The Weil pairing on elliptic ..."
Cited by 1123 (24 self)
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We propose a fully functional identity-based encryption scheme (IBE). The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational
Diffie-Hellman problem. Our system is based on bilinear maps between groups. The Weil pairing on elliptic curves is an example of such a map. We give precise definitions for secure identity based
encryption schemes and give several applications for such systems.
- In New Security Paradigms Workshop , 2001
"... The growth of the Internet has triggered tremendous opportunities for cooperative computation, where people are jointly conducting computation tasks based on the private inputs they each
supplies. These computations could occur between mutually untrusted parties, or even between competitors. For exa ..."
Cited by 67 (1 self)
Add to MetaCart
The growth of the Internet has triggered tremendous opportunities for cooperative computation, where people are jointly conducting computation tasks based on the private inputs they each supplies.
These computations could occur between mutually untrusted parties, or even between competitors. For example, customers might send to a remote database queries that contain private information; two
competing financial organizations might jointly invest in a project that must satisfy both organizations' private and valuable constraints, and so on. Today, to conduct such computations, one entity
must usually know the inputs from all the participants; however if nobody can be trusted enough to know all the inputs, privacy will become a primary concern. This problem is referred to as Secure
Multi-party Computation Problem (SMC) in the literature. Research in the SMC area has been focusing on only a limited set of specific SMC problems, while privacy concerned cooperative computations
call for SMC studies in a variety of computation domains. Before we can study the problems, we need to identify and define the specific SMC problems for those computation domains. We have developed a
frame to facilitate this problem-discovery task. Based on our framework, we have identified and defined a number of new SMC problems for a spectrum of computation domains. Those problems include
privacy-preserving database query, privacy-preserving scientific computations, privacy-preserving intrusion detection, privacy-preserving statistical analysis, privacy-preserving geometric
computations, and privacy-preserving data mining. The goal of this paper is not only to present our results, but also to serve as a guideline so other people can identify useful SMC problems in their
own computation domains.
- In World Congress on Formal Methods , 1999
"... . We introduce the concept of a group principal and present a number of different classes of group principals, including thresholdgroup -principals. These appear to naturally useful concepts for
looking at security. We provide an associated epistemic language and logic and use it to reason about ..."
Cited by 66 (5 self)
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. We introduce the concept of a group principal and present a number of different classes of group principals, including thresholdgroup -principals. These appear to naturally useful concepts for
looking at security. We provide an associated epistemic language and logic and use it to reason about anonymity protocols and anonymity services, where protection properties are formulated from the
intruder's knowledge of group principals. Using our language, we give an epistemic characterization of anonymity properties. We also present a specification of a simple anonymizing system using our
theory. 1 Introduction Though principals are typically viewed as atomic, there is no reason we cannot consider the knowledge and actions taken by a group. Hence, the basic notion of a group
principal. This notion appears to be a useful concept for reasoning about various properties of electronic commerce and security protocols. One such principal is a threshold-group-principal. Such a
principal a...
- CryptoBytes , 1997
"... Dalit Naor y Proactive security provides a method for maintaining the overall security of a system, even when individual components are repeatedly broken into and controlled by an attacker. In
particular it provides for automated recovery of the security of individual components, avoiding the use of ..."
Cited by 57 (9 self)
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Dalit Naor y Proactive security provides a method for maintaining the overall security of a system, even when individual components are repeatedly broken into and controlled by an attacker. In
particular it provides for automated recovery of the security of individual components, avoiding the use of expensive and inconvenient manual processes (unless perhaps when an ongoing attack is
detected). The technique calls for the distribution of trust among several components (servers), together with periodic refreshments of the sensitive data held by the servers. This way, the proactive
approach guarantees uninterrupted security as long as not too many servers are broken into at the same time. We describe the proactive approach and review some algorithms, implementations, and
applications. We elaborate on two of the most important results: proactive signatures and proactive secure communication. Proactive signatures provide a solution for long-lived secret keys, such as
the key of a certi cation authority. Proactive secure communication ensures secrecy and authenticity ofcommunication, with automated refresh of the secret keys. 1
- Journal of Cryptology , 1998
"... We study the problem of maintaining authenticated communication over untrusted communication channels, in a scenario where the communicating parties may be occasionally and repeatedly broken
into for transient periods of time. Once a party is broken into, its cryptographic keys are exposed and pe ..."
Cited by 40 (6 self)
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We study the problem of maintaining authenticated communication over untrusted communication channels, in a scenario where the communicating parties may be occasionally and repeatedly broken into for
transient periods of time. Once a party is broken into, its cryptographic keys are exposed and perhaps modified. Yet, we want parties whose security is thus compromised to regain their ability to
communicate in an authenticated way aided by other parties. In this work we present a mathematical model for this highly adversarial setting, exhibiting salient properties and parameters, and then
describe a practically-appealing protocol for the task of maintaining authenticated communication in this model. A key element in our solution is devising proactive distributed signature (PDS)
schemes in our model. Although PDS schemes are known in the literature, they are all designed for a model where authenticated communication and broadcast primitives are available. We therefore show
how t...
, 2000
"... Suppose that Bob has a database D and that Alice wants to perform a search query q on D (e.g., “is q in D?”). Since Alice is concerned about her privacy, she does not want Bob to know the query
or the response to the query. How could this be done? There are elegant cryptographic techniques for solvi ..."
Cited by 40 (10 self)
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Suppose that Bob has a database D and that Alice wants to perform a search query q on D (e.g., “is q in D?”). Since Alice is concerned about her privacy, she does not want Bob to know the query or
the response to the query. How could this be done? There are elegant cryptographic techniques for solving this problem under various constraints (such as “Bob should know neither nor the answer to
the query ” and “Alice should learn nothing about D other than the answer to the query”), while optimizing various performance criteria (e.g., amount of communication). We consider the version of
this problem where the query is of the type “is approximately in �? ” for a number of different notions of “approximate”, some of which arise in image processing and template matching, while others
are of the string-edit type that arise in biological sequence comparisons. New techniques are needed in this framework of approximate searching, because each notion of “approximate equality”
introduces its own set of difficulties; using encryption is more problematic in this framework because the items that are approximately equal cease to be so after encryption or cryptographic hashing.
Practical protocols for solving such problems make possible new forms of e-commerce between proprietary database owners and customers who seek to query the database, with privacy.
- In USENIX Security Symposium , 2004
"... Shmatikov z SRI International ..."
- ADVANCES IN CRYPTOLOGY: EUROCRYPT '99, VOLUME 1592 OF LECTURE NOTES IN COMPUTER SCIENCE , 1999
"... This work describes schemes for distributing between n servers the evaluation of a function f which is an approximation to a random function, such that only authorized subsets of servers are
able to compute the function. A user who wants to compute f(x) should send x to the members of an authorize ..."
Cited by 29 (0 self)
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This work describes schemes for distributing between n servers the evaluation of a function f which is an approximation to a random function, such that only authorized subsets of servers are able to
compute the function. A user who wants to compute f(x) should send x to the members of an authorized subset and receive information which enables him to compute f(x). We require that such a scheme is
consistent, i.e. that given an input x all authorized subsets compute the same value f(x). The solutions we present enable the operation of many servers, preventing bottlenecks or single points of
failure. There are also no single entities which can compromise the security of the entire network. The solutions can be used to distribute the operation of a Key Distribution Center (KDC). They are
far better than the known partitioning to domains or replication solutions to this problem, and are especially suited to handle users of multicast groups.
, 2000
"... Abstract. We discuss the following problem: Given an integer φ shared secretly among n players and a prime number e, how can the players efficiently compute a sharing of e −1 mod φ. The most
interesting case is when φ is the Euler function of a known RSA modulus N, φ = φ(N). The problem has several ..."
Cited by 26 (0 self)
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Abstract. We discuss the following problem: Given an integer φ shared secretly among n players and a prime number e, how can the players efficiently compute a sharing of e −1 mod φ. The most
interesting case is when φ is the Euler function of a known RSA modulus N, φ = φ(N). The problem has several applications, among which the construction of threshold variants for two recent signature
schemes proposed by Gennaro-Halevi-Rabin and Cramer-Shoup. We present new and efficient protocols to solve this problem, improving over previous solutions by Boneh-Franklin and Frankel et al. Our
basic protocol (secure against honest but curious players) requires only two rounds of communication and a single GCD computation. The robust protocol (secure against malicious players) adds only a
couple of rounds and a few modular exponentiations to the computation. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=385178","timestamp":"2014-04-20T13:47:27Z","content_type":null,"content_length":"38681","record_id":"<urn:uuid:6c64de22-8538-4f2c-be59-743ec7771091>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00001-ip-10-147-4-33.ec2.internal.warc.gz"} |
While at WWDC 2012, I had the opportunity to sit down with Shane Crawford of Blue Lightning Labs and take a peak at his Mathemagics apps, Mental Math Tricks and Easy Algebra Fast. These app are
design to teach you special rarely taught tricks and tips that can be used when solving different types of math problems. Most of the techniques can be done mentally and make it possible so solve
math problems at lightning speeds. Both apps have the exact same design and method of use which is why I'm reviewing them together. The only real difference between them is the subject matter --
Mental Math Tricks is arithmetic and Easy Algebra Fast is algebra. There are three different modes in Mathematics: Lessons, Practice, and Play.
