content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Rahns, PA SAT Math Tutor
Find a Rahns, PA SAT Math Tutor
...I have worked on applications, essays and resumes for seniors in my high school for seven years. I've always taught senior English and am an MG mentor. Before returning to teaching in 2006, I
worked as a college coordinator for a non-profit agency.
17 Subjects: including SAT math, reading, writing, English
...In addition, I offer FREE ALL NIGHT email/phone support just before the “big" exam, for students who pull "all nighters". One quick note about my cancellation policy, as it's different than
most tutors: Cancel one or all sessions at any time, and there is NO CHARGE. Thank you for considering my services, and the best of luck in all your endeavors!
14 Subjects: including SAT math, calculus, physics, geometry
...For those four years of high school I played club ball as well. I love it and play whenever possible. Now, I play in local leagues and in work intramurals.
10 Subjects: including SAT math, geometry, ASVAB, algebra 1
...I've taught a few friends over the years, but never professionally. I took lessons for four years and have a pretty good grasp on music theory, but I believe you should learn to play the
guitar first before you get to all the theory stuff. Playing is more fun anyway.
14 Subjects: including SAT math, calculus, physics, algebra 1
...I am always aware that every student is a unique individual, and I try to unlock the key to each one's particular way of thinking and learning. Most students who struggle with geometry think
it's over their head. I help build confidence and try to make it understandable.
28 Subjects: including SAT math, reading, English, writing
Related Rahns, PA Tutors
Rahns, PA Accounting Tutors
Rahns, PA ACT Tutors
Rahns, PA Algebra Tutors
Rahns, PA Algebra 2 Tutors
Rahns, PA Calculus Tutors
Rahns, PA Geometry Tutors
Rahns, PA Math Tutors
Rahns, PA Prealgebra Tutors
Rahns, PA Precalculus Tutors
Rahns, PA SAT Tutors
Rahns, PA SAT Math Tutors
Rahns, PA Science Tutors
Rahns, PA Statistics Tutors
Rahns, PA Trigonometry Tutors
Nearby Cities With SAT math Tutor
Charlestown, PA SAT math Tutors
Congo, PA SAT math Tutors
Creamery SAT math Tutors
Delphi, PA SAT math Tutors
Eagleville, PA SAT math Tutors
Englesville, PA SAT math Tutors
Fagleysville, PA SAT math Tutors
Gabelsville, PA SAT math Tutors
Graterford, PA SAT math Tutors
Gulph Mills, PA SAT math Tutors
Linfield, PA SAT math Tutors
Morysville, PA SAT math Tutors
Trappe, PA SAT math Tutors
Valley Forge SAT math Tutors
Zieglersville, PA SAT math Tutors | {"url":"http://www.purplemath.com/rahns_pa_sat_math_tutors.php","timestamp":"2014-04-16T10:17:29Z","content_type":null,"content_length":"23846","record_id":"<urn:uuid:3fa0c895-5b3f-4d5f-b187-d9a31aa900be>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00432-ip-10-147-4-33.ec2.internal.warc.gz"} |
24 Entries
• C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals, Chapter 1 of 13
Welcome to a new technical series on Channel 9 folded into a different kind of 9 format: C9 Lectures. These are what you think they are, lectures. They are not conversational in nature (like most
of what you're used to on 9), but rather these pieces are entirely focused on education,…
• Simon Peyton-Jones and John Hughes - It's Raining Haskell
Ever wonder what would happen if you happened upon Simon Peyton-Jones, author of the Glasgow Haskell Compiler (GHC) and a key contributor to the Haskell functional programming language, and John
Hughes, fellow Haskellite, computer scientist, creator of QuickCheck, and author of…
• C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals, Chapter 2 of 13
In Chapter 2, Dr. Meijer introduces Haskell syntax and notation (via a Haskell implementation called Hugs, to be precise, which is based on Haskell 98) and we learn about the Haskell syntax that
represents the fundamental construct of functional programming:functions. It's not like you're used…
• C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals Chapter 13 of 13
Well, my friends, the day has arrived. For thirteen weeks, you have been provided all the conceptual tools to take the leap into the deep end of the functional programming pool and float safely.
The great Dr. Erik Meijer has generously given his value time to teach us the fundamentals as delivered…
• C9 Lectures: Dr. Ralf Lämmel - Going Bananas
Dr. Ralf Lämmel returns for an exploration of folds, aka bananas. This is lecture 5 in his C9 Lecture series covering advanced functional programming topics. Welcome back, Ralf! We're so happy to
have you here! Why bananas, Ralf? Banana is functional programming slang…
• C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals, Chapter 3 of 13
In Chapter 3, Dr. Meijer explores types and classes in Haskell. A type is a collection of related values and in Haskell every well-formed expression has a type. Using type inference, these types
are automatically calculated at run time. Ifexpression e returns a type t, then e is of type t, e :: t. A…
• C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals Chapter 8 of 13
In Chapter 8, Functional Parsers, it's all about parsing and parsers. A parser is a program that analyses a piece of text to determine its syntactic structure. In a functional language such as
Haskell, parsers can naturallybe viewed as functions. type Parser = String -> TreeA parser is…
• C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals Chapter 9 of 13
In Chapter 9, Interactive Programs, Dr. Meijer will teach us how to make programs in Haskell that are side-effecting:interactive. Haskell programs are pure mathematical functions with no side
effects. That said, you want to be able to write Haskell programs that can read input from the keyboard and…
• C9 Lectures: Dr. Erik Meijer - Functional Programming Fundamentals Chapter 5 of 13
In Chapter 5, Dr. Meijer introduces and digs into List Comprehensions. In mathematics, comprehension notation is used to construct new sets from old sets. In Haskell, you can create new lists
from old lists using a similarcomprehension syntax:[x^2 | x <- [1..5]]The above notation represents the…
• C9 Lectures: Graham Hutton - How To Be More Productive
It's been far too long since we've had some meaty functional programming content on C9. Luckily, none other than Graham Hutton dropped off a present on our doorstep! Dr. Hutton graciously
provided Channel 9 with his latest self-filmed lecture—thank you, Graham! We're honored. This is certainly… | {"url":"http://channel9.msdn.com/Tags/haskell?sort=viewed","timestamp":"2014-04-17T10:22:39Z","content_type":null,"content_length":"53099","record_id":"<urn:uuid:b49354aa-3b23-41f3-aa9f-74fb8813b4f7>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00495-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is a Vector?
Date: 01/04/2002 at 12:25:09
From: Patrick Dornian
Subject: Vectors
Dear Dr. Math,
I am having trouble understanding exactly what a vector is and cannot
seem to find a simple, straightforward explanation. Please help!
Thank you,
Patrick Dornian
Date: 01/04/2002 at 23:29:18
From: Doctor Peterson
Subject: Re: Vectors
Hi, Patrick.
Vectors can be formally defined in several complicated ways, but I can
give a basic introduction to the concept.Simply put, a vector is a
directed quantity.
We'll start with a one-dimensional vector. This is just the same as a
number. Draw a number line, and draw an arrow starting at zero and
ending at 5. This is the vector (5). Draw another arrow starting at
the 5 and ending at the 8; this is the vector (3). It doesn't matter
where a vector starts; all that matters is how long it is and how far
it goes. So this second vector is 3 units long and points to the
right, making it identical to a vector starting at 0 and going to 3.
By putting two vectors end to end, as I did, I just added the vectors
(5) and (3) to get the vector (8):
If you are familiar with negative numbers, you can see that a vector
pointing to the left would correspond to a negative number. If we add
the vectors (5) and (-5), we get the vector (0), a vector with no size
at all. For any vector you draw, if you move it so that it starts at
0, it will point to its name.
Since a one-dimensional vector is nothing but a (signed) number, it's
nothing new. But this introduces the essential concept: only size and
direction (left or right in this case) count, not position. Now we can
look at two-dimensional vectors, in a plane, where things start to get
Draw an arrow on a piece of paper, pointing in any direction, and you
have a vector: a length with a direction. Draw another arrow starting
at the tip of the first one, and you have added two vectors:
/ |\
u+v / \v
/ \
/ \
If you draw vectors on a coordinate grid, you can give them names:
V(-2,3) ^ W(3,3)
+ | +
\ | / \
v\ | u+v / \v
\ | / \
\| / \
u U(5,0)
Place each vector so that it starts at the origin (0,0), and name it
for the point where it ends, just as we did on the number line. Our
vectors are u = (5,0) and v = (-2,3), since they end at points U
and V as shown. Their sum w = u+v is (3,3). Do you see how to add two
vectors? You just add their x coordinates and their y coordinates;
u goes 5 to the right and v goes 2 to the left, so w goes 5-2 = 3 to
the right. (Actually, we use the word "coordinate" only for points;
for vectors, we use the word "component.")
You can do the same for vectors in three-dimensional space, but I
won't bother drawing that. The important thing is that vectors give us
a way to talk about anything that has both size and direction, but not
position - things like velocities, wind speeds, forces, and so on.
If I row my boat in the direction of vector u, but the water itself
is moving along vector v, then I will actually be moving along vector
u+v, so the sum of the two vector velocities tells me how fast, and in
what direction, I am really going.
For an introduction to vectors with nicer pictures, try this:
Vectors - Gene Klotz, The Math Forum
Here's another; look under "Basic Stuff":
Maths Help - Working with vectors - Jenny Olive
If you have any further questions, feel free to write back.
- Doctor Peterson, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/55466.html","timestamp":"2014-04-17T10:07:48Z","content_type":null,"content_length":"9081","record_id":"<urn:uuid:3f06b0f7-61a6-443c-a508-757b5c520d02>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00379-ip-10-147-4-33.ec2.internal.warc.gz"} |
CLAIRAULT (Alexis-Claude)
, a celebrated French mathematician and academician, was born at Paris the 13th of May 1713, and died the 17th of May 1765, at 52 years of age. His father, a teacher of mathematics at Paris, was his
sole instructor, teaching him even the letters of the alphabet on the figures of Euclid's Elements, by which he was able to read and write at 4 years of age. By a similar stratagem it was that
calculations were rendered samiliar to him. At 9 years of age he put into his hands Guisnée's Application of Algebra to Geometry; at 10 he studied l'Hopital's Conic Sections; and between 12 and 13 he
read a memoir to the Academy of Sciences concerning four new Geometrical curves of his own invention. About the same time he laid the first foundation of his work upon curves that have a double
curvature, which he finished in 1729, at 16 years of age. He was named Adjoint-Mechanician to the Academy in 1731, at the age of 18, Associate in 1733, and Pensioner in 1738; during his connection
with the Academy, he had a great multitude of learned and ingenious communications inserted in their Memoirs, beside several other works which he published separately; the list of which is as
1. On Curves of a Double Curvature; in 1730, 4to.
2. Elements of Geometry; 1741, 8vo.
3. Theory of the Figure of the Earth; 1743, 8vo.
4. Elements of Algebra; 1746, 8vo.
5. Tables of the Moon; 1754, 8vo.
His papers inserted in the Memoirs of the Academy are too numerous to be particularised here; but they may be found from the year 1727, for almost every year till 1762; being upon a variety of
subjects, astronomical, mathematical, optical, &c. | {"url":"http://words.fromoldbooks.org/Hutton-Mathematical-and-Philosophical-Dictionary/c/clairault-alexis-claude.html","timestamp":"2014-04-16T16:49:05Z","content_type":null,"content_length":"6923","record_id":"<urn:uuid:cd7d5a82-7e81-44b7-865c-e368e6a65b60>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00067-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Good Absorber is a Good Emitter
A Good Absorber is a Good Emitter
According to the Stefan-Boltzmann law, the energy radiated by a blackbody radiator per second per unit area is proportional to the fourth power of the absolute temperature and is given by
For hot objects other than ideal radiators, the law is expressed in the form:
where e is the emissivity of the object (e = 1 for ideal radiator). If the hot object is radiating energy to its cooler surroundings at temperature T[c], the net radiation loss rate takes the form
In this relationship the term with T[c] represents the energy absorbed from the environment. This expression explicitly assumes that the same coefficient e applies to both the emission into the
environment and the absorption from the environment. That is, a good emitter is a good absorber and vice versa; the same coefficient can be used to characterize both processes. Why is that true?
Perhaps the most fundamental conceptual way to approach this question is to observe that a hot object placed in a room must ultimately come to thermal equilibrium with the room. The hot object will
initially emit more energy into the room than it absorbs from the room, but that will cause the temperature of the room to rise and the temperature of the object to drop. But when they reach the same
temperature, we can conclude that the amount of energy absorbed on average is exactly the same as the energy emitted. That is, the expression above for net energy radiated to the environment must
give us zero when T=T[c].
The above argument is based upon the Second Law of Thermodynamics in the form that states that heat will not spontaneously flow from a cold object to a hot object. If the absorption coefficient were
higher than the emission coefficient for the object, then it could absorb net energy from the room even when its temperature were higher than the room.
But suppose you wanted to argue that a good absorber must be a good emitter based on the microscopic processes involving the atoms in the surface of an object. Then it becomes quantum question and
involves the following ideas:
1. All electromagnetic radiation can be considered to be quantized, existing as photons with energy given by the Planck hypothesis, E=hf.
2. In order for a solid (or any matter, but I am assuming we are talking about solids) to absorb a photon of given energy hf , it must have a pair of energy levels separated by that amount of energy
hf, so that the photon elevates the system from the lower member of the pair to the upper.
3. For visible light or near visible, then energy level pairs involved in most absorption are electron energy levels, so that when you absorb a green photon of photon energy 2.2 eV, you are causing
an electron very near the surface of the solid to jump upward 2.2 eV. It can't do it unless there is a level at 2.2 eV up to receive it.
4. If a surface is an ideal absorber in the visible, this implies that there is an abundance of available electron states so that a photon of any color in the visible spectrum can interact with
electrons in the solid to elevate them to an available upper level. The implication is that any color in the visible spectrum can be readily absorbed, hence it is an ideal absorber, a perfectly black
5. The next step is not so obvious. If a pair of electron energy levels is available for absorption of a photon, it is also available for emission of a photon, i.e. , radiation. If it is available
for an upward jump, it is available for a downward jump. One of EinsteinŐs contributions was to show that for a given radiation, the probability for emission is the same as the probability of
absorption. This fact is described in terms of the Einstein A and B coefficients and is very important in laser theory. The implication for the current question is that it constrains a good absorber
to also be a good emitter. If the solid has lots of available electron levels for absorption, they will also be equally available for emission. To bring the discussion full circle, Einstein first
derived his A and B coefficients from a thermodynamic argument like the argument above about thermal equilibrium before the actual development of the quantum mechanical ideas.
The bottom line of this reasoning is that a good absorber of radiation will also be a good emitter. But the above is just the tip of the iceberg. For a black object, all visible colors are absorbed
by electron jumps, but the elevated electrons usually follow a different path downward, cascading down in smaller jumps associated with perhaps infrared radiation. So we say that the light is
absorbed and heats the object, associating heat with the infrared range of the electromagnetic spectrum. Nevertheless, it is a good emitter, just taking the light in as visible and reradiating it as
infrared. For such a radiator at equilibrium, the fraction of light it emits in any wavelength range depends upon the temperature and for the ideal radiator is distributed in wavelength according to
the blackbody relationship. The great complexity of dealing with all the quantum mechanisms and processes in a solid is the kind of thing that might have led Einstein to his purported preference for
approaching problems from a thermodynamic perspective whenever possible. | {"url":"http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/absrad.html","timestamp":"2014-04-19T04:21:20Z","content_type":null,"content_length":"7178","record_id":"<urn:uuid:2db9a5a7-5a3e-4e54-8816-a812673de5ce>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00030-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Total # Posts: 18
white liht is produced by blue light and what other liht? I think it's yellow, but I'm not even sure behind my reasoning. I could really just use some help on this; I've been trying to find a good
website to explain the whole color theory, but so far what I've ...
Is warm water heating a person swimming in it, convection or conduction; what about warm water heating the air above it?
Is warm water heating a person swimming in it, convection or conduction; what about warm water heating the air above it?
Is warm water heating a person swimming in it, convection or conduction; what about warm water heating the air above it?
How do I solve 2x^2-5x=0 by factoring, I tried taking the two out, but that doesn't seem to be working.
Math - urgent
oh okay, yeah, that's the answer I got, but I'm just still really confused as to how they even got sqrt3. Thank you, though.
Math - urgent
When solving for x by completing the square as in the equation 3(x-1)^2-1=0 should I add one to both sides then divide by three or add one to both sides then square root both sides? It makes sense to
me to first divide by three, but if I do this, I'm not getting the correc...
In school we're learning to use the quadratic formula. I understand how to use it and what its applications are, but I don't understand why sometime it's okay to simplify and sometimes it's not. For
example, I got the result (-2 ± 2 square root of 6)/2. ...
In school we're learning to use the quadratic formula. I understand how to use it and what its applications are, but I don't understand why sometime it's okay to simplify and sometimes it's not. For
example, I got the result (-2 ± 2 square root of 6)/2. ...
oh, okay.
why isn't it a good idea to spread salt on the road in really cold climates?
thanks! :D
no, I don't think I do. I tried completing the square and I'm just getting really weird numbers. :(
Two positive integers are in ratio 1:3. If their sum is added to their product, the result is 224. Find the integers. On my own, I came up with the formula 224 = x + 3 x + 3x^2 but it's not really
working out. Could someone please help me?
Please help me.
I really don't understand this, I don't mean to be a pain, but I really need to understand this.
I have a test tomorrow and I really don't get this, please help me. "Use transformations and the zeros of the quadratic function f(x)=(x-6)(x+4) to determine the zeros of each of the following
functions a) y=2f(x) b) y=f(2x) c) y=f(0.5x)" I am so lost, I don'...
Please, I still need help with this. I need to make tables of values for both horizontal and vertical stretching, but I wasn't in class and missed the lesson. I don't really understand how to get the
values; at first I figured you'd get the value of y by just plugi...
I need to make tables of values for both horizontal and vertical stretching, but I wasn't in class and missed the lesson. I don't really understand how to get the values; at first I figured you'd get
the value of y by just pluging in whatever value of x you're ... | {"url":"http://www.jiskha.com/members/profile/posts.cgi?name=Isobelle","timestamp":"2014-04-18T22:43:26Z","content_type":null,"content_length":"10055","record_id":"<urn:uuid:de1f8f96-0d3e-4f3b-bee4-53304c06e97e>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00610-ip-10-147-4-33.ec2.internal.warc.gz"} |
check my answers plz
February 28th 2009, 02:04 PM #1
Feb 2009
check my answers plz
Find the local extreme points, concavity and inflecton points using first and second derivative test.
1. $h(x)=xlnx$
My answer: Local Minimim (0.37, -0.37), concave up, no infection points
2. $g(x)=\frac1{t}$
My answer: no extreme points, no concavity, no inflection points
3. $m(x)=2x+\frac1{x}$
My answer:
Local Maxium $(-\frac{1}{\sqrt2}, -2-\frac2{\sqrt2})$ concave down
Local Minimum $(\frac{1}{\sqrt2}, 2+\frac2{\sqrt2})$ concave up
No inflection points
4. $w(x)=x^4-2x^2+x+1$
My answer: no extreme points, no concavity, no inflection points
5. $w(s)=s^2e^s$
My answer:
Local Minimum $(0,0)$ concave down
Local Maximum $(-2,\frac1{2})$ concave up
No inflection points
Find the local extreme points, concavity and inflecton points using first and second derivative test.
1. $h(x)=xlnx$
My answer: Local Minimim (0.37, -0.37), concave up, no infection points
2. $g(x)=\frac1{t}$
My answer: no extreme points, no concavity, no inflection points
3. $m(x)=2x+\frac1{x}$
My answer:
Local Maxium $(-\frac{1}{\sqrt2}, -2-\frac2{\sqrt2})$ concave down
Local Minimum $(\frac{1}{\sqrt2}, 2+\frac2{\sqrt2})$ concave up
No inflection points
4. $w(x)=x^4-2x^2+x+1$
My answer: no extreme points, no concavity, no inflection points
5. $w(s)=s^2e^s$
My answer:
Local Minimum $(0,0)$ concave down
Local Maximum $(-2,\frac1{2})$ concave up
No inflection points
Don't you have answers to check against?
(If these are questions from a graded task, then it's not appropriate to be asking us to check your answers).
What is your problem? I am here to receive help! If you don't know how to do these problems, you don't have to reply. In fact, I tried my best to do all these problems and I just want to know
whether I did it correctly or not. I am not asking you to give me the correct answer, you only have to tell me which question I did was wrong, so I can do it again until I get it right.
Last edited by jkami; February 28th 2009 at 04:33 PM.
What is your problem? I am here to receive help! If you don't know how to do these problems, you don't have to reply. In fact, I tried my best to do all these problems and I just want to know
whether I did it correctly or not. I am not asking you to give me the correct answer, you only have to tell me which question I did was wrong, so I can do it again until I get it right.
My problem is that it takes time and effort to check answers. That's why I asked if you have access to answers. If these are practise questions then surely it is the responsibility of your
teacher to provide you with this.
And if you want these answers checked because the questions are part of a graded assessment that you are handing in, then I thought I'd do you the courtesy of letting you know what MHF policy is
on that.
If you post your working, together with your answer, it will take much less effort and time (things that you are getting for free I might add) to check what you've done.
So if you would care to answer the question I asked ......
By the way, all the questions I have asked at all your threads have been designed to help you answer the questions yourself rather than spoon feeding you the solution. Your replies have suggested
that you could actually do most of them ....
February 28th 2009, 02:19 PM #2
February 28th 2009, 04:21 PM #3
Feb 2009
February 28th 2009, 06:06 PM #4 | {"url":"http://mathhelpforum.com/calculus/76209-check-my-answers-plz.html","timestamp":"2014-04-20T13:12:30Z","content_type":null,"content_length":"45938","record_id":"<urn:uuid:f4ec9d6b-85da-4773-a9fa-5ba8867b1206>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00198-ip-10-147-4-33.ec2.internal.warc.gz"} |
Crown Point, IN Calculus Tutor
Find a Crown Point, IN Calculus Tutor
...I won the Botany award for my genetic research on plants as an undergraduate, and I have done extensive research in Computational Biology for my Ph.D. dissertation. I was a teaching assistant
for both undergraduate and graduate students for a variety of Biology classes. I am fluent in a range of Science and History disciplines.
41 Subjects: including calculus, chemistry, physics, English
...In 2007, I received an award from the Hobart Chamber of Commerce as the outstanding secondary educator of the year. I feel that my experience has allowed me to reach students of different
ability levels in an effective manner. I believe that I have been able to note the common pitfalls and struggles that students have with topics and can guide them through these difficulties.
12 Subjects: including calculus, physics, geometry, algebra 1
...I feel that I am very proficient in all areas and levels of History and Math, but I can help in other subjects as well. Before college, I attended Homewood-Flossmoor High School, finished in
the top 10% of my class, and scored a 30 on my ACT. During my time there, I took AP Calculus, Physics, Chemistry, and Biology, and played on the baseball team.
28 Subjects: including calculus, chemistry, geometry, algebra 1
...My expertise is tutoring any level of middle school, high school or college mathematics. I can also help students who are preparing for the math portion of the SAT or ACT. When teaching
lessons, I put the material into a context that the student can understand.
12 Subjects: including calculus, geometry, algebra 1, algebra 2
I've taught Algebra 1, Algebra 2, Geometry, and Pre-Calculus at the high school level for 6 years. In addition, I've completed a BS in Electrical Engineering and I am quite knowledgeable of
advance mathematical concepts. (Linear Algebra, Calculus, Differential Equations) I create an individualized...
12 Subjects: including calculus, geometry, algebra 1, trigonometry | {"url":"http://www.purplemath.com/crown_point_in_calculus_tutors.php","timestamp":"2014-04-20T23:41:48Z","content_type":null,"content_length":"24334","record_id":"<urn:uuid:30e7cd3e-141c-4bda-97a3-08cc2c0929d3>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00105-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: [TowerTalk] PIPE ANCHORS
On 10/30/11 10:39 AM, Grant Saviers wrote:
> A couple more comments, some not covered before: (all comments w/o
> engineering calcs - YMMV)
> 1. Square tube is stiffer/stronger than round pipe or wide flange beams
> (I-beams) of same #/ft..
Hmm. I'm not sure this is true. Bending strength goes as the radius to
the 4th power, and for a given perimeter, a circle has the section
moment. (sort of like a circle has the most included area for a given
Cross sections with right angles are easier to use in construction,
especially for bolted connections, and a I beam typically has more metal
farther from the center than a square tube (which is why they use that
However, in "practical sizes commonly sold" it might be true that
squares are stronger than circles.
TowerTalk mailing list | {"url":"http://lists.contesting.com/_towertalk/2011-10/msg00487.html?contestingsid=lc360g1f0jlfbge4si3u1rsm31","timestamp":"2014-04-21T00:30:40Z","content_type":null,"content_length":"10277","record_id":"<urn:uuid:ac7bed26-225d-41d6-836e-93d33360ae4a>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00098-ip-10-147-4-33.ec2.internal.warc.gz"} |
Solid shapes --- cube --- tetrahedron --- octahedron --- icosahedron --- dodecahedron --- other shapes --- Euler's formula --- glossary --- for teachers
Platonic solids
There are five Platonic solids: cube, tetrahedron, octahedron, icosahedron and dodecahedron. These are convex regular polyhedra. Convex means that the vertices (corners) stick out rather than in. A
regular polyhedron has all its faces and angles between them the same.
There are other solids which are not so regular which are well-known.
The regular pyramid is a tetrahedron, which is made entirely of triangles, even its base. However, the pyramids in Egypt at Giza are square pyramids. Here is a net to make one for yourself. It is
not a regular polyhedron, since it uses a square as well as triangles.
The volume of a pyramid is a third the area of the base times the height.
The cuboid is similar to a cube, but is made of rectangles rather than squares. It is also known as a right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or
rectangular parallelepiped. (Sometimes the word 'cuboid' has a more general meaning.) There are 6 cuboid faces. Opposite sides are identical rectangles. A lot of food packages are cuboids. Here
is a net to make a cuboid for yourself. It is not a regular polyhedron, since it uses rectangles which are not regular shapes. While all the angles of a rectangle are the same, it does not have
all sides the same. Only opposite sides are.
If the sides of a cuboid are a, b and c, then the volume is abc.
Cube Octahedron A cube octahedron is an attractive shape with faces that are squares and triangles.
It has only 14 faces (6 squares and 8 triangles), so it is quite easy to make.
Here is its net. Print it out, stick it on thin card, score along the lines and fold them, form the shape, then stick it together with small amounts of glue.
For more details, see the notes for the net of a cube.
A cube octahedron makes a good base for a star.
Buckyball or Truncated Icosahedron
There is a story that a scientist discovered what the molecule of a new form of carbon looked like. He found that it was an interesting shape, a bit like a ball, but made of hexagons and pentagons
arranged in a regular pattern. He was very excited and rang up a friend who was a mathematician to boast of this new shape that he'd found. The mathematician told him to look at a soccer ball! Even
footballers can't get away from mathematics.
A buckyball has 32 faces, that is, 20 hexagons and 12 pentagons.
This shape is called a buckyball after Richard Buckminster Fuller, who invented the geodesic dome. If you look at the football, you will see that it is not really a polyhedron with flat faces. It
is made of leather which stretches slightly. So when it is stuffed or blown up, the centres of each face bulge out slightly. This makes a better sphere.
Here is a net of a buckyball. See the notes for the net of a cube to see how to print this net and make your own buckyball. I'm afraid that I've left the tabs out of this one. Add them on every other
side of the edges of the net. I suggest that you do NOT start on this net first! Try a simpler one to get used to the idea.
It is easy to make an attractive star. Start with a shape such as an octahedron or a cube octahedron. Make this shape up (the nets are provided on this site) and wait for it to dry.
Now make the points. You will need one for each face. Here are the nets for the cube octahedron, but you will need 6 of the four-sided points and 8 of the 3-sided points. These don't make a solid,
and there are tabs round the hole at the bottom. If you want, you can make a taller point, which will make a more pointy star. Experiment for yourself! Remember that the base of the point must
match the edge of the original shape, and the triangles must be isosceles (with two sides the same). Once you have made all the points, and allowed the glue to dry, carefully glue each one to each
face of the original solid. Once finished and dry, you can paint it, or stick shiny foil on each point, or cover it with glitter.
Any shape can be used as a base, but very simple shapes will not give a particularly convincing star, and complicated shapes will take a lot of work and gluing! Here is a very complicated star
indeed, but I must admit that it was made from a kit from Tarquin.
It is impossible to make a perfect sphere (ball or globe) from a flat sheet of paper. Paper can curve in one direction, but cannot curve in two directions at the same time. So all spheres made from
paper or card will be approximations. Probably the best way to make a sphere is to make a polyhedron with a large number of sides. A football is a buckyball, for example, and you can make a ball from
a dodecahedron or an icosahedron. In these cases, the material of the surface stretches a little to make a better sphere, since the faces are not flat but bulge out in the centre.
Another way to make a sphere is with pointed ellipses. Globes can be made this way, since the edges of the net run along longitudes. This would be easier if you were sticking the map of the globe
onto an existing ball, but I think it would be tricky to make a sphere like this with just this net. All those points meeting at the 'poles' would be very difficult to stick together. It would be a
good idea to have a small disc of paper to stick over each pole to hold them together. I've left out the tabs as well, as I'm not sure where they would go.
If you want to make a globe, here are some websites to help you. | {"url":"http://gwydir.demon.co.uk/jo/solid/other.htm","timestamp":"2014-04-19T17:21:45Z","content_type":null,"content_length":"10209","record_id":"<urn:uuid:59fc8f4d-e7b9-495f-b3e9-1fbf590f8a2a>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00000-ip-10-147-4-33.ec2.internal.warc.gz"} |
A survey of graphical languages for monoidal categories
Results 1 - 10 of 35
- Linguistic Analysis (Lambek Festschrift
"... We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely
on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the ..."
Cited by 23 (5 self)
Add to MetaCart
We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on
the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a well-typed sentence from the meanings of its constituents. Concretely, the type
reductions of Pregroups are ‘lifted ’ to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (well-typed) whole. Importantly, meanings of whole
sentences live in a single space, independent of the grammatical structure of the sentence. Hence the inner-product can be used to compare meanings of arbitrary sentences, as it is for comparing the
meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in
order to make up the meaning of the whole sentence. A variation of our ‘categorical model ’ which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a
Montague-style Boolean-valued semantics. 1
, 2010
"... first steps in infinite-dimensional ..."
, 2010
"... This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic.
It is based on the recently introduced away from much of the typical bureaucracy of proofs. We make cruci ..."
Cited by 4 (2 self)
Add to MetaCart
This work belongs to a wider effort aimed at eliminating syntactic bureaucracy from proof systems. In this paper, we present a novel cut elimination procedure for classical propositional logic. It is
based on the recently introduced away from much of the typical bureaucracy of proofs. We make crucial use of the path breaker, an atomic-flow construction that avoids some nasty termination problems,
and that can be used in any proof system with sufficient symmetry. This paper contains an original 2-dimensional-diagram exposition of atomic flows, which helps us to connect atomic flows with other
known formalisms.
"... Abstract We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to
specializations of other quantum knot polynomials. These identities can also be explained by coincidences between smal ..."
Cited by 4 (2 self)
Add to MetaCart
Abstract We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations
of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the D2n planar algebras. We discuss the origins
of these coincidences, explaining the role of SO level-rank duality, Kirby-Melvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves G2 and does not appear to be
related to level-rank duality. AMS Classification 18D10; 57M27 17B10 81R05 57R56
- In the proceedings of QPL 5 , 2008
"... We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces. Keywords: Dagger compact closed categories,
Hilbert spaces, completeness. ..."
Cited by 4 (0 self)
Add to MetaCart
We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces. Keywords: Dagger compact closed categories, Hilbert
spaces, completeness.
, 2010
"... Abstract. In this paper, we consider a non-posetal analogue of the notion of involutive quantale [MP92]; specifically, a (planar) monoidal category equipped with a covariant involution that
reverses the order of tensoring. We study the coherence issues that inevitably result when passing from posets ..."
Cited by 3 (0 self)
Add to MetaCart
Abstract. In this paper, we consider a non-posetal analogue of the notion of involutive quantale [MP92]; specifically, a (planar) monoidal category equipped with a covariant involution that reverses
the order of tensoring. We study the coherence issues that inevitably result when passing from posets to categories; we also link our subject with other notions already in the literature, such as
balanced monoidal categories [JS91] and dagger pivotal categories [Sel09]. 1.
"... Abstract—We present a diagrammatic system for constructing and presenting readable program proofs in separation logic. A program proof should not merely certify that a program is correct; it
should explain why it is correct. By examining a proof, one should gain understanding of both the program bei ..."
Cited by 3 (2 self)
Add to MetaCart
Abstract—We present a diagrammatic system for constructing and presenting readable program proofs in separation logic. A program proof should not merely certify that a program is correct; it should
explain why it is correct. By examining a proof, one should gain understanding of both the program being considered and the proof technique being used. To
- In Proceedings of 24th International Workshop on Description Logics
"... Abstract. Linked Data makes one central addition to the Semantic Web principles: all entity URIs should be dereferenceable to provide an authoritative RDF representation. URIs in a linked
dataset can be partitioned into the exported URIs for which the dataset is authoritative versus the imported URI ..."
Cited by 3 (1 self)
Add to MetaCart
Abstract. Linked Data makes one central addition to the Semantic Web principles: all entity URIs should be dereferenceable to provide an authoritative RDF representation. URIs in a linked dataset can
be partitioned into the exported URIs for which the dataset is authoritative versus the imported URIs the dataset is linking against. This partitioning has an impact on integrity constraints, as a
Closed World Assumption applies to the exported URIs, while a Open World Assumption applies to the imported URIs. We provide a definition of integrity constraint satisfaction in the presence of
partitioning, and show that it leads to a formal interpretation of dependency graphs which describe the hyperlinking relations between datasets. We prove that datasets with integrity constraints form
a symmetric monoidal category, from which the soundness of acyclic dependency graphs follows. 1
"... We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices.
These graphs have an interface made of half-edges (edges which are drawn with an unconnected end) and enjo ..."
Cited by 2 (1 self)
Add to MetaCart
We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These
graphs have an interface made of half-edges (edges which are drawn with an unconnected end) and enjoy rich compositional principles by connecting graphs along these half-edges. In particular, this
allows equations and rewrite rules to be specified between graphs. Particular computational models can then be encoded as an axiomatic set of such rules. Further rules can be derived graphically and
rewriting can be used to simulate the dynamics of a computational system, e.g. evaluating a program on an input. Examples of models which can be formalised in this way include traditional electronic
circuits as well as recent categorical accounts of quantum information. 1
"... Abstract. In our previous work, we developed a reversible programming language and established that every computation in it is a (partial) isomorphism that is reversible and that preserves
information. The language is founded on type isomorphisms that have a clear categorical semantics but that are ..."
Cited by 2 (2 self)
Add to MetaCart
Abstract. In our previous work, we developed a reversible programming language and established that every computation in it is a (partial) isomorphism that is reversible and that preserves
information. The language is founded on type isomorphisms that have a clear categorical semantics but that are awkward as a notation for writing actual programs, especially recursive ones. This paper
remedies this aspect by presenting a systematic technique for developing a large and expressive class of reversible recursive programs, that of logically reversible smallstep abstract machines. In
other words, this paper shows that once we have a logically reversible machine in a notation of our choice, expressing the machine as an isomorphic interpreter can be done systematically and does not
present any significant conceptual difficulties. Concretely, we develop several simple interpreters over numbers and addition, move on to tree traversals, and finish with a meta-circular interpreter
for our reversible language. This gives us a means of developing large reversible programs with the ease of reasoning at the level of a conventional smallstep semantics. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=10659816","timestamp":"2014-04-16T07:09:45Z","content_type":null,"content_length":"35671","record_id":"<urn:uuid:4713fc6c-4a13-483c-920f-dfc338276d16>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00265-ip-10-147-4-33.ec2.internal.warc.gz"} |
area defined as variable resedue question..
January 16th 2010, 05:13 AM #1
MHF Contributor
Nov 2008
area defined as variable resedue question..
i need to calculate this integral from plus to minus infinity $f(z)=\frac{z}{e^{2\pi iz^2}-1}\\$
in this area
$<br /> \gamma _r=\left \{ |z|=r \right \},n<r^2<n+1<br />$
i need to find the points which turn to zero in the denominator
and non zero in the numerator.
i got two such points
$z=\pm \sqrt{n}$
by using this formula
$res(\sqrt{n})=\frac{1}{4\pi i}$
$res(-\sqrt{n})=\frac{1}{4\pi i}$
the third point is z=0 but for it we have both numerator and denominator 0
i calculated the residium for it by $res(f(x),a)=\lim_{x->a}(f(x)(x-a))$ formula
but then
my prof says some stuff that involves the area
he says that my points are 0 +1 -1 +2^(0.5) -2^(0.5) etc.. because the denominator goes to zero
for each point have a residiu and i need to sum the residiums inside.
but here the area is not defined
its not like (by radius 3)
i dont know what point are inside the area
Last edited by transgalactic; January 16th 2010 at 05:30 AM.
Follow Math Help Forum on Facebook and Google+
This simply makes no sense. You say you want to integrate "from plus to minus infinity", but then you say that |z|= r. They can't both be true.
Perhaps you mean that you want to integrate around the circle z= |r|
And saying that " $n< r^2< n+1$" simply says that $\sqrt{n}< r< \sqrt{n+1}$
i need to find the points which turn to zero in the denominator
and non zero in the numerator.
i got two such points
$z=\pm \sqrt{n}$
by using this formula
$res(\sqrt{n})=\frac{1}{4\pi i}$
$res(-\sqrt{n})=\frac{1}{4\pi i}$
$e^{2\pi i z^2}- 1$ is 0 only when $e^{2\pi iz^2}= 1$ which happens only when $2\pi i z^2$ is a multiple of $2\pi i$: $z^2= m$ for some integer m so the singularities are at $-\sqrt{m}$ and $\
sqrt{m}$. If you are integrating around the path |z|= r, with $<br /> \sqrt{n}< r< \sqrt{n+1}$, then there are two poles for every positive integer $m\le n$
the third point is z=0 but for it we have both numerator and denominator 0
i calculated the residium for it by $res(f(x),a)=\lim_{x->a}(f(x)(x-a))$ formula
but then
my prof says some stuff that involves the area
he says that my points are 0 +1 -1 +2^(0.5) -2^(0.5) etc.. because the denominator goes to zero
for each point have a residiu and i need to sum the residiums inside.
but here the area is not defined
its not like (by radius 3)
i dont know what point are inside the area
Last edited by HallsofIvy; January 17th 2010 at 03:56 AM.
Follow Math Help Forum on Facebook and Google+
January 17th 2010, 03:07 AM #2
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/calculus/124013-area-defined-variable-resedue-question.html","timestamp":"2014-04-23T21:39:05Z","content_type":null,"content_length":"41226","record_id":"<urn:uuid:06b6375c-fdb6-40a2-9735-e45d8dac66bb>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00026-ip-10-147-4-33.ec2.internal.warc.gz"} |
Determinism and Chaos
Richard H. Bube
Department of Materials Science & Engineering
Stanford University
Stanford, CA 94305
From: Perspectives on science and Christian Faith 41 (December 1989)
The nature of the interaction between "determinism" and "chance" has been the subject of continued debate in one form or another from the early days of recorded human thought. Theologically, it is
well known as the "predestination vs. free will" debate, although the actual connection between the theological debate and the scientific debate is tenuous at best. The development of science in the
last few centuries has given it a new intensity, since scientific descriptions must fall either into the category of deterministic or chance (probabilistic), neither of which as an isolated world
view is compatible with biblical concepts of human responsibility. A thumbnail sketch of the question and its implications has recently been given in this journal.^1 I believe it is fair to say that
evangelical Christians with a commitment to both authentic science and authentic biblical theology largely follow the lead of those like MacKay who maintain the existence of a reality in which both
determinism and chance are intricately and sometimes even mysteriously interrelated.^2
A curious and fairly dramatic twist to this debate has been given in recent years by the scientific recognition of the state known as "chaos." In the popular mind, chaos is what one would expect in a
completely random or chance-oriented environment; we have been delivered from chaos by the existence of order (deterministic relationships). Some of the early ideas of creation dealt with God's
overcoming chaos with order, again emphasizing the common expectation that these two kinds of description are mutually exclusive.
The contention that we ought to expect complex interactions between determinism and chance, or between order and chaos, has found a rather dramatic expression in recent discovery of those specific
effects that have come to be known as "chaos." It is the purpose of this communication to illustrate the type of effect observed (in one of its simplest manifestations).
A recent insert, in Science magazine, entitled "A Simple Model of Chaos,"^3 describes a model based on population biology that illustrates nicely how "chaos" can proceed from a deterministically
described process. The population of a particular insect species in one year N[t] is related to the population in the following year N[t+1] approximately by the following relation:
N[t+1] = alphaN[t](1 - N[t]) (1)
Here N[t] is expressed in appropriate units so that its numerical values fall between 0 and 1 (a negative value for [t+1] would imply extinction), and alpha is a constant that controls the specific
form that the results of Eq. (1) take over a number of generations. In order to express the implications of Eq. (1) it is necessary to choose a value for alpha and an initial value for N[t]. It is
the extreme sensitivity of Eq. (1) to small variations in the initial value of N[t] for certain values of alpha that characterizes chaotic behavior.
Figure 1 shows the variation of the "population" with the number of generations from 1 to 100, for values of alpha between 2.9 and 3.5, and an initial value of N[t] = 0.50. It can be seen directly
from Eq. (1) that if alpha = 2.0 when N[t] = 0.50, N[t+1] = N[t], and the population is unchanged with successive generations. When alpha = 2.9, early generations show alternating values which
quickly decay down to a "population" of about 0.65 within 20 to 30 generations. When alpha = 3.0, the decay of the two alternating values is much slower and persists out to 100 generations, so that
the 99th generation shows N[t] = 0.68 and the 100th generation shows N[t] = 0.64. When alpha = 3.3, there are again two alternative values (0.82 and 0.47) but these are stable in alternate years over
the range from 1 to 100 generations. When alpha = 3.5, the number of alternative values (0.87, 0.82, 0.50 and 0.38) jumps to four, and these are unchanging from 1 to 100 generations. Although these
four cases show increasing complexity, they also give the appearance of an ordered and structured complexity.
When alpha = 3.57, this ordered behavior gives way to chaos. The data points in Figure 2 show thealpha = 3.9 and for three cases in which N[t] = 0.49, 0.50 and 0.51, three numbers differing by only
2% from one another. Right from the first generation on, the points jump around in a random fashion. For the first 10 generations the points for the three different initial values of N[t] are
approximately the same, but after 20 or 30 generations, major differences between the three sets of data arising from different initial values of N[t] are evident. Table 1 lists the specific
"population" values shown in Figure 2 for a few selected later generations, showing the very strong influence of the small difference in the initial values of N[t].
These results illustrate how the condition of chaos can be generated from a deterministic relationship. Counter
^1R.H. Bube, "Penetrating the Word Maze: Determinism/Chance," Perspectives on Science and Christian Faith, 41, March (1989), p. 37.
^2D.M. MacKay, The Open Mind and Other Essays: A Scientist in God's World, edited by Melvin Tinker. (Leicester, England: Inter-Varsity Press, 1988).
^3R. Pool, "A Simple Model of Chaos," Science 243, 311 (1989). | {"url":"http://www.asa3.org/ASA/PSCF/1989/PSCF12-89Bube.html","timestamp":"2014-04-18T03:23:33Z","content_type":null,"content_length":"8621","record_id":"<urn:uuid:baaf155a-1179-42e6-94ee-1f6c22ff1827>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00590-ip-10-147-4-33.ec2.internal.warc.gz"} |
188 helpers are online right now
75% of questions are answered within 5 minutes.
is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
• Problem Solving 19 Hero
• Engagement 19 Mad Hatter
• You have blocked this person.