Vertical Format This Multiplication worksheet may be configured for 2, 3, or 4 digit multiplicands being multiplied by 1, 2, or 3 digit multipliers. You may vary the numbers of problems on each
worksheet from 12 to 25. This multiplication worksheet is appropriate for Kindergarten, 1st Grade, 2nd Grade, 3rd Grade, 4th Grade, and 5th Grade. You may add a memo line that will appear on the
worksheet for additional instructions. Multiplication Worksheets | Multiple - Digit Multiplication Worksheets
Multiplication Worksheets | Multiplication Worksheets for Lesson Plans Dynamically Created Multiplication Worksheets for Teachers Here is a graphic preview for all of the multiplication worksheets.
You can select different variables to customize these multiplication worksheets for your needs.
Free 5th grade math worksheets. Randomly generated, printable from your browser!
We have multiplication sheets for timed tests or extra practice, as well as flashcards and games. Includes Multiplication Flashcards, Multiplication Bingo, Multiplication Tables, Multiplication I
Have - Who has, and lots more! To see the Common Core Standards associated with each multiplication worksheet, select the apple core logo ( ) below the worksheet's description. Games Multiplication
Game: I Have / Who HasFree | {"url":"http://www.pearltrees.com/sdnls/math/id3546355","timestamp":"2014-04-17T19:00:43Z","content_type":null,"content_length":"19383","record_id":"<urn:uuid:94425942-5d96-42ba-ad32-28807d96812c>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00285-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: finding directions in two Cartesian systems
Replies: 1 Last Post: Nov 22, 2007 1:15 PM
Messages: [ Previous | Next ]
t finding directions in two Cartesian systems
Posted: Nov 6, 2007 3:35 PM
Posts: 1
Registered: 11/6/07 hi, all,
run into one geometry problem: I have two 3-D Cartesian systems, A and B. they share the same original. and they have the same scale.
(2,3,3) in B system is the same direction as (118,2090,1000)in A system; then what is the direction in B corresponding to (89,2600,1000) in A?
I know there should a standard formula for this kind of conversion. but I didn't take this class, would anyone recommend any book on this? is there a software doing this work?
Date Subject Author
11/6/07 finding directions in two Cartesian systems t
11/22/07 Re: finding directions in two Cartesian systems Charles Lazo | {"url":"http://mathforum.org/kb/thread.jspa?threadID=1649890","timestamp":"2014-04-19T10:39:08Z","content_type":null,"content_length":"17433","record_id":"<urn:uuid:aa4da94a-2aff-4787-b570-9acb53ebca1e>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00533-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Middle School Math Madness!
This year, I teaching combining like terms really early with my 7th graders.
(Normally, we get to it towards the end of the year.)
It's now in the first unit. First, we discuss what like terms are and create examples of like terms and unlike terms. I've always done visuals to help students. For example, if you have 4x + 8x:
Once we do a few visuals, we then practice with larger expressions, highlighting or drawing different shapes around the like terms.
Make sure students include the sign in front of each term!
My interactive notebook page looks like this:
Page 10 will be for practicing problems. I'm came up with a couple of activities for centers that will work for combining like terms. The first was inspired from part of this week's
Made 4 Math Monday post
on the Radical Rational's blog. I'm going to write a bunch of like terms on two dice. One will have all terms with x; the other will have x^2 terms. Students will roll both dice twice, writing down
all 4 terms. They they will combine the like terms. I created a worksheet for students to record their answers.
You can download the worksheet from either of my stores:
Teachers Notebook
I also made a matching card game for small groups to play as a center activity to practice identifying like terms. I'm trying to figure out another way to use the cards. Any ideas?
You can get the cards and directions
The last item I created was a Race to the Top Triangle game. The triangles in the bottom row each have a term in them. Students will combine the two like terms next to each other and write the
simplified expression in the yellow triangle directly above the triangles they combined. They will continue until all yellow triangles are filled in.
The worksheet
(and answer key)
are in both my stores:
. I'm thinking of making this a center game where the students race to finish fastest. We'll see!
7 comments:
1. I absolutely love your race to the top activity. I've never seen this set up before and I'm definitely going to be using this for many different types of practice from now on.
I also have to tell you that I used it today with my high school kids and they told me that they REALLY liked it (getting them to actually say this is a very big compliment!) They actually asked
where I got it from and said they want to do more. So glad you posted this!!
1. I'm so glad to hear they liked it. You really can use it for so many topics! Did you have them race against each other?
2. I didn't have them race because it would be too much pressure for a few of them. But I have four boys that LOVE competition and they got 4 different answers (1 was correct) so I had them go
sit together and check their papers off each other line by line to try to figure out who was right. They love doing stuff like that and they work so well together when they all want to be
right. They were the ones that said they really liked it :)
2. LOVE your ideas and lessons for combining like terms. Thank you so much for sharing!!!
1. You are welcome!
3. are there any games on here because that's what I need
1. Yes, there is a matching game as well as a race to the top game. | {"url":"http://msmathmadness.blogspot.com/2012/08/combining-like-terms.html","timestamp":"2014-04-16T22:29:13Z","content_type":null,"content_length":"138964","record_id":"<urn:uuid:58608d47-e52d-4197-b2e1-03dc23cdc0fa>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00558-ip-10-147-4-33.ec2.internal.warc.gz"} |
Issues and Opinion on Structural Equation Modeling
Structural Equation Modeling in IS Research - Understanding the LISREL and PLS perspective
Wynne W. Chin
University of Houston
As in many other social science areas, the IS field has seen a substantial increase in the number of submissions and publications using structural equation modeling (SEM) techniques. This is likely
due to the proliferation of software packages to perform covariance-based (e.g., LISREL, EQS, AMOS, SEPATH, RAMONA, MX, and CALIS) and component-based (e.g., PLS-PC, PLS-Graph) analysis. The SEM
approach is integrative in the sense that it combines the perspective of two research traditions:
1. an econometric perspective focusing on prediction and,
2. a psychometric emphasis that models concepts as latent (unobserved) variables that are indirectly inferred from multiple observed measures (alternately termed as indicators or manifest
This resulting combination allows researchers to perform path analytic modeling with latent variables (LVs). Specifically, SEM provides the researcher with the flexibility to: (a) model relationships
among multiple predictor and criterion variables, (b) construct unobservable LVs, (c) model errors in measurements for observed variables, and (d) statistically test a priori substantive/theoretical
and measurement assumptions against empirical data (i.e., confirmatory analysis).
SEM involves generalizations and extensions of earlier first-generation procedures. By applying certain constraints or assumptions on an SEM analysis, a researcher can end up performing the
equivalent of techniques such as canonical correlation, multiple regression, multiple discriminant analysis, analysis of variance or covariance, or principle components analysis.
Naturally, along with the benefits comes the complexity. This virtual discussion will cover various issues that often appear among researchers. The primary focus will be on both the covariance based
approach often equated generically as a LISREL analysis and the Partial Least Squares approach. Hopefully, we will not only cover matters generic to social scientists, but also specific to the IS
field. To guide the questions, we might look at it from various frames.
One standard approach is to examine the stages in the traditional SEM lifecycle. They are:
☆ Model Specification,
☆ Identification,
☆ Estimation,
☆ Testing Fit, and
☆ Model Modification or Respecification.
Another approach is to examine common mistakes that are made. We can discuss such issues as:
☆ Critical Missing Information. Information that should be included but are often left out from research articles thereby preventing other researcher from reproducing the analysis and
building a cumulative tradition.
☆ Mismatch of questionnaire items and subsequent analysis. Survey questions analyzed are often formative in nature or a composite of formative and reflective measures. A LISREL analysis and
use of internal consistency measures such as Cronbach’s alpha would be incorrect.
☆ Sole reliance of overall goodness of fit measures. Using only covariance based goodness of fit measures as the primary arbiter of confirmation while ignoring other important measures of
model adequacy.
☆ Analyzing second order factors without a purpose. Demonstrating second order factor models without providing an underlying rationale for its subsequent usage.
☆ Lack of Empirical over-identification. Many empirical studies do not perform a strong test of the model/latent variables.
☆ Ignoring the statistical power of models.
☆ Ignoring equivalent models.
☆ Falling into an exploratory mode via initial exploratory factor analysis or using information from the statistical package to modify initial models for better fit.
☆ Premature or inappropriate approach of analysis when either substantive or theoretical knowledge is relatively new.
Finally, we can approach it from an applied perspective. Example questions might include:
☆ When should I consider using Partial Least Squares as opposed to LISREL?
☆ More importantly - how does Partial Least Square differ from LISREL?
☆ How does LISREL compare to path analysis using multiple regression?
☆ Does it make sense to do an exploratory factor analysis prior to using SEM?
☆ How about a confirmatory factor analysis first?
☆ What are the advantages to using SEM for multi-sample or cross-cultural analysis?
To start off, we might consider the following model (Figure 1) as a basis for discussion. The model is a simple two factor model where F1 is hypothesized to affect F2. The data in the form of a
correlation matrix is provided in Table 1 for the four measures/indicators (x1,x2,y1,and y2) of their respective factors.
Figure 1. Two factor model with two indicator/measures for each factor.
│ │ x1 │ x2 │ y1 │ y2 │
│ x1 │ 1.00 │ │ │ │
│ x2 │ .087 │ 1.00 │ │ │
│ y1 │ .140 │ .080 │ 1.00 │ │
│ y2 │ .152 │ .143 │ .272 │ 1.00 │
Table 1. Sample Data Set (n=1000).
All correlations, given the sample size, are significant but quite low ranging from 0.087 to 0.272. If we use the theoretical model as depicted in Figure 1, what would be the path estimate p linking
F1 and F2? The covariance based estimate using software such as LISREL would result in a standardized estimate of p at 0.83 whereas the PLS estimate was 0.22. The standardized loadings of a, b, c, d
using the covariance procedure were 0.33, 0.26, 0.46, and 0.59. The PLS estimates resulted in loadings of 0.81, 0.66, 0.73, and 0.85 with corresponding weights of 0.75, 0.60, 0.54, and 0.71. In the
case of the covariance estimates, the path estimate of 0.83 is much larger than the observed correlations between the x and y variables where the highest is 0.152. In the case of PLS, the estimate of
0.22 is much closer to the observed correlations. Which estimate should we place confidence in? | {"url":"http://disc-nt.cba.uh.edu/chin/ais/","timestamp":"2014-04-18T10:34:36Z","content_type":null,"content_length":"10066","record_id":"<urn:uuid:20278d4c-6d8b-404e-a2b1-ec2d638087de>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00506-ip-10-147-4-33.ec2.internal.warc.gz"} |
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May 25th 2009, 01:18 AM #1
Apr 2009
Not sure if this is the right area for combinations/pascal triangle questions...
A)What is a constant in Binomial expansion? Example, I have to state whether or not a constant appears in the expansion of (x+y)^11, and explain why. A quick explanation, pls?
B) Sandra is testing to see if she can tell the difference between diet pop, and regular pop. She knows there are 3 cups of diet pop, and 2 cups of regular pop, and she'll write the order which
she believes the pop to be in. How many ways can record D(for Diet) and R(for Regular) if order matters?
C)There are 8 non-fiction books, and 7 fiction books. You are to recieve two of them as a gift, how many combinations are possible if you are to recieve AT LEAST 1 fiction book?
-Is this as simple just figuring out how many combinations of 2 fiction books there are, and multiplying it by the combinations of 1 fiction book, 1 non-fiction book(which would be 56 right?)?
D) How do you solve for a constant tern in the expansion of (x+2)^9?
Mucho Gracias to any and all help!
Last edited by Blahdkm; May 25th 2009 at 02:13 AM.
Hello Blahdkm
A constant term is one that doesn't involve any variables - like $x$ and $y$, for instance. In the example you quote, there are no constant terms - every term involves $x$ or $y$ or both. But in
an expression like
$\Big(x + \frac{2}{x^2}\Big)^6$
there will a term that doesn't involve $x$. You find it as follows:
$\Big(x + \frac{2}{x^2}\Big)^6 = \Big(\frac{1}{x^2}\Big)^6 ( x^3 + 2)^6$
Since the term outside the bracket is $\frac{1}{x^{12}}$, the term in $x^{12}$ in the expansion of $(x^3+2)^6$ will result in the cancellation of the $x$'s, and hence a constant term. This term
$\binom64(x^3)^4.2^2 = 60x^{12}$
So the constant term in the expansion of $\Big(x + \frac{2}{x^2}\Big)^6$ is $60$.
B) Sandra is testing to see if she can tell the difference between diet pop, and regular pop. She knows there are 3 cups of diet pop, and 2 cups of regular pop, and she'll write the order which
she believes the pop to be in. How many ways can record D(for Diet) and R(for Regular) if order matters?
The D's can be placed in 3 positions out of 5, and the R's in the remaining positions. So the number of possible arrangements is $\binom53 = 10$.
C)There are 8 non-fiction books, and 7 fiction books. You are to recieve two of them as a gift, how many combinations are possible if you are to recieve AT LEAST 1 fiction book?
-Is this as simple just figuring out how many combinations of 2 fiction books there are, and multiplying it by the combinations of 1 fiction book, 1 non-fiction book(which would be 56 right?)?
This is only part of the answer - where you have exactly one fiction book. You need to add to this the number of ways of choosing 2 fiction books from 7, which is $\binom72 = 21$. So the total
number is $56+21=77$.
D) How do you solve for a constant tern in the expansion of (x+2)^9?
Mucho Gracias to any and all help!
Answer $2^9 = 512$. See A above!
Many Many thanks friend.
Last Question, I have a 3d square connected at one point. It has a height of 3 squares, a width of 1 square, and length of 3 squares.
I'm supposed to find the amount of paths from S(Upper left corner), to U(bottom right corner), with no backtracking. The fact that it's a 3d object is seriously throwing me for a loop.
How do you determine the amount of paths on a 3d object(I'm assuming you have to use pascals triangle in some way)?
Don't add a new question
Hello Blahdkm
Many Many thanks friend.
Last Question, I have a 3d square connected at one point. It has a height of 3 squares, a width of 1 square, and length of 3 squares.
I'm supposed to find the amount of paths from S(Upper left corner), to U(bottom right corner), with no backtracking. The fact that it's a 3d object is seriously throwing me for a loop.
How do you determine the amount of paths on a 3d object(I'm assuming you have to use pascals triangle in some way)?
Post this as a new thread - you're more likely to get a response that way. And you'll be sticking to the rules as well! (See Rule 14)
May 25th 2009, 04:59 AM #2
May 25th 2009, 05:43 AM #3
Apr 2009
May 25th 2009, 10:23 AM #4 | {"url":"http://mathhelpforum.com/discrete-math/90385-combinations.html","timestamp":"2014-04-17T08:55:38Z","content_type":null,"content_length":"47325","record_id":"<urn:uuid:4357134e-780a-4b59-9072-7d95332301c7>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00417-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Programming with DirectX : Projection Transformations
Projection transformations affect how a rendered scene looks when displayed to the screen. The two main types of projections are orthogonal projections and perspective projections, both of which are
supported by Direct3D. A projection is a matrix that stores projection information. To apply the projection to the geometry in the scene, we multiply the projection matrix and vertices of the
geometry, which is a process known as transformation. A projection is a representation of how objects are viewed when rendered. The second type, orthogonal projection, will be discussed next.
Orthogonal Projection
Orthogonal projection causes all objects to be rendered to the screen at the same size regardless of how far away an object is. In the real world, as objects move farther away from you, they appear
smaller. An example of this is shown in Figure 1.
Figure 1. An example of objects getting smaller as they move farther away.
In orthogonal projection, the size of the objects does not change due to distance. Many times, this effect is desired, but for most 3D scenes in modern video games it is often important to have a
different type of projection. Orthogonal projection can be a great type of projection for 2D elements such as menus, heads-up displays, and any other type of rendering where the geometry is not to
change in size with distance. An example of orthogonal projection is shown in Figure 2.
Figure 2. An example of orthogonal projection.
In Direct3D there are four different functions for creating an orthogonal matrix. The first two functions are D3DXMatrixOrthoLH() and D3DXMatrixOrthoRH(). The LH version creates a left-handed
projection matrix, while RH creates a right-handed projection matrix. Their function prototypes are as follows.
D3DXMATRIX * D3DXMatrixOrthoLH(
D3DXMATRIX *pOut,
FLOAT w,
FLOAT h,
FLOAT zn,
FLOAT zf
D3DXMATRIX * D3DXMatrixOrthoRH(
D3DXMATRIX *pOut,
FLOAT w,
FLOAT h,
FLOAT zn,
FLOAT zf
The parameters of the functions start with the D3DXMATRIX object, which is the structure that represents matrices in Direct3D, which will be created from the function call, the width and height of
the desired view volume, and the near and far plane. The near plane determines how close to the viewer an object can be before it is seen, while the far plane determines how far away an object can be
before it disappears. The width and height, which is normally the width and height of the screen or rendering area, in addition to the near and far values collectively represent the view volume. The
view volume is an area in which objects are visible. (see Figure 3)
Figure 3. An example of a view volume.
The left- and right-handedness of the functions refer to coordinate systems. A coordinate system essentially tells the graphics API which direction, left or right, the positive X axis travels and
which direction, toward or away, the positive Z axis travels. An illustration is shown in Figure 4.
Figure 4. Left- and right-handed coordinate systems.
OpenGL uses a right-handed coordinate system, while Direct3D traditionally used a left-handed system.
Direct3D allows developers to use either left- or right-hand coordinates. Using a right-handed system allows developers to use the same data in OpenGL and Direct3D applications without modification
of the geometry’s X and Z axes. This is the reason behind the multiple versions of the orthogonal projection functions. The last two orthogonal projection functions are as follows.
D3DXMATRIX * D3DXMatrixOrthoOffCenterLH(
D3DXMATRIX *pOut,
FLOAT l,
FLOAT r,
FLOAT b,
FLOAT t,
FLOAT zn,
FLOAT zf
D3DXMATRIX * D3DXMatrixOrthoOffCenterRH(
D3DXMATRIX *pOut,
FLOAT l,
FLOAT r,
FLOAT b,
FLOAT t,
FLOAT zn,
FLOAT zf
The l and r parameters represent the minimum and maximum width, while b and t represent the minimum and maximum height. zn and zf are the near and far values. The D3DXMatrixOrthoLH() and
D3DXMatrixOrthoRH() functions are special cases of D3DXMatrixOrthoOffCenterLH() and D3DXMatrixOrthoOffCenterRH(). The off center functions allow more customizability than the other two seen earlier
in this section.