• ✔ You're a fan Checking fan status...
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.
You haven't written a testimonial for Owlfred. | {"url":"http://openstudy.com/users/uditkulka/answered","timestamp":"2014-04-19T01:51:30Z","content_type":null,"content_length":"119326","record_id":"<urn:uuid:a8202792-fcf5-4aab-a6f7-172dc6df09b3>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00347-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Soliton Trajectories for the Kadomtsev-Petviashvili Equation
This Demonstration shows a particle-based fluid trajectory description of multi-soliton interactions. The Kadomtsev-Petviashvili (KP) equation is the two-dimensional version (2+1 dimensions,
including time) of the Korteweg–de Vries (KdV) equation and supports multi-soliton-solutions. These solutions produce a pattern of straight lines in the plane at time . In particular, the KP
equation describes approximately the slow evolution of waves in shallow water of uniform depth. Solitons are stable waves in space and time, in which the velocity depends on the amplitude. When
solitons interact with other solitons, their shapes do not change, but their phase shifts. The trajectory method for quantum motion, developed by Louis de Broglie and David Bohm, is applied to the
KP equation. In this method the motion of idealized particles is governed by the current flow, which is derived directly from the continuity equation; is proportional to the density of the wave in
which idealized particles are positioned at the point at time . The current flow divided by the density establishes the guiding equation (velocity field) for the individual path of the particles in
the wave. With , it yields the starting points of possible trajectories inside the wave, which lead to single trajectories governed by the velocity of the wave. Only the dynamic of the wave
influences the motion of the particles. The trajectories are the streamlines of the wave, regarded as paths of idealized particles, because the particles themselves do not interact and do not
influence the wave. Here the trajectories show the particle transfer of the wave in the fluid medium. The graphic shows the wave density, the trajectories, and the initial and actual position of
idealized particles in the wave.
The nonlinear KP equation can be transformed into a bilinear form through a variable transformation. By applying a perturbation technique on the bilinear equation, multi-soliton solutions can be
derived. This is called Hirota's direct method. From the KP equation , where the subscripts , , and denote partial derivatives, the continuity equation, representing conservation of mass, is given
by . Here gives the divergence of the current vector field with , which yields
The partial derivative of the above equation with respect to gives the KP equation again. The velocity field is deduced from the current vector via :
From classical mechanics, the path versus time dependence is obtained by integrating the velocity , which leads, together with a starting point for , to a trajectory in - space. The soliton solution
is time reversible. The -line-soliton is constructed by Hirota's direct method. With
you get the analytic solution
The trivial (vacuum) solution corresponds to .
R. Hirota,
The Direct Method in Soliton Theory
, Cambridge, UK: Cambridge University Press, 2004.
P. Holland,
The Quantum Theory of Motion
, Cambridge, UK: Cambridge University Press, 1993. | {"url":"http://demonstrations.wolfram.com/SolitonTrajectoriesForTheKadomtsevPetviashviliEquation/","timestamp":"2014-04-18T00:17:33Z","content_type":null,"content_length":"48530","record_id":"<urn:uuid:80dda359-fc34-4ce1-badf-230f9ac405d7>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00217-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: st: Dependent var is a proportion, with large spike in .95+
[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
Re: st: Dependent var is a proportion, with large spike in .95+
From David Airey <david.airey@Vanderbilt.Edu>
To statalist@hsphsun2.harvard.edu
Subject Re: st: Dependent var is a proportion, with large spike in .95+
Date Thu, 4 Sep 2008 06:56:53 -0400
Here is an article I used for a spiked distribution. It is probably not the same situation as yours, however.
Genetics. 2003 Mar;163(3):1169-75.
Mapping quantitative trait loci in the case of a spike in the phenotype
Broman KW.
Department of Biostatistics, Johns Hopkins University, Baltimore, Maryland 21205,
USA. kbroman@jhsph.edu
A common departure from the usual normality assumption in QTL mapping concerns a
spike in the phenotype distribution. For example, in measurements of tumor mass,
some individuals may exhibit no tumors; in measurements of time to death after a
bacterial infection, some individuals may recover from the infection and fail to
die. If an appreciable portion of individuals share a common phenotype value
(generally either the minimum or the maximum observed phenotype), the standard
approach to QTL mapping can behave poorly. We describe several alternative
approaches for QTL mapping in the case of such a spike in the phenotype
distribution, including the use of a two-part parametric model and a
nonparametric approach based on the Kruskal-Wallis test. The performance of the
proposed procedures is assessed via computer simulation. The procedures are
further illustrated with data from an intercross experiment to identify QTL
contributing to variation in survival of mice following infection with Listeria
PMCID: PMC1462498
PMID: 12663553 [PubMed - indexed for MEDLINE]
On Sep 3, 2008, at 3:22 PM, Dan Weitzenfeld wrote:
Hi Statalist,
I am trying to determine which testing factors drive a proportion
dependent variable, PercentNoise.
In searching the archives, I came across -betafit-, and a link to the
FAQ: "How do you fit a model when the dependent variable is a
proportion?" In that response, Allen McDowell and Nic Cox write, "In
practice, it is often helpful to look at the frequency distribution: a
marked spike at zero or one may well raise doubt about a single model
fitted to all data."
That describes my situation exactly: I have a marked spike in my
histogram at the top bin, roughly .95 - 1. I am wondering how to
account for this.
Does -betafit- take such a possibility into account?
Can someone briefly describe how I could use multiple models to fit
all the data, as implied in the FAQ response?
My fallback is setting a pass/fail bar and converting my proportions
to a binary, then using probit/logit. But the obvious drawback is
that I am throwing away information by collapsing the continuous
(albeit bounded) proportion variable to a binary.
Thanks in advance for any suggestions,
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2008-09/msg00170.html","timestamp":"2014-04-18T05:42:44Z","content_type":null,"content_length":"8499","record_id":"<urn:uuid:f59606c3-9bf5-4c9e-abda-6ddd62ad9d90>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00425-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: The modular degree,
congruence number, and
multiplicity one
Amod Agashe
Florida State University
joint work with K. Ribet and W. Stein
October 4, 2007
Slides and paper available at:
Elliptic curves
Let E be an elliptic curve over Q, i.e., an
equation of the form y2 = x3 + ax + b, where
a, b Q
Example: The graph of y2 = x3 - x over R:
If p is a prime, then we can "think of" the
equation for E modulo p
Let ap(E) = 1 + p - #solutions to E mod p.
Modular curves and modular forms | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/705/1437482.html","timestamp":"2014-04-20T09:05:39Z","content_type":null,"content_length":"7576","record_id":"<urn:uuid:c51676f6-bd88-487f-817f-997d820afc64>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00156-ip-10-147-4-33.ec2.internal.warc.gz"} |
Invasion fitness in moment-closure treatments (part 4)
Continued from Part 3.
Example 3: Evolution of Altruism
One application of this machinery has been to understand how evolution can produce altruistic behaviour. A behaviouristic definition of “altruism” would go something like, “Acting to increase the
reproductive success of another individual at the expense of one’s own”. This can be written out using game theory; one defines a parameter $b$ to stand for benefit, $c$ to stand for cost and a
payoff function which depends on $b$, $c$ and the strategy employed by an organism.
We shall consider a lattice of sites, each one of which can be in one of three states: empty, denoted by 0; occupied by a selfish organism, denoted by $S$; and occupied by an altruist, denoted by $A$
In the reproduction process, an organism spawns an offspring into an adjacent empty lattice site. This turns a pair of type $S0$ into a pair of type $SS$. At what rate should the transition $S0\to
SS$ occur? If we presume some baseline reproductive rate, call it ${b}_{0}$, then the presence of altruistic neighbours should augment that rate. We’ll say that if the number of nearby altruists is $
{n}_{A}$, then selfish individuals will reproduce at a rate ${b}_{0}+B{n}_{A}/n$, where the parameter $B$ specifies how helpful altruists are. The reproduction process for altruists, which we can
write $A0\to AA$, occurs at a rate ${b}_{0}+B{n}_{A}/n-C$. Here, the parameter $C$ is the cost of altruism: it’s how much an altruist gives up to help others.
In the differential equation for $d{p}_{S0}/dt$, the $S0\to SS$ transition contributes a term proportional to the density of $S0$ pairs:
$-\left[\left(1-\varphi \right){b}_{S}+\left({b}_{S}+m\right)\varphi {q}_{S\mid 0S}\right]{p}_{S0},$-[(1 - \phi) b_S + (b_S + m) \phi q_{S|0 S}] p_{S 0},
where we have written $\varphi$ for $1-1/z$, to save a little ink. All things told, the rate of change of ${p}_{S0}$ is given by
$\begin{array}{rcl}\frac{d{p}_{S0}}{dt}& =& \left({b}_{s}+m\right)\varphi {q}_{S\mid 00}{p}_{00}\\ & & -\left[{d}_{S}+m\varphi {q}_{0\mid S0}\right]{p}_{S0}\\ & & -\left[\left(1-\varphi \right){b}_
{s}+\left({b}_{S}+m\right)\varphi {q}_{S\mid 0S}\right]{p}_{S0}\\ & & +\left[{d}_{S}+m\varphi {q}_{0\mid SS}\right]{p}_{SS}\\ & & -\left({b}_{A}+m\right)\varphi {q}_{A\mid 0S}{p}_{S0}\\ & & +\left
[{d}_{A}+m\varphi {q}_{0\mid AS}\right]{p}_{SA}.\end{array}$\begin{array}{rcl} \frac{d p_{S 0}}{d t} &= &(b_s + m) \phi q_{S|0 0} p_{0 0}\\ & & - [d_S + m \phi q_{0|S 0}] p_{S 0}\\ & & - [(1 - \phi)
b_s + (b_S + m) \phi q_{S|0 S}] p_{S 0}\\ & & + [d_S + m \phi q_{0|S S}] p_{S S}\\ & & - (b_A + m) \phi q_{A|0 S} p_{S 0}\\ & & + [d_A + m \phi q_{0|A S}] p_{S A}. \end{array}
Yuck! After writing a few equations like that, it’s easy to wonder if maybe we should look for new mathematical ideas which could help us better organise our thinking.
The next steps follow the general plan we laid out above. We write differential equations for the pairwise probabilities ${p}_{ab}$, which depend on triplet quantities ${q}_{a\mid bc}$. Then, we
impose a pair approximation, declaring that ${q}_{a\mid bc}={q}_{a\mid b}$, which gives us a closed system of equations. Next, we find the fixed point with ${p}_{A}=0$, and we perturb around that
fixed point to see what happens when a strain of altruists is introduced into a selfish population. The dominant eigenvalue $\lambda$ of the time-evolution matrix $T$ tells us, in this approximation,
whether the altruistic strain will invade the lattice or wither away. The condition $\lambda >0$ can be written in the form
$B\varphi {\stackrel{˜}{q}}_{A\mid A}-C>0.$B \phi \tilde{q}_{A | A} - C \gt 0.
Here, we’ve written ${\stackrel{˜}{q}}_{A\mid A}$ for the conditional probability of altruists contacting altruists which obtains as the local densities equilibrate. That is to say, an attempted
invasion by altruists will succeed if a measure of benefit, $B$, multiplied by an indicator of “assortment” among genetically similar individuals, is greater than the cost of altruistic behaviour,
After all our mucking with eigenvalues, we have found a condition which is strongly reminiscent of a classic and influential idea from mid-twentieth-century evolutionary theory. In biology, the
$\left(\mathrm{benefit}\right)×\left(\mathrm{relatedness}\right)>\left(\mathrm{cost}\right)$(benefit) \times (relatedness) \gt (cost)
is known as Hamilton’s rule (Van Dyken et al., 2011). This is a rule-of-thumb for when natural selection can favour altruistic behaviour: altruists can prosper when the inequality is satisfied.
Hamilton’s rule was originally derived for unstructured populations, with no network topology or spatial arrangement to them. We can understand Hamiltion’s rule in this context in the following way:
How well an organism fares in the great contest of life depends on the environment it experiences. During the course of its life, an individual member of a species will interact with a set of others,
which we could call its “social circle”. The composition of that social circle affects how well an individual will propagate its genetic information to the next generation — its fitness. In an
unstructured population, we can think of such circles being formed by taking random samples of the population. An altruist, by our definition, sacrifices some of its own potential so that offspring
of other individuals can prosper. A social circle of altruists can fare better than a social circle of selfish individuals, increasing the chances that social circles which form in the next
generation will contain altruists (Van Dyken et al., 2011).
It’s common to treat “benefit” and “cost” as parameters of the system. We could potentially derive them from more fundamental dynamics, if we looked more closely at the interactions within a
particular ecosystem, but right now, they’re just knobs we can turn. What about the remaining quantity in Hamilton’s rule: what does “relatedness” mean? Excellent question! We can get a feel for
where the term came from by taking a gene’s-eye view: copies of many of my particular genetic variants will be sitting inside the cells of my close relatives. Consequently, as far as my genes are
concerned, if my relatives survive, that’s almost as good as my surviving. When reckoning the benefit of altruism against its cost, then, the aid one organism brings to another ought to be weighted
by how “related” they are.
So, we can say that we have “recovered Hamilton’s rule as an emergent property of the spatial dynamics” — if we are willing to draw a circle around the middle of our formula and declare those terms
to be the “relatedness”.
Knowing where our invasion condition came from, we can appreciate some of the caveats which scientists have raised in connection with Hamilton’s rule.
In particular, $r$ is often taken to be the average relatedness of interacting individuals, as compared to the average relatedness in the population, in which case inequality (1) $\left[rB>C\
right]$ is referred to as Hamilton’s rule. It is important to note that inequality (1) is only a description of whether the current level of assortment as subsumed in the parameter $r$ is
sufficient to favour cooperation, but not a description of the mechanisms that would lead to such assortment. It has been suggested repeatedly that the problem of cooperation can be understood
entirely based on Hamilton’s rules of the form (1). Even though often taken as gospel, this claim is wrong in general, for two reasons.
First, and foremost, even if a rule of the form (1) predicts the direction of selection for cooperation at a given point in time, the long-term evolution of cooperation cannot be understood
without having a dynamic equation for the quantity $r$, i.e., without understanding the temporal dynamics of assortment. The dynamics of $r$ in turn cannot be understood based solely on the
current level of cooperation, and hence expressions of type (1) are in general insufficient to describe the evolutionary dynamics of cooperation. Second, the quantity $r$, which measures the
average relatedness among interacting individuals, is insufficient to construct Hamilton’s rule in models that account for variable individual-level death rates and/or group-level events.
Damore and Gore (2012) have more to say on this point:
Contrary to the popular use of the word, “relatedness” describes a population of interacting individuals, where $r$ refers to how assorted similar individuals are in the population.
And in more detail:
every definition of relatedness must take into account the population. Therefore, relatedness is not the percent of genome shared, genetic distance, or any extent of similarity between two
isolated individuals in a larger population. Also, because horizontal gene transfer is commonplace between microbes and selection is strong, phylogenetic distance or any other indirect genetic
measure is likely to be inaccurate. Many of these false definitions live on partly because ambiguous heuristics like ”$\frac{1}{2}$ for brothers, $\frac{1}{8}$ for cousins,” which require very
specific assumptions, are repeated in the primary literature. Also, most non-theoretical papers simply define relatedness as “a measure of genetic similarity” and do not elaborate or instead
leave the precise definition to the supplemental information $\left[\dots \right]$ Unfortunately, scientists can easily misinterpret this “measure of genetic similarity” to be anything that is
empirically convenient such as genetic distance or percent of genome shared. Largely because of this confusion, we support the more widespread use of the term “assortment,” which is harder to
misinterpret $\left[\dots \right]$ For similar reasons of reader understanding, we also encourage authors to make calculations more explicit, either in the main or supplemental text, and to avoid
repeating previous results without giving the assumptions that went into deriving them.
It is for this reason that we called $\varphi {\stackrel{˜}{q}}_{A\mid A}$ a measure of “assortment” earlier. Of course, even with this careful choice of terminology, the limitations of our
Hamilton-esque rule still apply: we know that because we derived it from the condition that the dominant eigenvalue be positive, it will miss any effects which a fixed-point eigenvalue analysis is
not sensitive to.
Stepping back for a moment, notice that although the terms and coefficients started to proliferate on us, we haven’t introduced any remarkably “advanced” or “esoteric” mathematics. Derivatives,
matrices, eigenvalues — this is undergraduate stuff! The amount of algebra we’ve been able to stir up without really even trying is, however, a little worrying. We can invent a mathematical model for
some particular biological scenario, and we might even be able to solve it, or at least tell how it’ll behave in certain interesting circumstances. But what if we want general results which extend
across models, or ideas which will help us identify the common features and the key disparities among a host of examples?
With that attitude, then, a thought towards “higher” mathematics:
A Petri net specifies a symmetric monoidal category (Lerman et al. 2011). Each truncation of the moment-dynamics hierarchy for a system yields a Petri net, and so successive truncations of the
moment-dynamics hierarchy yield mappings between categories. Going from a pair approximation to a mean-field approximation, for example, transforms a Petri net whose circles are labelled with pair
states to one labelled by site states. Category theory might be able to say something interesting here. Anything which can tame the horrible spew of equations which arises in these problems would be
great to have. Ought we be considering, say, the strict 2-category whose objects are moment-closure approximations to an ecosystem, and whose morphisms are symmetric monoidal functors between them?
• H. Matsuda, N. Ogita, A. Sasaki, and K. Sato (1992), “Statistical mechanics of population”, Progress of Theoretical Physics 88, 6: 1035–49 (web).
• U. Dieckmann, R. Law, and J. A. J. Metz, eds., The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Cambridge University Press, 2000.
• T. Gross, C. J. Dommar D’Lima and B. Blasius (2006), “Epidemic dynamics on an adaptive network”, Physical Review Letters 96, 20: 208701 (web). arXiv:q-bio/0512037.
• P. Bijma and M. J. Wade (2008), “The joint effects of kin, multilevel selection and indirect genetic effects on response to genetic selection” Journal of Evolutionary Biology 21: 1175–88,
DOI:10.1111/j.1420-9101.2008.01550.x (web).
• A.-L. Do and T. Gross (2009), “Contact processes and moment closure on adaptive networks”, in T. Gross and H. Sayama, eds., Adaptive Networks: Theory, Models and Applications. Springer.
• B. Allen (2010), Studies in the Mathematics of Evolution and Biodiversity. PhD thesis, Boston University (web).
• J. A. Damore and J. Gore (2012), “Understanding microbial cooperation”. Journal of Theoretical Biology 299: 31–41, DOI:10.1016/j.jtbi.2011.03.008 (pdf). PMID:21419783.
• B. Simon, J. A. Fletcher and M. Doebeli (2012), “Hamilton’s rule in multi-level selection models” Journal of Theoretical Biology 299: 55–63 PMID:21820447.
• B. C. Stacey, A. Gros and Y. Bar-Yam (2014), “Eco-Evolutionary Feedback in Host–Pathogen Spatial Dynamics”, arXiv:1110.3845 [nlin.AO].
• J. D. Van Dyken, T. A. Linksvayer and M. J. Wade (2011), “Kin Selection–Mutation Balance: A Model for the Origin, Maintenance, and Consequences of Social Cheating” The American Naturalist 177, 3:
288–300. JSTOR:10.1086/658365 (pdf). | {"url":"http://www.azimuthproject.org/azimuth/show/Invasion+fitness+in+moment-closure+treatments+(part+4)","timestamp":"2014-04-19T19:32:52Z","content_type":null,"content_length":"37495","record_id":"<urn:uuid:0cc47ef7-67b1-4c72-a01f-15e11821d89a>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00040-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cryptology ePrint Archive: Report 2003/232
The Statistical Zero-knowledge Proof for Blum Integer Based on Discrete LogarithmChunming Tang and Zhuojun Liu and Jinwang LiuAbstract: Blum integers (BL), which has extensively been used in the
domain of cryptography, are integers with form $p^{k_1}q^{k_2}$, where $p$ and $q$ are different primes both $\equiv 3\hspace{4pt}mod\hspace{4pt}4$ and $k_1$ and $k_2$ are odd integers. These
integers can be divided two types: 1) $M=pq$, 2) $M=p^{k_1}q^{k_2}$, where at least one of $k_1$ and $k_2$ is greater than 1.\par
In \cite{dbk3}, Bruce Schneier has already proposed an open problem: {\it it is unknown whether there exists a truly practical zero-knowledge proof for $M(=pq)\in BL$}. In this paper, we construct
two statistical zero-knowledge proofs based on discrete logarithm, which satisfies the two following properties: 1) the prover can convince the verifier $M\in BL$ ; 2) the prover can convince the
verifier $M=pq$ or $M=p^{k_1}q^{k_2}$, where at least one of $k_1$ and $k_2$ is more than 1.\par
In addition, we propose a statistical zero-knowledge proof in which the prover proves that a committed integer $a$ is not equal to 0.\par
Category / Keywords: cryptographic protocols / cryptography, Blum integer, statistical zero-knowledgeDate: received 3 Nov 2003, last revised 7 Nov 2003Contact author: ctang at mmrc iss ac cnAvailable
format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation Version: 20031108:072704 (All versions of this report) Discussion forum: Show discussion | Start new discussion[
Cryptology ePrint archive ] | {"url":"http://eprint.iacr.org/2003/232/20031108:072704","timestamp":"2014-04-20T20:59:30Z","content_type":null,"content_length":"3085","record_id":"<urn:uuid:fb5b18b9-c968-4d30-b314-c9254877eef0>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00570-ip-10-147-4-33.ec2.internal.warc.gz"} |
NAG Library
NAG Library Routine Document
Note: this routine uses optional parameters to define choices in the problem specification. If you wish to use
settings for all of the optional parameters, then the option setting routine F12FDF need not be called. If, however, you wish to reset some or all of the settings please refer to Section 10 in F12FDF
for a detailed description of the specification of the optional parameters
1 Purpose
F12FCF is a post-processing routine in a suite of routines which includes
. F12FCF must be called following a final exit from
2 Specification
SUBROUTINE F12FCF ( NCONV, D, Z, LDZ, SIGMA, RESID, V, LDV, COMM, ICOMM, IFAIL)
INTEGER NCONV, LDZ, LDV, ICOMM(*), IFAIL
REAL (KIND=nag_wp) D(*), Z(LDZ,*), SIGMA, RESID(*), V(LDV,*), COMM(*)
3 Description
The suite of routines is designed to calculate some of the eigenvalues, $λ$, (and optionally the corresponding eigenvectors, $x$) of a standard eigenvalue problem $Ax = λx$, or of a generalized
eigenvalue problem $Ax = λBx$ of order $n$, where $n$ is large and the coefficient matrices $A$ and $B$ are sparse, real and symmetric. The suite can also be used to find selected eigenvalues/
eigenvectors of smaller scale dense, real and symmetric problems.
Following a call to
, F12FCF returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or an orthonormal basis for the associated approximate invariant subspace.
The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by real symmetric matrices. There is negligible additional cost to obtain
eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
F12FCF is based on the routine
from the ARPACK package, which uses the Implicitly Restarted Lanczos iteration method. The method is described in
Lehoucq and Sorensen (1996)
Lehoucq (2001)
while its use within the ARPACK software is described in great detail in
Lehoucq et al. (1998)
. An evaluation of software for computing eigenvalues of sparse symmetric matrices is provided in
Lehoucq and Scott (1996)
. This suite of routines offers the same functionality as the ARPACK software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and
to simplify some of the interfaces.
F12FCF, is a post-processing routine that must be called following a successful final exit from
. F12FCF uses data returned from
and options, set either by default or explicitly by calling
, to return the converged approximations to selected eigenvalues and (optionally):
– the corresponding approximate eigenvectors;
– an orthonormal basis for the associated approximate invariant subspace;
– both.
4 References
Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562
Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory
Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821
Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia
5 Parameters
1: NCONV – INTEGEROutput
2: D($*$) – REAL (KIND=nag_wp) arrayOutput
3: Z(LDZ,$*$) – REAL (KIND=nag_wp) arrayOutput
4: LDZ – INTEGERInput
5: SIGMA – REAL (KIND=nag_wp)Input
6: RESID($*$) – REAL (KIND=nag_wp) arrayInput
7: V(LDV,$*$) – REAL (KIND=nag_wp) arrayInput/Output
8: LDV – INTEGERInput
9: COMM($*$) – REAL (KIND=nag_wp) arrayCommunication Array
10: ICOMM($*$) – INTEGER arrayCommunication Array
11: IFAIL – INTEGERInput/Output
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by
Errors or warnings detected by the routine:
On entry, $LDZ < max1,N$ or $LDZ < 1$ when no vectors are required.
On entry, the option $Vectors = Select$ was selected, but this is not yet implemented.
The number of eigenvalues found to sufficient accuracy prior to calling F12FCF, as communicated through the parameter
, is zero.
The number of converged eigenvalues as calculated by
differ from the value passed to it through the parameter
Unexpected error during calculation of a tridiagonal form: there was a failure to compute all the converged eigenvalues. Please contact
The routine was unable to dynamically allocate sufficient internal workspace. Please contact
An unexpected error has occurred. Please contact
7 Accuracy
The relative accuracy of a Ritz value,
, is considered acceptable if its Ritz estimate
$≤ Tolerance × λ$
. The default
used is the
machine precision
given by
8 Further Comments
9 Example
This example solves $Ax = λBx$ in regular mode, where $A$ and $B$ are obtained from the standard central difference discretization of the one-dimensional Laplacian operator $d2u dx2$ on $0,1$, with
zero Dirichlet boundary conditions.
9.1 Program Text
9.2 Program Data
9.3 Program Results | {"url":"http://www.nag.com/numeric/FL/nagdoc_fl24/html/F12/f12fcf.html","timestamp":"2014-04-16T10:31:54Z","content_type":null,"content_length":"27046","record_id":"<urn:uuid:926462c3-fbbb-4b2d-8d4d-9ccc1798ef46>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00142-ip-10-147-4-33.ec2.internal.warc.gz"} |
Binary - Older Than You Think
Binary - Older Than You Think Follow
Written by Mike James
Tuesday, 17 December 2013
Binary is the number system of the computer age, so it is a big surprise to learn that it's more than 600 years old and was used by Polynesians as part of their counting system.
You can do arithmetic in any base you care to choose. Humans tend to prefer a base 10 system presumably because this is what you get when you start by counting on your fingers.
After you get to the tenth finger you have to remember one group of ten and start counting on your fingers all over again.
Base ten or decimal is a great system for writing down numbers but it doesn't make arithmetic as easy as in binary. In binary the multiplication tables are just
1x0=0 and
which should be compared to the tables we all have to learn at school. The big problem with binary is that it doesn't make writing numbers easy - for example 7 is 111 and so on.
Ease of implementing arithmetic and the natural way binary can be stored in a two state memory makes it the best thing to use for computer arithmetic but it was invented long before
the computer. The mathematician Gottfried Leibniz knew about number bases and had worked out that binary had advantages back in 1703 some 300 years ago.
Now it seems that we can push back that date when binary-like systems were used another 300 years earlier on the Pacific island of Mangareva. It is "binary like" because it is very
unlikely that anyone would invent pure binary counting for simple tasks. You would have to count zero, one, one lot of two, one lot of two and one, two lots of two and so on. You
just don't seem to get enough done in the early stages of counting.
There is lots of evidence that groupings of two have been important much earlier than the Mangareva counting system - the I Ching for example that was the inspiration for Liebniz's
interest in binary.
The Mangareva people however took another option. As well as pure base systems you can count in mixed base. For example the old UK monetary system used base 12 and 20 - 12 pence to
a shilling and 20 shillings to a pound. You can count using different bases for different sizes of number. Andrea Bender and Sieghard Beller of the University of Bergen in Norway
have just published a paper (behind a paywall unfortunately) that explains how the Mangareva counted using a mixed 10 and 2 base system. They had words for values from 1 to 10 but
after this groupings of powers of 2 are used. For eample takau (K) is 10, paua (P) is 20 and tataua (T) is 40. Thus a number like 70 is TPK and 75 is TPK5. This avoids the problem
of "small values" in binary but you still have some simple rules for arithmetic e.g.
2 x K =P and 2 x P =T.
It is argued that these rules made doing mental arithmetic easier.
Python Tools for Visual Studio 2.1 Beta
Microsoft is improving the support for Python in Visual Studio with the release of Python Tools for VS 2.1 beta.
+ Full Story
More Ties Than We Thought Or Ties Of The Matrix
The Matrix Reloaded started something when "The Merovingian" wore a number of very flashy ties. The problem was that we thought we knew how many ways you can tie a tie, but the
enumeration didn't incl [ ... ]
+ Full Story
More News
Last Updated ( Wednesday, 18 December 2013 )
RSS feed of news items only
Copyright © 2014 i-programmer.info. All Rights Reserved. | {"url":"http://i-programmer.info/news/82-heritage/6724-binary-older-than-you-think.html","timestamp":"2014-04-18T06:45:36Z","content_type":null,"content_length":"39748","record_id":"<urn:uuid:311e8958-500b-467e-9b20-9c52df17e1a3>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00539-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pathways to CyberInfrastructure
Shodor > CyberPathways > Workshops 2007 > WeaverStreet
• Week 3
• Week 4
• Week 5
Search Engines
Topic: How to Use a Search Engine
Time Duration: 40 minutes of lessons, 1 hr 20 min of class participation.
Grades: 4 to 8th grades
The students were introduced to different search engines and Web vocabulary. The instructor explained the concept behind web browsers. The students were able to give different examples of a web
browser. The instructor gave the students Internet Scavenger Hunts to get use to using search engines in a different way. The students tried to answer the questions without help from the instructor.
After the students were given time to find the answers, the instructor gave them some short cuts to help narrow down the search. The students tried to find more answers using some of the short cuts.
The students worked hard to answer the questions; therefore, the class played a game of Internet "Simon says." The instructor would say Simon says "jump up and down pat your head." The students would
perform both actions. Another example, "Simon says" kick you leg pat you head - kick you leg." The students would only pat their heads (See lesson plans). The class ended with fun and games.
Not Logged In. Login
©1994-2014 Shodor
Search Engines
Topic: How to Use a Search Engine
Time Duration: 40 minutes of lessons, 1 hr 20 min of class participation.
Grades: 4 to 8th grades
The students were introduced to different search engines and Web vocabulary. The instructor explained the concept behind web browsers. The students were able to give different examples of a web
browser. The instructor gave the students Internet Scavenger Hunts to get use to using search engines in a different way. The students tried to answer the questions without help from the instructor.
After the students were given time to find the answers, the instructor gave them some short cuts to help narrow down the search. The students tried to find more answers using some of the short cuts.
The students worked hard to answer the questions; therefore, the class played a game of Internet "Simon says." The instructor would say Simon says "jump up and down pat your head." The students would
perform both actions. Another example, "Simon says" kick you leg pat you head - kick you leg." The students would only pat their heads (See lesson plans). The class ended with fun and games.
Topic: How to Use a Search Engine Time Duration: 40 minutes of lessons, 1 hr 20 min of class participation. Grades: 4 to 8th grades
The students were introduced to different search engines and Web vocabulary. The instructor explained the concept behind web browsers. The students were able to give different examples of a web
browser. The instructor gave the students Internet Scavenger Hunts to get use to using search engines in a different way. The students tried to answer the questions without help from the instructor.
After the students were given time to find the answers, the instructor gave them some short cuts to help narrow down the search. The students tried to find more answers using some of the short cuts.
The students worked hard to answer the questions; therefore, the class played a game of Internet "Simon says." The instructor would say Simon says "jump up and down pat your head." The students would
perform both actions. Another example, "Simon says" kick you leg pat you head - kick you leg." The students would only pat their heads (See lesson plans). The class ended with fun and games. | {"url":"http://www.shodor.org/cyberpathways/archive2007/summer/WeaverStreet/overviews/week4","timestamp":"2014-04-18T16:03:41Z","content_type":null,"content_length":"7799","record_id":"<urn:uuid:4efbca7b-4185-4f18-97c8-b48404bb3580>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00278-ip-10-147-4-33.ec2.internal.warc.gz"} |
Heat transfer equation puzzling results
Using heat transfer equation to find out heat transfer rate but reached a puzzling result, where did I do wrong?
Problem statement:
Heat transfer rate of a heated plastic plate in air, considering only one side, the size of the plate is 15cm X 15cm, assuming temperature difference is 1K.
Formula to use:
Q = k × A × ΔT / d
Q: heat transfer rate in Watts
k: heat conductivity of plastic, 0.2 W/(mK)
A: area, 15cm x 15cm
ΔT: temperature difference, 1K
d: thickness of plastic, 2mm
Pluging in the numbers, Q=2.25W. For 1K temperature diff. the palm size plastic can transfer 2W???
The result is not reasonable. Since for 10K temperature diff. that palm size plastic plate can transfer 22.5W of power. How to include the low conductivity of air into calculation?
Thank you very much for your help. | {"url":"http://www.physicsforums.com/showthread.php?p=3868183","timestamp":"2014-04-18T13:46:55Z","content_type":null,"content_length":"22591","record_id":"<urn:uuid:db2c5584-4e3c-45a9-a22f-ae9cf858607d>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00333-ip-10-147-4-33.ec2.internal.warc.gz"} |
Read Practice: Word Problems text version
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
A Plan for Problem Solving
Use the four-step plan to solve each problem.
SKATEBOARDING For Exercises 1 and 2, use the table at the right. It shows the results of a recent survey in which teenagers were asked who the best professional skateboarder is.
Skater Bob Burnquist Danny Way Bam Margera Arto Saari
Votes 18 15 11 9
1. Estimate the total number of teenagers who voted.
2. How many more teenagers preferred Burnquist to Saari?
3. HISTORY The area of Manhattan Island is 641,000,000 square feet. According to legend, the Native Americans sold it to the Dutch for $24. Estimate the area that was purchased for one cent.
4. TRAVEL Britney's flight to Rome leaves New York City at 5:15 P.M. on Wednesday. The flight time is 7.5 hours. If Rome is 6 hours ahead of New York City, use Rome time to determine when she is scheduled to arrive.
5. OFFICE SUPPLIES At an office supply store, pens are $1.69 per dozen and note pads are $4.59 per dozen. Can Shirley buy 108 pens and 108 note pads for $50? Explain your reasoning.
6. SHOPPING Yoshi bought two pairs of shoes. The regular price of each pair was $108. With the purchase of one pair of shoes at regular price, the second pair was half price. How much did Yoshi pay altogether for the two pairs of shoes?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 11
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Variables, Expressions, and Properties
FOOTBALL For Exercises 1 and 2, use the table that shows statistics from the 2002 Super Bowl.
Team New England St. Louis
Touchdowns Extra Points Field Goals 2 2 2 2 2 1
1. Each team's final score for a football game can be found using the expression 6t e 3f, where t is the number of touchdowns, e is the number of extra points, and f is the number of field goals. Find New England's final score in the 2002 Super Bowl.
2. Use the expression 6t e 3f to find St. Louis's final score in the 2002 Super Bowl.
3. GEOMETRY The expression 6s2 can be used to find the surface area of a cube, where s is the lengh of an edge of the cube. Find the surface area of a cube with an edge of length 10 centimeters.
4. VERTICAL MOTION The height of an object dropped from the top of a 300foot tall building can be described by the expression 300 16t2, where t is the time, in seconds, after the ball is dropped. Find the height of the object 3 seconds after it is dropped.
10 cm
5. MOVIE RENTALS Mario intends to rent 10 movies for his birthday party. He can rent new releases for $4 each, while older ones are $2 each. If he rents n new releases, the total cost, in dollars, of the 10 movies is represented by the expression 4n 2(10 n). Evaluate the expression to find the total cost if he rents 7 new releases.
6. CIRCULAR MOTION Pelipa is able to spin her yo-yo along a circular path. The yo-yo is kept in this path by a force which can be described by the
2 expression mv . Evaluate the
expression to find the force when m 12, v 4, and r 8.
mv 2 r
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Integers and Absolute Value
GOLF For Exercises 1 and 2, use the table that lists ten players and
their scores in the 2002 Ladies Master Golf Tournament. Player Brooky, Lynnette Hjorth, Maria Jeong Jang King, Betsy Moodie, Janice Score 4 15 5 8 5 Player Neumann, Liselotte Park, Grace Se Ri Pak Sorenstam, Annika Tinning, Iben Score 0 10 14 19 3
1. Order the scores in the table from least to greatest.
2. Who had the lowest score?
3. LONGITUDE London, England, is located at 0° longitude. Write integers for the locations of New York City whose longitude is 74° west and Tokyo whose longitude is 140° east. Assume that east is the positive direction.
4. STOCK MARKET Your stock loses 53 points on Monday and 23 points on Tuesday, but gains 67 points on Wednesday. Write an integer for each day's change.
5. SOLAR SYSTEM The average temperature of Saturn is 218°F, while the average temperature of Jupiter is 162°F. Which planet has the lower average temperature?
6. OCEAN TRENCHES The elevation of the Puerto Rican Trench in the Atlantic Ocean is 8,605 meters, the elevation of the Mariana Trench in the Pacific Ocean is 10,924 meters, and the elevation of the Java Trench in the Indian Ocean is 7,125 meters. Which trench has the the lowest elevation?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 13
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Adding Integers
1. FOOTBALL A football team loses 5 yards on one play and then loses 8 yards on the next play. Write an addition expression that represents the change in position of the team for the two plays. Then find the sum. 2. ELEVATOR You park in a garage 3 floors below ground level. Then you get in the elevator and go up 12 floors. Write an addition expression to represent this situation. Then find the sum.
3. GOLF In 2002, Tiger Woods won the Masters Tournament. His scores were 2, 3, 6, and 1 for four rounds. Write an addition expression that represents his final score. Then find the sum.
4. INVENTORY A local bookstore has 30 copies of a bestseller when it opens Monday morning. On Monday, it sells 6 copies of the book. On Tuesday, it sells 3 copies. On Wednesday, it receives a shipment containing 24 copies of the book and also sells 8 copies. Write an addition expression that represents the number of copies of the book that store has at the end of the day on Wednesday. Then find the sum.
5. OCEANOGRAPHY A research team aboard an underwater research vessel descends 1,500 feet beneath the surface of the water. They then rise 525 feet and descend again 350 feet. Write an addition expression to represent this situation. Then find the sum.
6. SPORTS Peter weighs 156 pounds, but he would like to wrestle in a lower weight class. He loses 4 pounds one week, gains back 2 pounds the next week, loses 5 pounds the third week, and loses 3 pounds the fourth week. Write an addition expression to represent this situation. Then find the sum.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Subtracting Integers
elevations of several places on Earth. Place Mt. McKinley Puerto Rican Trench Mt. Everest Dead Sea Death Valley 1. Find the difference in elevation between the top of Mt. McKinley and and the top of Mt. Everest. Elevation (feet) 20,320 28,232 29,035 1,348 282 2. Find the difference in elevation between Death Valley and the Dead Sea.
3. TEMPERATURE The highest recorded temperature on Earth was recorded in Africa at 136°F, while the lowest was 129°F in Antarctica. What is the range of temperatures recorded on Earth?
4. WEATHER If the overnight temperature at the Arctic Circle was 14°F, but the temperature rose to 8°F during the day, what was the difference between these high and low temperatures?
5. WATER The boiling point of water is 212°F, while 460°F is its absolute lowest temperature. Find the difference between these two temperatures.
6. STOCK MARKET During the course of one day, the price of a stock fluctuated between a high of $3 above the previous day's closing price and a low of $2 below the previous day's closing price. What was the difference between the high and low prices for that day?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 15
GEOGRAPHY For Exercises 1 and 2, use the table. The table shows the
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Multiplying and Dividing Integers
1. STOCK MARKET The price of a stock decreased $2 per day for four consecutive days. What was the total change in value of the stock over the four-day period? 2. EVAPORATION The height of the water in a tank decreases 3 inches each week due to evaporation. What is the change in the height of the water over a fiveweek period due to evaporation?
3. FOOTBALL A football team lost 9 yards on each of three consecutive plays. What was the team's total change in position for the three plays?
4. HIKING A group of hikers is descending a mountain at a rate of 400 feet per hour. What is the change in the elevation of the hikers after 6 hours?
G 10 20 30 40 50 40 30 20 10 G
5. WEATHER On a certain day, the temperature changed at a rate of 2ºF per hour. How did long did it take for the change in temperature to be 14ºF?
14 F
6. GEOLOGY The length of an island is changing at the rate of 17 inches per year. How long will it take for the change in the length of the island to be 255 inches?
7. DEPRECIATION The value of a piece of office equipment is changing at a rate of $175 per year. How long will it take for the change in value to be $1,050?
8. POPULATION The population of a small town is changing at a rate of 255 people per year. How long will it take for the change in population to be 2,040 people?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Writing Expressions and Equations
1. AGE Julia is 3 years younger than Kevin. Define a variable and write an expression for Julia's age. 2. CIVICS In the 2000 presidential election, Texas had 21 more electoral votes than Tennessee. Define a variable and write an expression for the number of Texas's electoral votes.
number of curium; a
5. LIBRARIES The San Diego Public Library has 44 fewer branches than the Chicago Public Library. Define a variable and write an expression for the number of branches in the San Diego Public Library.
6. ASTRONOMY Saturn is 6 times further from the Sun than Mars. Define a variable and write an expression for the distance of Saturn from the Sun.
7. POPULATION The population of Oakland, California, is 5,417 less than the population of Omaha, Nebraska. Define a variable and write an expression for the population of Oakland.
8. GEOGRAPHY Kings Peak in Utah is 8,667 feet taller than Spruce Knob in West Virginia. Define a variable and write an expression for the height of Kings Peak.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 17
3. ENERGY One year, China consumed 4 times as much energy as Brazil. Define a variable and write an expression for the amount of energy China used that year.
4. CHEMISTRY The atomic number of cadmium is half the atomic number of curium. Define a variable and write an expression for the atomic number of cadmium.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Solving Addition and Subtraction Equations
1. AGE Walter lived 2 years longer than his brother Martin. Walter was 79 at the time of his death. Write and solve an addition equation to find Martin's age at the time of his death. 2. CIVICS New York has 21 fewer members in the House of Representatives than California. New York has 33 representatives. Write and solve a subtraction equation to find the number of California representatives.
3. GEOMETRY Two angles are supplementary if the sum of their measures is 180°. Angles A and B are supplementary. If the measure of angle A is 78°, write and solve an addition equation to find the measure of angle B.
4. BANKING After you withdraw $40 from your checking account, the balance is $287. Write and solve a subtraction equation to find your balance before this withdrawal.
B A
5. WEATHER After the temperature had risen 12°F, the temperature was 7°F. Write and solve an addition equation to find the 7 F starting temperature.
6. CHEMISTRY The atomic number of mercury is the sum of the atomic number of aluminum and 67. The atomic number of mercury is 80. Write and solve an addition equation to find the atomic number of aluminum.
7. ELEVATION The lowest point in Louisiana is 543 feet lower than the highest point in Louisiana. The elevation of the lowest point is 8 feet. Write and solve a subtraction equation to find the elevation of the highest point in Louisiana.
8. POPULATION The population of Honduras is the population of Haiti decreased by 618,397. The population of Honduras is 6,249,598. Write and solve a subtraction equation to find the population of Haiti.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Solving Multiplication and Division Equations
1. WAGES Felipe earns $9 per hour for helping his grandmother with her yard work. Write and solve a multiplication equation to find how many hours he must help his grandmother in order to earn $54. 2. SHOPPING Chocolate bars are on sale for $0.50 each. If Brad paid $5 for chocolate bars, write and solve a multiplication equation to find how many bars he bought.
3. EXERCISE Jasmine jogs 3 miles each day. Write and solve a multiplication equation to find how many days it will take her to jog 57 miles.
4. TRAVEL On a trip, the Rollins family drove at an average rate of 62 miles per hour. Write and solve a multiplication equation to find how long it took them to drive 558 miles.
5. ROBOTS The smallest robot can travel 20 inches per minute through a pipe. Write and solve a multiplication equation to find how long it will take this robot to travel through 10 feet of pipe.