Perspective Projection
The other type of projection is perspective projection. This type of projection adds perspective to scenes. Perspective projection allows objects to shrink as they move farther away from the viewer.
Objects also distort as they are viewed at an angle. Figure 5 shows an example of perspective projection. Perspective projection is a type of projection that can be seen in all modern 3D video games.
Figure 5. Perspective projection.
Perspective projection is the same as the idea behind perspective in drawing art.
The perspective projection matrix functions are as follows, where the parameters match those of the orthogonal counterparts.
D3DXMATRIX * D3DXMatrixPerspectiveLH(
D3DXMATRIX *pOut,
FLOAT w,
FLOAT h,
FLOAT zn,
FLOAT zf
D3DXMATRIX * D3DXMatrixPerspectiveRH(
D3DXMATRIX *pOut,
FLOAT w,
FLOAT h,
FLOAT zn,
FLOAT zf
D3DXMATRIX * D3DXMatrixPerspectiveOffCenterLH(
D3DXMATRIX *pOut,
FLOAT l,
FLOAT r,
FLOAT b,
FLOAT t,
FLOAT zn,
FLOAT zf
D3DXMATRIX * D3DXMatrixPerspectiveOffCenterRH(
D3DXMATRIX *pOut,
FLOAT l,
FLOAT r,
FLOAT b,
FLOAT t,
FLOAT zn,
FLOAT zf | {"url":"http://programming4.us/multimedia/1626.aspx","timestamp":"2014-04-20T18:22:54Z","content_type":null,"content_length":"46705","record_id":"<urn:uuid:16244105-d331-43c4-8648-2816bf175d2a>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00442-ip-10-147-4-33.ec2.internal.warc.gz"} |
Merrimack Science Tutor
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Wolfram Demonstrations Project
Inversive Geometry V: Circle through Three Points
This Demonstration shows how to use a sequence of three inversions to construct a circle passing through three different given points A, B, and C. Let us call its center Q. Let the fixed circle of
inversion have center at A and pass through B (orange) and let C' be the inverse of C. Reflect A in the line BC' to get the point A'. Then Q is the result of inverting A'. You can drag the points B
or C. Note that this construction still applies if the given points are collinear. | {"url":"http://demonstrations.wolfram.com/InversiveGeometryVCircleThroughThreePoints/","timestamp":"2014-04-20T03:32:14Z","content_type":null,"content_length":"41857","record_id":"<urn:uuid:91e309b8-ae57-4d4e-8951-4c3042be7d26>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00249-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathFiction: Life and Fate (Vasily Grossman)
A Russian nuclear physicist flirts with the wife of his mathematician colleague and makes an important mathematical discovery, all during the Nazi invasion of the Soviet Union.
I had not heard of this important work of Soviet fiction, a criticism of the Stalinist era, until it was brought to my attention this evening at the Notre Dame math department's "fluid dynamics
seminar". Several mathematicians were challenged by my presence to attempt to name a work of mathematical fiction not already on my list. Nearly all of the suggestions were ones I had already added
(or already considered and rejected). Then, François Ledrappier proposed Life and Fate, specifically emphasizing the amazing mathematical discovery that the protagonist makes.
Of course, I have not yet had time to read the novel myself, but the little I can find out from its Wikipedia entry and from browsing it on Amazon.com confirm that it certainly should be included in
my database of mathematical fiction.
Here is a brief passage from the portion of the novel where he makes his great discovery:
(quoted from Life and Fate)
His head had been full of mathematical relationships, differential equations, the laws of higher algebra, number and probability theory. These mathematical relationships had an existence of their own
in some void quite outside the world of atomic nuclei, stars, and electromagnetic or gravitational fields, outside space and time, outside the history of man and the geological history of the earth.
And yet these relationships existed inside his own head.
And at the same time his head had been full of other laws and relationships: quantum interactions, fields of force, the constants that determined the process undergone by nuclei, the movement of
light, and the expansion and contraction of space and time. To a theoretical physicist, the processes of the real world were only a reflection of laws that had been born in the desert of mathematics.
It was not mathematics that reflected the world; the world itself was a projections of differential equations, a reflection of mathematics. | {"url":"http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf1125","timestamp":"2014-04-16T04:20:47Z","content_type":null,"content_length":"10655","record_id":"<urn:uuid:1d01969c-880c-40d9-a041-30927769d9e8>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00012-ip-10-147-4-33.ec2.internal.warc.gz"} |
Convert quarts to half gallon - Conversion of Measurement Units
›› Convert quart [US, liquid] to half US gallon
Did you mean to convert quart [US, liquid] to half gallon
quart [US, dry]
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quart [Germany]
quart [UK]
›› More information from the unit converter
How many quarts in 1 half gallon? The answer is 1.99999998943.
We assume you are converting between quart [US, liquid] and half US gallon.
You can view more details on each measurement unit:
quarts or half gallon
The SI derived unit for volume is the cubic meter.
1 cubic meter is equal to 1056.68820497 quarts, or 528.344105275 half gallon.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between quarts and half US gallons.
Type in your own numbers in the form to convert the units!
›› Definition: Quart
The quart is a US customary unit of volume equal to a quarter of a gallon.
›› Metric conversions and more
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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Mathematical Interest Theory
Mathematical Interest Theory gives an introduction of how investments grow over time. This is done in a mathematically precise manner. The emphasis is on practical applications that give the reader a
concrete understanding of why the various relationships should be true. Among the modern financial topics introduced are: arbitrage, options, futures, and swaps. The content of the book, along with
an understanding of probability, will provide a solid foundation for readers embarking on actuarial careers.
On the other hand, Mathematical Interest Theory is written for anyone who has a strong high school algebra background and is interested in being an informed borrower or investor. The content is
suitable for a mid-level or upper-level undergraduate course or a beginner graduate course.
Table of Contents
An Introduction to the Texas Instruments BA II Plus
The Growth of Money
Equations of Value and Yield Rates
Annuities (annuities certain)
Annuities with Different Payment and Conversion Periods
Loan Repayment
Stocks and Financial Markets
Arbitrage, Term Structure of Interest Rates, and Derivatives
Interest Rate Sensitivity
About the Authors
About the Authors
Leslie Vaaler was born in Providence, Rhode Island. She is the daughter of Leila Federer and geometric measure theorist Herbert Federer. Leslie’s undergraduate degree is from MIT, and she received
her Masters and PhD from Princeton University. Her thesis concerned problems in Iwasawa theory, an area of algebraic number theory.
Leslie’s first faculty positions were at the University of Michigan in Ann Arbor, Oklahoma State University, and Washington and Lee University. She is currently at the University of Texas at Austin
where she serves as the Buck Consultants Associate Director of Actuarial Studies and Senior Lecturer in Mathematics. She has been teaching actuarial classes since 2000.
James W. Daniel was born in Indianapolis, IN. He received a BA from Wabash College, an MS and PhD in mathematics from Stanford, and an honorary ScD from Wabash College. Daniel is a life member of the
MAA, where he has served on its Executive Committee of the Board of Governors as Chair of the Audit, the Budget, and the Building Committees and of the Publications Council. He has been an Associate
of the Society of Actuaries (ASA) since 1991 and has served on various of its education-oriented committees. Since 1989 Jim has been the Director of the actuarial program at the University of Texas,
where he is a member of the Academy of Distinguished Teachers. He is also known in the actuarial community as the teacher of intensive exam-prep review short courses for some of the actuarial
professional exams. Daniel co-founded the Actuarial Faculty Forum and for many years organized the sessions on actuarial education at the Joint Mathematics Meetings.
Jim is the author of the books Elementary Linear Algebra and its Applications and of The Approximate Minimization of Functionals; he is the co-author of the second and third editions of Applied
Linear Algebra (with Ben Noble) and of Computation and Theory in Ordinary Differential Equations (with Ramon Moore).
Student Solutions Manual
By Leslie Jane Federer Vaaler
Catalog Code: MIT-SM
Print ISBN: 978-0-88385-755-7
Electronic ISBN: 978-1-61444-603-3
120 pp., Paperbound, 2009
List Price: $35.95
MAA Member: $28.95
Series: MAA Textbooks
This manual is written to accompany Mathematical Interest Theory. It includes detailed solutions to the odd-numbered problems. There are solutions to 239 problems, and sometimes more than one way to
reach the answer is presented. In keeping with the presentation of the text, calculator discussion for the Texas Instrument BA II Plus or BA II Plus Professional calculators is typeset in a different
font from the rest of the text.
Table of Contents
An Introduction to the Texas Instruments BA II Plus
The Growth of Money
Equations of Value and Yield Rates
Annuities (annuities certain)
Annuities with Different Payment and Conversion Periods
Loan Repayment
Stocks and Financial Markets
Arbitrage, Term Structure of Interest Rates, and Derivatives
Interest Rate Sensitivity | {"url":"http://www.maa.org/publications/books/mathematical-interest-theory?device=mobile","timestamp":"2014-04-17T15:12:56Z","content_type":null,"content_length":"25467","record_id":"<urn:uuid:1958fd03-6c80-4633-ad2b-04bbf496dcfb>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00294-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Pie is greater than not pie.