6. BANKING Nate withdraws $40 from his checking account each day. Write and solve a multiplication equation to find how long it will take him to withdraw $680.
7. AGE The product of Bart's age and 26 is 338. Write and solve a multiplication equation to find Bart's age.
8. POPULATION The population of a small town is increasing at a rate of 325 people per year. Write and solve a multiplication equation to find how long it will take the population to increase by 6,825.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 19
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Fractions and Decimals
1. ASTRONOMY The pull of gravity on the surface of Mars is 0.38 that of Earth. Write 0.38 as a fraction in simplest form. 2. ENERGY Nuclear power provided 76% of the energy used in France in 2000. Write 0.76 as a fraction in simplest form.
3. WEIGHTS AND MEASURES One pint is about 0.55 liter. Write 0.55 liter as a fraction in simplest form.
4. WEIGHTS AND MEASURES One inch is 25.4 millimeters. Write 25.4 millimeters as a mixed number in simplest form.
5. EDUCATION A local middle school has 47 computers and 174 students. What is the number of students per computer at the school? Write your answer as both a mixed number in simplest form and a decimal rounded to the nearest tenth.
6. BASEBALL In the 2002 season, the Atlanta Braves won 101 out of 162 games. What was the ratio of wins to total games? Write your answer as both a fraction in simplest form and a decimal rounded to the nearest thousandth.
101 or 0.623 win per game 162
7. COLLEGES AND UNIVERSITIES Recently, a small college had an enrollment of 1,342 students and a total of 215 faculty. What was the student-faculty ratio for this college? Write your answer as both a mixed number in simplest form and a decimal rounded to the nearest hundredth.
8. BASKETBALL In the 20002001 season, Shaquille O'Neal made 813 field goals out of 1,422 attempts. What was Shaquille O'Neal's ratio of successful field goals to attempts? Write your answer as both a fraction in simplest form and a decimal rounded to the nearest thousandth.
271 or 0.572 success per 474
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 21
25 2 mm
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Comparing and Ordering Rational Numbers
1. BASKETBALL In the last ten games, Percy made 7 of his free throws. For 12 the same period, Tariq made 4 of his 7 free throws. Which player has the better free throw record? 2. SPORTS Central's baseball team won
53 of its games last year, while 78 Southern's team won 55 of its games. 81
Which team had the better record?
3. MEASUREMENT Beaker A contains 4 1 fluid ounces of water, while beaker 3 B contains 4 3 fluid ounces of water. 10 Which beaker has the smaller amount of water?
4. NATURE The two trees in Opal's back yard have circumferences of 12 5 inches and 12 3 inches. Which circumference is
12 5 in.
5. EXERCISE On Monday, Rob averaged 3.75 laps per minute. On Tuesday, he averaged 3 4 laps per minute. On which
6. FOOD Hector and Carla both gave apples to their teacher. Hector's apple weighed 6 7 ounces, while Carla's
day did Rob run faster?
apple weighed 6.65 ounces. Which apple weighed more?
7. SPORTS Christina ran one lap in 83.86 seconds, while Della's time for one lap was 83 7 seconds. Which runner had
the faster time?
8. STATISTICS The median of a set of numbers can be found by first putting the numbers in order from least to greatest, then choosing the middle number. Find the median of 5.79, 5 3 , 5 7 , 5.9, and 5 4 .
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Multiplying Rational Numbers
2. ELECTIONS In the last election, 3 of the
1. NUTRITION Maria's favorite candy bar has 230 Calories. The nutrition label states that 7 of the Calories come from
fat. How many Calories in the candy bar come from fat?
voters in Afton voted for the incumbent mayor. If 424 people voted in Afton in the last election, how many voted for the incumbent mayor?
3. HOBBIES Jerry is building a 1 scale
4. COOKING Enola's recipe for cookies calls for 2 1 cups of flour. If she wants
2 3 to make of a batch of cookies, how 4
model of a race car. If the tires on the actual car are 33 inches in diameter, what is the diameter of the tires on the model?
much flour should she use?
17 c
5. TRANSPORTATION Hana's car used 3 of
a tank of gas to cross Arizona. The gas tank on her car holds 15 1 gallons. How
6. GEOMETRY The area of a rectangle is found by multiplying its length times its width. What is the area of a rectangle with a length of 2 1 inches
many gallons of gas did it take to cross Arizona?
and a width of 1 5 inches? 9 1 2 3 in
7. COOKING A recipe for ice cream calls for 3 1 cups of heavy cream. If Steve
8. ADVERTISING A jewelry advertisement shows a diamond at 6 2 times its actual
wants to make 2 1 times the normal 2 amount, how much heavy cream should he use?
size. If the actual diameter of the diamond is 5 3 millimeters, what is
the diameter of the diamond in the photograph?
33 11 mm
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 23
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Dividing Rational Numbers
1. CONTAINER GARDENING One bag of potting soil contains 8 1 quarts of soil.
2. MUSIC Doug has a shelf 9 3 inches long
for storing CDs. Each CD is 3 inch
How many clay pots can be filled from one bag of potting soil if each pot holds
3 quart? 4
wide. How many CDs will fit on one shelf?
3. SERVING SIZE A box of cereal contains 15 3 ounces of cereal. If a bowl holds 5 2 2 ounces of cereal, how many bowls of 5 cereal are in one box?
4. HOME IMPROVEMENT Lori is building a path in her backyard using square paving stones that are 1 3 feet on each
side. How many paving stones placed end-to-end are needed to make a path that is 21 feet long?
5. GEOMETRY Given the length of a rectangle and its area, you can find the width by dividing the area by the length. A rectangle has an area of 6 2
6. GEOMETRY Given the length of a rectangle and its area, you can find the width by dividing the area by the length. A rectangle has an area of 4 5
square inches and a length of 2 1 2 inches. What is the width of the rectangle?
square feet and a length of 3 2 feet. 3 What is the width of the rectangle?
1 2 ft
8. YARD WORK Leon is mowing his yard, which is 21 2 feet wide. His lawn
7. HOBBIES Dena has a picture frame that is 13 1 inches wide. How many pictures 2 that are 3 3 inches wide can be placed 8 beside each other within the frame?
mower makes a cut that is 1 2 feet wide
on each pass. How many passes will Leon need to finish the lawn?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Adding and Subtracting Like Fractions
1. GEOMETRY Find the perimeter of a rectangle with a length of 4 2 inches 3 1 and a width of 3 inches. 3 2. PETS Pat wants to find out how much her dog Hunter weighs. Pat steps on the scale and reads her weight as 126 3
pounds. The combined weight of Pat and Hunter is 137 7 pounds. How much
does Hunter weigh?
11 1 lb
3. MEASUREMENTS Tate fills a 13 1 ounce
4. DECORATING Jeri has two posters. One is 4 7 feet wide and the other is 5 1
glass from a 21 2 ounce bottle of juice.
How much juice is left in the bottle?
feet wide. Will the two posters fit beside each other on a wall that is 10 feet wide? Explain.
Yes; 4 7
5. AGE Nida is 11 1 years old, while her
12 5 sister Yoki is 8 years old. What is 12
6. GEOMETRY A triangle has sides of 1 1 inches, 1 3 inches, and 1 5 inches.
the sum of the ages of the sisters?
What is the perimeter of the triangle?
4 1 in.
7. HUMAN BODY Tom's right foot measures 10 2 inches, while Randy's 5 right foot measures 9 4 inches. How 5 much longer is Tom's foot than Randy's?
8. COMPUTERS Trey has two data files on his computer that he is going to combine. One file is 1 4 megabytes,
while the other file is 3 8 megabytes.
What will be the size of the resulting file?
5 1 megabytes
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 25
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Adding and Subtracting Unlike Fractions
1. GEOMETRY Two line segments have lengths of 3 1 inches and 1 1 inches. 4 3 What is the sum of the lengths of the two line segments?
2. COMPUTERS The biology class has created two data files on the computer. One file is 2 1 megabytes, while the
other file is 4 1 megabytes. How much
larger is the second file than the first?
2 7 megabytes
3. HUMAN BODY The index finger on Pablo's right hand measures 3 3 inches,
4. DECORATING Sugi has two pictures that she wants to put beside each other in a frame. One is 3 1 inches wide and the
2 1 other is 5 inches wide. How wide 8
while the index finger on his left hand measures 3 5 inches. Which hand has 16 the longer index finger? How much longer is it?
must the frame be to fit both pictures?
8 5 in.
5. PETS Laura purchased two puppies from a litter. One of the puppies weighs 4 5 pounds and the other puppy weighs 6 5 1 pounds. How much more does the 2 second puppy weigh than the first?
6. AGE Alma is 6 3 years old, while her
brother David is 3 5 years old. What is
the sum of the ages of Alma and David?
10 7 years
7. MEASUREMENT Ned pours 7 2 ounces of
8. GEOMETRY A triangle has sides of 1 1 inches, 1 1 inches, and 1 2 inches.
water from a beaker containing 10 1 ounces. How much water is left in
What is the perimeter of the triangle?
the beaker?
4 1 in.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Solving Equations with Rational Numbers
1. NATURE The height of a certain tree is 12.85 meters. The length of its longest branch can be found using the equation 3.23 12.85. Solve the equation. 2. SHOPPING Kristen went shopping and spent $84.63 on books and CDs. The equation 84.63 b 43.22 can be used to determine the amount b that she spent on books. Solve the equation.
3. ENERGY PRICES Suppose regular unleaded gasoline costs $1.40 per gallon. The price p of premium gasoline can be found using the equation
p 1.2
1.40. What is the price of the
premium gasoline?
5. AUTOMOBILES The bed of Julian's truck is 2 1 yards long. The length 3 equation 24
6. SPORTS Leo and Ted both ran in a race. Leo's time was 9 minutes, which was 3
of the
truck can be found by solving the 2 1 . What is the length of the truck?
of Ted's time. Using t for Ted's time, write a multiplication equation to represent the situation.
3 t 4
7. SPEED Ella rode the bus to work today. The distance she traveled was 4 1 miles 4 and the ride took 1 of an hour. The 3 1 1 equation s 4 can be used to find 3 4 the average speed s of the bus. What was the average speed of the bus?
8. GEOMETRY A rectangle has area 6 2 square inches and length 2 1 inches.
The equation 6 2 3
2 1 2 w can be used to 2
find the width w of the rectangle. Solve the equation.
2 2 in.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 27
4. DRIVING TIME Sam went for a drive last Sunday. His average speed was 46 miles per hour and he drove 115 miles. The equation 115 46t can be used to find the time t that he spent driving. Solve the equation.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Powers and Exponents
1. SPORTS In the first round of a local tennis tournament there are 25 matches. Find the number of matches.
2. GEOMETRY The volume of a box can be found by multiplying the length, width, and height of the box. If the length, width, and height of the box are all 5 inches, write the volume of the box using an exponent.
3. MONEY An apartment complex has 3 buildings. Each building has 3 apartments. There are 3 people living in each apartment, and each person pays 3 dollars per month for pool maintenance. The expression 34 denotes the amount paid each month for pool maintenance. Find this amount.
4. ACTIVISM A petition drive is being held in 10 cities. In each city, 10 people have collected 10 signatures each. The expression 103 denotes the number of signatures that have been collected altogether. Find this number.
5. MEASUREMENT There are 106 millimeters in a kilometer. Write the number of millimeters in a kilometer.
6. NATURE Suppose a certain forest fire doubles in size every 12 hours. If the initial size of the fire was 1 acre, how many acres will the fire cover in 2 days?
7. BANKING Suppose that a dollar placed into an account triples every 12 years. How much will be in the account after 60 years?
8. BIOLOGY Suppose a bacterium splits into two bacteria every 15 minutes. How many bacteria will there be in 3 hours?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Scientific Notation
1. MEASUREMENT There are about 25.4 millimeters in one inch. Write this number in scientific notation. 2. POPULATION In the year 2000, the population of Rahway, New Jersey, was 26,500. Write this number in scientific notation.
3. MEASUREMENT There are 5,280 feet in one mile. Write this number in scientific notation.
4. PHYSICS The speed of light is about 1.86 105 miles per second. Write this number in standard notation.
5. COMPUTERS A CD can store about 650,000,000 bytes of data. Write this number in scientific notation.
6. SPACE The diameter of the Sun is about 1.39 109 meters. Write this number in standard notation.
7. ECONOMICS The U.S. Gross Domestic Product in the year 2000 was 9.87 1012 dollars. Write this number in standard notation.
8. MASS The mass of planet Earth is about 5.98 1024 kilograms. Write this number in standard notation.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 29
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Square Roots
1. PLANNING Rosy wants a large picture window put in the living room of her new house. The window is to be square with an area of 49 square feet. How long should each side of the window be? 2. GEOMETRY If the area of a square is 1 square meter, how many centimeters long is each side?
3. ART A miniature portrait of George Washington is square and has an area of 169 square centimeters. How long is each side of the portrait?
5. ART Cara has 196 marbles that she is using to make a square formation. How many marbles should be in each row?
6. GARDENING Tate is planning to put a square garden with an area of 289 square feet in his back yard. What will be the length of each side of the garden?
7. HOME IMPROVEMENT Al has 324 square paving stones that he plans to use to construct a square patio. How many paving stones wide will the patio be?
8. GEOMETRY If the area of a square is 529 square inches, what is the length of a side of the square?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 31
4. BAKING Len is baking a square cake for his friend's wedding. When served to the guests, the cake will be cut into square pieces 1 inch on a side. The cake should be large enough so that each of the 121 guests gets one piece. How long should each side of the cake be?
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Estimating Square Roots
1. GEOMETRY If the area of a square is 29 square inches, estimate the length of each side of the square to the nearest whole number. 2. DECORATING Miki has an square rug in her living room that has an area of 19 square yards. Estimate the length of a side of the rug to the nearest whole number.
3. GARDENING Ruby is planning to put a square garden with an area of 200 square feet in her back yard. Estimate the length of each side of the garden to the nearest whole number.
4. ALGEBRA Estimate the solution of c2 40 to the nearest integer.
5. ALGEBRA Estimate the solution of x2 138.2 to the nearest integer.
6. ARITHMETIC The geometric mean of two numbers a and b can be found by evaluating a b. Estimate the geometric mean of 5 and 10 to the nearest whole number.
7. GEOMETRY The radius r of a certain circle is given by r 71. Estimate the radius of the circle to the nearest foot.
8. GEOMETRY In a triangle whose base and height are equal, the base b is given by the formula b 2A, where A is the area of the triangle. Estimate to the nearest whole number the base of this triangle if the area is 17 square meters.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
The Real Number System
1. GEOMETRY If the area of a square is 33 square inches, estimate the length of a side of the square to the nearest tenth of an inch. 2. GARDENING Hal has a square garden in his back yard with an area of 210 square feet. Estimate the length of a side of the garden to the nearest tenth of a foot.
3. ALGEBRA Estimate the solution of a2 21 to the nearest tenth.
4. ALGEBRA Estimate the solution of b2 67.5 to the nearest tenth.
5. ARITHMETIC The geometric mean of two numbers a and b can be found by evaluating a b. Estimate the geometric mean of 4 and 11 to the nearest tenth.
6. ELECTRICITY In a certain electrical circuit, the voltage V across a 20 ohm resistor is given by the formula V 20P, where P is the power dissipated in the resistor, in watts. Estimate to the nearest tenth the voltage across the resistor if the power P is 4 watts.
7. GEOMETRY The length s of a side of a cube is related to the surface area A of the cube by the formula s
A . If the 6
8. PETS Alicia and Ella are comparing the weights of their pet dogs. Alicia's reports that her dog weighs 11 1
surface area is 27 square inches, what is the length of a side of the cube to the nearest tenth of an inch?
pounds, while Ella says that her dog weighs 125 pounds. Whose dog weighs more?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 33
NAME ________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
The Pythagorean Theorem
1. ART What is the length of a diagonal of a rectangular picture whose sides are 12 inches by 17 inches? Round to the nearest tenth of an inch. 2. GARDENING Ross has a rectangular garden in his back yard. He measures one side of the garden as 22 feet and the diagonal as 33 feet. What is the length of the other side of his garden? Round to the nearest tenth of a foot.
3. TRAVEL Troy drove 8 miles due east and then 5 miles due north. How far is Troy from his starting point? Round the answer to the nearest tenth of a mile.
4. GEOMETRY What is the perimeter of a right triangle if the hypotenuse is 15 centimeters and one of the legs is 9 centimeters?
5. ART Anna is building a rectangular picture frame. If the sides of the frame are 20 inches by 30 inches, what should the diagonal measure? Round to the nearest tenth of an inch.
6. CONSTRUCTION A 20-foot ladder leaning against a wall is used to reach a window that is 17 feet above the ground. How far from the wall is the bottom of the ladder? Round to the nearest tenth of a foot.
7. CONSTRUCTION A door frame is 80 inches tall and 36 inches wide. What is the length of a diagonal of the door frame? Round to the nearest tenth of an inch.
8. TRAVEL Tina measures the distances between three cities on a map. The distances between the three cities are 45 miles, 56 miles, and 72 miles. Do the positions of the three cities form a right triangle?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Using The Pythagorean Theorem
1. RECREATION A pool table is 8 feet long and 4 feet wide. How far is it from one corner pocket to the diagonally opposite corner pocket? Round to the nearest tenth. 2. TRIATHLON The course for a local triathlon has the shape of a right triangle. The legs of the triangle consist of a 4-mile swim and a 10-mile run. The hypotenuse of the triangle is the biking portion of the event. How far is the biking part of the triathlon? Round to the nearest tenth if necessary.
3. LADDER A ladder 17 feet long is leaning against a wall. The bottom of the ladder is 8 feet from the base of the wall. How far up the wall is the top of the ladder? Round to the nearest tenth if necessary.
4. TRAVEL Tara drives due north for 22 miles then east for 11 miles. How far is Tara from her starting point? Round to the nearest tenth if necessary.
5. FLAGPOLE A wire 30 feet long is stretched from the top of a flagpole to the ground at a point 15 feet from the base of the pole. How high is the flagpole? Round to the nearest tenth if necessary.
6. ENTERTAINMENT Isaac's television is 25 inches wide and 18 inches high. What is the diagonal size of Isaac's television? Round to the nearest tenth if necessary.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 35
NAME ________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Distance on the Coordinate Plane
1. ARCHAEOLOGY An archaeologist at a dig sets up a coordinate system using string. Two similar artifacts are found--one at position (1, 4) and the other at (5, 2). How far apart were the two artifacts? Round to the nearest tenth of a unit if necessary. 2. GARDENING Vega set up a coordinate system with units of feet to locate the position of the vegetables she planted in her garden. She has a tomato plant at (1, 3) and a pepper plant at (5, 6). How far apart are the two plants? Round to the nearest tenth if necessary.
3. CHESS April is an avid chess player. She sets up a coordinate system on her chess board so she can record the position of the pieces during a game. In a recent game, April noted that her king was at (4, 2) at the same time that her opponent's king was at (7, 8). How far apart were the two kings? Round to the nearest tenth of a unit if necessary.
4. MAPPING Cory makes a map of his favorite park, using a coordinate system with units of yards. The old oak tree is at position (4, 8) and the granite boulder is at position ( 3, 7). How far apart are the old oak tree and the granite boulder? Round to the nearest tenth if necessary.
5. TREASURE HUNTING Taro uses a coordinate system with units of feet to keep track of the locations of any objects he finds with his metal detector. One lucky day he found a ring at (5, 7) and a old coin at (10, 19). How far apart were the ring and coin before Taro found them? Round to the nearest tenth if necessary.
6. GEOMETRY The coordinates of points A and B are ( 7, 5) and (4, 3), respectively. What is the distance between the points, rounded to the nearest tenth?
7. GEOMETRY The coordinates of points A, B, and C are (5, 4), ( 2, 1), and (4, 4), respectively. Which point, B or C, is closer to point A?
8. THEME PARK Tom is looking at a map of the theme park. The map is laid out in a coordinate system. Tom is at (2, 3). The roller coaster is at (7, 8), and the water ride is at (9, 1). Is Tom closer to the roller coaster or the water ride?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Ratios and Rates
1. COOKING In a bread dough recipe, there are 3 eggs for every 9 cups of flour. Express this ratio in simplest form. 2. WILDLIFE Dena counted 14 robins out of 150 birds. Express this ratio in simplest form.
3. INVESTMENTS Josh earned dividends of $2.16 on 54 shares of stock. Find the dividends per share.
4. TRANSPORTATION When Denise bought gasoline, she paid $18.48 for 11.2 gallons. Find the price of gasoline per gallon.
5. WATER FLOW Jacob filled his 60-gallon bathtub in 5 minutes. How fast was the water flowing?
6. TRAVEL On her vacation, Charmaine's flight lasted 4.5 hours. She traveled 954 miles. Find the average speed of the plane.
7. HOUSING Mr. And Mrs. Romero bought a 1,200 square-foot house for $111,600. How much did they pay per square foot?
8. SHOPPING A breakfast cereal comes in two different sized packages. The 8-ounce box costs $2.88, while the 12-ounce box costs $3.60. Which box is the better buy? Explain your reasoning.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 41
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Rate of Change
ELECTIONS For Exercises 13, use the table that shows the total number
of people who had voted in District 5 at various times on election day. Time Number of Voters 8:00 A.M. 141 10:00 A.M. 351 1:00 P.M. 798 4:30 P.M. 1,008 7:00 P.M. 1,753
1. Find the rate of change in the number of voters between 8:00 A.M. and 10:00 A.M. Then interpret its meaning.
2. Find the rate of change in the number of voters between 10:00 A.M. and 1:00 P.M. Then interpret its meaning.
3. During which of these two time periods did the number of people who had voted so far increase faster? Explain your reasoning.
4. MUSIC At the end of 1999, Candace had 47 CDs in her music collection. At the end of 2002, she had 134 CDs. Find the rate of change in the number of CDs in Candace's collection between 1999 and 2002.
5. FITNESS In 1992, the price of an annual membership at Mr. Jensen's health club was $225. In 2002, the price of the same membership was $319.50. Find the rate of change in the price of the annual membership between 1992 and 2002.
6. HIKING Last Saturday Fumio and Kishi went hiking in the mountains. When they started back at 2:00 P.M., their elevation was 3,560 feet above sea level. At 6:00 P.M., their elevation was 2,390 feet. Find the rate of change of their elevation between 2:00 P.M. and 6:00 P.M. Then interpret its meaning.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
FLOWERS For Exercises 1 and 2, use the graph that shows the depth of the water in a vase of flowers over 8 days. LONG DISTANCE For Exercises 36, use
the graph that compares the costs of long distance phone calls with three different companies.
Long Distance Charges
2.50 y 2.00 Company A Company B
Depth of Water in Vase
10 y 9 8 7 6 5 4 3 2 1 0
Depth (in.)
Cost ($)
1.50 1.00 0.50
Company C
Length of Call (minutes)
1. Find the slope of the line.
2. Interpret the meaning this slope as a rate of change.
3. Find the slope of the line for Company A. Then interpret this slope as a rate of change.
4. Find the slope of the line for Company B. Then interpret this slope as a rate of change.
5. Find the slope of the line for Company C. Then interpret this slope as a rate of change.
6. Which company charges the least for each additional minute? Explain your reasoning.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 43
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Solving Proportions
1. USAGE A 12-ounce bottle of shampoo lasts Enrique 16 weeks. How long would you expect an 18-ounce bottle of the same brand to last him? 2. COMPUTERS About 13 out of 20 homes have a personal computer. On a street with 60 homes, how many would you expect to have a personal computer?
3. SNACKS A 6-ounce package of fruit snacks contains 45 pieces. How many pieces would you expect in a 10-ounce package?
4. TYPING Ingrid types 3 pages in the same amount of time that Tanya types 4.5 pages. If Ingrid and Tanya start typing at the same time, how many pages will Tanya have typed when Ingrid has typed 11 pages?
5. SCHOOL A grading machine can grade 48 multiple-choice tests in 1 minute. How long will it take the machine to grade 300 tests?
6. AMUSEMENT PARKS The waiting time to ride a roller coaster is 20 minutes when 150 people are in line. How long is the waiting time when 240 people are in line?
7. PRODUCTION A shop produces 39 wetsuits every 2 weeks. How long will it take the shop to produce 429 wetsuits?
8. FISH Of the 50 fish that Jim caught from the lake, 14 were trout. The estimated population of the lake is 7,500 fish. About how many trout would you expect to be in the lake?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Similar Polygons
1. JOURNALISM The editor of the school newspaper must reduce the size of a graph to fit in one column. The original graph is 2 inches by 2 inches, and the scale factor from the original to the reduced graph is 8:3. Find the dimensions of the graph as it will appear in one column of the newspaper. 2. PHOTOCOPIES Lydia plans to use a photocopy machine to increase the size of a small chart that she has made as part of her science project. The original chart is 4 inches by 5 inches. If she uses a scale factor of 5:11, will the chart fit on a sheet of paper 8 1 inches 2 by 11 inches? Explain.
No; it will be 8 4 in. by 11 in.
3. MICROCHIPS The image of a microchip in a projection microscope measures 8 inches by 10 inches. The width of the actual chip is 4 millimeters. How long is the chip?
4. PROJECTIONS A drawing on a transparency is 11.25 centimeters wide by 23.5 centimeters tall. The width of the image of drawing projected onto a screen is 2.7 meters. How tall is the drawing on the screen?
5. GEOMETRY Polygon ABCD is similar to polygon FGHI. Each side of polygon ABCD is 3 1 times longer than the 4 corresponding side of polygon FGHI. Find the perimeter of polygon FGHI.
C B
6. KITES A toy company produces two kites whose shapes are geometrically similar. Find the length of the missing side of the smaller kite.
25 in. 25 in.
30 in. 30 in. 2 in. H 5 in. 3 in. 22.5 in.
G A
3 in.
F I D
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 45
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Scale Drawings and Models
CAMPUS PLANNING For Exercises 13, use the following information.
View of Campus from Above Gymnasium New Building
The local school district has made a scale model of the campus of Engels Middle School including a proposed new building. The scale of the model is 1 inch 3 feet.
Academic Building
1. An existing gymnasium is 8 inches tall in the model. How tall is the actual gymnasium?
2. The new building is 22.5 inches from the gymnasium in the model. What will be the actual distance from the gymnasium to the new building if it is built?
3. What is the scale factor of the model?
4. MAPS On a map, two cities are 5 3 4 inches apart. The scale of the map is 1 inch 3 miles. What is the actual 2 distance between the towns?
34 1 mi
5. TRUCKS The bed of Jerry's pickup truck is 6 feet long. On a scale model of the truck, the bed is 8 inches long. What is the scale of the model?
6 ft
6. HOUSING Marta is making a scale drawing of her apartment for a school project. The apartment is 28 feet wide. On her drawing, the apartment is 7 inches wide. What is the scale of Marta's drawing?
28 ft
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Indirect Measurement
1. HEIGHT Paco is 6 feet tall and casts a 12-foot shadow. At the same time, Diane casts an 11-foot shadow. How tall is Diane? 2. LIGHTING If a 25-foot-tall house casts a 75-foot shadow at the same time that a streetlight casts a 60-foot shadow, how tall is the streetlight?
3. FLAGPOLE Lena is 5 1 feet tall and casts 2 an 8-foot shadow. At the same time, a flagpole casts a 48-foot shadow. How tall is the flagpole?
5. NATIONAL MONUMENTS A 42-foot flagpole near the Washington Monument casts a shadow that is 14 feet long. At the same time, the Washington Monument casts a shadow that is 185 feet long. How tall is the Washington Monument?
6. ACCESSIBILITY A ramp slopes upward from the sidewalk to the entrance of a building at a constant incline. If the ramp is 2 feet high when it is 5 feet from the sidewalk, how high is the ramp when it is 7 feet from the sidewalk?
2 ft 5 ft
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson X1 47
4. LANDMARKS A woman who is 5 feet 5 inches tall is standing near the Space Needle in Seattle, Washington; she casts a 13-inch shadow at the same time that the Space Needle casts a 121-foot shadow. How tall is the Space Needle?
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. EYES Dave's optometrist used medicine to dilate his eyes. Before dilation, his pupils had a diameter of 4.1 millimeters. After dilation, his pupils had a diameter of 8.2 millimeters. What was the scale factor of the dilation?
2. BIOLOGY A microscope increases the size of objects by a factor of 8. How large will a 0.006 millimeter paramecium appear?
3. PHOTOGRAPHY A photograph was enlarged to a width of 15 inches. If the scale factor was 3 , what was the width 2 of the original photograph?
4. MOVIES Film with a width of 35 millimeters is projected onto a screen where the width is 5 meters. What is the scale factor of this enlargement?
1,000 7
5. PHOTOCOPYING A 10-inch long copy of a 2.5-inch long figure needs to be made with a copying machine. What is the appropriate scale factor?
6. MODELS A scale model of a boat is going to be made using a scale of 1 . 50 If the original length of the boat is 20 meters, what is the length of the model?
7. MODELS An architectural model is 30 inches tall. If the scale used to build the model is 1 , what is the height of 120 the actual building?
8. ADVERTISING An advertiser needs a 4-inch picture of a 14-foot automobile. What is the scale factor of the reduction?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Ratios and Percents
1. PETS Three out of every 20 dogs in the U.S. are Golden Retrievers. Write this ratio as a percent. 2. GEOGRAPHY About 29% of the world's surface is covered by land. Write this percent as a fraction.
3. BASKETBALL Shaquille O'Neal of the L.A. Lakers hit 11 out of 20 free throws in a 5-game series. Write this number as a percent.
4. EDUCATION In 2000, about 44% of 21-year-olds in the United States were enrolled in school. Write this percent as a fraction.
5. HEALTH CARE In 2000, 14% of Americans did not have health insurance. Write this percent as a fraction.
6. ENERGY In 2001, Japan accounted for about 8% of the world's petroleum consumption. Write this percent as a fraction.
7. GEOGRAPHY The federal government owns about 13 of the land in the state 20 of Utah. Write this fraction as a percent.
8. POPULATION In 2000, 11 out of every 50 people in the United States were age 65 or older. Write this ratio as a percent.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 51
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Fractions, Decimals, and Percents
1. BASKETBALL In the 20012002 season, Susan Bird of the WNBA team the Seattle Storm made 27% of her 3-point shots. Write this percent as a decimal. 2. POPULATION From 1990 to 2000, the population of Las Vegas, Nevada, increased by 85%. Write this percent as a decimal.
3. BASEBALL In the 2001 season, the Chicago White Sox had a team batting average of 0.268. Write this decimal as a percent.
4. HEALTH In 2000, 11.6% of Americans under the age of 18 were without health insurance. Write this percent as a decimal.
5. INTERNET Internet access in the U.S. has increased dramatically in recent years. In 2000, 83 out of every 200 households had Internet access. What percent of households had Internet access?
6. VOTING The rate of voter turnout in the 1932 U.S. presidential election was 0.524. Write this decimal as a percent.
7. ECONOMICS Consumer prices in the U.S. rose at a rate of 0.034 from 1999 to 2000. Write this decimal as a percent.
8. SPORTS In the 2001 season, the WNBA Cleveland Rockets won 22 of their
games. Write this fraction as a percent.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
The Percent Proportion
1. COMMUTING On his trip across town, Mark was stopped by a red light at 9 out of 15 intersections. At what percent of intersections was Mark stopped by a red light? 2. CLIMATE In Las Vegas, Nevada, the skies are clear on 92% of the days. How many days in the month of June would you expect the skies to be clear in Las Vegas? Round the answer to the nearest day.
3. POLLING A recent poll shows that 65% of adults are in favor of increased funding for education. The number of adults surveyed for the poll was 140. How many of the adults surveyed were in favor of increased funding for education?
4. FLOWERS Mika's rosebush had 24 blooms in the first week of May. This was 80% as many blooms as Tammy's rosebush had during the same period. How many blooms did Tammy's rosebush have?
5. SPORTS In the 2002 regular season, the San Francisco Giants won 95 out of 161 games. What percent of their games did they win? Round to the nearest tenth if necessary.
6. GOLF On a recent round of golf, Shana made par on 15 out of 18 holes. On what percent of holes did Shana make par? Round to the nearest tenth if necessary.
7. DRIVING TEST On the written portion of her driving test, Sara answered 84% of the questions correctly. If Sara answered 42 questions correctly, how many questions were on the driving test?
8. EDUCATION In a certain small town, 65% of the adults are college graduates. How many of the 240 adults living in the town are college graduates?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 53
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Finding Percents Mentally
1. ELECTIONS In a certain small town, 80% of the adults voted in the last election. How many of the 600 adults living in the town voted in the last election? 2. FISH POPULATION Fish and game managers have determined that 10% of the approximately 3,400 fish in Avondale Lake are catfish. How many catfish are there in Avondale Lake?
3. SURVEYS In a recent survey, 1% of the people had no opinion on the topic. How many of the 1,100 people surveyed had no opinion on the topic?
4. BAND In a local middle school, 33 1 % of
the students are in the band. There are 240 students in the school. How many middle school students are in the band?
5. AIR TRAVEL At one large international airport in the U.S., 20% of the arriving flights are from other countries. On a recent day, 240 flights arrived at the airport. How many of these flights were from other countries?
6. TELEPHONE Ramona likes to keep track of her incoming calls. Last month, 25% of the 132 calls Ramona received were from telemarketers. How many calls did Ramona get from telemarketers last month?
7. FARMING Jake grows corn and soybeans on his farm. He has corn growing on 66 2 % of his 330 acres. How many acres 3 are being used for corn?
8. ENERGY The U.S. has 25% of the nuclear power plants in the world. How many of the world's 416 nuclear power plants are in the U.S.?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Percent and Estimation
1. FITNESS At the office where Michael works, 8 out of 17 employees work out at least twice a week. Estimate the percent of employees that work out at least twice a week. 2. PETS Niki asked 25 of her classmates about what pets they have at home. Eleven of the 25 said they had both a cat and a dog. Estimate the percent of Niki's classmates that have both a cat and a dog.
3. BOOKS Jorge has read 19 novels this year, 4 of which were science fiction. Estimate the percent of novels that were science fiction.
4. PARKS The students in Kara's eighth grade science class determined that 9 out of 33 trees at a local park are pine trees. Estimate the percent of pine trees at the park.
5. BAND The marching band at Durango High School has 120 members. Of these, 18% are ninth-grade students. Estimate the number of ninth-grade students in the marching band.
6. RESTAURANTS In one east-coast city, 35% of the restaurants in the city are on the bay. The city has 180 restaurants. Estimate the number of restaurants that are on the bay.
7. HOTELS At the Westward Inn hotel, 48% of the rooms face the courtyard. The hotel has 91 rooms. Estimate the number of rooms that face the courtyard.
8. FARMING Roy has planted soybeans on 68% of his farm this year. Roy's farm has 598 acres of land. Estimate the number of acres of soybeans that Roy has this year.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 55
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
The Percent Equation
1. DINING OUT Trevor and Michelle's restaurant bill comes to $35.50. They are planning to tip the waiter 20%. How much money should they leave for a tip? 2. CHESS The local chess club has 60 members. Twenty-four of the members are younger than twenty. What percent of the members of the chess club are younger than twenty?
3. TENNIS In the city of Bridgeport, 75% of the parks have tennis courts. If 18 parks have tennis courts, how many parks does Bridgeport have altogether?
4. COLLEGE There are 175 students in twelveth grade at Silverado High School. A survey shows that 64% of them are planning to attend college. How many Silverado twelveth grade students are planning to attend college?
5. BASEBALL In the 2001 season, the Chicago Cubs won 88 out of 162 games. What percent of games did the Cubs win? Round to the nearest tenth if necessary.
6. HOUSING In the Lakeview apartment complex, 35% of the apartments have one bedroom. If there are 63 one bedroom apartments, what is the total number of apartments at Lakeview?
7. FOOTBALL In the 2000 season, quarterback Jeff Blake of the New Orleans Saints had 9 passes intercepted out of 302 attempts. What percent of Jeff Blake's passes were intercepted? Round to the nearest tenth if necessary.
8. SPACE On Mars, an object weighs 38% as much as on Earth. How much would a person who weighs 150 pounds on Earth weigh on Mars?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Percent of Change
1. CLUBS Last year the chess club had 20 members. This year the club has 15 members. Find the percent of change, and state whether the percent of change is an increase or a decrease. 2. READING During Todd's junior year in high school, he read 15 books. In his senior year, he read 18 books. Find the percent of change, and state whether the percent of change is an increase or a decrease.
3. COMPUTERS The computer store pays $250 each for flat screen monitor. The store uses a 30% markup. Find the selling price for each flat screen monitor.
5. CLOTHING Sandy's Clothing Shop has a markup of 45% on dresses. How much will Sandy's charge for a dress that costs the shop $48?
6. AUDIO The audio store is having a 20% off sale. What will be the sale price on a pair of speakers that normally sell for $280.00?
7. FURNITURE Leta is planning to buy a new sofa as soon as it goes on sale. The regular price for the sofa is $899.95. How much will the sofa cost if it goes on sale for 40% off? Round to the nearest cent.
8. AUTO REPAIR Don is getting a new set of tires for his car. The tires normally sell for $319.96, but they are on sale for 10% off. How much will Don pay for the new tires? Round to the nearest cent.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 57
4. SHOES A popular brand of running shoes costs a local store $68 for each pair. Find the selling price for a pair of running shoes if the store has a markup of 75%.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Simple Interest
1. SAVINGS ACCOUNT How much interest will be earned in 3 years from $730 placed in a savings account at 6.5% simple interest? 2. INVESTMENTS Terry's investment of $2,200 in the stock market earned $528 in two years. Find the simple interest rate for this investment.
3. SAVINGS ACCOUNT Lonnie places $950 in a savings account that earns 5.75% simple interest. Find the total amount in the account after four years.
4. INHERITANCE William's inheritance from his great uncle came to $225,000 after taxes. If William invests this money in a savings account at 7.3% interest, how much will he earn from the account each year?
5. RETIREMENT Han has $410,000 in a retirement account that earns $15,785 each year. Find the simple interest rate for this investment.
6. COLLEGE FUND When Melissa was born, her parents put $8,000 into a college fund account that earned 9% simple interest. Find the total amount in the account after 18 years.
7. MONEY Jessica won $800,000 in a state lottery. After paying $320,000 in taxes, she invested the remaining money in a savings account at 4.25% interest. How much interest will she receive from her investment each year?
8. SAVINGS Mona has an account with a balance of $738. She originally opened the account with a $500 deposit and a simple interest rate of 5.6%. If there were no deposits or withdrawals, how long ago was the account opened?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Line and Angle Relationships
1. SIGN The support wire for a sign meets the wall and the overhang as shown below. If m 2 42°, find m 1. Explain your reasoning. 2. AIRPORTS The runways at a local airport are laid out as shown below. Runways A and B are parallel, and runway C cuts across A and B. If m 1 55°, find m 2. Explain your reasoning.
m 2
1 and
2 are
3. RAILROADS East of the town of Rockport, the railroad tracks intersect Highway 67 as shown below. If m 1 133°, find m 2. Explain your reasoning.
4. CAMPING Jonna and Elizabeth found a level campsite and pitched their tent as shown below. If m 1 120°, find m 2. Explain your reasoning.
m 2
1 and
2 are
5. ALPHABET The top and bottom segments of the letter Z are parallel as shown below. If m 1 43°, find m 2. Explain your reasoning.
6. FLOORING Garret is designing a floor with diamond-shaped tiles as shown below. If m 1 125°, find m 2. Explain your reasoning.
m 2
1 and
2 are
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson X1 61
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Triangles and Angles
MAPS For Exercises 1 and 2, use the figure that shows the towns of Lakeview, Peoria, and Alton.
20 mi
Alton 58° Lakeview
20 mi
58° Peoria
1. The three towns form a triangle. Classify the triangle by its angles and by its sides.
2. Find the value of x in the figure. 64
3. FITNESS The running path around the lake shown in the figure is triangular. Classify the triangle by its angles and by its sides.
4. FITNESS Refer to the triangular running track shown in Exercise 3. Find the value of x.
335 ft
500 ft
372 ft
5. HIKING The trail shown in the figure is triangular. Find the value of x in the figure.
overlook 29° 29°
6. HIKING Refer to the triangular trail shown in Exercise 5. Classify the triangle by its angles and by its sides.
trail head
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Special Right Triangles
1. SHADOWS The shadow cast by a pole forms a 30°-60° right triangle, as shown below. What is the length of the shadow? 2. SHADOWS Refer to the figure shown in Exercise 1. What is the height h of the pole? Round to the nearest tenth.
18 ft
3. MAPS The towns of Oakland and Summit are linked by a highway and by a railroad, as shown below. What is the length of the section of highway between Oakland and the intersection?
Oakland 45°
4. MAPS Refer to the figure in Exercise 3. What is the length d of the section of railroad linking the towns? Round to the nearest tenth.
45° Summit 3 mi
Platform 60° 4 ft Ramp 30°
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 63
5. ACCESSIBILITY A ramp is to be constructed to a platform 4 feet above the ground. The triangle formed by the ramp, the ground, and the platform is a 30°-60° right triangle. Find the length of the ramp.
6. ACCESSIBILITY Refer to the ramp described in Exercise 5. What is the distance d from the foot of the ramp to the platform? Round to the nearest tenth.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Classifying Quadrilaterals
1. CAMPING The outline for a piece of canvas used to make a tent is shown below. What is the value of x in the quadrilateral?
140° 40°
2. CAMPING Refer to the figure in Exercise 1. Classify the quadrilateral using the name that best describes it.
3. ART The figure shows part of the pattern from a piece of stained glass. What is the measure of C?
B A
120° 60°
4. ART Refer to the figure in Exercise 3. The sides of quadrilateral ABCD are all congruent. Classify the quadrilateral using the name that best describes it.
5. HOME IMPROVEMENT The cross section of a wheelbarrow is shown below. What is the value of x in the figure?
6. HOME IMPROVEMENT Refer to the figure in Exercise 5. Classify the quadrilateral using the name that best describes it.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Congruent Polygons
AIRPLANES The diagram at the right is of an airplane
as seen from above. The wings of the airplane form congruent quadrilaterals, so quadrilateral ABCD quadrilateral EFGH. Use this figure for Exercises 1 and 2. 1. Name an unlabeled wing part whose length is 3 meters. Explain your answer.
2. Explain how a quality control person could find out if m DCB was correct?
3. WHALES The flukes of the Beluga whale are shaped like triangles. Determine whether these triangles are congruent. If so, name the corresponding parts and write a congruence statement. (Hint: RQ is a side of each triangle.)
P R S Q
4. PATTERNS Mandy is making name tags in the shape of triangles. They all should be the same size. Explain how she can use a pattern to make 25 name tags. How does she know they are all congruent?
5. ALGEBRA Find the value of x in the two congruent triangles.
2x 8 cm 14 cm
6. NATURE Part of a spider's web is shown in the figure. Determine whether the two marked triangles are congruent. If so, name the corresponding parts and write a congruence statement.
10 cm
E B C D
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 65
D 10 m A 3m B 9m C
120° E 110° F G 80°
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. FLAGS The flag of the Bahamas is shown below. Determine whether the flag has line symmetry. If it does, draw all lines of symmetry. If not, write none. 2. FLAGS Refer to the flag in Exercise 1. Determine whether the flag has rotational symmetry. Write yes or no. If yes, name its angles of rotation.
3. FLAGS The flag of Scotland is shown below. Determine whether the flag has line symmetry. If it does, draw all lines of symmetry. If not, write none.
4. FLAGS Refer to the flag in Exercise 3. Determine whether the flag has rotational symmetry. Write yes or no. If yes, name its angles of rotation.
5. LOGOS Discuss all of the properties of symmetry that the logo below has.
6. FLOWER OF LIFE This design has been found on Native American pots, in caves, and on buildings worldwide. Explain how to determine how many lines of symmetry it has. How many lines of symmetry are there?
Sample answer: Look for lines
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. ALPHABET The figure shows the letter V plotted on a coordinate system. Find the coordinates of points C and D after the figure is reflected over the y-axis.
D C
2. GREEK The figure shows the Greek letter gamma plotted on a coordinate system. Find the coordinates of points P and Q after the figure is reflected over the x-axis. Then draw the reflected image.
P´( 4,
3. CRAFTS Candace is making a pattern for star-shaped ornaments. Complete the pattern shown so that the completed star has a vertical line of symmetry.
4), Q´( 2,
4. FLOORING The Turners are replacing the flooring in their dining room. Complete the design shown so that the completed floor has a horizontal line of symmetry.
5. FLAG Macedonia is a country near Greece and Albania. The national flag of Macedonia has both vertical and horizontal symmetry. Complete the flag of Macedonia.
6. COYOTE Dasan is preparing a presentation on animal safety. Finish the drawing of a coyote's footprint so that it has vertical symmetry.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 67
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. BUILDINGS The figure shows an outline of the White House in Washington, D.C., plotted on a coordinate system. Find the coordinates of points C´ and D´ after the figure is translated 2 units right and 3 units up.
2. BUILDINGS Refer to the figure in Exercise 1. Find the coordinates of points C´ and D´ after the figure is translated 1 unit left and 4 units up.
C D
3. ALPHABET The figure shows a capital "N" plotted on a coordinate system. Find the coordinates of points F´ and G´ after the figure is translated 2 units right and 2 units down.
4. ALPHABET Refer to the figure in Exercise 3. Find the coordinates of points F´ and G´ after the figure is translated 5 units right and 6 units down.
5. QUILT The beginning of a quilt is shown below. Look for a pattern in the quilt. Copy and translate the quilt square to finish the quilt.
6. BEACH Tylia is walking on the beach. Copy and translate her footprints to show her path in the sand.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. ALPHABET Draw a figure on the grid below so that the figure together with its image after a 180° rotation will form a letter of the alphabet. 2. ALPHABET Draw a figure on the grid below so that the figure together with its images after 90°, 180°, and 270° counterclockwise rotations will form a letter of the alphabet.
3. QUILTS Complete the pattern for a quilt square by rotating the design 180° about the given point. What does the completed figure resemble?
4. QUILTS Complete the pattern for a quilt square by rotating the figure 90°, 180°, and 270° counterclockwise about the given point.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 69
5. SNOWFLAKE Mr. Ai is cutting paper snowflakes to decorate his classroom. Complete the snowflake below so that the completed figure has symmetry with 90°, 180°, and 270° as its angles of rotation.
6. LOGO The local swimming pool is having a contest, and all students are welcome to enter. The pool officials want a new logo that has rotational symmetry with 120° and 240° as its angles of rotation. The student whose logo is chosen will win a one-year pass to the pool. In the space below, draw an entry for the contest.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Area of Parallelograms, Triangles, and Trapezoids
1. PARKING A parking lot is constructed in the shape of a parallelogram. What is the area of the parking lot?
120 ft 200 ft 140 ft
2. DANCE FLOOR For a school dance, a section of the gymnasium has been designated as the dance floor. Ms. Picciuto needs to determine the area of the dance floor so she will know how many students can dance at one time. What is the area of the dance floor?
70 ft 40 ft 80 ft
3,000 ft2
3. SWIMMING POOLS The triangular swimming pool shown is surrounded by a concrete patio. Find the area of the patio. Round to the nearest tenth if necessary.
14 m
4. GARAGE BAND Sherice plays the bass in a garage band. Sherice's parents let her and her friends use a section of their garage in the shape of a parallelogram for rehearsals. How much space in square feet does Sherice's band have to practice in?
10 ft
10 m
8.7 m
12.1 m
8 ft 6 ft
5. CONSTRUCTION A 7-foot by 3-foot doorway is to be cut into the trapezoidshaped wall shown. What is the area of the wall, without the doorway?
6. CONSTRUCTION The wall in Exercise 5 is to be painted. If one can of paint covers 110 square feet, how many cans of paint will be needed if only one coat of paint is applied?
18 ft 7 ft 23 ft
3 ft
22 ft
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 71
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Circumference and Area of Circles
1. FOUNTAINS The circular fountain in front of the courthouse has a radius of 9.4 feet. What is the circumference of the fountain? Round to the nearest tenth.
2. PETS A dog is leashed to a point in the center of a large yard, so the area the dog is able to explore is circular. The leash is 20 feet long. What is the area of the region the dog is able to explore? Round to the nearest tenth.
3. GARDENING A flowerpot has a circular base with a diameter of 27 centimeters. Find the circumference of the base of the flowerpot. Round to the nearest tenth.
4. WINDOWS Find the area of the window shown below. Round to the nearest tenth.
36 in.
5. BICYCLES A bicycle tire has a radius of 13 1 inches. How far will the bicycle 4 travel in 40 rotations of the tire? Round to the nearest tenth.
6. LANDSCAPING Joni has a circular garden with a diameter of 14 1 feet. If
13 4 in.
she uses 2 teaspoons of fertilizer for every 25 square feet of garden, how much fertilizer will Joni need for her entire garden? Round to the nearest tenth.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Area of Complex Figures
LANDSCAPING For Exercises 1 and 2 use the diagram of a
15 ft 20 ft
yard and the following information. The figure shows the measurements of Marcus's yard which he intends to sod.
30 ft 50 ft
1. Find the area of the yard.
2. One pallet of sod covers 400 square feet. How many full pallets of sod will Marcus need to buy to have enough for his entire yard?
3. ICE CREAM Leeor was asked to repaint the sign for his mother's ice cream shop, so he needs to figure out how much paint he will need. Find the area of the ice cream cone on the sign. Round to the nearest tenth.
6 in.
4. HOME IMPROVEMENT Jim is planning to install a new countertop in his kitchen, as shown in the figure. Find the area of the countertop.
6 ft 2 ft 2 ft 3 ft 2.5 ft 3 ft 3 ft 2 ft 2.5 ft
12 in.
5. SCHOOL PRIDE Cindy has a jacket with the first letter of her school's name on it. Find the area of the letter on Cindy's jacket.
6 in. 2 in. 10 in. 6 in. 2 in.
6. SWIMMING POOLS The Cruz family is buying a custom-made cover for their swimming pool, shown below. The cover costs $2.95 per square foot. How much will the cover cost? Round to the nearest cent.
25 ft 15 ft
2 in.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 73
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Three-Dimensional Figures
ARCHITECTURE For Exercises 13, refer to the architectural drawing of
a table.
1 unit = 5 in.
1. Draw and label the top, front, and side views of the table.
2. Find the overall height of the table in feet.
2 1 ft
3. Find the area of the shaded region.
4. NAVIGATION Sailing ships once used deck prisms to allow sunlight to reach below the main deck. One such deck prism is shown below. Identify the solid. Name the number and shapes of the faces. Then name the number of edges and vertices.
5. PUBLIC SPEAKING A pedestal used in an auditorium is shaped like a rectangular prism that is 1 unit high, 5 units wide, and 5 units long. Sketch the pedestal using isometric dot paper.
6. PETS Lisa has four pet fish that she keeps in an aquarium. The aquarium is shaped like a triangular prism that is 4 units high. Sketch what this aquarium might look like using isometric dot paper.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Volume of Prisms and Cylinders
1. CAMPING A tent used for camping is shown below. Find the volume of the tent. 2. CONSTRUCTION The dimensions of a new tree house are shown below. How many cubic feet of space will the tree house contain?
2m 5 ft 8 ft 6 ft 6m 5m 33 m
3. FOAM The figure below shows a piece of foam packaging. Find the volume of the foam.
1 ft 2 ft 2 ft 1 ft
4. DONATIONS Lawrence is donating some outgrown clothes to charity. The dimensions of the box he is using are shown below. How many cubic feet of clothes will fit in the box?
7 ft 3 ft 2.5 ft
2 ft 3 ft
5. FARM LIFE A trough used for watering horses is shown in the figure. The trough is half of a cylinder. How many cubic feet of water will the trough hold? Round to the nearest tenth.
6. FARM LIFE If the volume of the water in the trough in Exercise 5 decreases by 5.6 ft3 per day, after how many days will the trough be empty? Round to the nearest tenth if necessary.
15 ft
1 ft
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 75
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Volume of Pyramids and Cones
1. DESSERT Find the volume of the ice cream cone shown below. Round to the nearest tenth if necessary.
1 in.
2. SOUVENIRS On a trip to Egypt, Myra bought a small glass pyramid as a souvenir. Find the volume of the glass used to make the pyramid. Round to the nearest tenth.
4 in.
4 in.
4 in. 4 in.
3. AUTO REPAIR A funnel used to fill the transmission on a car. Find the volume of the funnel. Round to the nearest tenth.
2 in.
4. ART An artist created a commemorative marker in the shape of a triangular pyramid. Find the volume of the stone used to make the marker. Round to the nearest tenth.
9 in.
12 ft
A = 15.6 ft3
5. FARMING The top of a silo is a cone, as shown in the figure. Find the volume of the cone. Round to the nearest tenth.
10 ft 7 ft
6. CONSTRUCTION The attic of a house is shaped like a rectangular pyramid, as shown. Calculate the volume of the attic.
15 ft
25 ft
35 ft
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Surface Area of Prisms and Cylinders
1. BAKING The top and sides of the cake shown below are to be covered in frosting. Calculate the area that will be covered with frosting.
2 in. 9 in. 12 in. 7 in. 14 in. 10 in.
2. GIFTS A birthday gift is placed inside the box shown below. What is the minimum amount of wrapping paper needed to wrap this gift?
616 in2
24 ft 24 ft 12 ft 41.6 ft 27 ft
5. LIGHT SHOW A mirrored cylinder used in a light show is shown below. Only the curved side of the cylinder is covered with mirrors. Find the area of the cylinder covered in mirrors. Round to the nearest tenth.
22 cm
6. SOUP Emily has the flu, so she decides to make chicken noodle soup. How many square inches of metal were used to make Emily's can of soup? Round to the nearest tenth.
3 in.
30 cm
4 2 in.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 77
3. FARMING Phil is planning to shingle the triangular roof on his barn shown below. How many square feet will he be shingling?
4. FARMING Refer to Exercise 3. If one package of shingles covers 325 square feet, how many packages will Phil need to buy?
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Surface Area of Pyramids and Cones
1. ROOFS A farmer is planning to put new roofing material on the conical roof of his silo shown below. Calculate the number of square feet of roofing material needed. Round to the nearest tenth. 2. ROOFS Refer to Exercise 1. If the roofing material costs $1.45 per square foot, how much will it cost to put new roofing material on the silo? Round to the nearest cent.
15 ft
7 ft
3. HOBBIES When the butterfly net shown below is fully extended, it forms the shape of a cone with a diameter of 12 inches and a slant height of 26 inches. Calculate the amount of mesh material needed to make the butterfly net.
12 in. 26 in.
4. HORTICULTURE The local college has a greenhouse that is shaped like a square pyramid, as shown below. The lateral faces of the greenhouse are made of glass. Find the surface area of the glass on the greenhouse.
12 m
9m 9m
5. VOLCANOES Find the surface area of the cinder-cone volcano shown below.
3 mi
6. COSTUMES The top of a costume hat is shaped like a triangular pyramid, as shown below. How much black felt is needed to cover the sides of the pyramid?
9 in. 11 in.
4.4 mi 11 in.
11 in.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Precision and Significant Digits
1. HOME IMPROVEMENT Which is the more precise measurement for the height of a door: 2 meters or 213 centimeters? Explain your reasoning. 2. CONSTRUCTION A rectangular window measures 108.2 inches long and 56.7 inches high. What is the area of the window? Round to the correct number of significant digits.
3. PETS Tara's two dogs, Cody and Tiger, weigh 34.4 pounds and 27.75 pounds, respectively. What is the difference in the weights of the two dogs? Write the difference using the correct precision.
4. GEOMETRY A rectangle has a length of 34.913 centimeters and a width of 18.43 centimeters. Write the perimeter of the rectangle using the correct precision.
5. REAL ESTATE An empty lot is rectangular in shape with a length of 62.4 feet and a width of 61.2 feet. Find the area of the lot. Round to the correct number of significant digits.
6. GEOMETRY A rectangular prism is 3.48 inches long, 1.56 inches wide, and 2.1 inches tall. Find the volume of the prism. Round to the correct number of significant digits.
7. HEALTH Last night, Niki used an electronic thermometer to find out that her temperature was 100.34 degrees. This morning, she used a mercury thermometer and got a reading of 98.9 degrees. How much did Niki's fever go down overnight? Write the answer using the correct precision.
8. LIFTING Andy is carrying three bags of groceries into the house. Individually the bags weigh 4.76 pounds, 7.4 pounds, and 9.12 pounds. What is the total weight that Andy is carrying? Write the answer using the correct precision.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 79
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Probability of Simple Events
FOOD For Exercises 1 and 2, use the table that shows the results of a survey that asked students in a classroom to choose their favorite fruit.
Fruit Number of Students
Orange 3
Apple Banana Strawberry Other 8 11 6 4
1. Suppose a student in the classroom is picked at random. Explain how to find the probability that the student's favorite fruit is strawberry. Then find the probability. Write it as a decimal.
2. Suppose a student in the classroom is picked at random. Explain how to find the probability that the student's favorite fruit is not an apple or a banana. Then find the probability. Write it as a decimal to the nearest thousandth.
GAMES For Exercises 3 and 4, use the board game spinner that determines how many spaces to move during each player's turn.
3. Explain how to find the probability of spinning a number that is greater than or equal to 4. Then find the probability.
4. What is the probability of spinning a number that is not a 2 or a 3?
5. CARPENTRY Hiromi builds a wooden birdhouse that is shaped like a cube. She paints 2 sides red, 1 side green, and 3 sides black. If she picks a side at random for the front, what is the probability that she will not pick a red side?
6. TRANSPORTATION In August 2002, 85% of an airline's flights arrived on time. What is the probability that one of its flights arrived late in August 2002?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 81
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Counting Outcomes
1. RESTAURANT An Italian restaurant offers mozzarella cheese, swiss cheese, sausage, ham, onions, and mushrooms for pizza toppings. For this week's special, you must choose one cheese, one meat, and one vegetable topping. On a separate sheet of paper, draw a tree diagram to find the number of possible outcomes. 2. TOYS Audra has a black and a white teddy bear. Cindy has a black, a white, a brown, and a pink teddy bear. Each girl picks a teddy bear at random to bring to a sleepover party. How many different combinations can the girls bring?
3. FOOD A candy maker offers milk, dark, or white chocolates with solid, cream, jelly, nut, fruit, or caramel centers. How many different chocolates can she make? Explain how you found your answer.
4. LOTTERY In a lottery game, balls numbered 0 to 9 are placed in each of four chambers of a drawing machine. One ball is drawn from each chamber. How many four-number combinations are possible?
GAMES Each of the spinners at the right is spun once to determine how a player's piece is moved in a board game.
Red White
Green Black Blue
5. Jason needs to spin a red and a blue to move to the last square and win the game. What is the probability that Jason will win? Explain how you found your answer.
6. If Jason spins a green or a white on either spinner, he will land on a "take an extra turn" square. What is the probability that Jason will get an extra turn?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. LACROSSE The United States Club Lacrosse Association has three divisions in the northeastern United States. The teams of the Empire Division are listed below. Empire Division CNY Brine Reebok Zbonis Binghamton DeBeer Tri-City 2. GAMES At lunchtime recess, 12 students race each other across the playground. In how many ways can students finish in first, second, third, and fourth places?
If there are no ties for placement in the division, how many ways can the teams finish the season from first to last place?
ENTERTAINMENT For Exercises 3 and 4, use the following information.
A music festival features 5 jazz bands, 9 rock bands, and 11 school bands. The bands play at various times over a long holiday weekend. 3. In how many ways can the first 4 rock bands be selected to play? 4. In how many ways can the first 3 school bands be selected to play? Explain how you found your answer.
5. FOOD Latesha buys a small box of 12 different assorted chocolates. She lets her sister have her 2 favorite chocolates, and then she has just enough left to give one chocolate to each girl attending basketball practice. In how many ways can Latesha give out the chocolates to the basketball players?
6. SCHEDULING A plumber has 8 jobs to schedule in the next week. One of the jobs is high priority and must be done first. In how many ways can the next 4 jobs be scheduled?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 83
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. ENTERTAINMENT During one month, a movie theater is planning to show a collection of 9 different Cary Grant movies. How many different double features (two-film showings) can they choose to show from this collection? 2. SCHOOL For a history test, students are asked to write essays on 4 topics. They must choose from a list of 10 topics about the European countries they have been studying. Is this situation a permutation or a combination? Explain. How many ways can a student choose 4 topics?
3. MARKET RESEARCH A taste test of 11 different soft drinks is held at a shopping mall. Each taster is randomly given 5 of the drinks to taste. How many combinations of soft drinks are possible?
4. BOOK FAIR A school book fair is offering a package deal on the opening day. For a special price, students may purchase any 6 different paperback books from a list of 30 books that have won the Newbery Medal. How many packages are possible?
GARDENING For Exercises 5 and 6,
use the shipping list at the right that shows the rosebushes Mrs. Lawson ordered for her front yard. She wants to plant 9 of them along the walkway from her driveway to her front porch.
Shipping List (1 each) Aquarius Purple Tiger Candy Apple Roundelay Desert Dawn Scarlet Knight Fragrant Plum Shining Hour Golden Girl Sonia Supreme Linda Ann Sundowner Mount Shasta Viceroy Pink Parfait Winifred 6. How many ways can she plant the rosebushes along the walkway if order is important?
5. How many ways can she plant the rosebushes along the walkway if order is not important?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Probability of Compound Events
1. CHECKERS In a game of checkers, there are 12 red game pieces and 12 black game pieces. Julio is setting up the board to begin playing. What is the probability that the first two checkers he pulls from the box at random will be two red checkers? 2. CHECKERS What is the probability that the first two pieces are a red followed by a black? Explain how you found your answer.
are 12 red out of 24 total, and on the probabilities 12 and 12 .
6 ; on the first selection, there 23
CHESS For Exercises 3 and 4, use the following information.
Ingrid keeps her white and black chess pieces in separate bags. For each color, there are 8 pawns, 2 rooks, 2 bishops, 2 knights, 1 queen, and 1 king. 3. Are the events of drawing a knight from the bag of white pieces and drawing a pawn from the bag of black pieces dependent or independent events? Explain. Find the probability of this compound event. 4. Are the events of drawing a bishop from the bag of white pieces and then drawing the queen from the same bag dependent or independent events? Explain. Find the probability of this compound event.
piece; 1 .
5. GAMES A blackjack hand of 2 cards is randomly dealt from a standard deck of 52 cards. What is the probability that the first card is an ace and the second card is a face card?
6. SPORTS During the 2002 soccer season, Maren Meinert of the Boston Breakers made approximately 2 goal points for every 5 of her shots on goal. What is the probability that Maren Meinert would make 2 goal points on two shots in a row during the 2002 season?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 85
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Experimental Probability
ENTERTAINMENT For Exercises 1 and 2, use the results of a survey of 120 eighth grade students shown at the right.
Video Game Playing Time Per Week Hours Number of Participants 0 18 13 43 36 35 more than 6 24 2. Out of 400 students, how many would you expect to play video games more than 6 hours per week?
1. Explain how to find the probability that a student plays video games more than 6 hours per week. Then find the probability.
3. DINING Only 6 out of 100 Americans say they leave a tip of more than 20% for satisfactory service in a restaurant. Out of 1,500 restaurant customers, how many would you expect to leave a tip of more than 20%?
4. PLANTS Jason has a packet of tomato seeds left over from last year. He plants 36 of the seeds and only 8 sprout. What is the experimental probability that a tomato seed from this packet will sprout?
SPORTS For Exercises 5 and 6, use the
results in the table at the right. In a survey, 102 people were asked to pick their favorite spectator sport.
Favorite Spectator Sport Sport Number professional football 42 professional baseball 27 professional basketball 21 college football 12 6. Out of 10,000 people, how many would you expect to say that professional baseball is their favorite spectator sport? Round to the nearest person.
5. What is the probability that a person's favorite spectator sport is professional baseball? Is this an experimental or a theoretical probability? Explain.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Using Sampling to Predict
FUND-RAISING For Exercises 1 and 2, use the survey results in the table at the right. Members of the Drama Club plan to sell popcorn as a fund-raiser for their Shakespeare production. They survey 75 students at random about their favorite flavors of popcorn.
Flavor butter cheese caramel
Number 33 15 27
1. What percent of the students prefer caramel popcorn?
2. If the club orders 400 boxes of popcorn to sell, how many boxes of caramel popcorn should they order? Explain how you found your answer.
DINING OUT For Exercises 3 and 4, use the following information. As
people leave a restaurant one evening, 20 people are surveyed at random. Eight people say they usually order dessert when they eat out. 3. What percent of those surveyed say they usually order dessert when they eat out? 4. If 130 people dine at the restaurant tomorrow, how many would you expect to order dessert?
RECREATION For Exercises 5 and 6, use the table at
the right which shows the responses of 50 people who expect to purchase a bicycle next year.
Bicycle Type mountain touring comfort juvenile other
Number 11 8 9 19 3
5. What percent of those planning to buy a bicycle next year think they will buy a mountain bike?
6. If Mike's Bike Shop plans to order 1,200 bicycles to sell next year, how many mountain bikes should be ordered?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 87
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
EXAMS For Exercises 13, use the MOVIES For Exercises 46, use the
histogram below that shows data about scores on a history test.
Exam Scores
histogram below that shows data about movie revenues in 2000.
Revenues of the 25 Top Grossing Movies of 2000
Number of Students
Number of Movies
61 7 0 71 8 0 81 9 0 91 1 00
51 6 0
61 1 00 10 1 14 0 14 1 18 0 18 1 22 0 22 1 26 0
1. How many students scored at least 81 on the test? Explain how you found your answer.
2. How many students scored less than 81 on the exam? Explain how you found your answer.
3. Can you determine the highest grade from the histogram? Explain.
4. How many movies grossed at least $141 million? Explain how you found your answer.
5. How many movies grossed between $61 million and $180 million? Explain how you found your answer.
6. Can you determine how many movies grossed between $121 and $140 million from the histogram? Explain.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 91
Revenue (millions)
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Circle Graphs
MUSIC For Exercises 1 and 2, use the circle graph below that shows data about music sales in 2001. Music Sales, 2001
2.4% Singles 3.4% Full-Length Cassettes 5% Others
INVESTMENTS For Exercises 36, use the table below that shows how Mr. Broussard has invested his money.
Investments Savings Account Money Market Account Mutual Funds $60,000 $100,000 $140,000 $500,000 $200,000
89.2% Full-Length CDs
Stocks Bonds
1. What angle corresponds to the sector labeled "Others" in the circle graph? Explain how you found your answer.
2. Use the circle graph to describe music sales in 2001.
Full-length CDs account for nearly all music sales. Other
3. Explain how a circle graph could help you visualize the data in the table.
4. Determine the percent of Mr. Broussard's total investments that each type of investment represents.
5. Draw a circle graph to represent the data.
Mr. Broussard's Investments
6. Use the circle graph you made in Exercise 5 to describe Mr. Broussard's investments.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Choosing an Appropriate Display
AGE For Exercises 14, use the following information. Cosmic, Inc. is a software company with 30 employees. The ages of the employees are displayed below using both a histogram and a stem-and-leaf plot.
Employee Age
Stem 1 2 3 4 5
50 5 9
Leaf 9 1224444556689 00012337889 2577 3 1|9 19
Number of Employees
10 1 9
20 2 9
30 3 9
1. Can you tell from the stem-and-leaf plot how many employees are between the ages of 20 and 29? If so, how many are there? If not, explain your reasoning. 3. Can you tell from the stem-and-leaf plot how many employees are between the ages of 36 and 43? If so, how many are there? If not, explain your reasoning.
40 4 9
2. Can you tell from the histogram how many employees are between the ages of 30 and 39? If so, how many are there? If not, explain your reasoning.
4. Can you tell from the histogram how many employees are between the ages of 36 and 43? If so, how many are there? If not, explain your reasoning.
Colors of Compact/Sports Cars Sold in the U.S., 2000 Color Silver Black White Percent 22% 14% 11% Color Red Blue Others Percent 16% 7% 30%
Colors of Compact/Sports Cars Sold in the U.S., 2000
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 93
5. CARS What percent of compact/sports cars sold in the year 2000 were red, white, or blue? Explain how you found your answer.
6. CARS Make a circle graph using the data in the table in question 5. What benefit does the circle graph have?
The circle graph shows how
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Measures of Central Tendency
ANIMALS For Exercises 14, use the information in the table below that shows the lifespan of selected mammals. Round to the nearest tenth if necessary. FOOTBALL For Exercises 5 and 6,
use the information in the table below. Round to the nearest tenth if necessary. 2001 NFL Season, Games Won Team Atlanta Carolina Denver Kansas City New Orleans Oakland St. Louis San Diego San Francisco Seattle Games Won 7 1 8 6 7 10 14 5 12 9
Average Lifespan for Mammals Mammal Baboon Camel Chimpanzee Cow Goat Gorilla Moose Pig Average Lifespan 20 yr 12 yr 20 yr 15 yr 8 yr 20 yr 12 yr 10 yr
1. Explain how to find the mean of the lifespans listed in the table. Then find the mean.
2. Explain how to find the median of the set of data. Then find the median.
3. Explain how to find the mode of the set of data. Then find the mode.
4. Which measure of central tendency is most representative of the data? Explain.
5. What are the mean, median, and mode of the number of games won by the teams in the table?
6. Which measure of central tendency is most representative of the data? Explain.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Measures of Variation
Lesson 95
FOOTBALL For Exercises 14, use the table below that shows the winning scores in the Super Bowl from 1994 through 2003.
Winning Super Bowl Scores, 19942003 1994 30 1995 49 1996 27 1997 35 1998 31 1999 34 2000 23 2001 23 2002 20 2003 48
1. Explain how to find the range of the data. Then find the range.
2. Find the median, the upper and lower quartiles, and the interquartile range of the winning scores.
3. Describe how to find the limits for outliers. Then find the limits.
4. Are there any outliers among the winning Super Bowl scores? If so, what are they? Explain.
GRADES For Exercises 5 and 6, use the
stem-and-leaf plot at the right showing the scores on the midterm exam in English.
Stem 7 8 9
Leaf 57 01456899 7 7|5
5. Find the range, median, upper and lower quartiles, and the interquartile range of the exam scores.
6. Are there any outliers in this data? Explain.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Box-and-Whisker Plots
U.S. SENATE For Exercises 14, use the Ages of U.S. Senators, 2002
box-and-whisker plot at the right.
1. Explain how to determine from the box-and-whisker plot whether there are any outliers in the data. Then identify any outliers.
2. Describe the distribution of the data. What can you say about the ages of U.S. senators?
3. What percent of U.S. senators are at least 54 years old? Explain how you found your answer.
4. Can you determine from the box-and-whisker plot whether there are any U.S. Senators exactly 65 years old? Explain.
box-and-whisker plot are the
HOCKEY For Exercises 5 and 6,
use the box-and-whisker plot at the right.
Goals Made by the Top 10 All-Time NHL Scorers
5. Identify any outliers in the data.
6. Describe the distribution of the data. What can you say about the number of goals made by the top 10 all-time leading NHL scorers?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Misleading Graphs and Statistics
1. AMUSEMENT PARKS The average wait times for the 10 different rides at an amusement park are 44, 37, 22, 11, 17, 25, 34, 17, 21, and 28 minutes. Find the mean, median, and mode of the average wait times for the rides. Round to the nearest tenth if necessary. 2. Use the data in Exercise 1. Which measure of central tendency would the amusement park use to encourage people to come the park? Explain.
5. Use the data in Exercise 4. Which measure of central tendency would the food company use to encourage people on a diet to try their cereal? Explain.
6. Use the data in Exercise 4. Which measure or measures of central tendency would be more representative of the data?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 97
3. Use the data in Exercise 1. Which measure or measures of central tendency would be more representative of the data?
4. CALORIES The number of Calories in one serving of 7 different kinds of breakfast cereal made by one food company are 80, 120, 190, 240, 100, 130, and 190. Find the mean, median, and mode of the number of Calories in one serving of each kind of cereal. Round to the nearest tenth if necessary.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
CITIES For Exercises 1 and 2, use the following information.
City Oak Hill Elm Grove Cedar Fork
Diners 19 11 12
Gas Stations 30 24 22
Theaters 3 2 4
Hotels 4 6 3
1. Make a matrix for the information in the table.
2. Explain what is meant by the dimensions of the matrix. What are the dimensions of the matrix?
FOOTBALL For Exercises 36, use the following information.
2002 NFL Season, Week 1 Team 49ers Giants Vikings Bears Points 16 13 23 27 First Downs 13 21 19 20 Completed Passes 16 28 16 20
2002 NFL Season, Week 2 Team Points 49ers Giants Vikings Bears 14 26 39 14 First Downs 18 16 31 13 Completed Passes 27 22 25 12
3. Make a matrix for the information for the first game of the 2002 season.
4. Make a matrix for the information for the second game of the 2002 season.
5. Explain the conditions necessary to be able to add two matrices.
6. Use the addition of matrices to find the totals in each category for each team in the two games. Write as a matrix.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Simplifying Algebraic Expressions
1. GAMES At the Beltway Outlet store, you buy x computer games for $13 each and a magazine for $4. Write an expression in simplest form that represents the total amount of money you spend. 2. TENNIS Two weeks ago James bought 3 cans of tennis balls. Last week he bought 4 cans of tennis balls. This week he bought 2 cans of tennis balls. The tennis balls cost d dollars per can. Write an expression in simplest form that represents the total amount that James spent.
3. AMUSEMENT PARKS Sari and her friends are going to play miniature golf. There are p people in the group. Each person pays $5 for a round of golf and together they spend $9 on snacks. Write an expression in simplest form that represents the total amount that Sari and her friends spent.
4. BICYCLING The bicycle path at the park is a loop that covers a distance of m miles. Jorge biked 2 loops each on Monday and Wednesday and 3 loops on Friday. On Sunday Jorge biked 10 miles. Write an expression in simplest form that represents the total distance that Jorge biked this week.
5. GEOMETRY Write an expression in simplest form for the perimeter of the triangle below.
2x 2x
6. SIBLINGS Mala is y years old. Her sister is 4 years older than Mala. Write an expression in simplest form that represents the sum of the ages of the sisters.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 101
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Solving Two-Step Equations
1. SHOPPING Jenna bought 5 reams of paper at the store for a total of $21. The tax on her purchase was $1. Solve 5x 1 21 to find the price for each ream of paper. 2. CARS It took Lisa 85 minutes to wash three cars. She spent x minutes on each car and 10 minutes putting everything away. Solve 3x 10 85 to find how long it took to wash each car.
3. EXERCISE Rick jogged the same distance on Tuesday and Friday, and 8 miles on Sunday, for a total of 20 miles for the week. Solve 2x 8 20 to find the distance Rick jogged on Tuesday and Friday.
4. MOVING Heather has a collection of 26 mugs. When packing to move, she put the same number of mugs in each of the first 4 boxes and 2 mugs in the last box. Solve 4x 2 26 to find the number of mugs in each of the first four boxes.
5. TELEVISION Burt's parents allow him to watch a total of 10 hours of television per week. This week Burt is planning to watch several two-hour movies and four hours of sports. Solve 2x 4 10 to find the number of movies Burt is planning to watch this week.
6. TRAVEL Lawrence drives the same distance Monday through Friday commuting to work. Last week Lawrence drove 25 miles on the weekend, for a total of 60 miles for the week. Solve 5x 25 60 to find the distance Lawrence drives each day commuting to work.
7. MONEY McKenna had $32 when she got to the carnival. After riding 6 rides, she had $20 left. Solve 32 6x 20 to find the price for each ride.
8. GARDENING Jack has 15 rosebushes. He has the same number of yellow, red, and pink bushes, and 3 multicolored bushes. Solve 3x 3 15 to find the number of yellow rosebushes Jack has.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Writing Two-Step Equations
Solve each problem by writing and solving an equation. 1. CONSTRUCTION Carlos is building a screen door. The height of the door is 1 foot more than twice its width. What is the width of the door if it is 7 feet high? 2. GEOMETRY A rectangle has a width of 6 inches and a perimeter of 26 inches. What is the length of the rectangle?
3. EXERCISE Ella swims four times a week at her club's pool. She swims the same number of laps on Monday, Wednesday, and Friday, and 15 laps on Saturday. She swims a total of 51 laps each week. How many laps does she swim on Monday?
4. SHOPPING While at the music store, Drew bought 5 CDs, all at the same price. The tax on his purchase was $6, and the total was $61. What was the price of each CD?
5. STUDYING Over the weekend, Koko spent 2 hours on an assignment, and she spent equal amounts of time studying for 4 exams for a total of 16 hours. How much time did she spend studying for each exam?
6. FOOD At the market, Meyer buys a bunch of bananas for $0.35 per pound and a frozen pizza for $4.99. The total for his purchase was $6.04, without tax. How many pounds of bananas did Meyer buy?
7. HOME IMPROVEMENT Laura is making a patio in her backyard using paving stones. She buys 44 paving stones and a flowerpot worth $7 for a total of $73. How much did each paving stone cost?
8. TAXI A taxi service charges you $1.50 plus $0.60 per minute for a trip to the airport. The distance to the airport is 10 miles, and the total charge is $13.50. How many minutes did the ride to the airport take?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 103
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Solving Equations with Variables on Each Side
Solve each problem by writing and solving an equation. 1. PLUMBING A1 Plumbing Service charges $35 per hour plus a $25 travel charge for a service call. Good Guys Plumbing Repair charges $40 per hour for a service call with no travel charge. How long must a service call be for the two companies to charge the same amount? 2. EXERCISE Mike's Fitness Center charges $30 per month for a membership. All-Day Fitness Club charges $22 per month plus an $80 initiation fee for a membership. After how many months will the total amount paid to the two fitness clubs be the same?
3. SHIPPING The Lone Star Shipping Company charges $14 plus $2 a pound to ship an overnight package. Discount Shipping Company charges $20 plus $1.50 a pound to ship an overnight package. For what weight is the charge the same for the two companies?
4. MONEY Julia and Lise are playing games at the arcade. Julia started with $15, and the machine she is playing costs $0.75 per game. Lise started with $13, and her machine costs $0.50 per game. After how many games will the two girls have the same amount of money remaining?
5. MONEY The Wayside Hotel charges its guests $1 plus $0.80 per minute for long distance calls. Across the street, the Blue Sky Hotel charges its guests $2 plus $0.75 per minute for long distance calls. Find the length of a call for which the two hotels charge the same amount.
6. COLLEGE Jeff is a part-time student at Horizon Community College. He currently has 22 credits, and he plans to take 6 credits per semester until he is finished. Jeff's friend Kila is also a student at the college. She has 4 credits and plans to take 12 credits per semester. After how many semesters will Jeff and Kila have the same number of credits?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. SPORTS Colin's time in the 400-meter run was 62 seconds. Alvin was at least 4 seconds ahead of Colin. Write an inequality for Alvin's time in the 400-meter run. 2. RESTAURANTS Before Valerie and her two friends left Mel's Diner, there were more than 25 people seated. Write an inequality for the number of people seated at the diner after Valerie and her two friends left.
3. FARM LIFE Reggie has 4 dogs on his farm. One of his dogs, Lark, is about to have puppies. Write an inequality for the number of dogs Reggie will have if Lark has fewer than 4 puppies.
4. MONEY Alicia had $25 when she arrived at the fair. She bought some ride tickets and she spent $6.50 on games. Write an inequality for the amount of money Alicia had when she left the fair.
5. HEALTH Marcus was in the waiting room for 26 minutes before being called. He waited at least another 5 minutes before the doctor entered the examination room. Write an inequality for the amount of time Marcus waited before seeing the doctor.
6. POPULATION The population of Ellisville was already less than 250 before Bob and Ann Tyler and their three children moved away. Write an inequality for the population of Ellisville after the Tyler family left.
7. HOMEWORK Nova spent one hour on Thursday, one hour on Saturday, and more than 2 hours on Sunday working on her writing assignment. Write an inequality for the amount of time she worked on the assignment.
8. YARD WORK Harold was able to mow more than 3 of his lawn on Saturday
night. Write an inequality for the fraction of the lawn that Harold will mow on Sunday.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 105
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Solving Inequalities by Adding or Subtracting
1. DRIVING Michael is driving from Lakeview to Dodge City, a distance of more than 250 miles. After driving 60 miles, Michael stops for gas. Write and solve an inequality to find how much farther Michael has to drive to reach Dodge City. 2. ENTERTAINMENT David and Marsha are going to dinner and a movie this evening. David wants to have at least $70 cash in his wallet. He currently has $10. Write and solve an inequality to find how much cash David should withdraw from the bank.
3. CLUBS The charter for the Spartan Club limits the membership to 85. Currently the club has 47 members. Write and solve an inequality to find how many more members can be recruited.
4. GROWTH Akira hopes that he will someday be more than 71 inches tall. He is currently 63 inches tall. Write and solve an inequality to find how much more Akira must grow to fulfill his wish.
5. MUSIC Jamie is preparing to burn a music CD. The CD holds at most 70 minutes of music. Jamie has 52 minutes of music already selected. Write and solve an inequality to find how many more minutes of music Jamie can select.
6. TELEVISION Dario limits his TV watching to no more than 11 hours a week. This week, he has already watched 6 hours of TV. Write and solve an inequality to find how much more time Dario can spend watching TV this week.
7. CARS At the gas station, Elena bought a quart of oil for $1.50, and she filled her car with gas. Her total was less than $20. Write and solve an inequality to find how much she spent on gas.
8. HOMEWORK Peter must write an essay with more than 500 words for his English class. So far, he has written 245 words. Write and solve an inequality to find how many more words Peter needs to write for his essay.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Solving Inequalities by Multiplying or Dividing
1. PLANTS Monroe needs more than 45 cubic feet of soil to fill the planter he built. Each bag of soil contains 2.5 cubic feet. Write and solve an inequality to find how many bags of soil Monroe will need. 2. ART Lois is making a rectangular collage. The area of the rectangle is 255 square inches, and the area of each photo is 15 square inches. She will overlap the photos so the total area of the photos is more than 255 square inches. Write and solve an inequality to find how many photos Lois will need.
5. PIZZA Trent and three of his friends are ordering a pizza. They plan to split the cost, and they want to spend at most $3.50 per person. Write and solve an inequality to find the cost of the pizza they should order.
6. GEOMETRY You are asked to draw a rectangle with a length of 6 inches and an area less than 30 square inches. Write and solve an inequality to find the width of the rectangle.
7. CONSTRUCTION Melinda wants to have a picture window in the shape of a regular hexagon in her new home. She wants the perimeter of the hexagon to be at least 9 feet. Write and solve an inequality to find the length of each side of the hexagon.
8. COOKING Len wants to make several batches of cookies. He is starting with less than 2 cups of raisins, and each batch takes 1 of a cup. Write and
solve an inequality to find how many batches of cookies Len can make.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 107
3. CAR WASH Jason's class is having a car wash to raise money for a project. They want to raise at least $120, and they are charging $5 to wash a car. Write and solve an inequality to find how many cars must be washed to raise $120.
4. PETS Kendra wants to buy some goldfish for her fish tank. She can spend no more than $18, and the fish cost $3 each. Write and solve an inequality to find how many goldfish Kendra can buy.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
GEOMETRY For Exercises 1 and 2, use the sequence of rectangles below.
4 units 2 units
5 units
6 units
7 units
3 units
4 units
5 units
1. Write a sequence for the perimeters of the rectangles. Is the sequence arithmetic, geometric, or neither? Explain how you know. If it is arithmetic or geometric, state the common difference or common ratio. Find the next four terms of the sequence.
2. Write a sequence for the areas of the rectangles. Is the sequence arithmetic, geometric, or neither? If it is arithmetic or geometric, state the common difference or common ratio. Explain how to find the next four terms of the sequence. Then find the next four terms.
the differences between consecutive
3. PIZZA A large pizza at Joe's Pizza Shack costs $7 plus $0.80 per topping. Write a sequence of pizza prices consisting of pizzas with no toppings, pizzas with one topping, pizzas with two toppings, and pizzas with three toppings. Is the sequence arithmetic, geometric, or neither? How do you know?
4. SAVINGS The ending balances in Carissa's savings account for each of the past four years form the sequence $1,000, $1,100, $1,210, $1,331, . . . . Is the sequence arithmetic, geometric, or neither? Explain how you know. Find the next two terms of the sequence.
5. PAYMENT PLAN A family purchased furniture on an interest-free payment plan with a fixed monthly payment. Their balances after each of the first four payments were $1,925, $1,750, $1,575, and $1,400. Is the sequence of the balances arithmetic, geometric, or neither? Explain how you know. If it is arithmetic or geometric, state the common difference or common ratio.
6. MONEY Continue to find the terms of the sequence of balances in Exercise 5 until you get a term of 0. After how many payments will the balance be $0?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 111
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
1. JOBS Strom works as a valet at the Westside Mall. He makes $48 per day plus $1 for each car that he parks. The total amount that Strom earns in one day can be found using the function f(x) x 48, where x represents the number of cars that Strom parked. Make a function table to show the total amount that Strom makes in one day if he parks 25 cars, 30 cars, 35 cars, and 40 cars. 2. PLUMBING Rico's Plumbing Service charges $40 for a service call plus $30 per hour for labor. The total charge can be found using the function f(x) 30x 40, where x represents the number of hours of labor. Make a function table to show the total amount that Rico's Plumbing Service charges if a job takes 1 hour, 2 hours, 3 hours, and 4 hours.
3. GEOMETRY The perimeter of an equilateral triangle equals 3 times the length of one side. Write a function using two variables for this situation.
4. GEOMETRY Explain how to use the function that you wrote in Exercise 3 to find the perimeter of an equilateral triangle with sides 18 inches long. Then find the perimeter.
5. LIBRARY FINES The amount that Sunrise Library charges for an overdue book is $0.25 per day plus a $1 service charge. Write a function using two variables for this situation.
6. LIBRARY FINES Explain how to find the amount of the fine the library in Exercise 5 will charge for a book that is overdue by 12 days. Then find the amount.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Graphing Linear Functions
1. FUEL CONSUMPTION The function d 18g describes the distance d that Rick can drive his truck on g gallons of gasoline. Graph this function. Explain why it is sufficient to graph this function in the upper right quadrant only. Use the graph to determine how far Rick can drive on 2.5 gallons of gasoline.
100 d 80
2. HOTELS The function c 0.5m 1 describes the cost c in dollars of a phone call that lasts m minutes made from a room at the Shady Tree Hotel. Graph the function. Use the graph to determine how much a 7-minute call will cost.
$5.00 d $4.00
Cost ($)
$3.00 $2.00 $1.00
Distance (mi)
Length of Call (min)
Gasoline (gal)
3. GIFTS Jonah received $300 in cash gifts for his fourteenth birthday. The function y 300 25x describes the amount y remaining after x weeks if Jonah spends $25 each week. Graph the function and determine the amount remaining after 9 weeks.
4. GIFTS What is the x-intercept of a graph? Find the x-intercept of the graph in Exercise 3 and interpret its meaning.
Amount Remaining ($)
5. GIFTS What is the y-intercept of a graph? Find the y-intercept of the graph in Exercise 3 and interpret its meaning.
6. GIFTS Explain how you can use your graph in Exercise 3 to determine during which week the amount remaining will fall below $190. Then find the week.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 113
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
The Slope Formula
1. MOVIES By the end of its first week, a movie had grossed $2.3 million. By the end of its sixth week, it had grossed $6.8 million. Graph the data with the week on the horizontal axis and the revenue on the vertical axis, and draw a line through the points. Then find and interpret the slope of the line.
Revenue (millions of dollars)
2. BASKETBALL After Game 1, Felicia had scored 14 points. After Game 5, she had scored a total of 82 points for the season. After Game 10, she had scored 129 points. Graph the data with the game number on the horizontal axis and the number of points on the vertical axis. Connect the points using two different line segments.