March 5, 2011 10:48 AM Subscribe
I'm trying to express my love for a certain dessert in a math/logic formula. I have this:
π > ∼π
, which I take to mean "Pi is greater than not Pi", and this:
π > ∞-π
, which I take to mean "Pi is greater than Everything but Pi". Do these make any sense or hold up in any legit way?
posted by TheCoug to Food & Drink (15 answers total) 5 users marked this as a favorite
Your notation is wrong. Here are some alternative formulations in TeX, with links to the rendering thereof.
This means that π is greater than anything that is not π:
x \neq \pi \Rightarrow \pi > x
This means that π is greater than all the things that are not π combined:
\pi > \bigcup_{x \neq \pi} xposted by grouse at 11:05 AM on March 5, 2011
Pretty much not. The “not” symbol is for Boolean logic, which doesn’t have a concept of “greater than”. And pi is most certainly not greater than an infinite quantity minus pi, which is still
If you wanted “pi is greater than not pi”, you could say π > 0π, since 0 multiplied by anything is 0, and π is most certainly greater than 0. Here 0π is “the absence of pi”.
For “pi is greater than everything but pi”, I don’t think there really is a concept that fits. “Everything but pi” is conceptually closer to a
than a number, and sets themselves aren’t automatically comparable in terms of “greater than” and “less than”. (The relevant concept here is the existence (or lack thereof) of a
for a set of sets of numbers.
On preview: if you just wanted things that use the correct symbols but weren’t actually true, go with what grouse said.
posted by spitefulcrow at 11:08 AM on March 5, 2011
On what should have been preview: “an infinite quantity minus pi” is still “an infinite quantity”, not “infinity”.
posted by spitefulcrow at 11:09 AM on March 5, 2011
∀ x, x + π > x + 0π
for all x, x plus pi > x plus no pi
rough translation: everything is improved by pie
posted by robtoo at 11:26 AM on March 5, 2011 [6 favorites]
My answers obviously aren't true for the real numbers. They presupposes a partially ordered set where π is the greatest element. Including an element that is the set of all of the other elements, as
is done in my second equation, seems like a contradiction, though.
posted by grouse at 11:34 AM on March 5, 2011
Other versions:
∀ x. π > x ("pie is greater than everything" - suffers from the awkward "is pie greater than itself?")
∀ x. x ≠ π → π > x (improved version of grouse's first formula above, makes explicit we mean it for eveythring that is not pie - in absence of quantifiers, one can't really claim so in a technical
sense, though it IS standard mathematical practice to assume variables are universally quantified so this is mostly nitpicking)
∀ x,y. Pie(x) ∧ ¬Pie(y) → x>y (for any x that is pie and any y that is not, x is greater than y)
posted by Iosephus at 11:42 AM on March 5, 2011
My stab at "pi is greater than everything that is not pi": ∀ x ∈ {x | x ≠ π}, π > x
Maybe a little plodding and redundant, though.
posted by en forme de poire at 11:44 AM on March 5, 2011
Another option: ∀x ∈ {x : x ≠ π}, x < π.
This is equivalent to the first expression grouse supplied. Of course, without an explicit link between π and pie, this is nonsense. Might want to replace π with the image of a pie, you know, to make
sure you're talking about some set of things that includes pie, rather than just the set of real numbers.
posted by Nomyte at 11:44 AM on March 5, 2011
Thanks guys, I think that gives me enough to go on, but feel free to keep 'em coming!
posted by TheCoug at 11:46 AM on March 5, 2011
Risking being annoying by now (professional deformation, heh), yes there is a risk that the logic one works in falls into paradoxes when everything works at the same level (here, elements and sets of
elements). Solving this typically involves some kind of type theory so the levels get separated.
And yes, while ditching the pi symbol would diminish the joke, more logic representations of your notions can be found in the style of my last formula. In fact, this is the most straightforward way
one goes around doing KR ("knowledge representation") in computer science. In such work, one even gets insufferable and does things like:
∀ x,y. Pie(x) ∧ Food(y) ∧ ¬Pie(y) → x>y
if one wants to say that pie is the greatest food of all, but not that pie is, say, greater than kittens, movies, or Mt Vesuvius.
posted by Iosephus at 11:55 AM on March 5, 2011
I think grouse said it best when correctly presupposed that I was living in an unordered set where Pie is the greatest element. I mean, it's just the greatest!
posted by TheCoug at 12:01 PM on March 5, 2011
Partially ordered set. Unordered means there is no greatest element.
robtoo's formulation has the advantage that it actually makes sense with the standard conception of π and real numbers, and requires less advanced math.
posted by grouse at 12:12 PM on March 5, 2011
There's a complication... if (F, ≤) is the set of foods (as represented by real numbers) partially-ordered by greatness, and π = max(F), then we have the problem that π < sup(F), where sup is the
supremum (least upper bound). This could be interpreted to mean that pie is not as good as a soup made out of all possible foods. While we can't actually check this, it doesn't seem like a very
desirable property of our theory. There's also the problem that sup(F) is not in F - i.e. said soup is not a food. In culinary terms this may be reasonable, but mathematically it is counter-intuitive
at best.
To fix this, I suggest that we allow combinations of foods to be considered separately - so that π < 2π < 3π < ... as we'd expect. Two pies are clearly better than one so this may in fact be a better
model. The downside is that soup no longer exists, and a single pie is not the greatest any more. But thanks to Archimedes, we at least have that for any collection of food you care to take, you can
do better simply by using a large enough number of pies.
Hope that helps.
posted by d11 at 12:18 PM on March 5, 2011 [3 favorites]
d11, I think that is not correct. If my memory serves, any subset of the reals that has a greatest element has that same element as the supremum of the subset.
posted by Iosephus at 12:51 PM on March 5, 2011 [1 favorite]
(∀ q : (q ∉ J{ π })) ∧ (∀ p : (p ∈ J{ π })) • p > q
for all q such that q is not a member of the set of all sets {pi}
for all p such that p is a member of the set of all sets {pi}
it holds that p is greater than q
or to put it another way
for all things q that are not in the set of things that are pie,
for all things p that are in the set of things that are pie,
p is greater than q
posted by tel3path at 1:45 PM on March 5, 2011
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Maths Question
You could try this one
solve for x
We're required to take 30 hours of math courses. We end in a minor of math. I'm in my third year of university, though haven't really taken much of the math yet. We're allowed to pretty much make our
own schedule and are responsible for knowing material prior to a class. I've taken one (out of three required) 5 hour calculus course, one (out of two required) discrete math course, and college
algebra and trig my first year, which I don't really count as those are pretty simple. But yea, I haven't gotten much of my math in yet and this particular topic isn't one that has like ever come up.
At least not quite this way. So kind of both.
This seems different than how other schools (such as Helios') handle this, and start you off with a lot of math. And my university had a more "practical" approach to CS than some schools which are
more research and theory oriented. But we are still a certified computer science program by whatever organization handles those standards (I wanna say IEEE but I think that's wrong).
+1 quarkonium's first post.
I actually noticed the same thing L B.
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Committee Seating permutation problem
June 16th 2008, 04:42 PM #1
Apr 2008
Committee Seating permutation problem
A committee has two presidents, two vice presidents, and two secretaries. In how many distinct ways can they sit around a circular table? Each office-holder must face across the table a person
who holds a different office. This means one president cannot sit across from another president, etc. Assume that members of the same office are indistinguishable.
I began by finding the total number of arrangements, which I *think* is 120/(2!2!2!) which is a circle permutation for 6 objects divided by 2! three times because the three offices have 2 members
each that can be switched around to produce the same thing. But now that I think about it I could also switch the presidents AND vp's or even all of the members.
Then, finding the number of ways we can make it so at least one office sits across from each other, there's 2 ways we can have all 3 pairs matched up, and if we have 2 pairs matched up there has
to be 3 matched up. If we want only one pair matched up there are 2 ways to do it for the presidents matched up, 2 ways for the vp's and 2 ways for the secretairs. So I took 15-2-2-2-2 and got 7.
However when I drew it out I have at least 8! Any thoughts?
Remember, arrangements around a circle are (n-1)!
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Algebraic Topology and Category Theory
Proseminar on
Algebraic Topology and Category Theory
Spring 2009
Time and Place: Tuesdays and Thursdays 1:30-3:00 in Eckhart Room 203. Talks will be in the fields of
topology, category theory, and their intersection.
Audience: Graduate students in their second year or higher, with an interest in algebraic topology and/or category theory.
Postdocs and other faculty interested in these topics will also attend. Talks will be contributed by people at all levels.
Email List: http://zaphod.uchicago.edu:8080/mailman/listinfo/topology
Proseminar Website: This one. www.math.uchicago.edu/~fiore/1/proseminar.html
Seminar Website: UC Topology Seminar
Visitors: Andrew Blumberg May 17-June 10, Mike Mandell May 10-June 10
Course Plan:
Tues 7 Justin Noel on Nilpotency in Stable Homotopy
Thur 9 Justin Noel on Nilpotency in Stable Homotopy
Tues 14 Claire Tomesch on Complete Segal Spaces
Thur 16 Tom Fiore on the Leinster-Weber Nerve Theorem and Joachim Kock's work on Trees
17, 18, 19 Graduate Student Topology Conference
Tues 21 John Lind on Goodwille Calculus
Thur 23 No Meeting
Tues 28 Mike Shulman on Topos Theory
Thur 30 Mike Shulman on Topos Theory
2, 3 Midwest Topology Seminar
Tues 5 Pretalk by José Gómez
Thur 7 Anna Marie Bohmann on Generalizing Freyd's Generating Hypothesis
Tues 12 Mike Mandell?