Number of Points
3. BASKETBALL Find the slope of each line segment in your graph from Exercise 2 and interpret it. Which part of the graph shows the greater rate of change? Explain.
4. GEOMETRY The figure shows triangle ABC plotted on a coordinate system. Explain how to find the slope of the line through points A and B. Then find the slope.
B(2, 4)
A( 3, 2)
C(2, 2)
5. Use the figure in Exercise 4. What is the slope of the line through points A and C? How do you know?
6. Use the figure in Exercise 4. What is the slope of the line through points B and C? How do you know?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Slope-Intercept Form
Lesson 115
CAR RENTAL For Exercises 1 and 2, use the following information. Ace Car Rentals charges $20 per day plus a $10 service charge to rent one of its compact cars. The total cost can be represented by the equation y 20x 10, where x is the number of days and y is the total cost.
1. Graph the equation. What do the slope and y-intercept represent?
2. Explain how to use your graph to find the total cost of renting a compact car for 7 days. Then find this cost.
Cost ($)
Number of Days
TRAVEL For Exercises 3 and 4, use the following information. Thomas is driving from Oak Ridge to Lakeview, a distance of 300 miles. He drives at a constant 60 miles per hour. The equation for the distance yet to go is y 300 60x, where x is the number of hours since he left.
3. What is the slope and y-intercept? Explain how to use the slope and y-intercept to graph the equation. Then graph the equation.
4. What is the x-intercept? What does it represent?
Distance (mi)
Time (h)
5. WEATHER The equation y 0.2x 3.5 can be used to find the amount of accumulated snow y in inches x hours after 5 P.M. on a certain day. Identify the slope and y-intercept of the graph of the equation and explain what each represents.
6. SALARY Janette's weekly salary can be represented by the equation y 500 0.4x, where x is the dollar total of her sales for the week. Identify the slope and y-intercept of the graph of the equation and explain what each represents.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Scatter Plots
WAGES For Exercises 1 and 2, use
the table at the right.
Year 1998 1999 2000 2001 2002 2003
Average Hourly Wage $11.43 $11.82 $12.28 $12.78 $13.24 $13.75
1. Explain how to draw a scatter plot for the data. Then draw one.
2. Does the scatter plot show a positive, negative, or no relationship? Explain.
Wage ($)
RESALE VALUE For Exercises 36, use the scatter plot
at the right. It shows the resale value of 6 SUVs plotted against the age of the vehicle.
Value (thousands)
Age (years)
3. Does the scatter plot show a positive, negative, or no relationship? Explain what this means in terms of the resale value of a SUV.
4. The equation y 2,000x 25,000 is an equation of a best-fit line for the data. Explain what a best-fit line is.
5. Find the slope and y-intercept of the best-fit line and explain what each represents.
6. Explain how to use the equation in Exercise 4 to estimate the resale value of an 8-year-old SUV. Find the value.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Graphing Systems of Equations
TAXI SERVICE For Exercises 14, use the following information.
A-1 Taxi service charges $5 for pickup plus $1 per mile for a taxi ride. All-About-Town Taxi service charge $1 for pickup plus $2 per mile. 1. Write an equation for the total charge y for a ride that covers x miles in an A-1 Taxi. 2. Write an equation for the total charge y for a ride that covers x miles in an All-About-Town Taxi.
3. Explain how to solve a system of equations by graphing. Then solve the system by graphing.
Cost of Taxi Ride ($)
5. INCOME Robert and Leta each work at a bicycle shop selling bicycles. Leta makes $150 per week plus $20 for each bicycle she sells, and Robert makes $250 per week. The equations y 20x 150 and y 250 can be used to represent their weekly salaries. Explain how to solve the system of equations by substitution. Then solve the system by substitution. What does your solution represent?
6. FOOD Antonio's Pizza charges $8.00 for a large pizza and $1.50 for each topping. Zina's Pizzaria charges $10.00 for a large pizza and $1.00 for each topping. Write and solve a system of equations to determine the number of toppings for which the pizzas would cost the same. What is that cost?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 117
4. For what distance is the charge the same for both companies? What is the charge for a ride of this distance? Explain how you know this.
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Graphing Linear Inequalities
NUTRITION For Exercises 1 and 2, use the following information.
Carl is making his own sports drink by mixing orange juice and water in a 40 ounce container. 1. Make a graph showing all the different amounts of orange juice and water that Carl can use in his drink.
70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 Water (oz)
2. Give three possible amounts of orange juice and water that Carl can use.
GEOMETRY For Exercises 3 and 4, use the following information.
The formula for the perimeter, P, of a rectangle of length x and width y is P
y. 4. Give three possible measurements for the length and width of a rectangle that has a perimeter of less than or equal to 20 units.
3. Make a graph for all rectangles that have a perimeter of less than or equal to 20 units. 20 y
OJ (oz)
FOOD For Exercises 5 and 6, use the following information. At the local farmer's market, apples are $2 per pound and blueberries are $3 per pound. Rene wants to buy at least $12 worth of apples and blueberries.
5. Make a graph for all the weights of apples and blueberries that Rene can buy.
14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 Apples (lb)
6. Give three possible ways that Rene can buy the amount of fruit she wants.
Blueberries (lb)
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Linear and Nonlinear Functions
GEOMETRY For Exercises 1 and 2, use the
following information. Recall that the perimeter of a square is equal to 4 times the length of one of its sides, and the area of a square is equal to the square of one of its sides.
s s
1. Write a function for the perimeter of the square. Is the perimeter of a square a linear or nonlinear function of the length of one of its sides? Explain.
2. Write a function for the area of the square. Is the area of a square a linear or nonlinear function of the length of one of its sides? Explain.
3. BUSINESS The Devon Tool Company uses the equation p 150t to calculate the gross profit p the company makes, in dollars, when it sells t tools. Is the gross profit a linear or nonlinear function of the number of tools sold? Explain.
4. GRAVITY A camera is accidentally dropped from a balloon at a height of 300 feet. The height of the camera after falling for t seconds is given by h 300 16t2. Is the height of the camera a linear or nonlinear function of the time it takes to fall? Explain.
5. LONG DISTANCE The table shows the charge for a long distance call as a function of the number of minutes the call lasts. Is the charge a linear or nonlinear function of the number of minutes? Explain. Minutes Cost (cents) 1 5 2 10 3 15 4 20
6. DRIVING The table shows the cost of a speeding ticket as a function of the speed of the car. Is the cost a linear or nonlinear function of the car's speed? Explain. Speed (mph) 70 80 90 100 Cost (dollars) 25 50 150 300
Nonlinear; the rate of change
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 121
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Graphing Quadratic Functions
GEOMETRY For Exercises 13, use the following information.
The quadratic equation A
6x2 models the area of a triangle with base 3x and height 4x. 2. Explain how to find the area of the triangle when x 3 inches. Then find the area.
1. Graph the equation. Explain why you only need to graph the function in the upper right quadrant.
3. Explain how to use your graph to determine the value of x when the area is 24 square inches. Then find the base and height of the triangle when its area is 24 square inches.
4. BUSINESS The quadratic equation p 50 2r2 models the gross profit made by a factory that produces r ovens. Graph the equation.
Profit (dollars)
r 10
Number of Ovens
5. PHYSICS The quadratic equation K 500s2 models the kinetic energy in joules of a 1,000-kilogram car moving at speed s meters per second. Graph the equation.
6. CARS The quadratic equation d 20 models the stopping distance in feet of a car moving at a speed s feet per second. Graph the equation.
Stopping Distance (feet)
Kinetic Energy (joules)
40,000 30,000 20,000 10,000 0 2 4 6 8
s 10
s 50
Speed (m/s)
Speed (ft/s)
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Simplifying Polynomials
1. BAKING Mila baked 2 cakes and 3 pies yesterday. Today she baked 4 cakes and 1 pie. Each cake takes c cups of flour, and each pie takes p cups of flour. Write an expression with four terms that represents the total amount of flour Mila used. Then simplify your expression. 2. GARDENING You have 2 bags of potting soil and 1 bag of peat moss. You buy 4 more bags of potting soil and 2 bags of peat moss. Each bag of potting soil weighs s pounds and each bag of peat moss weighs m pounds. Write an expression with four terms that represents the total weight of the bags. Then simplify your expression.
3. FOOTBALL The table shows the numbers of touchdowns, extra points, and field goals earned by each team at a football game. If t represents the number of points for a touchdown, e the points for an extra point, and f the points for a field goal, write an expression with six terms for the total number of points scored during the game. Then simplify your expression. Extra Field Team Touchdowns Points Goals Huskies 2 1 1 Hornets 1 1 3
4. CELL PHONES The table shows the numbers of anytime minutes and night and weekend minutes that Celia used for three days. If a represents the cost per minute for an anytime minute and n represents the cost per minute for a night and weekend minute, write an expression with four terms for the total cost of Celia's cell phone usage for the three days. Then simplify your expression. Anytime Night and Day Minutes Weekend Minutes Thursday 25 0 Friday 34 15 Saturday 0 55
5. MONEY Suppose your coin jar contains 3 rolls of quarters, 2 rolls of dimes, and 5 rolls of nickels. Your sister's coin jar contains 1 roll of quarters, 4 rolls of dimes, and 3 rolls of nickels. Each roll of quarters is worth q dollars, each roll of dimes is worth d dollars, and each roll of nickels is worth n dollars. Write an expression for the total amount of money you and your sister have in your jars.
6. ART You are making a collage using red triangles, blue squares, and green rectangles. You have 4 squares and 6 triangles on the collage. You plan to add 5 squares, 2 more triangles, and 3 rectangles. Each square has an area of s square inches, each triangle has an area of t square inches, and each rectangle has an area of r square inches. Write an expression in simplest form for the total area of the squares, triangles, and rectangles that will make up your collage.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 123
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Adding Polynomials
GEOMETRY For Exercises 1 and 2, use the figure at the right.
A ( 2x
(x 5) °
1. Write an expression in simplest form for the sum of the angles of a triangle.
2. Explain how to find the measure of angle A. Then find the measure.
3. GIFTS For his birthday, Carlos's parents give him $5 for each year of his age plus $50. His grandmother gives him $10 for each year of his age. Let a represent Carlos's age in years. Write a polynomial expression for the amount that Carlos receives from his parents. Then write a polynomial expression for the amount that he receives from his grandmother.
4. Write a polynomial expression for the total amount that Carlos receives from his parents and grandmother in Exercise 3. How much will Carlos receive when he is 15 years old?
5. TAXIS Lydia took a taxi from her home to school that charged $2 plus $0.50 per mile. Her brother Luke took a taxi the same distance that charged $3 plus $0.30 per mile. Let d represent the distance in miles. Write a polynomial expression for the cost of Lydia's taxi. Then write a polynomial expression for the cost of Luke's taxi.
6. Find an expression in simplest form representing the total cost of Lydia and Luke's taxi rides in Exercise 5. What is the total cost if the distance is 20 miles?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Subtracting Polynomials
figures at the right.
1. Write polynomial expressions in simplest form that represent the perimeters of the two rectangles. Then write a polynomial expression in simplest form that represents the difference between the perimeter of the larger rectangle and the perimeter of the smaller rectangle.
2. Write a polynomial expression in simplest form that represents the difference between the area of the larger rectangle and the area of the smaller rectangle. Then find the difference when x 4.
3. SALARY The polynomial expression (300 0.4s) (500 0.3s) represents the difference between two salary options that Chuck has in his new position as a salesperson. Write this difference in simplest form.
4. SHOPPING Maria bought 7 CDs at
x dollars each and used a coupon for $20 off her purchase of more than 5 CDs. Ricky bought 4 CDs at x dollars each and redeemed a coupon for $10 off his purchase of more than 3 CDs. Write polynomial expressions representing how much each spent after the discount. Then write a polynomial expression in simplest form representing how much more Maria spent than Ricky.
5. TESTS On a test worth 100 points, Jerome missed 3 questions worth p points each but answered a bonus question correctly for an extra 5 points. Suni answered 4 questions incorrectly and did not get the bonus. Write polynomial expressions in simplest form representing each student's score on the test. Then write a polynomial expression in simplest form representing how many more points Jerome scored than Suni.
6. PIZZA Sal's Pizza Place charges $8 for a large pizza plus $0.75 for each topping, while Greco's Cafe charges $10 for the same size pizza plus $0.90 for each topping. Write a polynomial in simplest form that represents how much more a pizza with t toppings would cost at Greco's than at Sal's.
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 125
GEOMETRY For Exercises 1 and 2, use the
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Multiplying and Dividing Monomials
1. MONEY The number 10,000 is equal to 104. There are 100 or 102 pennies in each dollar. How many pennies are there in $10,000? Write the answer using exponents. 2. RABBITS Randall has 23 pairs of rabbits on his farm. Each pair of rabbits can be expected to produce 25 baby rabbits in a year. How many baby rabbits will there be on Randall's farm each year? Write the answer using exponents.
3. DEBT The U.S. national debt is about 1013 dollars. If the debt were divided evenly among the roughly 108 adults, how much would each adult owe? Write the answer using exponents.
4. BOOKS A publisher sells 1,000,000 or 106 copies of a new book. Each book has 100 or 102 pages. How many pages total are there in all of the books sold? Write the answer using exponents.
5. GEOMETRY Find the area of the rectangle in the figure.
3y 9y
6. GEOMETRY The area of the rectangle in the figure is 24ab3 square units. Find the width of the rectangle.
4b 2 units
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
Practice: Word Problems
Multiplying Monomials and Polynomials
1. GEOMETRY Write an expression in simplest form for the area of the rectangle. What is the area of the rectangle if c 5 units?
4c 5
2. GEOMETRY Write an expression in simplest form for the area of the triangle. What is the area of the triangle if z 2 units?
4z 5z 8
10z 2
16z; 72 units2
5. FUND-RAISING When the Science Club members charged p dollars to wash each car at their car wash, they had 8p customers. When they doubled their price, they had 12 fewer customers. Write expressions representing the new price and the new number of customers. Then write an expression in simplest form representing the amount of money they made at the new price. How much money did they raise at the new price if the original price was $5 for each car?
6. GROUP RATES If Mr. Casey buys t tickets for his class to see a play, each ticket will cost 0.5t 1 dollars. If he buys three times as many tickets so that all three eighth grade classes can go, the price for each ticket is 2 dollars less. Write an expression for the total cost of the tickets for all three classes. If there are 20 students in Mr. Casey's class, how much will the tickets for all three classes cost?
© Glencoe/McGraw-Hill
Mathematics: Applications and Concepts, Course 3
Lesson 127
3. SWIMMING POOLS The Marshalls' pool is 5 feet longer than twice its width w. Write two expressions for the area of the pool. What is the area of the pool if it is 12 feet wide?
4. BUSINESS When a factory makes t bicycles in a month, the gross profit on each bicycle is 25 2t dollars. Write an expression in simplest form for the total gross profit the factory makes in a month that it produces t bicycles. What is the gross profit if the factory makes 40 bicycles?
Practice: Word Problems
95 pages
Our content is added by our users. We aim to remove reported files within 1 working day. Please use this link to notify us:
Report this file as copyright or inappropriate
You might also be interested in | {"url":"http://www.readbag.com/glencoe-sites-common-assets-workbooks-math-pdf-workbook-m3pwp","timestamp":"2014-04-19T22:08:05Z","content_type":null,"content_length":"173722","record_id":"<urn:uuid:41176ff5-6683-495d-a8c6-aa99506d0a03>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00663-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mean Value Theorem?
December 16th 2012, 10:34 PM #1
Junior Member
Nov 2012
New Jersey
Mean Value Theorem?
Given f(x) { a-bx and x is less than or equal to 3} on [0,4], Determine values of b and a so that the function will satisfy the conditions of the Mean Value Theorem. What is the value of x that's
guaranteed to exist by the Theorem?
Re: Mean Value Theorem?
Please, rewrite this question. Try to quote the original wording.
As it is now, it does not make much sense.
Re: Mean Value Theorem?
Did you mean $f(x) \leq 3$ ?
December 17th 2012, 09:10 AM #2
December 17th 2012, 10:09 AM #3 | {"url":"http://mathhelpforum.com/calculus/209967-mean-value-theorem.html","timestamp":"2014-04-18T08:40:43Z","content_type":null,"content_length":"36148","record_id":"<urn:uuid:f4c78b2e-1d42-403a-bcba-e2af2a6ae7d9>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00099-ip-10-147-4-33.ec2.internal.warc.gz"} |
Infinite and finite Gleason’s theorems and the logic of indeterminacy
, 2003
"... We develop a systematic approach to quantum probability as a theory of rational bettingin quantum gambles. In these games of chance, the agent is betting in advance on the outcomes of several
(finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and ..."
Cited by 12 (4 self)
Add to MetaCart
We develop a systematic approach to quantum probability as a theory of rational bettingin quantum gambles. In these games of chance, the agent is betting in advance on the outcomes of several
(finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and the money placed on the other measurements is returned to the agent. We show how the rules
of rational betting imply all the interesting features of quantum probability, even in such finite gambles. These include the uncertainty principle and the violation of Bell’s inequality amongothers.
Quantum gambles are closely related to quantum logic and provide a new semantics for it. We conclude with a philosophical discussion on the interpretation of quantum mechanics.
, 1965
"... theorem and the effectiveness of Gleason’s ..."
- CONTEMPORARY MATHEMATICS , 2002
"... The purpose of this note is to give a generalization of Gleason's theorem inspired by recent work in quantum information theory. For multipartite quantum systems, each of dimension three or
greater, the only nonnegative frame functions over the set of unentangled states are those given by the stand ..."
Cited by 3 (0 self)
Add to MetaCart
The purpose of this note is to give a generalization of Gleason's theorem inspired by recent work in quantum information theory. For multipartite quantum systems, each of dimension three or greater,
the only nonnegative frame functions over the set of unentangled states are those given by the standard Born probability rule. However, if one system is of dimension 2 this is not necessarily the
- Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 35, 177194 , 2004
"... Abstract. Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays
in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s th ..."
Cited by 3 (1 self)
Add to MetaCart
Abstract. Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in
Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s theorem itself has a constructive proof, based on a generic, finite, effectively generated set of
rays, on which every quantum state can be approximated. 1. Gleason’s Theorem and Logical Compactness Kochen and Specker’s (1967) theorem (KS) puts a severe constraint on possible hidden-variable
interpretations of quantum mechanics. Often it is considered an improvement on a similar argument derived from Gleason (1957) theorem (see, for example, Held. 2000). This is true in the sense that KS
provide an explicit construction of a finite set of rays on which no two-valued homomorphism exists. However, the fact that there is such a finite set follows from Gleason’s theorem using a simple
logical compactness argument (Pitowsky 1998, a similar point is made in Bell 1996). The existence of finite sets of rays with other interesting features | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=6575010","timestamp":"2014-04-16T13:56:09Z","content_type":null,"content_length":"22973","record_id":"<urn:uuid:cca71ecd-66b2-4b93-9cd8-34fb44536b83>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00195-ip-10-147-4-33.ec2.internal.warc.gz"} |
How the leopard got its spots
Issue 30
May 2004
Some Just So stories of animal patterning
Alan Turing is considered to be one of the most brilliant mathematicians of the last century. He helped crack the German Enigma code during the Second World War and laid the foundations for the
digital computer. His only foray into mathematical biology produced a paper so insightful that it is still regularly cited today, over 50 years since it was published.
Reticulated giraffe Rothschild's giraffe
Modelling an embryo
Turing's paper described how "reaction-diffusion equations" might be used by animals to generate patterned structure during their development as an embryo. Animals start as a single cell that divides
many times to create a full-size individual. During the early stages, the small ball of cells is completely uniform, or homogeneous, but out of this develop the dramatic patterns of a zebra, leopard,
giraffe, butterfly or angelfish. Turing was interested in how a spatially homogeneous system, such as a uniform ball of cells, can generate a spatially inhomogeneous but static pattern, such as the
stripes of a zebra. He managed to formulate a series of differential equations that, when solved, show very elegantly how the diversity of wonderful patterns on animals might be created.
Imagine an embryo with two types of chemical inside it. The two chemicals, as we will see, interact to generate patterns, and so are called morphogens (morpho from the Greek for "form", and gen from
the Greek for "to beget"). For the sake of this discussion, we can imagine the embryo as a one-dimensional line and look at the concentration of each of the two morphogens at each point along the
line. The chemicals can diffuse left and right along the line from a point of high concentration to lower concentration, and can also be produced afresh by cells along the embryo. One morphogen is an
"Inhibitor" and suppresses the production of both itself and the other chemical. The other, an "Activator", promotes the production of both morphogens.
At any time (t) and any point along the embryo (x), the concentrations of the Activator and Inhibitor are given by A(x,t) and I(x,t) respectively. But these concentrations change over time due to new
production (a reaction) and diffusion. The system is therefore known as a reaction-diffusion equation.
As we saw in Making the grade in Issue 27 of Plus, differentiation is a method of working out the gradient of a curve - how quickly one variable changes with respect to another. If, as in this case,
the function is of two variables (x and t) then calculating the gradient with respect to just one of them is known as partial differentiation.
So the change of concentration of Activator over time can be written as the partial differential equation
The first term on the right-hand side describes how much Activator is being produced. It is a function of Activator and Inhibitor concentrations because they both affect the reaction rate. The second
term is a second derivative describing how quickly the gradient of Activator is changing. It gives the rate of diffusion. The change of Inhibitor with respect to time is given by
The extra term, d, on the right-hand side is the diffusion coefficient - how much quicker Inhibitor diffuses than Activator. The Inhibitor being a faster diffuser was shown by Turing to be pivotal in
driving the process of pattern generation.
Very perturbing
Initially (i.e. when t=0), the two chemicals are in equilibrium -their concentrations do not change over time. The amount of Activator and Inhibitor is just right so that the reaction and diffusion
rates exactly balance. The situation is an "unstable equilibrium", however, and the first nudge, or perturbation in maths speak, knocks the system away from this equilibrium. It is like a pencil
poised on its tip - it might be perfectly balanced but the slightest nudge pushes the pencil over and it never recovers this equilibrium point.
Say that, for whatever reason, the concentration of Activator increases slightly at one point. Now the local concentration of Activator is greater than Inhibitor, so more Activator is produced, and
so on in a snowball effect. But Inhibitor is also being produced, and because it diffuses faster it quickly spreads to either side of the perturbation and decreases the concentration of Activator
there. So you end up with a region of high Activator concentration bordered on both sides by high Inhibitor.
This process can be seen in the video to the right. As the animation steps through time the concentration of Activator along the embryo organises into a series of peaks.
The reaction-diffusion equations can also be formulated for two dimensions. In this case an island of high Activator becomes surrounded by a moat of Inhibitor. Beyond this inhibitory halo, however,
the levels of Inhibitor drop again and so other seeds can produce an area of high activator concentration. In this way the symmetry of the uniform concentration is broken into roughly evenly spaced
regions of high Activator.
Revealing the pattern
The Activator and Inhibitor are not colour pigments themselves, just the morphogens that interact to create an underlying pattern. If the Activator also promotes the generation of a pigment in the
skin of the animal then this pattern can be made visible. Skin cells could produce yellow pigment unless they detect high levels of Activator instructing them to produce black. This would yield a
visible pattern similar to that of a cheetah.
The size of these spots will depend on what are known as thresholds. The concentration of Activator can be thought of as a landscape of hills, with a certain concentration of Activator (i.e.
altitude) required to turn ON the pigment. If this threshold is high, then only tiny spots at the very summit of the hills are seen, but if the threshold is lowered, then more of each hill is
coloured and the spots are larger with less space between them. Such a mechanism may explain the difference in markings between two subspecies of giraffe: the Rothschild's giraffe and the reticulated
giraffe (shown at the top of this page), the first of which has smaller, more widely-spaced spots than the other.
A low threshold for turning pigment ON A high threshold for turning pigment ON
Saturation can also be an important factor. If the concentration of Activator can reach a maximum value (ie. it is produced as fast as it breaks down or diffuses away) then the spots may join up into
stripes. This is believed to be what happens in the zebra.
Size matters
The size of the embryo at the time of pattern generation is also very important. If the Inhibitor diffuses quickly relative to the size of the domain then few spots will be able to form. In fact, the
stationary wave of Activator concentration is very similar to modes of vibration on a guitar string: only certain wavelengths can fit. The diagram below shows the reaction-diffusion simulation run on
"embryos" of different sizes: 5, 30, 150 and 1000 units long. No pattern at all can form on small animals, and on very large animals the spots are too small-scale and seem to blend together.
The cheetah has spots on its body and a stripy tail - but it's not possible for a stripy animal to have a spotted tail
Some developmental biologists have argued that this explains why neither small mice nor large elephants have any patterning. In between, however, more and more spots fit along the embryo. If d (the
diffusion constant) is assumed to be the same for all mammals, then this would explain why hamsters have only a few patches of colour whilst leopards have hundreds of small spots.
The size of the domain also affects the type of patterns that can form. An animal's tail can be thought of as a cylinder with a steadily decreasing radius. The top is large enough to support
two-dimensional patterns like spots, but down at the bottom the domain becomes too small. The region of high Activator spreads all the way around the tail and joins up with itself, so that a spot
becomes a stripe. The transition between spots and stripes is shown very well by a cheetah's tail. This aspect of the maths also explains why a spotted animal can have a striped tail, but a striped
animal can never have a spotted tail.
The process of pattern generation is completed in mammals during the embryonic stage. But some animals need to keep their markings up to date as they grow to full size. The stripes along the Marine
angelfish move very slowly over time as the domain size increases. The basic bands on a young fish move apart as the fish grows, with new stripes appearing or dividing off existing ones to fill in
any gaps.
Nature as Art?
The perturbations that trigger spots and stripes are usually statistical variations in the rate of morphogen production or diffusion. But physical disturbances from outside the embryo can have the
same effect. The beautiful eyespots on butterfly wings are thought to rely on the principles described above, although involving more morphogens. Marta de Menezes produces art with living objects by
pricking a butterfly wing with a pin while it is still developing in the chrysalis. This disrupts the concentration gradients and so alters the natural design.
Further reading
• A. Turing (1952). The Chemical Basis of Morphogenesis. Philosophical Transactions of the Royal Society of London. Available for free download, at participating institutions, from JSTOR.
• J.D. Murray (2001). Mathematical Biology. Published by Springer-Verlag New York Inc.
• D.S. Jones and B.D. Sleeman (2003). Differential Equations and Mathematical Biology. Published by Chapman & Hall.
About this article
Lewis Dartnell read Biological Sciences at Queen's College, Oxford. He took a year off to travel, but has now started a four-year MRes/PhD program in Bioinformatics at University College London's
Centre for multidisciplinary science, CoMPLEX. In 2003 he came second in the THES/OUP science writing competition with an article on the parallels between language and the structure of proteins. You
can read more of Lewis' work at www.ucl.ac.uk/~ucbplrd.
Submitted by Anonymous on May 16, 2013.
Thanks for the article. You may be interested in a project I've been tinkering with over the last few months - an online reaction diffusion laboratory. It simulates a few well known reaction
diffusion systems and recreates phenomina such as solitons, tip splitting and different waves.
It's an ongoing project and the blog link is: http://flexmonkey.blogspot.co.uk/search/label/ReDiLab | {"url":"http://plus.maths.org/content/how-leopard-got-its-spots","timestamp":"2014-04-17T01:21:41Z","content_type":null,"content_length":"40158","record_id":"<urn:uuid:8eb543b5-184a-4ed9-8439-30d0bad135c5>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00315-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts from August 2010 on Just Rakudo It
Masak recently posted on the series operator solution to Pascal’s Triangle. I’d like to focus on an important property of his final solution that hasn’t been much discussed yet.
But first, let me back up and consider another well-known problem with similar properties: the Fibonacci series. A straightforward recursive solution looks like this in Perl 6:
sub Fibonacci($n) {
return $n if $n == 0 | 1;
Fibonacci($n - 1) + Fibonacci($n - 2);
This nicely implements the definition, but it has one critical problem — it is incredibly inefficient. (And not just in Rakudo — in any language!) The reason is that calculating Fibonacci(N) requires
all the work of calculating Fibonacci(N-1) and Fibonacci(N-2). The number of subroutine calls involved grows as fast as the series does.
The standard recursive technique for dealing with this problem is called memoization. Basically, you cache the values which have already been computed somewhere, so you only need to compute them
once. That’s a bit awkward to implement by hand — it roughly doubles the complexity of the routine, I’d say. Luckily, lots of languages have automatic ways to apply memoization. Perl 6 is supposed
to, with the “is cached” modifier to a subroutine definition, but Rakudo doesn’t have it implemented yet.
That’s okay, because there is more than one way to do it, and there is a great way which is available already in Rakudo.
> my @Fibonacci := (0, 1, -> $a, $b { $a + $b } ... *); 1;
> say @Fibonacci[5]
> say @Fibonacci[10]
> say @Fibonacci[30]
(I’ve used parentheses instead of do because that’s how I roll, but otherwise this is exactly the same idea as Masak’s solution. The additional 1; is needed only because I typed this into the REPL,
and without it the REPL will try to evaluate the entire infinite series so it can print it out.)
You can’t see it here, of course, but calculating these numbers was very quick. That’s because this solution using the series operator is equivalent to writing a very smart memoization. Infinite
lists in Perl 6 are implemented (roughly speaking) with an array of values that have been already calculated, and an iterator which knows how to calculate further values as needed.
So when you say @Fibonacci[5] the first time, the list calls the iterator generated by the series operator to get the first six values, storing them in @Fibonacci[0] through @Fibonacci[5]. (Those
values are generated in a straightforward iterative way, so it is very efficient.) When you then say @Fibonacci[10] those first six values are still there, and only the next five values must be
calculated. If at that point you said @Fibonacci[8], that is just a simple array lookup of the already calculated value!
Think about it. It would take a very smart automatic memoizer to somehow figure out that the function would only be called when $n was a non-negative integer, so that an array could be used to
efficiently cache the results. Using a series operator this way gets you that kind of performance automatically already in Rakudo.
So it’s a double win. Using the series operator is not only the most succinct way to express this series in Perl 6. It’s also an extremely efficient way of calculating the series. | {"url":"http://justrakudoit.wordpress.com/2010/08/","timestamp":"2014-04-20T20:57:25Z","content_type":null,"content_length":"35129","record_id":"<urn:uuid:6e7ec08c-2ea6-4c84-8ccb-3c757fb7d5b2>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00562-ip-10-147-4-33.ec2.internal.warc.gz"} |
Riverdale, IL Algebra Tutor
Find a Riverdale, IL Algebra Tutor
...Before college, I attended Homewood-Flossmoor High School, finished in the top 10% of my class, and scored a 30 on my ACT. During my time there, I took AP Calculus, Physics, Chemistry, and
Biology, and played on the baseball team. I have worked at Flossmoor Country Club for 10 years, so I have met many of the South Suburbs' most influential people.
28 Subjects: including algebra 1, algebra 2, calculus, chemistry
...Beyond this academic instruction I have worked with students since I was in college on the side helping them optimize their own study habits and techniques for both their classwork but also
their approach to test prep. Often times to master a substantial amount of material in a limited amount of...
38 Subjects: including algebra 1, reading, algebra 2, statistics
...I also assisted them with homework problems. I am a current graduate student in Statistics, so I have been through both introductory and advanced courses. I received a 5 on the AP exam and have
succeeded in all classes related to Statistics.
5 Subjects: including algebra 1, statistics, prealgebra, probability
...I have 17 years teaching experience with K-8 students in the classroom and have also tutored students from ages 5-adult. I specialize in grades K-8 reading, writing, math, social studies, study
skills, test prep, and 6-12 reading, algebra, geometry, social studies, study skills, and test prep. ...
40 Subjects: including algebra 2, algebra 1, reading, English
...I work well with students from middle school through college and I can tutor all K-12 math, including college Calculus, Probability, Statistics, Discrete Math, Linear Algebra, and other
subjects. I have flexible days and afternoons, and I can get around Chicago without difficulty. I look forward to hearing from you.I took discrete math undergraduate at Tufts and received an A.
22 Subjects: including algebra 2, algebra 1, calculus, geometry | {"url":"http://www.purplemath.com/riverdale_il_algebra_tutors.php","timestamp":"2014-04-20T23:59:21Z","content_type":null,"content_length":"24170","record_id":"<urn:uuid:6ef4e56a-f3ad-489c-80e2-6efe14836c20>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00021-ip-10-147-4-33.ec2.internal.warc.gz"} |
integration constant
June 20th 2012, 01:53 AM #1
Jun 2012
integration constant
anyway let me ask something more basic...why do we have a integrational constant in calculus? if we sum ( in summation) up the thing do we need any constant like this??
Re: integration constant
Summation is equivalent to definite integral. In definite integral we don't need constant.
Re: integration constant
Consider the indefinite integral
$\int x^2 \, dx$
We can say that the anti-derivative of $x^2$ is $\frac{x^3}{3}$. However, another anti-derivative could be $\frac{x^3}{3} + 1$, or perhaps $\frac{x^3}{3} - \pi$. This is because the derivative of
a constant is zero.
To remove all this ambiguity we say that $\int x^2 \, dx = \frac{x^3}{3} + C$, where C is an arbitrary constant.
June 20th 2012, 02:29 AM #2
Jun 2012
June 20th 2012, 09:00 AM #3
Super Member
Jun 2012 | {"url":"http://mathhelpforum.com/calculus/200218-integration-constant.html","timestamp":"2014-04-18T22:01:15Z","content_type":null,"content_length":"34488","record_id":"<urn:uuid:47144758-4c61-40d8-8770-38765bf88278>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00118-ip-10-147-4-33.ec2.internal.warc.gz"} |
Got Homework?
Connect with other students for help. It's a free community.
• across
MIT Grad Student
Online now
• laura*
Helped 1,000 students
Online now
• Hero
College Math Guru
Online now
Here's the question you clicked on:
How much work is performed when 17 newtons are moved 3 meters?___n-m
• 11 months ago
• 11 months ago
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
• Problem Solving 19 Hero
• Engagement 19 Mad Hatter
• You have blocked this person.
• ✔ You're a fan Checking fan status...
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.
You haven't written a testimonial for Owlfred. | {"url":"http://openstudy.com/updates/518d5ba4e4b062a8d1da27be","timestamp":"2014-04-17T16:13:09Z","content_type":null,"content_length":"41669","record_id":"<urn:uuid:a87fd334-0b47-4e48-a5a0-12d870d240bc>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00469-ip-10-147-4-33.ec2.internal.warc.gz"} |
Functions and graphs...
February 26th 2006, 06:42 AM
Functions and graphs...
I am having trouble with this one, looked everywhere in the book but did not find what i was looking for. I understand what i have to do but not sure how to do it.
"The graph of f(x) is given below(i will describe it for you).
Use the graph of the given function F to sketch the graph of the function g(x)=f(-x-1)+2"
graph of f(x).... when x=-3, y=-3
x=-2, y=-1
x=-1, y=1
x=0, y=3
x=1, y=3
x=2, y=3
x=3, y=3
x=4, y=1
thanks in advance
February 26th 2006, 10:09 AM
MOODYtargi stop spamming!!!
-User has been warned- | {"url":"http://mathhelpforum.com/algebra/2014-functions-graphs-print.html","timestamp":"2014-04-20T06:18:09Z","content_type":null,"content_length":"4051","record_id":"<urn:uuid:a2ccb7ca-257d-4745-b24b-7f384deec1bb>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00186-ip-10-147-4-33.ec2.internal.warc.gz"} |
Happy birthday dear Rabbit
Period tripling bifurcation is similar to the period doubling one. You see here how Douady's rabbit appears out of a "single embryo cell" :)
At c = -0.125 + 0.65i on the border of the main cardioid the attracting fixed point loses its stability with multiplier λ = exp(2π /3). You see below this 120^ o rotation symmetry of the critical
orbit near the bifurcation point. Unstable period-3 orbit approch to the fixed point.
After the period tripling bifurcation attracting period-3 orbit and unstable fixed point (with three whiskers) appear. Contents Previous: Period doubling bifurcations Next: Windows of periodicity
updated 12 Sep 2013 | {"url":"http://www.ibiblio.org/e-notes/MSet/orbit3.html","timestamp":"2014-04-17T22:54:30Z","content_type":null,"content_length":"3177","record_id":"<urn:uuid:7590a445-7230-4646-a809-2c6bd995f2ac>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00117-ip-10-147-4-33.ec2.internal.warc.gz"} |
Lyapunov stability (idea)
Consider a continuous function f:G→R^N for a non-empty subset G⊂R^N and an autonomous differential equation dx/dt=f(x(t)), hereafter x'=f(x), with an equilibrium point 0∈G - that is, f(0)=0.
The equilibrium 0 is described as being stable in the sense of Lyapunov if ∀ε>0 ∃δ>0 such that
If x:I→G is a maximal solution of the differential equation with 0∈I satisfying ||x(0)||<δ
then [0,∞)⊂I
and ||x(t)||≤ε ∀t≥0.
Say what?
A differential equation describes the way in which a system of variables changes with respect to change in another variable, usually time. But, it may be that what is of particular interest is
preventing change- that is, examining conditions under which there is no change. In the formalisation above, the assertion that 0 is an equilibrium point (sometimes called a critical point) means
that our system will, on attaining a value of zero, stay there.
The question of stability then arises when we seek to examine what happens near, but not at, the equilibrium. Is it only 0 that stays settled, or are there other initial conditions that are somehow
drawn to a steady state, either that of 0 or another? If not, can we at least guarantee that nothing too outlandish happens whilst only a short distance away from the equilibrium?
If Lyapunov stability is achieved, then the system, whilst not necessarily settling on any one value, stays bounded, and can be described for all times. Moreover, if we have a particular bound in
mind (the ε), then there is a restriction on initial condition (being within δ of 0) that will ensure we don't exceed that bound.
Lyapunov's direct method
With the setup as described in the opening paragraph (autonomous system, continuity of f, 0 an equilibrium point), there is a relatively straightforward test for this form of stability.
Lyapunov Stability Theorem: If there is an open neighbourhood U of 0 contained in G, and there exists a continuously differentiable function V from that neighbourhood to the reals satisfying
□ V(0)=0
□ V(z)>0 ∀ z ∈ U\{0}
□ V[f](z):=<∇V(z),f(z)> ≤0 ∀ z ∈ U - where < · , · > denotes inner product
Then, 0 is a stable (in the sense of Lyapunov) equilibrium.
Proof of the Lyapunov Stability Theorem
We need only consider ε sufficiently small that the closed epsilon-ball about 0 is contained in the set U, since we can satisfy any larger bound by satisfying such an epsilon. Moreover, such an
epsilon exists since U is open and contains 0 and thus must contain some open neighbourhood of it. Thus we restrict our attention to
B:= { z ∈ R^N st ||z|| ≤ ε } ⊂ U
S:= ∂B = { z ∈ R^N st ||z|| = ε }
Now, the boundary S is closed and bounded, that is, compact, and V takes strictly positive values on S (since the only place it takes a value of zero is at 0, and it is never negative). V is
continuous, and such functions achieve their extrema on compact sets. Thus, μ, the minimum of V(z) for z in S is strictly greater than zero.
μ > 0
Since V does attain a value of 0 at 0, there must be some neighbourhood of 0 where V takes (positive) values less than μ, that is, since V(0)=0,
∃δ∈(0,ε) such that ||z|| < δ ⇒ V(Z) = V(Z)- 0 = |V(z)-V(0)| < μ
Given x:I→G a maximal solution of x'=f(x) with 0∈I and ||x(0)||≤δ, we claim that [0,∞) ⊂ I with ||x(t)||≤ε for all t≥0. Were this true, the theorem would hold. Let ω be the supremum of I and suppose
(for contradiction) the assertion were not true. Then it must fail- at some time τ ∈ (0,ω)⊂I, x escapes the bound, that is, ||x(τ)||>ε.
Since initially x is less than ε, and at τ is greater than epsilon, continuity and the intermediate value theorem force the existence of a σ∈(0,τ) such that ||x(σ)||=ε. Further, we can chose the
earliest such τ, so for all t∈[0,τ), ||x(t)||<ε.
Note by the chain rule that V[f](t) is dV(x(t))/dt, and V[f] is assumed to be less than or equal to 0 everywhere in U, so in particular
dV(x(t))/dt ≤ 0 ∀t∈[0,σ]
This means, however, that t→V(x(t)) is a non-increasing function on [0,σ], so
V(x(σ)) ≤ V(x(0))
But x(σ) is a point on S since ||x(σ)||=ε, so it is clearly at least the minimum on that set, so we obtain
μ ≤ V(x(σ)) ≤ V(x(0))
But this is no good; ||0||≤δ so V(0)<μ. Adding that in gives us
μ ≤ V(x(σ)) ≤ V(x(0)) < μ
or simply μ<μ
Which is absurd. So, ||x(t)||≤ε for any t∈[0,ω). By a property of maximal intervals of existence and compact sets which would be too much of a detour to prove here, this forces ω to be ∞, so we have
the desired result of [0,∞)⊂I. This completes the proof (in suitable handwaving fashion).
Lyapunov Functions
A function satisfying the criteria of the theorem is known as a Lyapunov function. Given the above theorem, the challenge (as Swap points out, often considerable) is to construct such a function for
a given system x'=f(x), but the reward is that finding any one such function suffices to ensure Lyapunov stability.
Asymptotic Stability
Lyapunov stability is not a very strong stability requirement; it does not mean that solutions 'near' to the equilibrium are 'pulled' to the equilibrium over time. This stronger notion is asymptotic
stability and requires an additional notion of attractivity (which does not itself imply stability). However, to tackle this concept requires a deeper notion of invariant sets and local flows, whilst
enforcing a local Lipschitz condition on f, so I leave it for another node.
Reference:MA40062 ODEs (University of Bath), my revision notes. | {"url":"http://www.everything2.com/index.pl?node_id=1724179","timestamp":"2014-04-20T15:53:18Z","content_type":null,"content_length":"28636","record_id":"<urn:uuid:2e7d8b4e-3c5b-435c-bb70-4d80e5a46691>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00228-ip-10-147-4-33.ec2.internal.warc.gz"} |
Binary - Older Than You Think
Binary - Older Than You Think Follow
Written by Mike James
Tuesday, 17 December 2013
Binary is the number system of the computer age, so it is a big surprise to learn that it's more than 600 years old and was used by Polynesians as part of their counting system.
You can do arithmetic in any base you care to choose. Humans tend to prefer a base 10 system presumably because this is what you get when you start by counting on your fingers.
After you get to the tenth finger you have to remember one group of ten and start counting on your fingers all over again.
Base ten or decimal is a great system for writing down numbers but it doesn't make arithmetic as easy as in binary. In binary the multiplication tables are just
1x0=0 and
which should be compared to the tables we all have to learn at school. The big problem with binary is that it doesn't make writing numbers easy - for example 7 is 111 and so on.
Ease of implementing arithmetic and the natural way binary can be stored in a two state memory makes it the best thing to use for computer arithmetic but it was invented long before
the computer. The mathematician Gottfried Leibniz knew about number bases and had worked out that binary had advantages back in 1703 some 300 years ago.
Now it seems that we can push back that date when binary-like systems were used another 300 years earlier on the Pacific island of Mangareva. It is "binary like" because it is very
unlikely that anyone would invent pure binary counting for simple tasks. You would have to count zero, one, one lot of two, one lot of two and one, two lots of two and so on. You
just don't seem to get enough done in the early stages of counting.
There is lots of evidence that groupings of two have been important much earlier than the Mangareva counting system - the I Ching for example that was the inspiration for Liebniz's
interest in binary.
The Mangareva people however took another option. As well as pure base systems you can count in mixed base. For example the old UK monetary system used base 12 and 20 - 12 pence to
a shilling and 20 shillings to a pound. You can count using different bases for different sizes of number. Andrea Bender and Sieghard Beller of the University of Bergen in Norway
have just published a paper (behind a paywall unfortunately) that explains how the Mangareva counted using a mixed 10 and 2 base system. They had words for values from 1 to 10 but
after this groupings of powers of 2 are used. For eample takau (K) is 10, paua (P) is 20 and tataua (T) is 40. Thus a number like 70 is TPK and 75 is TPK5. This avoids the problem
of "small values" in binary but you still have some simple rules for arithmetic e.g.
2 x K =P and 2 x P =T.
It is argued that these rules made doing mental arithmetic easier.
Python Tools for Visual Studio 2.1 Beta
Microsoft is improving the support for Python in Visual Studio with the release of Python Tools for VS 2.1 beta.
+ Full Story
More Ties Than We Thought Or Ties Of The Matrix
The Matrix Reloaded started something when "The Merovingian" wore a number of very flashy ties. The problem was that we thought we knew how many ways you can tie a tie, but the
enumeration didn't incl [ ... ]
+ Full Story
More News
Last Updated ( Wednesday, 18 December 2013 )
RSS feed of news items only
Copyright © 2014 i-programmer.info. All Rights Reserved. | {"url":"http://i-programmer.info/news/82-heritage/6724-binary-older-than-you-think.html","timestamp":"2014-04-18T06:45:36Z","content_type":null,"content_length":"39748","record_id":"<urn:uuid:311e8958-500b-467e-9b20-9c52df17e1a3>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00539-ip-10-147-4-33.ec2.internal.warc.gz"} |
Download munkres topology solutions manual
Download munkres topology solutions manual - 0 views
Filename: munkres topology solutions manual
Date: 13/9/2012
Type of compression: zip
Total downloads: 7873
Nick: pulreb
File checked: Kaspersky
Download speed: 47 Mb/s
Price: FREE
Topology answers and solutions to James Munkres textbooks. Cramster provides expert homework help plus ask a Topology question
munkres topology solutions manual
and get fast answers.
* pdf Solutions to Topology Homework #3, due Week 6. Problems: Munkres. Solutions to Topology Homework #3, due Week 6. Problems:
MATH u0026 STAT TEXTBOOKS Fall 2008... 534-40469-7 11 Swokowski/Cole Brooks/Cole optional: Student Solutions Manual. 978-0-471-43334 3rd Dummit …
Science Education > Academic Guidance. i think i've accelerated my learning enough, and now i'm going to start doing. guillemin - pollack, spivak calculus on.
Feb 03, 2011 · نگارستان کتاب آبی ISBN Title Author Rials YEAR Accounting & Finance 0470405422.
Math 651: Homework 3 Solutions February 8, 2007 1. Munkres 16.4. Englewood Cliffs, Topologies for hybrid solutions - 1 Introduction The... Available online at 'www.
Math 441/541{Topology II Spring 2011 Course Syllabus Math 441/541{Topology II Spring 2011 Course Syllabus Instructor: Anton Kaul Oce: 25-312 (Faculty Oces East) …
* pdf Solutions to Topology Homework #4, due Week 8. Problems: Munkres. Solutions to Topology Homework #4, due Week 8. Problems:
Updated: 2012-10-03 Infinity PowerPoint... exposed to and know: Biology - Algebra I, II Geometry. Kit The Complete Classroom Technology Solution.
Information on textbooks for all Undergraduate Mathematics courses at the New Brunswick and Piscataway campuses of Rutgers is collected here.
Subject Poster Date [26534] - [#msc_mtech_maths.pdf] - [University MSc Tech,_Mathematics_] - "UNIVERSITY OF PUNE.
Download The Solution Of Topology By James R Munkres Chapter 3 And 4 Hot Sponsored Downloads. munkres topology solutions manual download the solution of topology by james r munkres …
Book Description: The Clandistine Chemist's NoteBook by Zero Introduction Welcome to the very first version of The Clandestine Chemist's Notebook.
University MSc Tech,_Mathematics_ UNIVERSITY OF PUNE DEPARTMENT OF MATHEMATICS SYLLABUS M. Sc. (TECH.) (Industrial Mathematics with Computer …
Show that the subspace (a,b) of R is homeomorphic with (0,1) and the subspace [a,b] of R is homeomorphic with [0,1]
M 101 Illustrated Analysis Bedtime Stories Special Bounded Edition Book 6.58 MB | Ebook Pages: 209 [2] James Munkres, Topology, 2nd ed, Prentice Hall, (2000). [3]
Munkres, Topology, Chapter 2 - Free download as PDF File (.pdf), text file (.txt) or read online for free.
Jan 19, 2011 · VERY IMPORTANT INSTRUCTIONS: Kindly refer the official communication of the University in the B.A. R&S file . APPENDIX - 17 [R] UNIVERSITY OF MADRAS …
Sponsored Download Links motorola sp50 service schematic manual [Full Version] 5342 downloads @ 3211 KB/s motorola sp50 service schematic manual - Full … | {"url":"https://groups.diigo.com/group/calavane20/content/download-munkres-topology-solutions-manual-7349184","timestamp":"2014-04-20T16:45:41Z","content_type":null,"content_length":"28574","record_id":"<urn:uuid:e0ccee28-5a01-40a2-9180-4b87f270f577>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00567-ip-10-147-4-33.ec2.internal.warc.gz"} |
discrete function - onto and 1-1
October 29th 2008, 11:35 AM
discrete function - onto and 1-1
Give a example of a function from N to N (naturals) that is
a) one-to-one and onto
b) onto but not one-to-one
October 29th 2008, 12:03 PM
What about $f(n) = \left\lfloor {\frac{n}{2}} \right\rfloor$?
Which part does that answer?
October 29th 2008, 10:57 PM
November 2nd 2008, 03:17 PM
Would f(n) = n^2 be onto but not 1-1? | {"url":"http://mathhelpforum.com/discrete-math/56383-discrete-function-onto-1-1-a-print.html","timestamp":"2014-04-19T23:43:02Z","content_type":null,"content_length":"5119","record_id":"<urn:uuid:420db5e4-3ef9-48f6-844e-ae179ea4d42d>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00330-ip-10-147-4-33.ec2.internal.warc.gz"} |
Next Article
Contents of this Issue
Other Issues
ELibM Journals
ELibM Home
EMIS Home
Pick a mirror
Miroljub Jevtic and Miroslav Pavlovic
Matematicki fakultet, Studentski trg 16, Beograd, Serbia
Abstract: The solid hulls of the Hardy–Lorentz spaces $H^{p,q}$, $0<p<1$, $0<q\leq\infty$ and $H_0^{p,\infty}$, $0<p<1$, as well as of the mixed norm space $H^{p,\infty,\alpha}_0$, $0<p\leq1$, $0<\
alpha<\infty$, are determined.
Classification (MSC2000): 30D55; 42A45
Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 23 Apr 2009. This page was last modified: 22 Oct 2009.
© 2009 Mathematical Institute of the Serbian Academy of Science and Arts
© 2009 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition | {"url":"http://www.maths.soton.ac.uk/EMIS/journals/PIMB/099/6.html","timestamp":"2014-04-21T09:47:03Z","content_type":null,"content_length":"4266","record_id":"<urn:uuid:6390b95f-a201-4560-acf8-61ce9a581fb2>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00622-ip-10-147-4-33.ec2.internal.warc.gz"} |
new-typeable, new cast?
Shachaf Ben-Kiki shachaf at gmail.com
Mon Mar 4 08:43:35 CET 2013
This came up in #haskell -- rather than just provide coerce, cast (or
a primitive that cast can be built on, as well as other things) can
give a type equality witness of some kind. Some possible signatures
cast :: (Typeable a, Typeable b) => Maybe (p a -> p b) --
Leibniz-style equality
cast :: (Typeable a, Typeable b) => p a -> Maybe (p b) -- another form
cast :: (Typeable a, Typeable b) => Maybe (Is a b) -- GADT-style equality
cast :: (Typeable a, Typeable b) => MightBe a b -- another form
-- With
data Is a b where { Refl :: Is a a } -- standard equality GADT
data MightBe a b where -- version with built-in Maybe, for more
convenient types
Is :: MightBe a a
Isn't :: MightBe a b
Any of these lets you write any of the other ones, of course. But
"cast" doesn't let you write any of them, because it only goes in one
The GADT form can let you write code like
foo :: forall a. Typeable a => a -> ...
foo x
| Is <- cast :: MightBe a Int = ...x...
| Is <- cast :: MightBe a Char = ...x...
| otherwise = ...
Without coming up with a new name for x. It's possible to provide
other functions on top of this primitive, like
cast :: (Typeable a, Typeable b) => proxy a -> qroxy b -> ... --
to save some type annotations
cast :: (Typeable a, Typeable b) => (a ~ b => r) -> Maybe r -- to
use an expression directly
cast :: (Typeable a, Typeable b) => a -> Maybe b -- the original
Of the primitives I mentioned, the first two are H2010. The last two
are nice to use directly, but use extensions (this probably doesn't
matter so much for new-typeable). The Maybe form lets you say things
like `fmap (\Is -> ...) (cast :: ...)` -- unfortunately, there's no
"proper" Is type in base right now. The fourth version might save some
typing, and might be slightly better performance-wise, since it saves
a ⊥ (it's just represented like Bool at runtime)? Probably it all gets
inlined so that doesn't matter anyway.
Regardless of which particular primitive is used, it would be great
for new-typeable to provide something stronger than cast.
More information about the Libraries mailing list | {"url":"http://www.haskell.org/pipermail/libraries/2013-March/019465.html","timestamp":"2014-04-18T17:06:56Z","content_type":null,"content_length":"4672","record_id":"<urn:uuid:018e96dd-326f-4615-9fe3-bd6496e811fa>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00524-ip-10-147-4-33.ec2.internal.warc.gz"} |
Got Homework?
Connect with other students for help. It's a free community.
• across
MIT Grad Student
Online now
• laura*
Helped 1,000 students
Online now
• Hero
College Math Guru
Online now
Here's the question you clicked on:
the product of two consecutive positive even numbers is 1520. What are the numbers? when I try "n" and "n+2" i get an odd number
• one year ago
• one year ago
Best Response
You've already chosen the best response.
you can express those consecutive even numbers like "n" and "n+2" and their product is (n)(n+2) = 1520
Best Response
You've already chosen the best response.
guess and check for this one there is no other way
Best Response
You've already chosen the best response.
\(40\times 42=1680\) nope, too big. try again with smaller numbers
Best Response
You've already chosen the best response.
That's not quite the right equation...
Best Response
You've already chosen the best response.
@Albert0898 it is the PRODUCT not the sum
Best Response
You've already chosen the best response.
you cannot use an equation to solve this, only to rewrite the question, don't be fooled you just have to grind it until you find it
Best Response
You've already chosen the best response.
38 and 40
Best Response
You've already chosen the best response.
satellite was close ;)
Best Response
You've already chosen the best response.
hahah, sorry i used your guess to approximate.
Best Response
You've already chosen the best response.
yeah a calculator helps i started too high, went down a couple, got it in try two
Best Response
You've already chosen the best response.
n(n+2) = 1520 n^2 + 2n - 1520 = 0 (n-38)(n+40) = 0 Since the numbers are positive, the 2nd solution doesn't work. n must be 38 and n + 2 is 40.
Best Response
You've already chosen the best response.
(I couldn't factor that as easily as I made it sound... so a calculator helped me too :) )
Best Response
You've already chosen the best response.
You definitely do not have to grind it out and guess. But the equation is not shown correctly above. It is (2N)(2N + 2) = 1520. Becomes a simple quadratic. Solve for N. Answer will be the 2N and
2N + 2.
Best Response
You've already chosen the best response.
so my question is, how did you know how to factor ? you had to come up with two numbers whose product is 1520 and that are two apart in other words just rewrote the question, still had to guess
and check
Best Response
You've already chosen the best response.
x = even number x + 2 = consecutive even number x + 2 * x = 1520 x + 2 = 1520/x x = 1518/x x * x = 1518 Find the square root of 1518. Approximate and Closest Even Number: 38. x = 38 x + 2 = 40
Best Response
You've already chosen the best response.
i guess if you wanted to be ridiculous you could write \[x^2+2x=1520\] and complete the square \[(x+1)^2=1521\] \[x+1=\sqrt{1521}=39\]\[x=38\] but that is a silly thing to do
Best Response
You've already chosen the best response.
@satellite73 Yes, I agree with you... that's what I meant... you either guess and check to factor, or just guess and check vs. 1520. My only point was that it is possible to write an equation to
solve something like this.
Best Response
You've already chosen the best response.
@JakeV8 yes you are right, but i maintain that changing a problem like this in to an equation and then solving it by solving the original problem in words is a dog chasing its tail
Best Response
You've already chosen the best response.
You guys are missing it all together. Set the variables as 2N and 2N + 2. If you set N to a positive integer, 2N and 2N + 2 are automatically even. Becomes a simple quadratic. See my previous
Best Response
You've already chosen the best response.
thanks all I think I found my problem
Best Response
You've already chosen the best response.
Ok, I'm here. Give me a second to refresh myself on this.
Best Response
You've already chosen the best response.
Ok, I'm up to speed. I still stand firmly on my answer methodology. The way to GUARANTEE that you have two even numbers is to use (2N)(2N + 2) = 1520 because 2N and 2N + 2 will be FORCED to be
even if N is a positive integer. Absolutely. I strongly suggest you use this methodology. If I understand, your current question has something to do with 3 positive even numbers now? Is that
Best Response
You've already chosen the best response.
And please, whatever you do, don't follow satellite73 on this problem. He is WAY off base suggesting that you guess. That's beyond ludicrous.
Best Response
You've already chosen the best response.
@tcarroll010, that is the equation I was using however, i am still unable to find two positive even numbers. No I do not have to find three evennumbers, I was seeing if the way I was working it
would give even numbers for three. hope that makes sence, sorry my computer is slow. . I only need two positive even numbers
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
• Problem Solving 19 Hero
• Engagement 19 Mad Hatter
• You have blocked this person.
• ✔ You're a fan Checking fan status...
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.
You haven't written a testimonial for Owlfred. | {"url":"http://openstudy.com/updates/508db124e4b078e4677b5718","timestamp":"2014-04-17T06:54:12Z","content_type":null,"content_length":"84846","record_id":"<urn:uuid:d2937b2a-4aab-48a4-a49e-f125b6664a34>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00293-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is a Klein Bottle?
Ever hear of a Möbius Loop -- a one sided, one edged surface? Give a strip of paper a half-twist, then tape the ends together. It's one side and one boundary, with delightful properties dear to
││In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. Its inside is its outside. It contains itself. │
││ │
││Take a rectangle and join one pair of opposite sides -- you'll now have a cylinder. Now join the other pair of sides with a half-twist. That last step isn't possible in our universe, sad to say. A│
││true Klein Bottle requires 4-dimensions because the surface has to pass through itself without a hole. │
││ │
││It's closed and non-orientable, so a symbol on its surface can be slid around on it and reappear backwards at the same place.You can't do this trick on a sphere, doughnut, or pet ferret -- they're│
││orientable. │
Alas, our universe has only 3 spatial dimensions, so even Acme's dedicated engineers can't make a true Klein Bottle.
A photograph of a stapler is a 2-dimensional immersion of a 3-dimensional stapler. The true stapler has been flattened into the flatland of the photo. In the same way, our glass Klein Bottles are 3-D
immersions of the 4-D Klein Bottle. Acme's Klein Bottle is a 3-dimensional photograph of a "true" Klein Bottle.
A Klein Bottle cannot be embedded in 3 dimensions, but you can immerse it in 3-D. (An immersion may have self-intersections; Embeddings have no self-intersections. Neither an embedding nor an
immersion has folds or cusps.)
We represent a Klein Bottle in glass by stretching the neck of a bottle through its side and joining its end to a hole in the base. Except at the side-connection (the nexus), this properly shows the
shape of a 4-D Klein Bottle. And except at the nexus, any small patch follows the laws of 2-dimensional Euclidean geometry.
Contrast this with a corked bottle -- say, a wine bottle. It has two sides: inside and outside. You can't get from one to the other without drilling a hole or popping the top. Once uncorked, it has a
lip which separates the inside from the outside. If you make the glass arbitrarily thin, that lip won't go away. It'll become more prominent. The lip divides one side of the bottle from the other. So
an uncorked bottle is topologically the same as a disc ... it has two sides, separated by a boundary -- an edge.
But a Klein Bottle does not have an edge. It's boundary-free, and an ant can walk along the entire surface without ever crossing an edge. This is true of both theoretical Klein Bottles and our glass
ones. And so, a Klein Bottle is one-sided.
A Klein Bottle has one hole. This, in turn, causes it to have one handle. The genus number of an object is the number of holes (well, it's more subtle than that, but I'm not allowed to tell you why).
Other genus-1 objects include innertubes, bagels, wedding rings, and teacups. A wine bottle has no holes and so is genus 0. (The genus of a human being is difficult to define because it depends on
what you consider a hole -- I'd estimate most people have a genus of 0 to 4, slightly higher when yawning. Pierce your ear and you'll increase your genus by one.)
As an alternative to buying an Acme Klein Bottle, you can save money by just memorizing this set of parametric equations, since it defines the surface of every Klein Bottle.:
x = cos(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
y = sin(u)*(cos(u/2)*(sqrt_2+cos(v))+(sin(u/2)*sin(v)*cos(v)))
z = -1*sin(u/2)*(sqrt_2+cos(v))+cos(u/2)*sin(v)*cos(v)
or in polynomial form:
Yep, no doubt about it: Your Acme's Klein Bottle is a real Riemannian manifold, just waiting for you to define a Euclidean metric at every point.
Acme is proud to be our universe's foremost supplier of immersed, boundary-free, nonorientable, one-sided surfaces. We make and sell Klein Bottles.
For more information, on Klein Bottles, visit the Topological Zoo or the Geometry Center. Or click here to see a diagram on how to make one in Japan Notice that topologists simulate Klein Bottles ...
but ACME makes 'em in glass!
Here's a few other topology links:
Konrad Polthier of the Zuse Institute / Berlin has written an outstanding article about the Klein Bottle in Plus Magazine (issue 26, Sept 2003). Several sweet applets let you assemble a Klein bottle
on your computer screen (but Acme's glass Klein Bottles are made slightly differently!)
Neil Strickland's What is Topology (Did you know that there must be some point on the earth without any wind?)
A friendly, happy view of topology, Klein Bottles, projective planes, spheres, Klein Bottles, curved space, Klein Bottles, manifolds, cosmology, Klein Bottles and more ... read Jeffrey Week's
splendid book, The Shape of Space. When Weeks cuts Klein Bottle into two Moebius loops, it looks amazing like how Acme does it.. This book's a perfect introduction to topology - for high school
students through postodocs. A perfect match to an Acme Klein Bottle.
Janna Levin's delightful presentation, Topology: From Doughnuts to the Universe connects flatland to tiling to space travel ... an infinite universe is similar to assuming that the earth is flat!
Here's a mathematical history of topology without any mention of the Klein Bottle!
If you're really into topology, look at this topology glossary for definitions of terms like automorphism, isometry, and suchnot.
The Cylinder, Mobius Band, Torus, and Klein Bottle ... a nice comparison of four happy manifolds.
You can play TicTacToe on a Klein Bottle.
Nathaniel Hellerstein (a friend of this author) created a Klein Bag from a sock. Works great as a change purse! Endorsed by Cliff Stoll.
For 7 to 10 year olds, here's a k-12 classroom project to make & explore Moebius Loops, a Moebius Cross, and Euler numbers.
Acme Klein Bottles - Exclusive Purveyors of the Non-Orientable
This page last updated 24-mar-14
Return to Acme's Home Page | {"url":"http://www.kleinbottle.com/whats_a_klein_bottle.htm","timestamp":"2014-04-18T18:26:39Z","content_type":null,"content_length":"11634","record_id":"<urn:uuid:bbe5d8ce-f68f-450b-88c0-612b22592cdf>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00586-ip-10-147-4-33.ec2.internal.warc.gz"} |
Word Problems
July 10th 2011, 01:30 PM #1
Junior Member
Jul 2011
Word Problems
If one outlet pipe can drain a tank in 24 hours and another pipe can drain the tank in 36 hours, how long will it take for both pipes to drain the tank?
Have no clue how to set this up? Help Thanks
Re: Word Problems
One way to look at this is to imagine both pipes connected to 2 identical tanks.
Consider the pipes to have started draining tanks at the same time.
In 36 hours, pipe B will drain one tank.
In the first 24 hours, pipe A will have drained the other tank
and would be able to drain another half tank in the last 12 hours.
Hence, the taps operating simultaneously could drain 2.5 tanks in 36 hours.
How long will it take to drain one tank ?
Yet another way is...
in 12 hours, one pipe will have drained half the tank
while the other pipe will have drained a third.
That's 3 sixths and 2 sixths.
In 12 hours, 5 sixths will be drained.
How much longer for another sixth ?
Last edited by Archie Meade; July 10th 2011 at 02:47 PM. Reason: added alternative
Re: Word Problems
setting up these "job type" problems ...
(combined rates to do the job)(time for job completion) = 1 job completed
$\left(\frac{1 \, tank}{24 \, hrs} + \frac{1 \, tank}{36 \, hrs}\right) \cdot (t \, hrs) = 1 \, tank \, drained$
solve for
$t \, hrs$
July 10th 2011, 01:42 PM #2
MHF Contributor
Dec 2009
July 10th 2011, 02:17 PM #3 | {"url":"http://mathhelpforum.com/algebra/184385-word-problems.html","timestamp":"2014-04-21T15:50:44Z","content_type":null,"content_length":"37856","record_id":"<urn:uuid:694c993f-7004-4a35-9eae-dd51413563b1>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00608-ip-10-147-4-33.ec2.internal.warc.gz"} |
How long is 2m in feet and inches?
You asked:
How long is 2m in feet and inches?
Say hello to Evi
Evi is our best selling mobile app that can answer questions about local knowledge, weather, books, music, films, people and places, recipe ideas, shopping and much more. Over the next few months we
will be adding all of Evi's power to this site.
Until then, to experience all of the power of Evi you can download Evi for free on iOS, Android and Kindle Fire. | {"url":"http://www.evi.com/q/how_long_is_2m_in_feet_and_inches","timestamp":"2014-04-18T19:09:34Z","content_type":null,"content_length":"59359","record_id":"<urn:uuid:de54c7f7-9380-430c-94ab-a6d580b081b8>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00264-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Neuroscience] Re: Op Amps - The Voltage Follower Circuit
J.A.Legris via neur-sci%40net.bio.net (by jalegris from sympatico.ca)
Sun Feb 15 08:51:58 EST 2009
On Feb 15, 3:48 am, Bill <connelly.b... from gmail.com> wrote:
> On Feb 14, 4:02 pm, r norman <r_s_norman from _comcast.net> wrote:
> > On Fri, 13 Feb 2009 17:16:27 -0800 (PST), Bill
> > <connelly.b... from gmail.com> wrote:
> > >I'm having a hard time understanding the classical Voltage Follower
> > >Circuit made by a single op amp. I wont bother trying to draw it in
> > >ASCII, just have a look here if you don't know what I'm talking about:
> > >http://hyperphysics.phy-astr.gsu.edu/Hbase/electronic/opampvar2.html
> > >This is who I'm thinking about it (which I'm sure is wrong):
> > >1) Lets imagine you have a 1mV input to the + input.
> > >2) At the instant this is switched on the - input is 0, so the op amp
> > >outputs 1mV
> > >3) Now the + input still gets 1mV and - input gets 1mV, so the amp
> > >outputs 0mV, sending us back 1)
> > >I appriciate that the op amp works 'instantaneously' so you don't get
> > >oscillations like I described, but I still don't know how one can
> > >conceptualize op amps without getting into these kind of oscillations.
> > >Thanks for anyone who can tell me the correct way to think about this
> > You forget that the op amp is really an amplifier with a very high
> > gain. Lets suppose it has a gain of one thousand. Actually it is
> > normally higher than that but this will do for calculation.
> > When you put 1 mv on the input, it responds by putting an output not
> > of 1 mv, but only about 999 microvolts, 1/1000 less than the desired 1
> > mv. Then there is 1 millivolt on the + input and 0.999 mv on the -
> > input for a difference of 0.001 mv which, times the 1000 gain, would
> > produce an output of 1 mv. Remember, I said "about" 999 microvolts.
> > The difference is the approximation error.
> > To be precise, if the gain is G, then the output is G times the
> > difference between the + and - inputs. Let the output be y and the
> > input x. Then y = G(x - y) since the output is applied to the -
> > input. Solve that to get y = x * G/(1+G). If G is very large, then
> > G/(1+G) is very close to one.
> > Oscillations are an entirely different story. For that you have to
> > describe the output in terms of the LaPlace (or Fourier) transform of
> > its "transfer function". But control theory tells you exactly how to
> > do it and to build a circuit that does not produce oscillations.
> Yes, I see how without the G term in that equation, it breaks down to
> output = input/2, i.e. halfway between those osscillation i described
> above.
> And so this is the importance of the "infinite open-loop gain" that
> the perfect op amp has.
The oscillations you described are a byproduct of your discrete, step-
by-step characterization of the op-amp. Real op-amps do not make
instantaneous changes. Rather, the voltage on the output rises or
falls until equilibrium is established through the feedback to the (-)
input. This all happens very quickly of course, but it is still a
continuous process. Oscillations occur when the delay and overall gain
of the feedback loop "reinforce" particular frequencies. It is
possible to get an op-amp to oscillate between discrete states, but
this requires special techniques, such as non-linear circuit elements
or topologies.
More information about the Neur-sci mailing list | {"url":"http://www.bio.net/bionet/mm/neur-sci/2009-February/062411.html","timestamp":"2014-04-16T22:36:26Z","content_type":null,"content_length":"6793","record_id":"<urn:uuid:3bc32feb-9ef1-450c-8919-bbbb4418691a>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00368-ip-10-147-4-33.ec2.internal.warc.gz"} |
IntroductionThe PCA AlgorithmObtaining the principal components of an imageDescription of the ArchitectureImplementing PCA on FPGAsGenerating the eigenvectorsCalculating the mean of the M images (Ψ)Obtaining the covariance matrix (CT)Calculating the eigenvectors of the matrix (V)Obtaining the eigenvectors of the matrix (Vt)Obtaining the eigenvectors of the matrix UtCalculating the norms of the eigenvectorsThe on-line stageDetecting new objects in the sceneConstructing the Map of Distances (MD) and the Map of average Distances (MDV)Detecting objects from the MDV mapResultsConclusionsReferencesFigures and Tables
Sensors Sensors 1424-8220 Molecular Diversity Preservation International (MDPI) 10.3390/s101009232 sensors-10-09232 Article An Intelligent Architecture Based on Field Programmable Gate Arrays
Designed to Detect Moving Objects by Using Principal Component Analysis BravoIgnacio* MazoManuel LázaroJosé L. GardelAlfredo JiménezPedro PizarroDaniel Electronics Department, University Alcala,
Escuela Politecnica, Campus Universitario, Ctra. Madrid Barcelona km. 33.6 28871, Alcala de Henares, Madrid, Spain; E-Mails: mazo@depeca.uah.es (M.M.); lazaro@depeca.uah.es (J.L.L.);
alfredo@depeca.uah.es (A.G.); pjimenez@depeca.uah.es (P.J.); pizarro@depeca.uah.es (D.P.) Author to whom correspondence should be addressed: E-Mail: ibravo@depeca.uah.es; Tel.: +34-918-856-580; Fax:
+34-918-856-540. 2010 15 10 2010 10 10 9232 9251 2 9 2010 1 10 2010 10 10 2010 © 2010 by the authors; licensee MDPI, Basel, Switzerland. 2010
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
This paper presents a complete implementation of the Principal Component Analysis (PCA) algorithm in Field Programmable Gate Array (FPGA) devices applied to high rate background segmentation of
images. The classical sequential execution of different parts of the PCA algorithm has been parallelized. This parallelization has led to the specific development and implementation in hardware of
the different stages of PCA, such as computation of the correlation matrix, matrix diagonalization using the Jacobi method and subspace projections of images. On the application side, the paper
presents a motion detection algorithm, also entirely implemented on the FPGA, and based on the developed PCA core. This consists of dynamically thresholding the differences between the input image
and the one obtained by expressing the input image using the PCA linear subspace previously obtained as a background model. The proposal achieves a high ratio of processed images (up to 120 frames
per second) and high quality segmentation results, with a completely embedded and reliable hardware architecture based on commercial CMOS sensors and FPGA devices.
FPGA PCA CMOS sensor object detection image processing
One of the main research areas in the field of computer vision is the automatic description of the features of a given scene [1,2]. The greater demand made by the performance of image processing
algorithms, together with improved spatial resolution and the increased rate of images per second from the new CMOS sensors, means that the need for computational power is continuously increasing. If
real time performance is to be achieved, the need to reduce algorithm execution time is even greater, requiring the incorporation of an operating system in the processor capable of executing
deterministic tasks, which in turn increases the cost of the products and makes it more difficult to program.
It is usual for the platforms chosen to carry out these algorithms based on sequential programs, in which the only improvements currently available consist in applying multi-threading programming
techniques so the power of the new multicore processors may be used. However, from a performance point of view these processing architectures are not so efficient in many applications like the
digital processing of images, which normally requires a high number of operations to be handled at the bit level as quickly as possible, by processing in parallel a small number of input samples. Due
to the sequential architecture of conventional computers, a notorious amount of operations cannot be performed concurrently. Another issue is the amount of data processed in each instruction, which
is limited by the type and width of used communication bus and the image capture board. For this reason, when a large amount of data must be handled, the system performs slowly. This has given rise
to solutions that make use of coprocessor systems that handle low level preprocessing tasks, where the amount of data to be processed is high but the operations to be carried out are simple [3]. Our
proposal is to create a hardware platform for a specific purpose (designed specifically for one application), as it can produce excellent results working in an ad-hoc low-cost platform. In fact the
FPGA used to validate the proposal could be considered as a FPGA with medium/low features (Xilinx V2P7).
The detection of both static and moving objects within a captured area is one of the more common tasks undertaken by many computer vision applications. Movement analysis is involved, among other
things, in real time applications such as navigation and tracking and obtaining information about static and moving objects within a scene [4]. Movement analysis, which is closely related to the
image transfer rate from the video sensor, is fundamental for addressing topics such as image sequence reconstruction, video compression, fixed image capture and multi-resolution, techniques, etc.
Within the field of image processing previous works have partially developed the processing algorithm of PCA using programmable devices. In [5] for example, all of the PCA is implemented on the FPGA,
however the calculation of eigenvalues is implemented on a PC due to it is mathematically too complex to be implemented on the FPGA. In [6] on the other hand, a variant of PCA called a Modular PCA,
applied to face recognition, has been implemented on an FPGA, as this version of PCA has a much lower volume of mathematical operations than the conventional PCA algorithm. In [7] a system based on
FPGA is proposed for detecting objects known a priori by comparing their eigenvectors. However, as far as the authors know no work has been found on the detection of moving objects employing PCA that
uses FPGAs as the processing element. It is important to point out that in none of the works found is PCA implemented exclusively on FPGAs, due mainly to the heavy data dependence and complex
mathematical operations involve within PCA. The data dependences cause several hazards which make difficult the implementation of efficient pipeline systems. On the other hand, the mathematical
operations needed by PCA algorithm, are not usual operations used for other algorithms (e.g., solving eigenproblems). Due to this fact, new specific mathematical cores have been designed for this
These situations make difficult to segment/divide the hardware processing of the different parts of PCA. For this reason, executing PCA is normally divided between an FPGA and a PC or microprocessor
[5], so that normally an ad-hoc HW/SW partition of the system is made, without adequately exploring the design space (HW/SW co-design methods).
One of the main contributions of this work is the FPGA implementation of the complete PCA algorithm on reconfigurable hardware; indeed it is the first work in the literature to do so. Classic
sequential execution of different parts of the PCA algorithm has been parallelized. This parallelization has led to the development and implementation of seldom used alternatives for the different
stages of PCA. One example is the calculation of eigenvalues and eigenvectors, matrix multiplication in hardware or calculation of a dynamic threshold for detecting moving objects. This latter issue
is another major contribution of the paper because the information generated by PCA is used to detect moving objects. In this work, PCA is implemented on an FPGA to detect moving objects within a
scene, based on the PCA algorithm. To achieve this, a specifically designed intelligent camera has been implemented based on a CMOS sensor and an FPGA [8]. Thanks to the design and implementation of
this new proposal it can be used in any situation requiring an autonomous system (without PC).
The other sections of this paper are as follows: Section 2 sets out the mathematical foundations of the PCA algorithm applied to image processing; Section 3 describes the platform design; Section 4
presents the implementation in VHDL of the PCA algorithm on an FPGA; and finally, Sections 5 and 6 set out the results and present the conclusions respectively.
Principal Component Analysis (PCA) is a method that is used in different fields, such as statistics, power electronics or artificial vision. The main feature of PCA is the reduction of redundant
information, retaining only information that is fundamental (principal components).
Artificial vision is a good example of a field where the PCA technique can be applied directly, as an image contains a large number of highly correlated variables (pixels). Therefore, applying the
PCA technique to image processing allows us to reduce the redundant information of the initial variables and determine the degree of similarity between two or more images by analyzing only the basic
features within the transformed space. This last feature is of interest as far as the detection of new objects within the scene is concerned.
The PCA algorithm can be applied to images using the following steps [9,10]:
Capturing M images to construct a reference model of the scene. We identify each of M references image by I[i] ∈ ℜ^N×N, with i = 1,...M and where it is assumed that the spatial resolution of the
images is N × N.
Each image is represented as a column vector of the dimensions N ^2 × 1
Calculating the mean image from the M reference images : Ψ ∈ ℜ^N^2×1 given for: Ψ = 1 M ⋅ ∑ i = 1 M I i = [ I 1 , 1 + I 2 , 1 + … + I M , 1 M I 1 , 2 + I 2 , 2 + … + I M , 2 M … I 1 , N 2 + I 2 , N 2
+ … + I M , N 2 M ] = [ Ψ 1 Ψ 2 ⋯ Ψ N 2 ] N 2 × 1 ; i = 1 , … , Mwhere I[i,j] is the j (j=1,…,N^2) element of I[i] image.
Form a matrix A ∈ ℜ^N^2×M (3) whose columns are the vectors Φ[j] = I[j] − Ψ (2): Φ i = I i − Ψ = [ I i , 1 − Ψ 1 I i , 2 − Ψ 2 .......... ........ I i , N 2 − Ψ N 2 ] N 2 × 1 = [ Φ i , 1 Φ i , 2
........ Φ i , N 2 ] N 2 × 1 ; i = 1 , ... , M A = [ Φ 1 .... Φ M ] = [ Φ 1 , 1 Φ 2 , 1 .... Φ M , 1 Φ 1 , 2 Φ 2 , 2 .... Φ M , 2 ............ ............ .... ............ Φ 1 , N 2 Φ 2 , N
2 .... Φ M , N 2 ] N 2 × M
Obtaining the covariance matrix, C ∈ ℜ^N^2×N^2 from the matrix A (4): C = 1 M ⋅ A ⋅ A T
Obtaining the associated eigenvalues and eigenvectors of the matrix C. Given that matrix A is of the size N^2 × M and generally N^2 >> M, to reduce the number of operations that must be performed,
the eigenvalues and eigenvectors of (A^T · A) are calculated (5): A T ⋅ A ⋅ V = λ I ⋅ Vwhere V ∈ ℜ ^M ^× ^M is the matrix of the eigenvectors of A^T · A. The eigenvalues of C match up with those of A
^T · A while the eigenvectors of C are obtained from (6): U = A ⋅ V
Obtaining the principal eigenvalues. From the eigenvalues obtained in point 6 the most significant eigenvalues t are selected, using for example, the criteria the normalized root mean square error
(RMSE) [10,11] given by (7), that is the eigenvalues of greatest value (λ[1] > λ[2] > …> λ[t]): RMSE = ∑ i = t + 1 M λ i ∑ i = 1 M λ i < Pwhere P is the percentage of necessary eigenvalues required
to achieved the most significant eigenvalues t.
The transformation matrix U[t] ∈ ℜ ^N^2×t is given by (8) where [u[1],u[2],....,u[t]] are the eigenvectors associated to the eigenvalues λ[1] > λ[2] > ….> λ[t]: U t = [ u 1 , u 2 , … , u t ]
An important issue is the quantification of the value M, which is the number of captured reference images used to build the background. Theoretically, it is a good idea to employ a high M value that
allows different lighting conditions of the same scene to be considered. However the use of a high M number implies a significant increase in computational load and memory storage. The features of
external memory in which background images will be loaded, will determinate the size of M. Due to this fact, the bus width of used external memory (128 bits) and according to the results shown in
[12] in our case, it has been chosen size of M = 8 associated to the same scene without moving obstacles and under soft natural lighting variations. Thanks to this size and CMOS features, it is
possible to read from the sensor 8 pixels in each clock period.
According to the results shown in [12] in our case, a size of M = 8 has been chosen. Once the transformation matrix U[t] has been obtained, the next step is to determine whether in a newly captured
image of the scene new objects have appeared. To do this the following steps must be performed:
Projection onto the transformed space. The first step is the projection onto the transformed space using (9): Ω = U t T ⋅ Φ j = U t T ⋅ ( I j − Ψ )where Ω ∈ ℜ^t^×1 is characterized by a vector of
dimension t (Ω = [ω[1], ω[2],…, ω[t]]^T), where each component ω[i] represents the contribution of each eigenvector in the representation of I[j].
Recovering the projected image. Once the image has been projected onto the transformed space using (9), then Φ̂ ∈ ℜ^N^2×1 is recovered using (10): Φ ^ j = U t ⋅ Ω
Determining the existence of new objects in the scene. Finally, the captured image is compared with the recovered image, thus obtaining what is termed the error recovery. If the result of the
comparison is above a determined threshold (Th[MD]) it implies the presence of new objects in the scene (11): ‖ Φ j − Φ ^ j ‖ ≤ Th MD → there are no new objects in the scene ‖ Φ j − Φ ^ j ‖ > Th MD →
there are new objects in the sceneThe threshold Th[MD] is a dynamically obtained value that is adjusted according to the conditions of the scene.
Spatial localization of the detected object. To detect the presence of new objects it is only necessary to apply expression (11). However, if we want to know in which part of the scene the new object
has appeared a localization method must be found.. With the aim of reducing the effect of noise, the value of each pixel of the captured image and that of the recovered image is averaged with that of
the adjacent pixels by means of a mask of q × q elements. As a result a matrix known as an average distance map is obtained (MD[V] ∈ ℜ^N^2×N^2), where every one of its elements (ε[w,i]) corresponds
to the Euclidean distance between the corresponding average pixels of the original and recovered image (12). ε w . i = ‖ Φ w , i − Φ ^ w , i ‖ w = 1 , 2 , … N i = 1 , 2 , .... , N
Once the MD[V] has been obtained, the next step is to threshold the map so that the new objects can be found easily. To do this, a new binary image is built (BW) where each element is the result of a
comparison between each MD[V] pixel and a Th[MD] threshold (13): BW i = 255 if ( ε wi ) = ‖ Φ wi − Φ ^ wi ‖ > Th MD BW i = 0 if ( ε wi ) = ‖ Φ wi − Φ ^ wi ‖ ≤ Th MD
The system proposed is based on a high speed CMOS sensor (up to 500 images per second with a maximum resolution of 1,280 × 1,024 [13]) and an FPGA in which a novel design has been implemented for
managing and capturing the images from the sensor, as well as executing the PCA algorithm. The system implemented on the FPGA is separated into the following logical blocks as shown in Figure 1 with
green color:
CMOS sensor controller: This block is responsible for implementing image demands to the CMOS sensor such as parameterizing its internal registers according to the desired configuration (images per
second, exposure time, etc.).
Image capture controller: the purpose of this block is to allow the user to select an area of interest within the image from the CMOS sensor.
External memory controller: the system is equipped with a 128 MB, SDRAM memory bank that is external to the FPGA. Images from the CMOS sensor are stored in this bank.
Communications Controller with the PC: this block controls the communication between the FPGA and the PC. This is used to transmit commands and results.
Head Controller: This block is responsible for synchronizing the entire system so that everything works correctly and at maximum speed.
PCA algorithm: This block implements the PCA algorithm and its implementation is the most important contribution of this work.
The mathematical complexity of the operations of the PCA algorithm presented in Section 2 (calculation of eigenvectors, matrix multiplication, square roots, etc) makes it impractical to implement
them directly on reconfigurable hardware. The proposal and selection of different hardware structures and computing alternatives in order to obtain an efficient solution to resolve these operations
on FPGAs is essential for the PCA implementation and, thus, constitutes one of the major contributions of this paper. This section presents the hardware solution found which permits the PCA algorithm
to be implemented on an FPGA. Figure 2 shows a block diagram of the PCA algorithm implemented on the FPGA, grouping the different modules into three stages: generation of eigenvectors (light yellow),
on-line (light orange), and object detection (light pink). The three phases in which the PCA algorithm is divided are now described.
The first phase of the PCA algorithm is the generation of the eigenvectors of the reduced transformation matrix U[t]. This first phase includes five stages:
Calculating the mean of the M images (Ψ ∈ ℜ^N^2×1) and the matrix A ∈ ℜ^N^2×M (3).
Obtaining the covariance matrix C^T ∈ ℜ^M^×^M (5).
Calculating the eigenvectors of the matrix V ∈ ℜ^M^×^M and the posterior matrix of the reduced eigenvectors V[t] ∈ ℜ^M^×^t.
Obtaining the eigenvectors of the matrix U[t] ∈ ℜ^N^2×t from V[t] ∈ ℜ^M^×^M where t < M.
Calculating the norms of matrix eigenvectors.
The hardware architecture that has been developed for this module stores the captured M images in an SDRAM external memory. The block shown in Figure 3 has been implemented on the FPGA, where the M =
8 images are stored in different memory components (B[1]). Once the eight pixels have been extracted, one for each image, the mean calculation process is initiated using a set of cascade adders (see
B[2] Figure 3). As this process takes three clock cycles and the aim is for the system to be as segmented as possible, the eight extracted pixels are inserted into a delay unit consisting of
flip-flops that synchronizes the subtraction process of each pixel with that of the corresponding mean (B[6]).
Generating C^T from matrix A, means the product of two matrices A^T · A, must be produced on an FPGA, which entails a complex process. In the case of the PCA the aim is to multiplex the matrix
multiplication module using it to: generate the covariance matrix, generate the eigenvector matrix (U[t]), project an image onto the transformed space and recover the projected image. Different
approaches to the matrix multiplication have been analyzed and developed by the authors [14]. After this study, an ad-hoc matrix multiplier system based on a semi-systolic array proposed by the
authors in [14] has been chosen because the maximum performance for PCA is achieved with this approach thanks to the possibility to reuse the system for the different types of matrix multiplication
that PCA needs.
The computation of eigenvalues and eigenvectors represents the greatest computational burden on the PCA algorithm. Different techniques have been proposed for obtaining the eigenvalues of a matrix
using specific hardware, all of them based on recurrent methods that look to diagonalize the matrix [15,16]. Once the matrix has been diagonalized, the eigenvalues coincide with the values of the
diagonal. The method proposed in [17] is the most interesting as it allows parallel processing hardware structures to be implemented [18]. For this reason, the solution developed in this work is
based on the Jacobi method. A previous article by the authors [19] describes the architecture developed.
The first step in determining the most significant t eigenvectors is to arrange the eigenvalues and their associated eigenvectors in either ascending or descending order. This step is necessary as
the Jacobi method does not generate the eigenvalues in order. To determine t, the largest t eigenvalues are found and then their associated eigenvectors are selected depending on how much bigger than
the eigenvalues that have been obtained the user wants them to be (7). In this work, bubble sort has been used as the sorting algorithm [20].
To obtain the matrix U[t], according to (6), the matrix A must be multiplied by V[t]. To do this, once again the semi-systolic array presented in [14] is used.