Thur 14 Mike Shulman Thesis Defense
Tues 19 Topological Field Theory at Northwestern
Thur 21 Topological Field Theory at Northwestern
Tues 26 Topological Field Theory at Northwestern
Thur 28 Topological Field Theory at Northwestern
Tues 2 ?
Thur 4 ?
Tues 9 Probably not meeting (finals)
Thur 11 Extra Topology Seminar: Thomas Noll
Winter 2009
Course Plan:
List of Topics
Tues 13 Peter May on Two Sided Bar Construction and the Classification of Bundles and Fibrations
Thur 15 Tom Fiore on the Homotopy Theory of n-Fold Categories
Tues 20 Tom Fiore on Homotopy Theory, Higher Categories, and Applications
Thur 22 Mike Shulman on Enriched Model Structures
Tues 27 Peter May on Classification of Bundles
Thur 29 Rina Anno on Segal Categories vs. Quasicategories
Tues 3 John Lind on Infinite Loop Space Theory of Diagram Spectra
Thur 5 Emily Riehl on Homotopy Colimits
Tues 10 John Lind
Thur 12 Emily Riehl on Enriched Homotopy Colimits
Tues 17 Vigleik Angeltveit
Thur 19 Mike Shulman on Homotopy Colimits
Tues 24 Vigleik Angeltveit
Thur 26 Tom Fiore on Cisinski Model Structures
Tues 3 Anna Marie Bohmann on an Introduction to the Adams Spectral Sequence
Thur 5 Tom Fiore on Cisinski Model Structures
Tues 10 Anna Marie Bohmann on an Introduction to the Adams Spectral Sequence
Thur 12 Claire Tomesch
Tues 17
Autumn 2008
Course Plan:
List of Topics
Thur 2 First Talk: Emily Riehl on Factorization Systems
Thur 2 Second Talk: Justin Noel on Generalized Witt Schemes
Tues 7 Peter May on Characteristic Classes [see references below on characteristic classes]
Thur 9 Mike Shulman on Enriched Categories [see references below by Kelly and Lawvere]
Tues 14 Anna Marie Bohmann on Spectral Sequences
Thur 16 Mike Shulman on Enriched Categories and Weighted Limits
Tues 21 Anna Marie Bohmann on the Cohomology of Stiefel Manifolds via Spectral Sequences
Thur 23 Mike Shulman on Generators for Categories Notes
Tues 28 John Lind on Cohomology Operations modulo 2
Thur 30 Niles Johnson on Bicategorical Morita Theory, Azumaya, Picard, and Brauer
November (subject to change):
Tues 4 John Lind on Cohomology Operations modulo 2
Thur 6 Mike Shulman on Enriched Categories
Tues 11 No meeting
Thur 13 No meeting
Tues 18 Justin Noel on K-theory
Thur 20 Mike Shulman on Cat-enriched Categories
Tues 25 Vigleik Angeltveit on Hopf Invariant One
Thur 27 Thanksgiving Break
Tues 2 Peter May on Classifying Spaces, Cohomology, Characteristic Classes
Thur 4 Rina Anno on Homotopy Categories of Model Categories
Tues 9 Emily Riehl on Quasicategories, and Claire Tomesch on Opetopic Approaches to Higher Categories, 1:00 in E203 and 3:00 in Barn (Ryerson Bridge)
Thur 11 No Meeting, Have a good break!
Cisinski, Denis-Charles. Les préfaisceaux comme modèles des types d'homotopie.
Astérisque Volume 308, Soc. Math. France (2006), xxiv+392 pages.
Fiore, Thomas M. Nerves of Algebras à la Leinster and Weber. Talk in 2009.
Fiore, Thomas M. Grothendieckian Homotopy Theory of Presheaves à la Cisinski. Talk in 2009.
Grothendieck, Alexander. Pursuing Stacks. 1983.
Jardine, J. F. Categorical homotopy theory. Homology, Homotopy Appl. 8 (2006), no. 1, 71--144.
Kono, Akira and Tamaki, Dai. Generalized cohomology. Translated from the 2002 Japanese edition by Tamaki.
Translations of Mathematical Monographs, 230. Iwanami Series in Modern Mathematics.
American Mathematical Society, Providence, RI, 2006.
Kelly, Gregory Maxwell. Basic concepts of enriched category theory. London Mathematical Society Lecture Note Series, 64.
Cambridge University Press, Cambridge-New York, 1982.
Lawvere, F. William. Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43 (1973), 135--166 (1974).
Mac Lane, Saunders. Homology. (Die Grundlehren der Mathematischen Wissenschaften, Bd. 114.)
Springer, Berlin, 1963; Academic Press, New York, 1963.
Maltsiniotis, Georges. La théorie de l'homotopie de Grothendieck. Astérisque No. 301 (2005), vi+140 pp.
May, J. P. Stable algebraic topology, 1945--1966. History of topology, 665--723, North-Holland, Amsterdam, 1999.
May, J. P. A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999.
[See Chapter 23 on Characteristic Classes]
May, J.P. Notes on Characteristic Classes.
May, J. P. Books Available Online.
Milnor, John W. and Stasheff, James D. Characteristic classes. Annals of Mathematics Studies, No. 76.
Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.
Riehl, Emily. Factorization Systems.
Riehl, Emily. Homotopy (Limits and) Colimits.
Shulman, Michael. Generators and Colimit Closures.
Previous Quarters:
Fall 2004, Winter 2005, Spring 2005, Fall 2005, Winter 2006, Spring 2006, Fall 2007.
Other Seminars:
Geometric Langlands Seminar
MIT Student Topology Seminar "Babytop"
MIT Student Topology Reading Seminar "Juvitop"
MIT Topology Seminar
Michigan Student Topology Seminar
This site is maintained by Tom Fiore, fiore AT math.uchicago.edu. The template was Jespers old site. | {"url":"http://www-personal.umd.umich.edu/~tmfiore/1/proseminar.html","timestamp":"2014-04-23T18:20:48Z","content_type":null,"content_length":"11060","record_id":"<urn:uuid:15e33b66-7092-4774-8363-ec99c7bbd407>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00570-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Supplementary Material for "Learning the Structure of Deep Sparse
Graphical Models"
Ryan Prescott Adams Hanna M. Wallach Zoubin Ghahramani
1 Proof of General CIBP Termination
In the main paper, we discussed that the cascading Indian buffet process (CIBP) for fixed and finite and
eventually reaches a restaurant in which the customers choose no dishes. Every deeper restaurant also has
no dishes. Here, we show a more general result, for IBP parameters that vary with depth: (m)
and (m)
Let there be an inhomogeneous Markov chain M with state space N. Let m index time and let the state
at time m be denoted K(m)
. The initial state K(0)
is finite. The probability mass function describing the
transition distribution for M at time m is given by the following equation:
= k | K(m)
, (m)
, (m)
) = | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/843/3127301.html","timestamp":"2014-04-19T13:04:06Z","content_type":null,"content_length":"8013","record_id":"<urn:uuid:aa0afce3-5219-48c2-baaa-6884a683bdac>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00111-ip-10-147-4-33.ec2.internal.warc.gz"} |
the first resource for mathematics
Global stability of single-species diffusion Volterra models with continuous time delays.
(English) Zbl 0627.92021
Consider an ecological system composed of multiple heterogeneous patches connected by discrete diffusion, and each patch is assumed to be occupied by a single species whose evolution equation for the
${i}^{th}$ patch is:
$\left(1\right)\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{i}={x}_{i}\left({e}_{i}-{a}_{i}{x}_{i}+{\gamma }_{i}{\int }_{-\infty }^{t}{F}_{i}\left(t-\tau \right){x}_{i}\left(\tau \right)d\tau \right)
+\sum _{\mu =1}^{n}{D}_{i\mu }\left({x}_{\mu }-{x}_{i}\right),\phantom{\rule{1.em}{0ex}}i\in N,$
where $N=\left\{1,···,n\right\}$, n is the number of patches and ${x}_{i}$ is the population density in the ${i}^{th}$ patch. (1) may be thought as a generalization of the Volterra
integral-differential equation to the n-patch case in which ${a}_{i},{e}_{i}\in {ℝ}^{+}$; ${\gamma }_{i}\in ℝ$ for all $i\in N$, where ${e}_{i}$, $i\in N$, are the intrinsic growth rates, and ${a}_
{i}$, $i\in N$, represent the intraspecific relationships.