The eigenvectors obtained in the previous stage do not possess a unit module so they must be normalized (14) U[tn] according to (15): n j = ∑ i = 1 N 2 ( u i , j ) 2 j = 1 , … , t U tn =
U t norms = ( u 1 n 1 u 2 n 2 .... u t n t ) norms = [ n 1 , n 2 , … , n t ] ∈ ℜ 1 × t
To implement in hardware the arithmetical operations shown in expressions (14) and (15) is extremely complex as a consequence of the square root, and it also uses a large amount of resources. To
avoid calculating the square root when calculating Φ̂[j] it is only necessary to express this matrix in accordance with the squared norm, as shown in (16): Φ ^ j = U tn ⋅ Ω = U tn ⋅ U tn T ⋅ Φ j = U t
⋅ U t T ⋅ Φ j norms 2
If there is a new object in the captured image, with respect to the reference scene, it is determined during the on-line stage. For this to be done, the new captured image is projected onto the
transformed space so that it can be recovered later and studied to determine whether or not there is a new object in the scene. To do this the following steps are followed:
Subtraction of the mean of the present image: If I[j] is the captured image, Φ[j] is obtained.
Projecting Φ[j] onto the transformed space and obtaining Φ̂[j]: With the aim of reaching the maximum concurrence possible when executing (16), first the U t T ⋅ Φ j product is performed and this
result is divided by the squared norms ( ( U t T ⋅ Φ j ) norms 2), and finally U t ⋅ ( ( U t T ⋅ Φ j ) norms 2 ) product is performed.
Determining the recovery error: In this final stage, the degree of similarity between Φ[j] and Φ̂[j]is evaluated.
Figure 4 shows the VHDL encoded modular design of this on-line stage. With respect to the internal workings of the system shown in Figure 4, this starts when a new vector image I[j] is captured and
later stored in the external memory so that the system has an initial latency of one image. As explained earlier, once Φ[j] has been obtained the next step is to produce the U t T ⋅ Φ j. To do this,
the semi-systolic array for matrix multiplication is used [14]. It is important to point out at this point that the execution time of the Matrix Multiplier depends on the number of significant
eigenvalues (t). In accordance with the percentage of significant eigenvalues (see (7)), a value of t equals 6 has been decided upon. This reduction introduces a recovery error (ε) (17), after
analyzing 1,000 images it could be seen that the induced error is approximately 1%: ε = ‖ Φ j − Φ ^ j ‖ = ∑ i = 1 N 2 ( Φ j i − Φ ^ j i ) 2
Once the first results from the U t T ⋅ Φ j product have been obtained, the next step is to divide these results by norms^2. As each component of U t T ⋅ Φ j is generated in one clock cycle, given
that they are output by the semi-systolic array, they are divided by the corresponding squared norm.
To perform the division operation on an FPGA, there are basically two possibilities: either design a division unit specifically for that purpose, or use a coordinate rotation digital computer
(CORDIC) algorithm [21]. In this work the latter option has been chosen, as it consumes fewer resources than the former. Dividing two numbers is feasible in CORDIC if it is used in vectorization mode
with a linear coordinate system [22]. To do so, a division module based on a parallel CORDIC architecture has been implemented.
When the first component of the division has been obtained, the next step to be performed in (16) is to obtain Φ̂[j]. Once again, to perform this fourth matrix multiplication, the semi-systolic array
described in [14] is used.
This section presents the solution developed for implementing an identification of new objects from the error recovery (ε) system in reconfigurable hardware (17). It proposes the building of an error
recovery map or Map of Distances (MD) that will permit the new objects to be located spatially. The size of this map of distances will coincide with the size of the image, where each of its positions
is the pixel to pixel Euclidean distance between Φ̂[j] and Φ[j]. A new Map of Distances (MD[V]) will be built in order to reduce the noise effect. The final detection of moving obstacles will be
obtained using the dynamic threshold Th[MD] (11). Calculating Th[MD] presents difficulties as it must be adaptable and its value depends on both the features of the scene under analysis and the
lighting conditions. For this reason, in this section we present a new method for dynamically calculating the threshold that minimizes the false detection of new objects within the scene of interest.
Figure 5 shows a block diagram of this proposal for detecting objects from Φ̂[j] and Φ[j] (green blocks). Next the hardware solution implemented in each block of Figure 5 is presented.
The Map of Distances MD is obtained from (18), ε′[j] being the square of the Euclidean distance between each component Φ[ji] ∈ Φ[j] and Φ̂[ji] ∈ Φ̂[j] i = 1,....N^2 (for images of the size N × N): ε i
′ = ‖ Φ j i − Φ ^ j i ‖ 2 i = 1 , … N 2
Working with the square of the Euclidean distance rather than the Euclidean distance (17), facilitates the design of hardware associated with this function, as it avoids the need to perform the
square root operation. As such, to obtain MD only requires one subtraction and one multiplication operation, so that with an adder/subtraction block and a multiplier connected in cascade the
segmented execution of (18) can be performed.
Once the initial components of MD have been generated, the generating of the map of average distances (MD[V]) can be started. The use of a mask of q × q components is proposed that averages the
pixels adjacent to MD, applying a 2D low-pass filter. The components that make up the map MD[V] are ε′[v[i,w]]; i, w = 1,....,N.
To provide a compromise value to the size of mask q, different sizes applied to different maps MD have been simulated; all of them are fixed point encoded. The size chosen for q is 3, given that it
provides the algorithm with a certain degree of robustness and reliability, and few hardware resources are required.
To implement the averaging function with masks of q × q (3 × 3) on the adjacent pixels the corresponding convolution function is implemented [23]. To select the best alternative for hardware
implementation, several proposals for convolutions have been designed [23], evaluating at all times the execution time as well as how much of the FPGA’s internal resources are consumed.
To perform the convolution between a matrix and a generic mask, nine multiplication operations and eight accumulation operations must be performed for each resulting component. However, when all the
coefficients of the mask have been identified, as happens in our case, another way of performing the convolution is according to (19), whereby one that reduces the number of multiplications to one.
In this way, to obtain each ε′[v[i,w]] component of the MD[V] it is necessary to perform a nine component sum backlog and one multiplication for the equivalent factor in fixed point: ε ′ ν i , w = 1
/ 9 ⋅ ( ε ′ i − 1 , w − 1 + ε ′ i − 1 , w + ε ′ i − 1 , w + 1 + ε ′ i , w − 1 + ε ′ i , w + ε ′ i , w + 1 + ε ′ i + 1 , w − 1 + ε ′ i + 1 , w + ε ′ i + 1 , w + 1 )
Once the map of average distances (MD[V]) has been obtained, the next step is to analyze the map to evaluate whether or not there are new objects in the scene of interest. To do so, a threshold Th
[MD] is obtained, which, when applied to MD[V] makes it possible to perform the segmentation and as a consequence detect the presence of new objects. The value of Th[MD] must be dynamic as its value
must adapt, amongst other factors, to changes in light within the scene. In order to obtain this dynamic Th[MD] different alternatives have been proposed, [12,24,25]. Our proposal calculates the
histogram (with f intervals) of the maximum Euclidean distances of each column of the MD[V] (Figure 6) and then obtains the dynamic threshold Th[MD] from the histogram. This algorithm, implemented on
an FPGA, generates excellent results, as will be seen later in the results section.
Analyzing the information supplied by the histogram on the maximums of the MD[V] columns, it can be seen how most of the maximum Euclidean distances represented are concentrated in the lower
intervals. However, when a new object appears in the scene being studied, the maximum Euclidean distances of the MD[V] columns where the object is located are expressed by a valley in the histogram.
If there is no new object in the scene, then the valley does not appear. On the basis of this last feature of the histogram, to threshold MD[V] it is necessary to find the value of Th[MD] that makes
it possible to discriminate between the new object and the background. The minimum value of Th[MD] needed to correctly detect new objects must be the same as the value of the histogram interval that
contains the valley associated with the new object.
The hardware to perform the threshold is shown in Figure 7. Each block in Figure 7 is described below:
Block 1: this block is responsible for calculating the maximum of each column of the map of distances MD[V]. Internally it consists of a single register that stores the maximum value and a comparator
that evaluates whether the new data is bigger or smaller than the stored temporal maximum.
Block 2: After calculating the maximums of the columns of the MD[V], Block 2 is responsible for building the histogram of the maximums of the columns. It is executed in parallel with Block 1 once the
maximum of the first column has been obtained. Every time a maximum is obtained the histogram interval that belongs to that maximum must be looked for and its accumulator increased by 1.
Block 3: This module, which is executed when Block 2 generates the first data, is responsible for calculating the maximum values of the histogram (V[MX] of Figure 6). This block works as follows:
every time the maximum of a column is obtained in Block 2, a new value is added to the corresponding histogram interval and the number of the histogram interval with the maximum accumulated value is
updated. At the same time, in Block 3 the increased value is evaluated to see whether it is the largest. If it proves to be so, then it is stored so that it can be compared with the following output
from Block 2 and its memory address, which gives the location of the new maximum, generated by Block 2 is also stored.
Block 4: Finally, this component is responsible for looking for valleys in the histogram once Block 2 and Block 3 have finished.
To find a valley, a hardware block has been designed to check the memory of Block 2, which contains the histogram of the maximums of the columns of MD[V]. The counter starts from the address stored
in Block 3, that is to say, the address of the histogram interval with the maximum accumulated value. To find a valley, it is only necessary to find a value in the memory that is bigger than the one
stored in the position before it. If no local minimum exists the system will increase the threshold (checking the intervals defined by the histogram) until it considers that the threshold is situated
in the extreme interval and then classifies all the pixels in the image as belonging to the background. The number of histogram intervals (f) has been empirically set at 10, as with this value the
developed proposal works correctly.
This section sets out the results obtained in detecting new objects with a FPGA running PCA algorithm. All the images presented in this work have been captured by an “intelligent camera” described in
From a quantitative point of view, in calculating the execution time of the entire proposal presented in this work (T[PCA_TOTAL]) from the moment the first M images are captured, the total time
consumed is given by (20), with Table 1 giving a description of each of the times in (20): T PCA _ TOTAL = T GEN _ WR _ U + T IMAGE + L MEM + T OBJ
When it comes to calculating the number of complete clock cycles employed by T[PCA_TOTAL], the value obtained is not constant as it depends on the number of significant eigenvectors, the size of the
matrix and the number of Jacobi algorithm iterations, as explained in [19]. Adjusting the expression (20) for six eigenvectors (worst case), capturing eight images (256 × 256 pixels) to build a
reference model (M = 8), an internal data width of 18 bits (n = 8) and 23 iterations for the Jacobi algorithm the value obtained in clock cycles is: T PCA _ TOTAL = 131076 T CLK _ CAMERA + 526939
T CLKwhere T[CLK_CAMERA] is the signal period of the CMOS sensor’s clock and T[CLK] the FPGA’s master clock. Clock Camera is generated by the FPGA using a DCM (digital clock management) block.
Thanks to this element and a bank register managed for a FSM (finite state machine), both clocks working rightly. To obtain a ratio of the number of images the system processes, if the CMOS sensor’s
clock (T[CLK_CAMERA]) is 66 MHz and the FPGA’s master clock is 100 MHz (frequency reached once the entire system has been implemented) a minimum of 121 images of 256 × 256 pixels have been processed
per second. This ratio increases notably if any of the following situations occur:
Number of significant eigenvectors (t) under four. In this case the number of matrix multiplication operations (6), (9) and (10) are notably reduced. In this way the new T[PCA_TOTAL] value would
reach an equivalent image per second ratio of 189.
Selective actualization. Cadence is another very important factor that conditions the number of images processed per second when updating the eigenvectors of the matrix (background model). If the
eigenvalues of the matrix are not continuously updated, but between one update and another b images pass, the new ratio of images per second obtained is shown in Figure 8.
As may be seen from this figure, from b = 100 onwards, independent of the number of significant eigenvectors, the system reaches its maximum value at around 250 images per second for t ≤ 3 and around
190 for t ≥ 4. This is because the system segmentation is at its most efficient at this number of images. In Table 2 a summary of the final amount of resources consumed by the different blocks
implemented on the Xilinx FPGA is presented. It is important to point out that due to the limited resources of the FPGA every attempt has been made to optimise the design at all times, with the aim
of reducing the use of internal resources. Thanks to this, from a number of BRAM (block RAM) components and slices point of view, it has been possible to implement the entire system on a medium to
low range FPGA like the Xilinx XC2VP7.
With respect to the frequency of the FPGA clock, according to the reports generated by the implementation tool, a maximum value of 1,124 MHz for the entire FPGA is assured. However, the master
frequency chosen for our design is 100 MHz as from this value all the other necessary frequencies can be generated (the camera and external memory frequencies).
As for the real results obtained, Figure 9 shows images captured with the developed platform [8] with an initial resolution of 1,280 × 1,023 reducing their size to 256 × 256 by applying a binning
process on the FPGA. This sequence was captured in the grounds of the University of Alcala where the distance between the objects to be detected, in this case people and the camera, is 25 meters.
Figure 10 shows the detection that was performed. The proposed design has been tested with a bank of 1,000 images captured under moderate lighting conditions in outside environments. The accuracy
achieved in the test was remarkable (around 97% of true matches). Despite the promising results for an embedded architecture, it is widely known that when using PCA for modelling strong illumination
changes in the intensity values of the image require a high amount of PCA vectors to train the background. Besides, due to the fact that illumination changes are non-linear variations of the
intensity, the PCA subspace cannot model such variations properly, which could increase the number of false detections. In a near future the proposal can be easily applied to other colour spaces,
such as the light invariant space proposed in [26], which maps a RGB image to a scalar image where same surfaces under different illuminations are mapped to the same intensity value.
This work presents a new image capture and processing system implemented on FPGAs for detecting new objects in a scene, starting from a reference model of the scene. To achieve this, the Principle
Component Analysis (PCA) technique has been used. The main objective is to parallelize it in order to achieve a concurrent execution which will enable processing speeds of around 120 images per
second to be reached. This processing speed, including all stages included in the PCA technique (calculating eigenvalues and eigenvectors, projection and recovery of images to/from the transformed
space, obtaining map of distances, etc.) responds to the requirements of many applications, where the goal is the detection of new objects in the scene, even in those cases where, for a variety of
reasons, (changes in lighting for example) a continuous update of the background model is required. The proposed solution is a significant improvement on other hybrid solutions based on the use of a
PC and an FPGA [5]. The complete integrated development of the PCA algorithm on an FPGA was a task that until now had not been achieved or performed, at least according to our thorough review of
related work done on this topic. Thanks to the designed solution new applications with PCA algorithm could be implemented for new proposals or applications.
This work was made possible thanks to the sponsorship of the Ministry of Education and Science (MEC) and the projects ESPIRA (REF-DPI2009-10143) and SIAUCON (REF-CCG08-UAH/DPI-4139), funded by the
University of Alcala and the Madrid Regional Government.
RathaNKJainAKComputer vision algorithms on reconfigurable logic arrays199910294310.1109/71.744833 MagdalenoERodríguezMRodríguez-RamosJMAn efficient pipeline wavefront phase recovery for the CAFADIS
camera for extremely large telescopes20101011510.1109/JSEN.2009.203928722399874 TaluHMIgciETekinMESevtekinHSGençBÇHeywoodMIReconfigurable computing implementation of binary morphological operators
using 4-, 6- and 8- connectivityProceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'00)Las Vegas, NV, USAJune 200033863389
ParraIFernándezDSoteloMABergasaLMRevengaPNuevoOMGarcíaMACombination of feature extraction methods for SVM pedestrian detection2007829230710.1109/TITS.2007.894194 FleuryMSelfRPDowntonACDevelopment of
a fine-grained parallel Karhunen-Loeve transform20046452053510.1016/j.jpdc.2004.03.003 GottumukkalRAsariVKAn improved face recognition technique based on modular PCA approach20042542943610.1016/
j.patrec.2003.11.005 ZhongFCapsonDWSchuurmanDCParallel architecture for PCA image feature detection using FPGAProceedings of Canadian Conference on Electrical and Computer Engineering (CCECE 2008)
Dundas, ON, CanadaMay 200813411344 BravoIJiménezPMazoMLázaroJLMartínEArchitecture based on FPGA’s for real-time image processing20063985153157 DuntemanGHSage PublicationsNewbury Park, CA, USA1989
JolliffeIT2nd edSpringer Series in StatisticsSpringer-VerlagNew York, NY, USA2002 SwetsDLWengJJUsing discriminant eigenfeatures for image retrieval19961883183610.1109/34.531802
VázquezJFLázaroJLMazoMLunaCASensor for object detection in railway environment2008669069810.1166/sl.2008.m104 Aptina ImagingAptina Imaging CorpSan Jose, CA, USA2004 BravoIJimenezPMazoMLazaroJLde las
HerasJJGardelADifferent proposals to matrix multiplication based on FPGAsProceedings of the IEEE International Symposium on Industrial Electronics (ISIE07)Vigo, SpainDecember 200717091714
AhmedsaidAAmiraABouridaneAAccelerating MUSIC method on reconfigurable hardware for source localizationProceedings of the 2004 International Symposium on Circuits and Systems (ISCAS'04)Tel Aviv,
IsraelDecember 2004369372 KimMIchigeKAraiHImplementation of FPGA based fast DOA estimator using unitary MUSIC algorithmProceedings of the IEEE 58th Vehicular Technology Conference (VTC 03)Jeju,
KoreaApril 2003213217 WilkinsonJHOxford University PressOxford, UK1999 BrentRPLukFTThe solution of singular-value and symmetric eigenvalue problems on multiprocessor arrays19856698410.1137/0906007
BravoIMazoMLazaroJLJimenezPGardelAMarronMNovel HW Architecture Based on FPGAs Oriented to Solve the Eigen Problem2008161722172510.1109/TVLSI.2008.2001939 ToddSAlgorithm and hardware for a merge sort
using multiple processors19782250951710.1147/rd.225.0509 WaltherJSA unified algorithm for elementary functionsProceedings of AFIPS Spring Joint Computer ConferenceAtlantic City, NJ, USAMay 1971379385
KarabernouSMTerrantiFReal-time FPGA implementation of Hough Transform using gradient and CORDIC algorithm20051110091017 BravoIHernándezAGardelAMateosRLázaroJLDíazVDifferent proposals to the
multiplication of 3 × 3 vision mask in VHDL for FPGAsProceedings of the IEEE Conference on Emerging Technologies and Factory Automation (ETFA)Lisbon, PortugalSeptember 2003208211
NakagawaYRosenfeldASome experiments on variable thresholding19791119120410.1016/0031-3203(79)90006-2 ManaySYezziAAnti-geometric diffusion for thresholding and fast segmentation2003121310132310.1109/
TIP.2003.81803918244690 FinlaysonGDHordleySDLuCDrewMSOn the removal of shadows from images200628596810.1109/TPAMI.2006.18
Block diagram of the internal architecture of the FPGA.
Block diagram of the PCA algorithm implemented on an FPGA.
Block diagram of the proposed circuit for calculating the mean (Ψ) of the M captured images.
Block diagram of the modules of the design in VHDL of the on-line stage of the PCA.
Proposal for the system consisting of the construction of the MD, detection of new objects and the updating of the background model.
Example of histogram construction of the maximum of the columns for an average map of distances (MD[V]).
Block diagram on an FPGA of the dynamic threshold calculating system for detecting new objects.
Ratio of images achieved per second with b ≠ 1.
Sequence of images captured to determine new objects.
Sequence of images detected to determine a new object from those captured in Figure 9.
Description of the partial times of T[PCA_TOTAL].
T[GEN_WR_U] Time the FPGA takes to generate and write in SDRAM the eigenvectors of the matrix U[t].
T[IMAGE] Time employed in capturing a new image and its subsequent writing in SDRAM.
L[MEM] Latency of the SDRAM memory, from the time it gives the order to read an image until the first data is received.
T[OBJ] Time consumed in detecting new objects after the recovered image (Φ̂[j]) has been obtained from the transformed space
Summary of all the resources consumed by the entire developed system on a XC2VP7.
Area (Slices) BRAM Multipliers f[CLKMAX]
4225 (86%) 40 (91%) 43 (98%) 112,4MHz | {"url":"http://www.mdpi.com/1424-8220/10/10/9232/xml","timestamp":"2014-04-19T12:00:38Z","content_type":null,"content_length":"107139","record_id":"<urn:uuid:ae4385ed-7f23-45c6-8c17-093a296482ec>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00387-ip-10-147-4-33.ec2.internal.warc.gz"} |
Title: Nonlinear extraction of 'Independent Components' of
elliptically symmetric densities using radial Gaussianization
Authors: Siwei Lyu and Eero P. Simoncelli
We consider the problem of efficiently encoding a signal by transforming it to a new
representation whose components are statistically independent (also known as factorial).
A widely studied family of solutions, generally known as independent components
analysis (ICA), exists for the case when the signal is generated as a linear transformation
of independent non-Gaussian sources. Here, we examine a complementary case, in
which the signal density is non-Gaussian but elliptically symmetric. In this case, no linear
transform suffices to properly decompose the signal into independent components,
and thus, the ICA methodology fails. We show that a simple nonlinear transformation,
which we call radial Gaussianization (RG), provides an exact solution for this case. We
then examine this methodology in the context of natural image statistics, demonstrating
that joint statistics of spatially proximal coefficients in a multi-scale image representation
are better described as elliptical than factorial. We quantify this by showing that
reduction in dependency achieved by RG is far greater than that achieved by ICA, for
local spatial neighborhoods. We also show that the RG transformation may be closely
approximated by divisive normalization transformations that have been used to model
the nonlinear response properties of visual neurons, and that have been shown to reduce
dependencies between multi-scale image coefficients. | {"url":"http://cs.nyu.edu/csweb/Research/TechReports/TR2008-911/TR2008-911.html","timestamp":"2014-04-18T05:34:02Z","content_type":null,"content_length":"1860","record_id":"<urn:uuid:44662991-a6e0-4bb7-9823-dec51c29830f>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00479-ip-10-147-4-33.ec2.internal.warc.gz"} |
Bootstrap-after-Bootstrap Model Averaging for Reducing Model Uncertainty in Model Selection for Air Pollution Mortality Studies
Steven Roberts, Michael A. Martin
School of Finance and Applied Statistics, College of Business and Economics, Australian National University, Canberra, Australian Capital Territory, Australia
Environ Health Perspect 118:131-136 (2009). http://dx.doi.org/10.1289/ehp.0901007 [online 17 September 2009]
Research Article
Background: Concerns have been raised about findings of associations between particulate matter (PM) air pollution and mortality that have been based on a single “best” model arising from a model
selection procedure, because such a strategy may ignore model uncertainty inherently involved in searching through a set of candidate models to find the best model. Model averaging has been proposed
as a method of allowing for model uncertainty in this context.
Objectives: To propose an extension (double BOOT) to a previously described bootstrap model-averaging procedure (BOOT) for use in time series studies of the association between PM and mortality. We
compared double BOOT and BOOT with Bayesian model averaging (BMA) and a standard method of model selection [standard Akaike’s information criterion (AIC)].
Method: Actual time series data from the United States are used to conduct a simulation study to compare and contrast the performance of double BOOT, BOOT, BMA, and standard AIC.
Results: Double BOOT produced estimates of the effect of PM on mortality that have had smaller root mean squared error than did those produced by BOOT, BMA, and standard AIC. This performance boost
resulted from estimates produced by double BOOT having smaller variance than those produced by BOOTand BMA.
Conclusions: Double BOOT is a viable alternative to BOOT and BMA for producing estimates of the mortality effect of PM.
Key words: air pollution, Bayesian, bootstrap, model averaging, mortality, particulate matter
Address correspondence to S. Roberts, School of Finance and Applied Statistics, Australian National University, Canberra, ACT 0200, Australia. Telephone: 61-2-6125-3470. Fax: 61-2-6125-0087. E-mail:
This work was supported by the Australian Research Council (DP0878988).
The authors declare they have no competing -financial interests.
Received 20 May 2009; accepted 17 September 2009; online 17 September 2009.
Over the past decade, time series studies that have investigated the association between daily variations in particulate matter (PM) air pollution and daily variations in mortality have become
commonplace (Breitner et al. 2009; Kelsall et al. 1997; Roberts 2004). Studies conducted in Europe and North America have found statistically significant associations between increases in daily PM
concentrations and increases in daily mortality (Samoli et al. 2008). One common feature of these time series studies is that myriad modeling choices must be made to arrive at an “optimal” model from
which an estimate of the association between PM and mortality can be obtained. This array of choices means there are potentially many candidate models for investigating the association between daily
PM and mortality. In some studies, models that are selected because they optimize a particular model selection criterion are used to infer a relationship between PM and mortality (Draper 1995;
Goldberg et al. 2006; Kelsall et al. 1997). In this context, concerns have been raised in the literature about statistical issues that may arise from the process of selecting a single model from
among a potentially large number of competing candidates (Clyde 2000; Koop and Tole 2004; National Research Council 1998). The procedure of selecting a single “best” model may ignore the model
uncertainty, which is inherently involved in searching through the set of candidate models to determine the best one. Ignoring model uncertainty is problematic because it reflects statistical
variation not captured within the single chosen model, and failure to account for this variation can increase the chance of erroneously concluding a statistically significant association between PM
and mortality (Clyde 2000; National Research Council 1998).
Model averaging in both Bayesian and frequentist forms has been proposed as a means of allowing for model uncertainty in time series studies of PM and mortality (Clyde 2000; Koop and Tole 2004, 2006;
Martin and Roberts 2006). Model-averaging procedures assign probabilities or weights to each candidate model that reflect the degree to which the model is supported by the data. These probabilities
can be used to produce “weighted” average estimates of the association between PM and mortality that explicitly incorporate information from each candidate model. This process of explicitly
incorporating each candidate model into the estimation process produces estimates that incorporate the variation inherent in the model selection process. Clyde (2000) and Koop and Tole (2004, 2006)
implemented Bayesian model-averaging (BMA) techniques to estimate the association between air pollution and mortality. Martin and Roberts (2006) implemented model averaging using a bootstrap-based
procedure and showed that it is competitive with BMA in that context. Previous investigations have also used the bootstrap in the context of time series studies of air pollution, including
investigations of the effect of concurvity in generalized additive models (Figueiras et al. 2005; Ramsay et al. 2003).
In this paper, we discuss a double bootstrap model-averaging (double BOOT) approach that extends and improves the bootstrap model-averaging (BOOT) procedure that was implemented in Martin and Roberts
Materials and Methods
Materials. The data used in this report were obtained from the publicly available National Morbidity, Mortality, and Air Pollution Study database (Zeger et al. 2006). The data consist of daily time
series of mortality, temperature, dew point temperature, and PM air pollution measures for five United States (U.S.) cities for the period 19992000. The mortality data are daily counts of
non-accidental deaths of individuals ≥ 65 years of age. The measure of ambient PM used is the ambient 24-hr concentration of PM of < 2.5 µm in aerodynamic diameter (PM[2.5]) measured in micrograms
per cubic meter.
The five U.S. cities included in this study—Birmingham, Alabama; Orlando, Florida; Seattle, Washington; St. Louis, Missouri; and Tampa, Florida—were selected because they had nearly complete PM[2.5]
data over the period of investigation. For these cities, the number of days missing PM[2.5] concentrations over the 730-day period of investigation ranged from 2 to 18 days. Missing PM[2.5]
concentrations were imputed using the average of the previous and subsequent days’ concentrations.
Methods. We investigated model averaging in the context of additive Poisson log-linear models. Under these models, the daily mortality counts are modeled as independent Poisson random variables with
mean µ[t] on day t
log(µ[t]) = confounders(α)[t] + θPM[2.5][t,j], [1]
where confounders(α)[t] represent other time-varying variables related to daily mortality, and PM[2.5][t][,][j] is the PM[2.5] concentration on day tj, for a specific time lag j; α is a tuning
parameter—as α increases, so too does the flexibility of the smooth functions used to adjust for the effects of the confounders. Adjusting for confounders is important to avoid spurious findings of
an association between PM[2.5] and mortality (Bell et al. 2004). Commonly used confounders include weather variables, such as temperature and dew point temperature, and time (Dominici et al. 2003).
Our focus in Model [1] on a PM[2.5] exposure measure, which corresponds to a specific lag of PM[2.5,] is consistent with recent time series studies (Dominici et al. 2006, 2007; Peng et al. 2009).
Models of the same general form as Model [1] are commonly used in time series studies of the adverse health effects of PM (Dominici et al. 2007; Peng et al. 2006; Roberts 2005).
Using Model [1] involves selecting a value of α and a lag of PM[2.5]. For example, if p values of α and q lags of PM[2.5] were thought plausible, then K = p × q candidate models could be fitted and
assessed with respect to some model selection criterion. If the K candidate models are fitted and a single “best” model chosen, the common practice of reporting the statistical characteristics of the
winning model effectively ignores the statistical variation suffered as a result of the model selection procedure itself.
In the paragraphs that follow, we describe Akaike’s information criterion (AIC; Akaike 1973) and outline the bootstrap (BOOT) method used by Martin and Roberts (2006) and our extension that refines
this method.
AIC is commonly used for model selection in time series studies of the association between PM and mortality (Goldberg et al. 2006; Samoli et al. 2008). It takes a measure of the lack of fit of a
model and adds a penalty for the number of parameters in the model. Specifically, AIC is defined as
AIC = 2(maximum log-likelihood)
+ 2(number of parameters). [2]
To use AIC for model selection, the model with the smallest value of AIC among the candidate models is selected. Further details on AIC, including a discussion of its derivation, can be found in
numerous articles (e.g., Burnham and Anderson 2004). In the context of the models considered in this paper, the number of parameters is an increasing function of α.
The BOOT method used by Martin and Roberts (2006) proceeds through the following steps:
1. Fit the K candidate models defined by Model [1]. Select as “best” the model with the smallest value of AIC, which is denoted M*. We also define M[i] to represent candidate model i fitted to the
observed mortality time series data, for i = 1, 2,…, K.
2. Extract the mean adjusted, standardized Pearson residuals (Davison and Hinkley 1997a) and the estimated mean mortality counts from the best model M*, which was obtained in step 1. In our context,
the mean adjusted, standardized Pearson residuals ξ[t] are defined as
1. where T is the length of the mortality time series, y[t] is the observed mortality count on day t, û[t] is the estimated mean mortality count on day t, and h[t] is the leverage for the
observation on day t.
3. Use the stationary bootstrap to generate B resamples of the residuals ξ[1],…, ξ[T] obtained in step 2. The stationary bootstrap is implemented using the approach of Politis and Romano (1994).
Under this approach, the stationary bootstrap resamples blocks of data of random length, where the length of each block has a geometric distribution.
4. Create B bootstrap replicate mortality time series by adding the estimated mean mortality counts from step 2 to each of the B resampled residual series generated in step 3. This process is
carried out using the following formula:
d[[t]]* = µ^ˆ[t] + [√]^—µ^ˆ[t]ξ[t]*,t = 1, …, T, [4]
1. where ξ[1]*,…, ξ[T]* is a resampled residual series, and d[1]*,…, d[T]* is the resultant bootstrap replicate mortality time series. The d[1]*,…, d[T]* are rounded to the nearest integer
before proceeding to step 5.
5. Using each of the B bootstrap replicate mortality time series, repeat step 1 with the observed mortality time series data replaced by the bootstrap replicate mortality time series, each time
tabulating which of the K models is “best” based on AIC.
6. Assign a weight w[i] equal to the proportion of the B times that the model was selected as best in step 5, to each of the K candidate models.
7. Use the weights obtained in step 6 to compute a “bootstrap weighted” estimate for the effect of PM[2.5] on mortality: w[1]θ[1]+…+ w[K]θ[K], where θ[i] is the estimated effect of PM[2.5] on
mortality obtained from M[i].
In step 3, the stationary bootstrap is used to allow the resampled residuals to mimic the dependence structure of the original residual process under the notion that, although adjacent data points
might suffer dependence, blocks of sufficient length may be close to independent of one another. Based on our earlier work, the stationary bootstrap is implemented using a mean block length of size
10 (Martin and Roberts 2006). Lahiri (2003) provides additional information on the use of resampling methods for dependent data. It is important to note that the replicate mortality time series
generated in step 4 are not Poisson distributed, but this issue is not of particular concern because the observed mortality time series will also not be Poisson distributed. Indeed, some studies
explicitly allow for the non-Poisson nature of the observed mortality time series via quasi-likelihood estimation (Goldberg et al. 2006; Samoli et al. 2008). In our context, the overdispersion
estimated within the framework of a Poisson generalized linear model was mild. Thus, we did not consider a quasi-likelihood approach necessary. Further information on residual-based resampling for
generalized linear models can be found in Davison and Hinkley (1997b).
Our extension to BOOT described above (termed “double BOOT”) uses a second bootstrap layer after step 6. The second bootstrap layer involves generating another B bootstrap replicate mortality time
series that are based on the weights w[i] found for each model in the first bootstrap layer. For each of the K candidate models, this procedure involves generating Bw[i] replicate mortality time
series using model M[i] as the basis for the bootstrap procedure described above, for each i = 1, 2,…, K. As before, based on this new set of B replicate mortality time series, updated weights are
constructed for each model based on the proportion of times it was selected as best based on AIC.
The procedure for implementing double BOOT is as follows:
1. Perform steps 16 above of the BOOT method.
2. For each of the i = 1 to K candidate models, construct Bw[i] replicate mortality time series using the procedure described in steps 24 of BOOT with M* replaced by M[i]. This process will produce
B = Bw[1] +…+ Bw[K] second-layer bootstrap replicate mortality time series.
3. Fit the K candidate models to each of the B replicate mortality time series, each time noting which of the K models is “best” based on AIC.
4. Assign a weight w[i]* to each of the K candidate models. For each model, the weights are calculated as the proportion of the B times the model was selected as best in the preceding step.
5. Use the weights w[i]* to compute a double-bootstrap weighted estimate for the effect of PM[2.5] on mortality: w[1]*θ[1] +…+ w[K]* θ[K], where θ[i] is the estimated effect of PM[2.5] on mortality
obtained from M[i].
A rationale for this proposed extension to BOOT can be provided through a simple example. Consider a setting where there are only two candidate models, and model 1 is judged as best based on AIC. Now
suppose the original BOOT procedure is implemented resulting in weights of w[1] = 0.51 and w[2] = 0.49 being assigned to models 1 and 2, respectively. The original BOOT procedure simply uses these
weights to produce an average effect estimate. However, the weights of 0.51 and 0.49 can be interpreted as the data providing essentially equal support for the two candidate models. This outcome
poses the question of whether it is desirable for the bootstrap replicate mortality time series to be constructed solely on the basis of model 1 when, in fact, according to the evidence given by the
weights, the two models are almost equally supported by the data. Double BOOT offers a solution to this problem by performing a second layer of bootstrapping that uses a bootstrap data-generating
process to weight each of the original candidate models according to their prevalence (measured through w[i]) as “best” models among the original B bootstrap replicate series. The logic used here
could be extended to the case of many competing models where it seems reasonable to perform a second layer of bootstrapping based on how well each candidate model is supported by the data, rather
than a single layer where the bootstrapping is based on a single model that essentially assumes full support from the available data. The difference in the double BOOT weights compared with the
original BOOT weights would depend on a number of factors, including the number of candidate models that are “close” in terms of support offered by the data and the similarity of these models in
terms of model structure. Irrespective of the change in the double BOOT weights, we believe the reweighting to be important—inherent to the success of the bootstrap is the premise that the
data-generating process should mimic the true underlying process as closely as possible. In the case of something as complex as a model-selection process, the weights effectively measure a state of
belief about the set of candidate models. Thus, our bootstrap resamples mimic that state of belief by generating data sets arising from a variety of candidate models in proportion to our confidence
that such models are the correct ones.
The use of the bootstrap to tune another initial bootstrap algorithm has a long history. For example, Efron (1983) used a second level of bootstrap resampling to reduce the bias of the apparent error
rate of a linear discriminant rule. Efron termed his method a “double bootstrap” because it involved a second layer of B resamples to bias correct an initial bootstrap bias-corrected estimate. Beran
(1987) and Hall (1986) discussed the use of second-level resampling to correct for coverage error in confidence intervals. Hall and Martin (1988) proposed a general framework for bootstrap iteration
for which the second-level resamples were used to estimate and correct for the error in the original bootstrap procedure. Loh (1987) also used a second layer of bootstrap resamples to correct
confidence interval endpoints. However, the methods of Beran (1987), Hall (1986), and Loh (1987) differ in the way the bootstrap critical points are modified. In our approach, the first-layer
bootstrap resamples are used to generate an initial set of weights for the set of candidate models. In one way, these weights can themselves be considered as outputs from the initial bootstrap
procedure. But, of course, these weights are not “correct” because of the way the bootstrap resamples are constructed in the generalized linear model context. Because the resampling is based on model
residuals, there is a tendency for the initial bootstrap step to favor (i.e., give higher weight to) the model from which the original residuals were obtained. Our second-layer bootstrap -resampling
is directed at addressing this problem, by using the information gleaned from the initial bootstrap step as a starting point to constructing second-level resamples based on residuals not from a
single model fit, but rather from a weighted set of plausible candidate models. Our method is a fully frequentist analog of the bootstrap-after-Bayesian model averaging approach proposed by Buckland
et al. (1997). In their paper, the authors had observed that a single-layer bootstrap model averaging approach tended to favor the initial model on which resamples were based. They suggested that an
initial Bayesian model averaging (BMA) step could be used to provide a weighted set of models from which resamples could be based in a second bootstrap model selection step. Our method takes a fully
frequentist approach by adopting bootstrap methods at both steps.
The form of BMA that will be used in our paper is based on AIC as described in Clyde (2000). In the context of Model [1], BMA based on AIC proceeds by assigning each candidate model i a posterior
probability given by the following formula:
where AIC[i] is the AIC for candidate model i and K is the number of candidate models.The estimated mortality effect is obtained by weighting the PM effect estimates obtained from each model by its
posterior probability.
In the context of our analyses, it is worth discussing the interpretation of the weighted average effect estimates obtained from BOOT, double BOOT, and BMA. These quantities, which are obtained by
weighting estimates of the effect of an increase in PM[2.5] on a single day’s mortality, may be viewed as weighted or model-averaged estimates of the effect of an increase in PM[2.5] on a single
day’s mortality. However, care should be taken when using model averaged estimates because the interpretation of particular parameters may change when other variables, such as copollutants, are added
to the model (Lukacs et al. 2009; Thomas et al. 2007). Indeed, not all researchers would agree with the process of averaging estimates obtained using different lags of PM[2.5]. Some advocate that
model averaging is best suited for making predictions (Thomas et al. 2007). In this regard, we also investigate the predictive performance of the three model-averaging procedures considered in this
We used the statistical package R along with packages “boot,” “splines,” and “tseries” for all the analyses (R Development Core Team 2009). Computational constraints meant that producing estimates of
the standard errors (SEs) for values presented in Tables 1 and 2 was not feasible, and the provision of SEs for simulated values is not common practice in studies of this kind.
Simulation study. We used the 730 days of data from Seattle, Washington, along with the specification of Model [1] to generate mortality time series where the effect of PM[2.5] on mortality was
known. Generating mortality time series was achieved by producing mortality counts on day t that were Poisson distributed with mean µ[t]
We considered three different specifications of confounders(α = 1.2)[t]
Specification A
S[t,][1](time, df = 8α) + S[t,][2](temp, df = 6α)
+ S[t,][3](dew, df = 3α),
Specification B
S[t,][1](time, df = 4α) + S[t,][2](temp, df = 6α)
+ S[t,][3](dew, df = 3α),
Specification C
S[t,][1](time, df = 8α) + S[t,][2](temp, df = 6α)
+ S[t,][3](dew, df = 3α)
+ S[t,][4](temp[13], df = 6α).
In the above equations, θ is the known PM[2.5] effect, and temp, temp[13], and dew represent the current day’s temperature, temperature of the previous 3 days, and current day’s dew point
temperature, respectively. The functions S[t][,][j]() are smooth natural cubic spline functions with the indicated degrees of freedom. To ensure that the degrees of freedom take integer values, the
values of 8α, 6α, 4α, and 3α are rounded to the nearest integer. To find realistic representations of the S[t][,][j](), we fitted Model [6], using each specification of confounders(α = 1.2)[t] to the
actual Seattle data using a Poisson log-linear generalized linear model with an offset term allowing the effect of PM[2.5] to be set equal to θ. The offset term allows a term to be included in a
generalized linear model with a known, rather than an estimated, coefficient value. We used the fitted values from these models to generate daily Poisson mortality estimates that incorporate a known
PM[2.5] effect θ. Three values of θ: 0, 0.0003, and 0.001 were considered.
To implement model averaging, a set of candidate models was required. We considered two sets of candidate models that were defined by Model [1] with α taking 10 equally spaced values ranging from α =
0.3 to α = 3, confounders(α)[t] as defined in specification A, and either three lags of PM[2.5] (PM[2.5][t][,0], PM[2.5][t][,1], PM[2.5][t][,2]) or one lag of PM[2.5] (PM[2.5][t][,1]). In the case of
three lags of PM[2.5], we have a set of 10 × 3 = 30 candidate models, and in the case of one lag of PM[2.5], a set of 10 × 1 = 10 candidate models. Similar methods for defining the tuning parameter α
for time and weather variables have been used in previous investigations (Dominici et al. 2004; Roberts 2004). The number of parameters estimated in each candidate model is equal to the total number
of degrees of freedom used in the S[t][,][j]() plus 1 for the intercept and 1 for the estimated PM[2.5] effect.
For mortality generated using the confounders(α = 1.2)[t] specification A, the “true” model is contained among both sets of candidate models, but for mortality generated using confounders(α = 1.2)[t]
specifications B and C, this is not the case. In specification B, the degrees of freedom used for time have been halved for each value of α compared with the candidate models; whereas, specification
C includes temp[13], a variable that is not included in any of the candidate models. These latter two situations are perhaps more realistic because in practice no candidate model would correspond to
the true model.
In the simulations, B = 1,000 was used in BOOT and for both layers of double BOOT. The simulations were conducted by generating sets of 1,000 mortality time series defined by Model [6] with α = 1.2,
one of the confounder specifications A, B, or C, and θ and then by applying BOOT, double BOOT, and BMA using the two sets of candidate models [i.e., with 3 lags of PM[2.5] (30 candidate models total)
or 1 lag of PM[2.5] (10 candidate models total)]. Table 1 contains the results of these simulations. In the simulations involving 30 candidate models, it is evident from the smaller root-mean-squared
error (RMSE) values that double BOOT has superior performance to that of both BOOT and BMA. The breakdown of RMSE into bias and SE components shows that the improvement in performance offered by
double BOOT is principally due to the lower SE of the estimates obtained by this method. In the simulations involving 10 candidate models, the methods offer similar performance.
For the simulations with 30 candidate models and confounders(α = 1.2)[t] = specification A, we investigated the use of standard AIC model selection (results not shown) by basing estimates on the
single model selected as “best” based on AIC. Performance, as measured by RMSE, was substantially worse than that of double BOOT, BOOT, and BMA, with the average values of RMSE of approximately 1.90
for each of the three scenarios considered.
As a final comparison we compared the predictive performance of the three methods using both simulated and actual mortality data. For each mortality time series, we randomly removed 100 observations
and applied BOOT, double BOOT, and BMA to the remaining data to obtain predictions for the removed observations. The predictions were computed as weighted averages of the predictions obtained from
each candidate model weighted by the weight or probability assigned to that model. The performance of each method was based on the predictive mean squared error (PMSE) computed as {(y[1] ˆy[1])^2 +…+
(y[100] ˆy[100])^2}/100, where y[i] and ˆy[i] are the actual and predicted mortality estimates, respectively. For a given mortality time series, we repeated the process of randomly removing 100 data
points and computing the PMSE 100 times. Table 2 reports the number of times (out of 100) that each method had a better predictive performance than alternative methods based on lower PMSE. It is
clear that double BOOT has predictive performance superior to that of BOOT, with double BOOT having a smaller PMSE about 70% of the time. The results also provide support for double BOOT versus BMA,
with double BOOT providing the same or better predictive performance in two of the three model-specific simulations and in three of the five city-specific simulations.