By introducing the supplementary functions ${x}_{i}^{\left(j\right)}$, $j=1,···,{k}_{i}$, $i\in N$, (1) is transformed into the expanded system of O.D.E.:
$\left(2\right)\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{i}={x}_{i}\left({e}_{i}-{a}_{i}{x}_{i}+{\gamma }_{i}\sum _{j=1}^{{k}_{i}}{C}_{i}^{\left(j\right)}{x}_{i}^{\left(j\right)}\right)+\sum _{\mu
=1}^{n}{D}_{i\mu }\left({x}_{\mu }-{x}_{i}\right),$
${\stackrel{˙}{x}}_{i}^{\left(j\right)}={\alpha }_{i}{x}_{i}^{\left(j-1\right)}-{\alpha }_{i}{x}_{i}^{\left(j\right)},\phantom{\rule{1.em}{0ex}}j=1,···,{k}_{i},\phantom{\rule{1.em}{0ex}}{x}_{i}^{\
left(0\right)}={x}_{i}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}i\in N·$
The dynamical behavior of (2) implies the same kind of dynamical behavior of (1). By applying homotopy function techniques [see e.g. C. B. Garcia and W. I. Zangwill, Pathways to solutions, fixed
points, and equilibria (1981; Zbl 0512.90070)] the authors give sufficient conditions for the existence of a positive equilibrium and for its global and local stability. The biological meanings of
the results are considered and compared with some known results.
92D25 Population dynamics (general)
34D20 Stability of ODE
45J05 Integro-ordinary differential equations
92D40 Ecology | {"url":"http://zbmath.org/?q=an:0627.92021","timestamp":"2014-04-19T12:36:26Z","content_type":null,"content_length":"27591","record_id":"<urn:uuid:7493df57-e1cd-4a24-a3a0-d2c01da4a8ab>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00405-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calhoun, GA Trigonometry Tutor
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If an object is accelerating at 2 m/s*2. If the net force is tripled and the mass is doubled, then what is the new acceleration?
In physics, classical mechanics and quantum mechanics are the two major sub-fields of mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under
the action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology.
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Besides
this, many specializations within the subject deal with gases, liquids, solids, and other specific sub-topics. Classical mechanics provides extremely accurate results as long as the domain of study
is restricted to large objects and the speeds involved do not approach the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to introduce the other
major sub-field of mechanics, quantum mechanics, which reconciles the macroscopic laws of physics with the atomic nature of matter and handles the wave–particle duality of atoms and molecules.
However, when both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom, quantum field theory (QFT) becomes applicable. QFT deals with
small distances and large speeds with many degrees of freedom as well as the possibility of any change in the number of particles throughout the interaction. To deal with large degrees of freedom at
the macroscopic level, statistical mechanics becomes valid. Statistical mechanics explores the large number of particles and their interactions as a whole in everyday life. Statistical mechanics is
mainly used in thermodynamics. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. General relativity unifies special
relativity with Newton's law of universal gravitation, allowing physicists to handle gravitation at a deeper level.
A physical quantity (or "physical magnitude") is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.
The following outline is provided as an overview of and topical guide to physics:
Physics – natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of
nature, conducted in order to understand how the universe behaves.
In physics, net force is the overall force acting on an object. In order to calculate the net force, the body is isolated and interactions with the environment or constraints are introduced as
forces and torques forming a free-body diagram.
The net force does not have the same effect on the movement of the object as the original system forces, unless the point of application of the net force and an associated torque are determined
so that they form the resultant force and torque. It is always possible to determine the torque associated with a point of application of a net force so that it maintains the movement of the
object under the original system of forces.
Specific force is defined as the non-gravitational force per unit mass.
Specific force (also called g-force and mass-specific force) is measured in meters/second² (m·s−2) which is the units for acceleration. Thus, specific force is not actually a force, but a type of
acceleration. However, the (mass-)specific force is not a coordinate-acceleration, but rather a proper acceleration, which is the acceleration relative to free-fall. Forces, specific forces, and
proper accelerations are the same in all reference frames, but coordinate accelerations are frame-dependent. For free bodies, the specific force is the cause of, and a measure of, the body's
proper acceleration.
A fictitious force, also called a phantom force, pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses whose motion is described using a non-inertial
frame of reference, such as a rotating reference frame.
The force F does not arise from any physical interaction between two objects, but rather from the acceleration a of the non-inertial reference frame itself. As stated by Iro:
Science of drugs including their origin, composition, pharmacokinetics,
pharmacodynamics, therapeutic use, and toxicology.
Pharmacology (from Greek φάρμακον, pharmakon, "poison" in classic Greek; "drug" in modern Greek; and -λογία, -logia "study of", "knowledge of") is the branch of medicine and biology concerned
with the study of drug action, where a drug can be broadly defined as any man-made, natural, or endogenous (within the body) molecule which exerts a biochemical and/or physiological effect on the
cell, tissue, organ, or organism. More specifically, it is the study of the interactions that occur between a living organism and chemicals that affect normal or abnormal biochemical function. If
substances have medicinal properties, they are considered pharmaceuticals.
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Published Books
K.T. Tang's Published Books
S.H. Patil and K.T. Tang
Asymptotic Methods in Quantum Mechanics, Applications to Atoms, Molecules and Nuclei
(Springer, 2000)
This book describes some general properties of wave functions, with an emphasis on their asymptotic behavior. The asymptotic region is particularly important since it is the wave function in the
outer region of an atom, a molecule or a nucleus, which is sensitive to external interaction. An analysis of these properties helps in construction simple and compact wave functions and in
developing a broad understanding of different aspects of the quantum mechanics of many partic e systems. As applications, wave functions with correct asymptotic forms are used to generate a large
data base for susceptibilities, polarizabilities, interatomic potentials, and unclear densities.
K.T. Tang
• Mathematical Methods for Engineers and Scientists 1
Complex Analysis, Determinants and Matrices
(Springer, 2006)
• Mathematical Methods for Engineers and Scientists 2
Vector Analysis, Ordinary Differential Equations and Laplace Transforms
(Springer, 2007)
• Mathematical Methods for Engineers and Scientists 3
Fourier Analysis, Partial Differential Equations and Variational Methods
(Springer, 2007)
Pedagogical insights gained through 30 years of teaching applied mathematics led the author to write this set of student oriented books. Topics such as complex analysis, matrix theory, vector and
tensor analysis, Fourier analysis, integral transforms, ordinary and partial differential equations, and variational methods are presented in a discursive style that is readable and easy to
follow. Numerous clearly stated, completely worked out examples together with carefully selected problems with answers are used to make students comfortable and confident in using advanced
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2pm pacific time in philipp
You asked:
2pm pacific time in philipp
Philipp, Mississippi
4:00:00pm Central Daylight Time
4:00:00pm Central Daylight Time (the American timezone UTC -5)
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Explanation of function
Explanation of function
1 bool inline KEYDOWN(int vkCode)
2 {
3 return (GetAsyncKeyState(vkCode) & 0x8000) ? true : false;
4 }
I know what it does, but I have no idea how it works, can someone explain it to me? I mean this part
(GetAsyncKeyState(vkCode) & 0x8000) ? true : false;
Last edited on
GetAsyncKeyState determines if a key is up or down.
(condition) ? (return type)
sort of like:
1 if (something)
2 return true
3 else
4 return false
Last edited on
i gathered that much but what is it doing with that memory address there?
There is no memory address there.
0x8000 is a value (in hexadecimal notation) that serves as a bitmask. In binary it would look like: 10000000000000000000000000000000 (which you can see has only the most signficant bit set.) Using
the bitwise and operator on this value and the value returned by GetAsyncKeyState results in another value that is not equal to 0 if the value returned by GetAsyncKeyState had it's most significant
bit set and is equal to 0 if the value returned by GetAsyncKeyState did not have it's most significant bit set.
thanks cire! I barely understand that but at least something :D
Last edited on
I think cire explained it as best it could be explained.
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[Haskell-cafe] progress reporting in a backtracking algorithm
Harald Bögeholz bo at ct.de
Fri Sep 21 08:29:07 CEST 2012
Dear Haskell Cafe,
I am playing around with a backtracking algorithm that can take quite a
while to compute all solutions to a problem.
I'd like to use Debug.Trace.trace to provisionally output something that
lets me estimate the total time it might take. But I just can't wrap my
head around it.
This is how I think I'd write it in a C-like language:
backtrack(int level, double position, double stepsize, misc...)
// with variations = number of variations to try on this level
double part = stepsize / variations // split time on this level
for (i=0; i<variations; ++i)
double current = position + part*i
// do the actual work
backtrack(level+1, current, part);
if (level < not_too_much_detail)
printf("progress: %f%%\n", current);
and start with backtrack(1, 0.0, 100.0).
And now for something completely Haskell:
I have a function
step :: State -> Index -> [State]
that on a certain level tries all allowable varaiants and returns a list
of those that can be further pursued on deeper levels.
Then solving the problem involves applying the step on all levels
(whicht are indexed by some array indices here):
solve :: Problem -> [State]
solve problem = foldM step start grid
where start = stateFromProblem problem
grid = indices (sLines start)
I am totally at loss at how I could accomplish some kind of progress
reporting in this lazily evaluated (I hope) backtracking scheme.
If anybody would like to review the full code (about 80 lines total vor
the solver, not counting I/O), this is where I am right now:
and this is the branch I am working on right now:
Or is there maybe a totally different and better way to approach this
kind of tree search in Haskell? I'm eager to learn.
Thanks for your attention
Harald Bögeholz <bo at ct.de> (PGP key available from servers)
Redaktion c't Tel.: +49 511 5352-300 Fax: +49 511 5352-417
int f[9814],b,c=9814,g,i;long a=1e4,d,e,h;
Affe Apfel Vergaser
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Permutations with no fixpoints
Think of a permutation on n elements as a function {1,...,n} → {1,...,n}. It makes sense to count how many fixpoints this function has. There are n! permutations. How many permutations have no
fixpoints? All the rotations, for a start, but obviously many more.
We may calculate a formula for this number using Poincaré's inclusion / exclusion principle. Note that the proportion of permutations fixing some specified point is 1/n, and, in general, 1/(n⋅(n-1)
⋅...⋅(n-k+1)) of the permutations fix some k specified points.
By the inclusion / exclusion principle, the number of permutations with no fixpoints is
n! - a[1]*(n-1)! + a[2]*(n-2)! + ... + (-1)^ka[k]*(n-k)! + ...,
where a
=n!/(k!(n-k)!) is the number of ways to select k points. Cancelling, we see that the number is precisely
n! * (1/0! - 1/1! + 1/2! - 1/3! + 1/4! - ... + (-1)^n/n!)
Note that the numbers in the brackets converges very rapidly to 1/e (where e=2.718281828459045... is Napier's constant, the base for natural logarithms). So it is fair to say that approximately 1/e
of all permutations have no fixpoint. In fact, the exact number is the closest integer to n!/e. | {"url":"http://everything2.com/title/Permutations+with+no+fixpoints","timestamp":"2014-04-21T15:58:08Z","content_type":null,"content_length":"20637","record_id":"<urn:uuid:a4f79f50-81a6-4fe7-accc-e666d353e70a>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00274-ip-10-147-4-33.ec2.internal.warc.gz"} |
Area of Regular Polygons
10.12: Area of Regular Polygons
Created by: CK-12
What if you were asked to find the distance across The Pentagon in Arlington, VA? The Pentagon, which also houses the Department of Defense, is composed of two regular pentagons with the same center.
The entire area of the building is 29 acres (40,000 square feet in an acre), with an additional 5 acre courtyard in the center. The length of each outer wall is 921 feet. What is the total distance
across the pentagon? Round your answer to the nearest hundredth. After completing this Concept, you'll be able to answer questions like this.
Watch This
CK-12 Foundation: Chapter10AreaofRegularPolygonsA
Brightstorm: Area of Regular Polygons
A regular polygon is a polygon with congruent sides and angles. Recall that the perimeter of a square is 4 times the length of a side because each side is congruent. We can extend this concept to any
regular polygon.
Perimeter of a Regular Polygon: If the length of a side is $s$$n$$P=ns$
In order to find the area of a regular polygon, we need to define some new terminology. First, all regular polygons can be inscribed in a circle. So, regular polygons have a center and radius, which
are the center and radius of the circumscribed circle. Also like a circle, a regular polygon will have a central angle formed. In a regular polygon, however, the central angle is the angle formed by
two radii drawn to consecutive vertices of the polygon. In the picture below, the central angle is $\angle BAD$$\triangle BAD$$n$$n$apothem.
The area of each triangle is $A_\triangle = \frac{1}{2} bh= \frac{1}{2} sa$$s$$a$$n$$n$
Area of a Regular Polygon: If there are $n$$s$$a$$A=\frac{1}{2} asn$$A=\frac{1}{2} aP$$P$
Example A
What is the perimeter of a regular octagon with 4 inch sides?
If each side is 4 inches and there are 8 sides, that means the perimeter is 8(4 in) = 32 inches.
Example B
The perimeter of a regular heptagon is 35 cm. What is the length of each side?
If $P=ns$$35 \ cm=7s$$s=5 \ cm$
Example C
Find the length of the apothem in the regular octagon. Round your answer to the nearest hundredth.
To find the length of the apothem, $AB$$m \angle CAD$$360^\circ$$m \angle CAD= \frac{360^\circ}{8}=45^\circ$$\triangle CAD$$m \angle ACB$$m \angle ADC$$\triangle CAD$
$m \angle CAD+m \angle ACB+m \angle ADC &= 180^\circ\\45^\circ+2m \angle ACB &= 180^\circ\\2m \angle ACB &= 135^\circ\\m \angle ACB &= 67.5^\circ$
To find $AB$
$\tan 67.5^\circ &= \frac{AB}{6}\\AB &= 6 \cdot \tan 67.5^\circ \approx 14.49$
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter10AreaofRegularPolygonsB
Concept Problem Revisited
From the picture below, we can see that the total distance across the Pentagon is the length of the apothem plus the length of the radius. If the total area of the Pentagon is 34 acres, that is
2,720,000 square feet. Therefore, the area equation is $2720000=\frac{1}{2} a(921)(5)$$72^\circ$
$\sin 36^\circ=\frac{460.5}{r} \rightarrow r=\frac{460.5}{\sin 36^\circ} \approx 783.45 \ ft.$
Therefore, the total distance across is $590.66 + 783.45 = 1374.11 \ ft$
Perimeter is the distance around a shape. The perimeter of any figure must have a unit of measurement attached to it. If no specific units are given (feet, inches, centimeters, etc), write “units.”
Area is the amount of space inside a figure. Area is measured in square units. The center and radius of a regular polygon is the center and radius of the circumscribed circle. An apothem is a line
segment drawn from the center of a regular polygon to the midpoint of one of its sides.
Guided Practice
1. Find the area of the regular octagon in Example C.
2. Find the area of the regular polygon with radius 4.
3. The area of a regular hexagon is $54 \sqrt{3}$
1. The octagon can be split into 8 congruent triangles. So, if we find the area of one triangle and multiply it by 8, we will have the area of the entire octagon.
$A_{octagon}=8 \left( \frac{1}{2} \cdot 12 \cdot 14.49 \right)=695.52 \ units^2$
2. In this problem we need to find the apothem and the length of the side before we can find the area of the entire polygon. Each central angle for a regular pentagon is $\frac{360^\circ}{5}=72^\
$\sin 36^\circ &= \frac{.5n}{4} && \ \cos 36^\circ=\frac{a}{4}\\4 \sin 36^\circ &= \frac{1}{2} n && 4 \cos 36^\circ=a\\8 \sin 36^\circ &= n && \qquad \quad a \approx 3.24\ &\approx 4.7$
Using these two pieces of information, we can now find the area. $A=\frac{1}{2}(3.24)(5)(4.7) \approx 38.07 \ units^2$
3. Plug in what you know into both the area and the perimeter formulas to solve for the length of a side and the apothem.
$P &= sn && \quad \ \ A= \frac{1}{2} aP\\36 &= 6s && 54 \sqrt{3}=\frac{1}{2} a(36)\\s &= 6 && 54 \sqrt{3}=18a\\& && \ \ 3 \sqrt{3}=a$
Use the regular hexagon below to answer the following questions. Each side is 10 cm long.
1. Each dashed line segment is $a(n)$
2. The red line segment is $a(n)$
3. There are _____ congruent triangles in a regular hexagon.
4. In a regular hexagon, all the triangles are _________________.
5. Find the radius of this hexagon.
6. Find the apothem.
7. Find the perimeter.
8. Find the area.
Find the area and perimeter of each of the following regular polygons. Round your answer to the nearest hundredth.
13. If the perimeter of a regular decagon is 65, what is the length of each side?
14. A regular polygon has a perimeter of 132 and the sides are 11 units long. How many sides does the polygon have?
15. The area of a regular pentagon is $440.44 \ in^2$
16. The area of a regular octagon is $695.3 \ cm^2$
A regular 20-gon and a regular 40-gon are inscribed in a circle with a radius of 15 units.
17. Challenge Derive a formula for the area of a regular hexagon with sides of length $s$$s$
18. Challenge in the following steps you will derive an alternate formula for finding the area of a regular polygon with $n$ We are going to start by thinking of a polygon with $n$$n$$\frac{1}{2} bh$
$\frac{1}{2} sa$$s$$a$$x$
1. The apothem, $a$$\frac{x^\circ}{2}$$\sin \left( \frac{x^\circ}{2} \right)$$\cos \left( \frac{x^\circ}{2} \right)$
2. Solve your $\sin$$s$$r$$x$
3. Solve your $\cos$$a$$r$$x$
4. Substitute these expressions into the equation for the area of one of the triangles, $\frac{1}{2} sa$
5. Since there will be $n$$n$
6. How would you tell someone to find the value of $x$
Use the formula you derived in problem 18 to find the area of the regular polygons described in problems 19-22. Round your answers to the nearest hundredth.
19. Decagon with radius 12 cm.
20. 20-gon with radius 5 in.
21. 15-gon with radius length 8 cm.
22. 45-gon with radius length 7 in.
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Help with math problem
September 21st 2009, 02:46 PM #1
Sep 2009
Help with math problem
Let F(n) be the number of occurrences of the digit zero in all the positive integers less than or equal to n. For example, F(9)=0, F(10)=1, and F(103)=14. With is F(800)?
Is there an easier way than to just count all of them to solve this problem? Please help!
Well basically, you have 10,20,30,40,50,60,70,80,90,100=11 zeros and that is going to happen 8 times (210,220,230,...,780,790,800)
Then you have 101,102,103,...,109 for a total of 9 zeros which happens 7 times
So 8*11+7*9=151
September 21st 2009, 03:12 PM #2
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