Application. Tables 3 and 4 show the results of applying the three model-averaging methods and standard AIC to the five cities described above. We calculated the SE values in Table 3 using equation 4
of Burnham and Anderson (2004). For these five cities, the estimates obtained from the three model-averaging methods were similar and the conclusions drawn about the association between PM[2.5] and
mortality would be essentially the same. However, the results also illustrate that the estimates obtained from standard AIC can be significantly different to those obtained from model averaging. The
SEs assigned to the estimates obtained from standard AIC are smaller because these SEs do not take into account the model selection process that was used to find the single best model.
The reason for the differences in the estimates obtained from the three model-averaging
methods based on 30 candidate models compared with standard AIC is a result of the model-averaging methods
assigning nonnegligible weights to a number of candidate models. Within each city, the three model-averaging techniques tended to assign nonnegligible weights to three models corresponding to the
three different lags of PM[2.5] but the same level of confounder adjustment α. Comparing the weights obtained from BOOT and double BOOT illustrates that the second bootstrap layer can result in
substantial changes to the weights assigned to each model. For example, for Seattle and Tampa in some situations the weights assigned to candidate models differ by approximately 40%.
We have illustrated that double BOOT model averaging can offer benefits over BMA and BOOT for both estimation and prediction. The benefits were particularly noticeable for double BOOT compared with
BOOT. This increased performance was attributable to a reduction in the variance of the estimates obtained from double BOOT compared with BOOT and BMA. An interesting observation was that the bias of
the estimates obtained from double BOOT was larger than the estimates obtained from BOOT and BMA when the “true” model was contained among the candidate models. This was not the case, however, when
the “true” model was not among the candidate models because the double BOOT procedure tended to give less weight to the true model as a consequence of the second bootstrap layer moving some of the
weight from the true model to other plausible models. Of course, this phenomenon could not occur in the simulations where the “true” model was not among the candidate models, and the result was that
double BOOT had slight improvements in terms of lower bias.
A report of particular relevance to the present study is that of Buckland et al. (1997) who investigated various forms of bootstrap model averaging, including the BOOT method in the present
investigation. Buckland et al. (1997) and Claeskens and Hjort (2008) each provide excellent introductory treatments of the issues surrounding model selection and model averaging. Burnham and Anderson
(2002) showed that AIC can be derived as a Bayesian result and that the AIC-based BMA weights used in the present paper correspond to posterior model probabilities. Unlike the implementation in this
report, BMA can also be implemented by explicitly assigning prior model probabilities (Hoeting et al. 1999; Koop and Tole 2004). In the present setting, AIC-based BMA has the advantage of using
objective prior distributions (Clyde 2000) and ease of implementation, compared with explicitly assigned prior model probabilities. An obvious disadvantage of AIC-based BMA is that is does not allow
for the incorporation of prior information about the importance of a variable.
It is important to note that the use of BMA applied to time series studies of air pollution and mortality, and in particular the approach of Koop and Tole (2004), has received some criticism in the
literature (Crainiceanu et al. 2008; Thomas et al. 2007). In this study we have attempted to avoid these same criticisms by ensuring that when illustrating our proposed averaging method we did so
over a range of plausible candidate models, ensuring that a measure of air pollution exposure is included in each candidate model, focusing on single-pollutant models, and also investigating
predictive performance. We are of the view that a carefully applied model-averaging procedure can provide useful insight into understanding air pollution health effects by, for example, providing
information on how much the data support various models, helping practitioners to appreciate and allow for the effects of model selection and uncertainty, and in some circumstances providing improved
estimators of air pollution health effects. However, we are also of the view that the use of model averaging does not negate the need for careful planning and data-gathering processes along with
detailed investigations of models arising from a suitably rich set of initial covariates to find an initial and sufficiently rich plausible set of candidate models. We also believe that future
comparisons of results obtained from model averaging with traditional methods such as standard AIC would prove valuable.
CEHN April 2014 Article of the Month
“Behavioral Sexual Dimorphism in School-Age Children and Early Developmental Exposure to Dioxins and PCBs: A Follow-Up Study of the Duisburg Cohort” [Winneke G, et al. Environ Health Perspect;
DOI:10.1289/ehp.1306533] has been selected by the Children’s Environmental Health Network (CEHN) for its April 2014 Article of the Month summary. These summaries discuss the potential policy
implications of current children’s environmental health research.
Sign Up to Receive E-mail Alerts | {"url":"http://ehp.niehs.nih.gov/0901007/","timestamp":"2014-04-18T15:53:51Z","content_type":null,"content_length":"91573","record_id":"<urn:uuid:8780c8c2-8ee7-4d81-a859-b027a44f3cee>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00287-ip-10-147-4-33.ec2.internal.warc.gz"} |
[plt-scheme] OT: not the halting problem, but what is it?
From: John Clements (clements at brinckerhoff.org)
Date: Tue May 4 00:55:18 EDT 2010
While I was out riding today, I came up with a formulation of something extraordinarily close to the halting problem that's entirely independent of halting, computation and decidability, and that is essentially a restatement of the cantor theorem (power set of A is bigger than A). I'm confident that many people in the world can tell me swiftly whether this is well-known or incorrect and how it fails to lead to the halting problem, and optimistic that one of them is reading this message and inclined to answer it.
1) As our model of a programming language, consider an injective mapping E (for "encoding") from the integers (a.k.a. the "programs") to functions in (int -> bool) (a.k.a. the "meanings"). Note that I don't claim that this map is a total function, or that it's surjective. There's also no "bottom" needed in this model. Also, I'm referring to real mathematical functions, and not their computational approximations.
2) Next, consider the set of numbers for which (E(n)) (n) returns true. That is, the numbers that represent functions which when called with their "program" form return true.
3) This set can be represented as a characteristic function, m, (in (int -> bool) that maps n to false when it's a member of the set described above, and otherwise produces true.
4) Q: Is there some number p such that E(p) = m ? That is, is there a program 'p' that corresponds to the meaning 'm'?
5) No, there can't be, for the usual paradox reasons. (If there were such a number, then if E(p) (p) = true then it belongs to the set and therefore E(p) (p) = false, and if E(p) (p) = false then it doesn't belong to the set and thus E(p) (p) should return true.)
What's the interpretation of this in computing? That for any programming language as defined here with at least two possible outputs, the program that determines whether another program produces one of the two values *when called with its own representation* is inexpressible in the language.
Note that (unlike the proof on the wikipedia page :)) this proof has nothing at all to do with halting, and doesn't even require a model of computability / turing completeness / whatever. In essence, it's just a restatement of the theorem that you can't put the programs (ints) into correspondence with the meanings (functions from ints->bools, a.k.a. the power set of the ints).
The only gap between this and the halting problem, AFAICT, is that it applies only to the expressibility of the function that computes whether a program produces true when called with itself, and not the expressibility of the function that decides whether a program produces true when called with, say, 4. In most of the pop proofs of the halting problem that I've read, this difference is glossed over, but it now appears to me that *this* is the point where you actually have to bring in the big heavy machinery so that you can talk about reducing the problem of determining programs' behavior on 4 to their behavior on their own representations.
Am I missing something obvious?
Lonely in rural CA,
John Clements
-------------- next part --------------
A non-text attachment was scrubbed...
Name: smime.p7s
Type: application/pkcs7-signature
Size: 4669 bytes
Desc: not available
URL: <http://lists.racket-lang.org/users/archive/attachments/20100503/f4f38616/attachment.p7s>
Posted on the users mailing list. | {"url":"http://lists.racket-lang.org/users/archive/2010-May/039367.html","timestamp":"2014-04-20T18:32:35Z","content_type":null,"content_length":"8967","record_id":"<urn:uuid:0fb5add0-e48d-4b41-858b-22e56d9d73f9>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00619-ip-10-147-4-33.ec2.internal.warc.gz"} |
Learning Multiplication Facts: with Skip Counting - MathPowerLine
YouTube version: Learning Multiplication Facts
“Skip Counting” is a great method for learning multiplication facts.
Take a look at this method. It will make more sense, tie in the idea of multiplies and division, and save a lot of time just memorizing the times table for young students.
Like this video? | {"url":"http://mathpowerline.com/learning-multiplication-facts-skip-counting/","timestamp":"2014-04-18T10:34:40Z","content_type":null,"content_length":"21529","record_id":"<urn:uuid:d9cb27a7-be28-4c9b-9c0e-6b849bbd51c8>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00528-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is a Cubit?
Date: 03/31/2002 at 17:26:31
From: Becky
Subject: Cubit
The Bible says that Moses was 9 cubits tall and Pharaoh was 1 cubit
tall. How do you calculate that in today's standard of height in
inches or feet?
Thank you.
Date: 04/01/2002 at 13:33:13
From: Doctor Sarah
Subject: Re: Cubit
Hi Becky - thanks for writing to Dr. Math.
Russ Rowlett's _How Many? A Dictionary of Units of Measurement_ is a
good place to look up things like a 'cubit'.
a historic unit of distance frequently mentioned in the Bible. The
word comes from the Latin cubitum, "elbow," because the unit
represents the length of a man's forearm from his elbow to the tip of
his outstretched middle finger. This distance tends to be about 18
inches or roughly 45 centimeters. In ancient times, the cubit was
usually defined to equal 24 digits or 6 palms. The Egyptian royal or
"long" cubit, however, was equal to 28 digits or 7 palms. In the
English system, the digit is conventionally identified as 3/4 inch;
this makes the ordinary cubit exactly 18 inches (45.72 centimeters).
The Roman cubit was shorter, about 44.4 centimeters (17.5 inches). The
ordinary Egyptian cubit was just under 45 centimeters, and most
authorities estimate the royal cubit at about 52.35 centimeters (20.61
- Doctor Sarah, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/58488.html","timestamp":"2014-04-17T18:31:58Z","content_type":null,"content_length":"6436","record_id":"<urn:uuid:bc3e3fcb-70b8-4bfb-8472-a038da9f10ab>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00086-ip-10-147-4-33.ec2.internal.warc.gz"} |
Simultaneous equations
August 16th 2007, 09:41 AM #1
Aug 2007
for what value of a will simultaneos equations of the form
fail to possess a unique solution?
Consider the matrix of coefficients and set its determinant equal to 0. This gives only one value for a.
August 16th 2007, 10:42 AM #2 | {"url":"http://mathhelpforum.com/algebra/17816-simultaneous-equations.html","timestamp":"2014-04-19T07:13:53Z","content_type":null,"content_length":"31162","record_id":"<urn:uuid:8f372f4e-8d9d-445e-b562-36199f14fc8a>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00322-ip-10-147-4-33.ec2.internal.warc.gz"} |
[FOM] The boundary of objective mathematics
joeshipman@aol.com joeshipman at aol.com
Thu Mar 12 11:22:41 EDT 2009
-----Original Message-----
>From: Paul Budnik <paul at mtnmath.com>
>I an dividing mathematics into absolute objective statements and
>relative ones that are sometimes treated as if they were absolute.
>The difference I want to make is between an an arbitrary path that is
>followed by a recursive process in a potentially infinite universe and
>completed infinite set. This is a distinction that goes back at
>to Aristotle. My position is perhaps close to constructivists, but I
>not demand a constructive proof of a statement. I only demand a
>constructive proof that all the events that determine the statement
>themselves determined by finite events.
The practical attitude many mathematicians seem to take is that
statements in the first-order language of arithmetic OR statements of
higher type which have arithmetical consequences are meaningful. For
example, Schanuel's conjecture cannot be formulated in arithmetic but
using it one can prove that (e^(e^n)) is never an integer when n is an
integer, a statement which can be given an arithmetical formulation in
terms of convergence of computations. The existence of a (countably
additive) real-valued measure on all subsets of the continuum is a
statement of even higher type which has useful arithmetical
consequences such as Con(ZFC).
Many mathematicians would also declare as meaningful statements those
which are set-theoretically absolute (have the same truth value in all
transitive models of ZFC). For example, the Invariant Subspace
Conjecture (all bounded linear operators on Hilbert space have
nontrivial invariant subspaces) does not have any arithmetical
consequences that I know of, but is considered to be one of the major
open problems in mathematics. (Of course one can't prove that the
Invariant Subspace conjecture has no arithmetical consequences without
proving it consistent, which might be no easier than proving it
My own view is that any statement about sets of bounded rank is
meaningful, and that statements like GCH which involve universal
quantification for sets of arbitrary rank are vague. In between these
two classes of statements are existential statements with no bound upon
the rank -- statements like "a measurable cardinal exists". (In other
words, the statement "no measurable cardinal exists" is vague while "a
measurable cardinal exists" is less so; this asymmetry is not
unreasonable and is analogous the asymmetry between the statements "the
Riemann Hypothesis is not provable" and "the Riemann Hypothesis is
provable" which have different epistemological statuses).
-- JS
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/2009-March/013471.html","timestamp":"2014-04-18T16:07:20Z","content_type":null,"content_length":"5286","record_id":"<urn:uuid:4fedaba8-4b13-4f9f-80fb-81a64a78b755>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00605-ip-10-147-4-33.ec2.internal.warc.gz"} |
EDUCATION (K-12)
Daniel Finkel is back this week with another original navigation challenge. (We last saw Dr. Finkel several weeks ago with Angle Mazes.) This week we have something a bit different: a nun on a
mission. Let’s help her out. Here’s —
The Pilgrim’s Puzzle
A pilgrim arrives in Duona, a sixteen-block town created by five streets running north-south that intersect with five streets running east-west. Like all pilgrims, she arrives in the northwest
corner of town, and needs to make her way to a shrine in the southeast corner. Unfortunately for the pilgrim, Duona imposes a tax system on visitors, charging 2 silver pieces for each block
walked to the east, and doubling what you owe every time you walk south. To make the payment system fairer and encourage longer stays, they subtract 2 silver pieces for each block walked to the
west, and halve what you owe when you walk a block north. The townsfolk keep track of your path, and you must pay in full on your arrival in the southeast corner. (Legend has it that certain
savvy travelers have planned trips through the town and ended up receiving silver by following the tax rules.)
The pilgrim has no money, and has no intention of leaving with any. How can she travel to the southeast corner of Duona without owing or receiving any silver?
For example — the pilgrim could certainly reach the southeast corner of Duona by walking straight east and then straight south, but this would be an expensive route, costing 128 silver pieces.
As an additional challenge, can the pilgrim make the trip without covering the same segment of street more than once?
Dr. Finkel continues by sharing a bit about why he created this particular challenge:
At Math for Love, we’re always thinking about how to motivate the math that kids have to learn in school. Arithmetic is tough, because even though everyone knows they have to learn it, it can be
very dry when it’s presented without context. I came up with this puzzle as I was messing around with different mechanisms for a deeper, more playful experience with arithmetic.
The Cupcake Puzzle
Our puzzle this week was suggested by Jeremy Copeland, an instructor for Art of Problem Solving, an online resource for high-performing math students.
You have 3 cupcakes and 5 students. You want to divide the cakes evenly, but no student wants a tiny sliver of cupcake. What division of the cupcakes maximizes the smallest piece?
This is a simple puzzle with a lot to explore. Here’s Dr. Copeland:
The thing that I think you’ll like most about the problem is that it opens such a large number of doors with all of its various generalizations and people seem to have remarkable intuition about
how hard a follow-up question to ask. The things a student might want to know next look much different than the things a math professor will ask. Incidentally, my two favorite follow-up questions
to the 3 cupcakes/5 students question are: “What happens when you have 5 cupcakes and 3 students?” and “What happens when you have 6 cupcakes and 10 students?”
The Cupcake Puzzle is a terrific problem — one of many used by Art of Problem Solving in its range of courses for elementary and high school students. It’s also the kind of problem found in Worldwide
Online Olympiad Training (WOOT), the Art of Problem Solving’s most most advanced course. WOOT is a three-year program that trains students to win the top high school contests in the world, and is
quite effective: All 12 winners of the 2012 USA Mathematical Olympiad were WOOT graduates.
Math for Love: The Angle Maze
This week’s puzzle was suggested by Daniel Finkel of Math for Love, the Seattle-based math duo dedicated to helping students and teachers fall in love with the beauty of mathematics. The challenge is
an original creation — an example of what Dr. Finkel and his Math for Love partner Katherine Cook call Angle Mazes. Get out your compass and get ready for —
The 10-Point Angle Maze
Consider 10 points equally spaced around a circle, labeled zero to nine as pictured. Starting and ending at zero, your goal is to draw a continuous straight-line path that hits every point
without creating any acute or right angles at those points.
For example, one answer is to make a regular 10-gon (or decagon). But it turns out there’s one more way to do it. Can you find it?
Just to clarify, you may cross your own path, and you need to make obtuse angles only at the original 10 points: New vertices created by overlapping lines don’t count. We’ll consider solutions to
be unchanged under reflections. Aside from starting and ending at zero, you can’t hit the same point twice or retrace your own path.
If you can find the second solution, can you prove there are no more?
I thought the angle maze concept was really cool, so I asked Dr. Finkel how he came up with the idea. He sent the following response by e-mail:
Here’s the creation story of Angle Mazes. As part of our professional development work at Math for Love, we lead demo lessons for kids as well as work with teachers. In some schools, teachers
have pretty strict guidelines on what lessons they’re supposed to teach, and when.
It didn’t seem fair — or helpful — to come in with a really exciting, novel math lesson. It’s fun for the kids, but we didn’t want the message for the teachers to be that great math teaching can
only happen when you do something that’s not normal; we wanted to show how normal math teaching could be engaging every day.
We told the teachers to bring us their most boring lessons, the ones they dreaded, and we’d see how it might be possible to teach them in an engaging way. One of them was about angles. A lesson
for third graders, it had kids draw examples of acute, obtuse and right angles. It was numbingly boring. Katherine, my wife and partner at Math for Love, was out of town, and while she was away I
just looked at this question I had yoked myself to, trying to figure out what to do, trying not to panic. A few days later, Katherine came back, glanced at the lesson, and said, “Let’s have them
find angles in circles.” I instantly saw what she saw, and the boring lesson became so interesting that I would later pursue it on my own time. (This is why we work well together.)
Triangle Mysteries
This week’s puzzle is based on a beautiful discovery by Steve Humble, who teaches primary and second school trainee teachers in the education department at Newcastle University in northeastern
England. The conundrum is based on Mr. Humble’s research that appears in the current issue of The Mathematical Intelligencer.
I asked Mr. Humble about the insight at the core of the puzzle. He responded by e-mail:
As for the discovery it was just that — I wanted to discuss randomness with families, and created this color game to play. The idea was to show how order came from initial disorder. You get these
patches of triangle color.
In working with this game I spotted that the top end two colors gave the bottom color. I ran some computer code and discovered not all cases worked but could not come up with a complete proof. I
described the problem to Ehrhard Behrends in the summer at a conference in Murcia. In a few days he came back to me with a proof — and then we put together the paper. Ehrhard’s genius was to
generalize to n colors and look at all the variety of ways we could set up the initial rules for creating the pattern.
Thank you, Mr. Humble. And now we present —
Triangle Mysteries
Start with a row of dots randomly colored red, blue and yellow. Underneath each pair in the row, place a third dot to form triangles according to the following rule.
If the two dots above are the same color, then the third one matches both, whereas if the two dots above are different colors, then the third one is different from either.
Our puzzle: take a row of ten dots colored as follows, and place a row of nine dots underneath it, coloring them according to our rule. Continue with eight dots underneath that and so on, all the way
to a single dot. What will be the color of that final dot?
Bonus Problem: What happens if we start with 28 dots?
Triangle Mysteries — The Paper
Springer, the publisher of The Mathematical Intelligencer, has generously granted permission to include a copy of Steven Humble’s and Dr. Ehrhard Behrends’ research paper with this Numberplay post.
Click here to download Triangle Mysteries in PDF form. Spoiler alert: the answer to this week’s puzzle is included.
As Mike and several other readers suggested, the solutions to the main puzzle and bonus problem are the same: the last dot in each case will be yellow. Start with any row of 3^n + 1 dots (2, 4, 10,
28 and so on) and the last dot in the resulting triangle will be determined by the first and last dots in the initial row.
The VIEWMONGOUS Puzzle
Our puzzle this week was suggested by Karim Ani, founder of Mathalicious, an online source for real-world math lessons for classroom teachers.
An advertisement for Sharp’s 80-inch LED TV claims it has “more than double the screen area of a 55-inch TV.” Is this true?
Mathalicious stands out in the world of online math resources. It’s practical, fun and it does make you think. I find it refreshing. I asked Mr. Ani about the Mathalicious difference:
We believe the world is an interesting place full of interesting questions, and that mathematics offers a way to explore them. For instance, in the Olympics, did Usain Bolt have an unfair height
advantage, and what would happen if sprinters ran distances based on their heights? Do people with small feet pay too much for shoes, and should Nike charge by weight? What does the Les
Miserables soundtrack tell us about the height of bridges in Paris…and did Javert really jump?
Students across the country think of math as a bunch of random skills to memorize and regurgitate. At Mathalicious, we help teachers teach math in a new way, and challenge their students to think
more critically about the world. Math class should be the most interesting part of the day. That’s why we do what we do.
Mathalicious contributor Matt Lane will be chiming in during the week as the discussion progresses.
Teachers: do you want to explore this lesson with your students? Click here to download the free lesson materials.
Here’s Matt Lane with a recap of last week’s discussion:
TV sizes inspired some great discussion during the week. The first to answer the original question was deo, who wrote:
George Carlin and the Integer Called Bleen
Our puzzle this week was suggested by educator Frank Potter, who introduces it this way.
I have visited middle school and high school classes, proposed this challenge, and was surprised at the variety of results. The challenge forces one to go back to the number line to understand how
the insertion of this new integer affects counting.
The idea originated with comedian George Carlin several decades ago — probably in the 1970s — when he announced that a new integer had been discovered by mathematicians.
The Integer Called Bleen
Suppose that a “new integer” is “discovered” between 5 and 6 called bleen, written as a capital B. That is, counting fingers, one says, “one, two, three, four, five, bleen, six, seven, eight, nine.”
There will be thirteen, fourteen, fifteen, bleenteen, sixteen, etc. Also fifty nine, bleenty, bleenty one, bleenty two, etc.
So, what is bleen plus three? Bleen plus bleen? Ten times bleen? Bleen times bleen?
That’s it.
Recap: The Ant on a Rubber Band Puzzle
Last week we expanded our minds with The Ant on a Rubber Band Puzzle — one of the more flexible puzzles seen on Numberplay.
The puzzle:
Your favorite stationery store is carrying a new product: the Super-Super Stretchy rubber band. “Infinite stretch without a break!” claims the box. What could be better than that? You buy one.
When you get home you decide to put the Super-Super Stretchy to the test. You nail one end of the rubber band to the wall, grab the other end, and start to walk away from the wall at a constant
speed. The rubber band starts to stretch.
At the same time, an adventurous ant crawls from the wall onto the rubber band and starts making its way toward you.
Let’s say the rubber band initially spans one meter — before you start stretching it — and that you maintain a pace of 1 meter per second as you walk away from the wall. The ant crawls on the rubber
band at 1 centimeter per second. Will the ant ever reach you?
The solution: Patrick C worked out the solution analytically:
Read more… | {"url":"http://wordplay.blogs.nytimes.com/tag/education-k-12/","timestamp":"2014-04-19T11:57:12Z","content_type":null,"content_length":"50934","record_id":"<urn:uuid:f3ebc5fb-5d42-48ce-8f0c-b19106575235>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00069-ip-10-147-4-33.ec2.internal.warc.gz"} |
Please help!
September 18th 2008, 02:31 PM
Please help!
How to prove that this below argument is valid, using established Rules of reasoning and Logical Equivalences (using vertical statement -reason format.)
{[(¬p ˅ q ) → r ] ˄ [ r →(s v t)] ˄ (¬s ˄ ¬u)˄(¬u → ¬t)}→ p ??????
p.s For those who don't see the symbols I will put this argument in words:
{[( not p v q) --> r] and [r --> (s v t)] and (not s and not u) and ( not u --> not t)} -->p
September 18th 2008, 03:51 PM
Ummm - keep simplifying using
a -> b = !a + b
!(a + b) = !a * !b
!(ab) = !a + !b
where ! means "not", + means "or", and * or multiplication means "and".
To prove that
{[( not p v q) --> r] and [r --> (s v t)] and (not s and not u) and ( not u --> not t)} -->p
= ((!p + q) -> r)(r -> (s+t))(!s)(!u)(!u -> !t)) -> p
= (p(!q)+r)(!r+s+t)(!s)(!u)(u+!t)) -> p
Now (!u)(u+!t) = !u !t
(!r+s+t)(!s)(!t) = !r !s !t
So LHS = (p !q + r)(!r)(!s)(!t)(!u) =
To prove that
p(!q)(!r)(!s)(!t)(!u) -> p or
!p + q + r + s + t + u + p
which is true since it has !p + p
September 19th 2008, 07:00 AM
How to prove that this below argument is valid, using established Rules of reasoning and Logical Equivalences (using vertical statement -reason format.)
{[(¬p ˅ q ) → r ] ˄ [ r →(s v t)] ˄ (¬s ˄ ¬u)˄(¬u → ¬t)}→ p ??????
p.s For those who don't see the symbols I will put this argument in words:
{[( not p v q) --> r] and [r --> (s v t)] and (not s and not u) and ( not u --> not t)} -->p
We want to prove that all the statements logically imply p
We notice that the statement having p is the statement (~pvq)--->r but in the proof that it will follow the way it is written down it will not help us to get p out of it.So through contrapositive
law we can convert it to:
........................~r-----> ~(~pvq)..........................................1
Now we convert r---->(svt) by contrapositive law again to:
.........................~(svt)----->~r............................................. .2
we are also given as assumptions :
...........................~u..................... .........................................3
..............................~s.................. ............................................4
From ....~u----> ~t and 3 and using M.Ponens we get :
...............................~t................. ............................................5
From 4 and 5 and using the law of logic called addition introduction we get :
..............................~s^~t............... ..........................................6
From ...6 and using the law called De Morgan we get:
................................~(svt)............ .............................................7
Now from 2 and 7 and using the law called M.Ponens we get:
.................................~r............... ...............................................8
And from 1 and 8 and using again M.Ponens we get :
.........................................~(~pvq )...........................................9
From 9 and using again De Morgan we get :
.........................................p^~ q..............................................10
And finally from 10 and using addition elimination we get :
...........................................p...... ..............................................11
NOTE the above logically implies ~q as well.
The above can be written in amore tabular form, like the following:
1) (~pvq)------>r................................................ ......assumption
2) r----->(svt)............................................ ...............assumption
3) ~s................................................ ........................assumption
4) ~u................................................ .......................assumption
5) ~u------>~t............................................... ...........assumption
6) ~r-----~(~pvq).................................... 1, contra positive
7) ~(svt)------>~r......................................2,contr a positive
8) ~t................................................ .......4,5 M.Ponens
9) ~s^~t............................................. ......3,8 addition intr.
10) ~(svt)............................................ .....9, De Morgan
11) ~r................................................ .......7,10 M.Ponens
12) ~(~pvq)........................................... .....6,11 M.Ponens
13) p^~q.............................................. .....12, De Morgan
14) ............p..................................... .........13 addition elimination
The laws use are:
a) contra positive:........(P----->Q) ------>(~Q----->~P) and also (~Q------>~P)-------->(P------>Q)
b) M.Ponens :..........(P----->Q AND P)------>Q
c) De Mrgan...........~(P ^Q) <-------> ~P v ~Q
d) addition introduction ...... P,Q --------> P^Q
e) addition elimination......P^Q------>P
THERE may be other shorter or longer proofs the above only shows the way
Also as hwhelper mentioned there can be a Boolean proof by substituting "a---->b" with ~avb.
Finally by the use of true tables there can be a semantical proof in which we must show that:
The conditional:
{[(¬p ˅ q ) → r ] ˄ [ r →(s v t)] ˄ (¬s ˄ ¬u)˄(¬u → ¬t)}→ p
IS A tautology
September 19th 2008, 02:30 PM
Thank you so much guys ! U R the best!!!!(Clapping)
September 19th 2008, 03:22 PM
I for one am not so sure about that. It is not standard.
$\begin{gathered} 1.\,\left( {eg p \vee q} \right) \to r \hfill \\<br /> 2.\,r \to \left( {s \vee t} \right) \hfill \\ 3.\,eg s \wedge eg u \hfill \\<br /> 4.\,eg u \to eg t \hfill \\ \therefore
\,p \hfill \\ \end{gathered}$
$\begin{gathered}<br /> 5.\,\left( {eg p \vee q} \right) \to \left( {s \vee t} \right)\;\left[ {1\,\& \,2} \right] \hfill \\<br /> 6.\,eg u\;\left[ 3 \right] \hfill \\<br /> 7.\,eg t\;\left[ 4 ,\
& \,6\right] \hfill \\<br /> 8.\,eg s\;\left[ 3 \right] \hfill \\<br /> 9.\,eg s \wedge eg t\left[ {7\,\& \,8} \right] \hfill \\ <br /> \end{gathered}$
$\begin{gathered} 10.\,eg \left( {s \vee t} \right)\left[ 9 \right] \hfill \\ 11.\,eg \left( {eg p \vee q} \right)\left[ {10\,\& \,5} \right] \hfill \\ 12.\,p \wedge eg q\left[ {11} \right] \
hfill \\ \therefore \,p\left[ {12} \right] \hfill \\ \end{gathered}$
I will leave it to you to supply the reasons by name. Names of the rules differ from text to text and author to author. That is the reason I don’t trust what you have handed. I have taught this
material for years in North America. I can tell you that I got confused each time we changed textbooks.
Notations and rules names are simply not standard.
However, the layout of the proof that I have given you is fairly standard.
September 19th 2008, 04:19 PM
Here is ashorter proof by contradiction:
Let assume ~p and start a contradiction,then by the law called disjunction introduction we have :
...........................~pvq................... ...........................................1
from the assumption ~pvq----> r and 1 and using M.Ponens we get:
...................................r.............. ................................................2
from the assumption r----->(svt) and 2 and using M.Ponens we get :
.........................................(svt).... ..............................................3
from the assumptions ~u------>~t and ~u and using M.Ponens we get:
...............................................~t. ...............................................4
from the assumption ~s and 4 and using addition introduction we get:
.......................................~s^~t...... ...........................................5
from 5 and using De MORGAN we get :
.....................................~(svt)....... ...........................................6
from 3 and 6 and using addition introduction we get :
....................................(svt) and ~(svt)......................................7
...................................a contradiction..................................... ...
HENCE .........................P........................ ..................................
September 19th 2008, 04:47 PM
I for one am not so sure about that. It is not standard.
$\begin{gathered} 1.\,\left( {eg p \vee q} \right) \to r \hfill \\<br /> 2.\,r \to \left( {s \vee t} \right) \hfill \\ 3.\,eg s \wedge eg u \hfill \\<br /> 4.\,eg u \to eg t \hfill \\ \therefore
\,p \hfill \\ \end{gathered}$
$\begin{gathered}<br /> 5.\,\left( {eg p \vee q} \right) \to \left( {s \vee t} \right)\;\left[ {1\,\& \,2} \right] \hfill \\<br /> 6.\,eg u\;\left[ 3 \right] \hfill \\<br /> 7.\,eg t\;\left[ 4 ,\
& \,6\right] \hfill \\<br /> 8.\,eg s\;\left[ 3 \right] \hfill \\<br /> 9.\,eg s \wedge eg t\left[ {7\,\& \,8} \right] \hfill \\ <br /> \end{gathered}$
$\begin{gathered} 10.\,eg \left( {s \vee t} \right)\left[ 9 \right] \hfill \\ 11.\,eg \left( {eg p \vee q} \right)\left[ {10\,\& \,5} \right] \hfill \\ 12.\,p \wedge eg q\left[ {11} \right] \
hfill \\ \therefore \,p\left[ {12} \right] \hfill \\ \end{gathered}$
I will leave it to you to supply the reasons by name. Names of the rules differ from text to text and author to author. That is the reason I don’t trust what you have handed. I have taught this
material for years in North America. I can tell you that I got confused each time we changed textbooks.
Notations and rules names are simply not standard.
However, the layout of the proof that I have given you is fairly standard.
The names may vary from book to book but the structure of the law is still the same .
for addition elimination for example you can write Q^P------> P e.t.c ,e.tc
And since you taught logic i suppose you will be able to give us a stepwise proof like the above where the laws of logic are explicitly mentioned of the following:
$\forall x$$\forall y$( x $\geq 0$ and y $\geq 0$--------> $\sqrt{xy}$= $\sqrt{x}$$\sqrt{y}$)
Also any theorems axioms definitions involved i.e justification of each step
......................................here and now............................................... .........................
September 19th 2008, 05:26 PM
IN propositional calculus you check each and every exercise if it s provable by using the true tables.
If the exercise is a TAUTOLOGY then it is provable.
Propositional calculus is
........................................ a DECIDABLE THEORY......................................
SO if youdo not trust what she handed out simply go to the true tables and see if what she handed out is a Tautology or not.
September 19th 2008, 05:38 PM
Triclino, If you had any idea of what is really going on here, I would take offence at your response.
But the fact is, you know so little about what you are talking about it is not worth the time a logician to response to your elementary errors.
While I applaud your efforts to understand the basics of logical arguments, I must tell you that your limited experience in this area can be a real determent to those who hope to learn from your
posts. Actually your own confusion is a determent to the understanding you hope to contribute by your postings.
September 19th 2008, 07:41 PM
Hello, olenka!
I forgot the name of this rule: . $\bigg[(p \vee q) \wedge (\sim p)\bigg] \to q$
. . I'll call it "Detchment", okay?
How to prove that this below argument is valid,
using established Rules of reasoning and Logical Equivalences
(using vertical statement - reason format.)
$\bigg\{ [(\sim p \vee q) \to r] \wedge [r \to (s \vee t)] \wedge (\sim s \wedge \sim u) \wedge (\sim u \to \sim t)\bigg\} \:\to\:p$
On the left side, we have:
. . $\bigg[(\sim p \vee q) \to r\bigg] \wedge \bigg[r \to (s \vee t)\bigg] \wedge (\sim s) \wedge (\sim u) \wedge \bigg[(\sim u) \to (\sim t)\bigg] \qquad\text{Given}$
. . $\bigg[\sim(\sim p \;\vee\; q) \;\vee\; r\bigg] \wedge \bigg[\sim r \;\vee (s \;\vee \;t)\bigg] \wedge \;(\sim s)\; \wedge\; (\sim u) \wedge\; \bigg[u \vee \sim t\bigg] \qquad \text{Equiv. of
. . $\bigg[(p \wedge \sim q) \vee r\bigg] \wedge \bigg[\sim r \vee(s \vee t)\bigg] \wedge (\sim s) \wedge (\sim u) \wedge \bigg[u \vee \sim t\bigg] \qquad \text{DeMorgan}$
. . $\bigg[r \vee (p \wedge \sim q)\bigg] \wedge \bigg[(s \vee t) \vee \sim r \bigg] \wedge (\sim s) \wedge \underbrace{(\sim u) \wedge \bigg[u \vee \sim t\bigg]} \qquad \text{Commutative}$
. . $\bigg[r \vee (p \wedge \sim q)\bigg] \wedge \bigg[(s \vee t) \vee \sim r\bigg] \wedge (\sim s)\quad \wedge\quad (\sim t)\qquad \text{Detachment}$
. . $\bigg[r \vee (p \vee \sim q)\bigg] \wedge \underbrace{\bigg[\sim(\sim s \wedge \sim t) \vee \sim r\bigg] \wedge \bigg[\sim s \wedge \sim t\bigg]} \qquad\text{ DeMorgan}$
. . . . . . $\underbrace{\bigg[r \vee (p\; \wedge \sim q)\bigg] \qquad\wedge \qquad \sim r} \qquad\qquad\text{ Detachment}$
. . . . . . . . . . . . . $p\; \wedge \sim q\qquad\qquad\text{ Detachment}$
And: . $(p \:\wedge \sim q) \to p\quad\hdots$ . Forgot the name of this rule, too
September 21st 2008, 08:03 PM
Thanks to all of you for your effort(Rofl)(Clapping)t!!!!!!! | {"url":"http://mathhelpforum.com/discrete-math/49661-please-help-print.html","timestamp":"2014-04-21T07:51:57Z","content_type":null,"content_length":"29607","record_id":"<urn:uuid:4048923b-2a49-437e-884c-49bccfb813ec>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00262-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions
Math Forum
Ask Dr. Math
Internet Newsletter
Teacher Exchange
Search All of the Math Forum:
Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
Topic: Basic GLM Question
Replies: 6 Last Post: Apr 9, 2013 2:53 PM
Messages: [ Previous | Next ]
Basic GLM Question
Posted: Mar 30, 2013 9:20 AM
Why is it that we don't try to predict individual values in Generalized Linear Model. But in a General Linear Model (Simple Linear Regression) we do try and predict individual response variable | {"url":"http://mathforum.org/kb/message.jspa?messageID=8789422","timestamp":"2014-04-17T07:59:33Z","content_type":null,"content_length":"22897","record_id":"<urn:uuid:d8d57409-4a5f-480a-967e-2b0e0c6c1524>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00074-ip-10-147-4-33.ec2.internal.warc.gz"} |
Explanation of the Formula for the Volume of a Pyramid
Date: 06/16/2005 at 00:59:22
From: Adriano
Subject: Reason why the volume of a pyramid works that way
OK, I know that the volume of a pyramid is Base*Height/3, but can you
please, please explain why? I couldn't find the answer on _why_ the
formula works in your archives.
I know that a pyramid is basically a prism with some of it "shaved"
off, so I guess this is why you have to divide Base*Height by 3.
Date: 06/16/2005 at 09:57:06
From: Doctor Greenie
Subject: Re: Reason why the volume of a pyramid works that way
Hello, Adriano --
Here is a link to a page in the Dr. Math archives where this question
is answered:
Volume of a Pyramid
I found this page by searching the archives using the key words
"volume pyramid".
On this page, there is a link to a site on the Internet containing a
proof of this formula based on dividing a prism into 3 pyramids. I
myself find that figure hard to visualize, and the proof hard to
Here is a different approach which uses a similar process and a
somewhat informal approach to argue that the formula is base times
height divided by 3.
Suppose we start with a cube, and from the center of the cube we
draw lines to the 8 corners of the cube. This divides the cube into
6 congruent pyramids with square bases.
If the side of the cube is s, then the volume of the cube is s^3.
In each of the pyramids, the base is a square of side s, and the
height is half of s. If we were to multiply the area of the base
times the height to get the volume of each pyramid, then we would find
that the volume of each pyramid is
(s^2)*(s/2) = (s^3)/2
But then the total volume of the 6 pyramids would be
6 * (s^3)/2 = 3s^3
But we know the total volume of the pyramids is the volume of the
cube, which is s^3.
Multiplying base times height to find the volume of a pyramid gives us
a volume for the cube which is 3 times as large as it is supposed to
be. From this we can conclude that the volume of each pyramid is not
base times height, but rather base times height divided by 3.
I hope all this helps. Please write back if you have any further
questions about any of this.
- Doctor Greenie, The Math Forum
Date: 06/17/2005 at 00:32:56
From: Adriano
Subject: Thank you (Reason why the volume of a pyramid works that way)
Thanks so much for answering my question! Your explanation was way
better than the website you showed me. I understand it better now. | {"url":"http://mathforum.org/library/drmath/view/66972.html","timestamp":"2014-04-19T17:48:49Z","content_type":null,"content_length":"7644","record_id":"<urn:uuid:8e155d77-7334-42e5-8ea0-2ce5c91097b0>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00083-ip-10-147-4-33.ec2.internal.warc.gz"} |
Need Help With This Exponential
austinallexis wrote:this one too: solve for x (2x-3 is exponent of 125)
They show how to solve exponential equations
. I would start by remembering the power that is necessary to turn the base into "1". So what must "2x - 3" be equal to? Then I'd set 2x - 3 equal to that value, and solve for x. | {"url":"http://www.purplemath.com/learning/viewtopic.php?p=7885","timestamp":"2014-04-20T21:11:54Z","content_type":null,"content_length":"18588","record_id":"<urn:uuid:21cd0400-c5e8-4776-b301-61d450f627e6>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00463-ip-10-147-4-33.ec2.internal.warc.gz"} |
unfold deforestation
Carsten Schultz carsten at codimi.de
Thu Sep 2 09:37:21 EDT 2004
[If this should have gone to a different list, please tell me and I
will subscribe to that.]
There is no rule in standard libs related to unfoldr. From short
googling I know that there may be several approaches to this, but as
long as nothing fancy is introduced, the following might be a simple
unfoldr :: (b -> Maybe (a,b)) -> b -> [a]
unfoldr u x = build (g x)
g x c n = f x
f x = case u x of
Just (h,t) -> h `c` f t
Nothing -> n
{-# INLINE unfoldr #-}
It makes unfoldr a good producer and works nicely with the following
example from `The Under-Appreciated Unfold' (Gibbons&Jones):
module Tree(Tree(..), bftf) where
import Unfold
data Tree a = Nd a [Tree a]
root (Nd r _) = r
kids (Nd _ ks) = ks
bftf = concat . levelf
levelf = unfold null (map root) (concat . map kids)
unfold :: (b -> Bool) -> (b -> a) -> (b -> b) -> b -> [a]
unfold p h t = unfoldr u
u x | p x = Nothing
| otherwise = Just (h x, t x)
Carsten Schultz (2:38, 33:47), FB Mathematik, FU Berlin
PGP/GPG key on the pgp.net key servers,
fingerprint on my home page.
More information about the Glasgow-haskell-users mailing list | {"url":"http://www.haskell.org/pipermail/glasgow-haskell-users/2004-September/007084.html","timestamp":"2014-04-21T12:59:45Z","content_type":null,"content_length":"3623","record_id":"<urn:uuid:f66bece6-8a5b-438b-b7ec-b9d8ea38082e>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00129-ip-10-147-4-33.ec2.internal.warc.gz"} |
passing assumed-shape arrays in two levels of subroutines (Fortran 90)
up vote 2 down vote favorite
I have had problems calling successive subroutines with assumed-shape arrays in Fortran 90. More specifically, I call two levels of subroutines, passing an assumed-shape array as a parameter, but in
the end the array is lost. To demonstrate it, one can follow the code below.
program main
subroutine sub1(x)
real, dimension(:):: x
real C
end subroutine sub1
subroutine sub2(x)
real, dimension(:):: x
real C
end subroutine sub2
real, dimension(:), allocatable:: x
allocate(x(1:10)) ! First executable command in main
x(1) = 5.
call sub1(x)
write(*,*) 'result = ',x(1)
end program main
subroutine sub1(x) ! The first subroutine
real, dimension(:):: x
real C
call sub2(x)
end subroutine sub1
subroutine sub2(x) ! The second subroutine
real, dimension(:):: x
real C
end subroutine sub2
Very shortly, main allocates x then call sub1(x). Then sub1 calls sub2(x). That means an allocated array is passed to a subroutine that passes it to another subroutine. I would expect to have in sub2
the same array that I've created in main, but no. Using gdb as a tool to explore it, I get this:
1) In main, just before calling sub1, the array x is perfectly defined:
(gdb) p x
$1 = (5, 0, 0, 0, 0, 0, 0, 0, 0, 0)
2) Within sub1, just before calling sub2, x is also well defined:
(gdb) p x
$2 = (5, 0, 0, 0, 0, 0, 0, 0, 0, 0)
3) Inside sub2, however, x have an unexpected value and even its dimension is absolutely wrong:
(gdb) p x
$3 = ()
(gdb) whatis x
type = REAL(4) (0:-1)
So, x has been successfully passed from main to sub1, but not from sub1 to sub2. I've been using Intel Fortran an gfortran with the same results.
I've struggling with that for a long time. Any help would be much appreciated.
arrays fortran fortran90
add comment
1 Answer
active oldest votes
The use of assumed-shape dummy arguments requires an explicit interface. In your main program you've provided explicit interfaces for the two subroutines, but these don't propagate into
the subroutines themselves. The subroutines are compiled as separate units, even if you've put all your code into one source file.
This means that sub1 doesn't have an explicit interface available for sub2, and thus uses an implicit interface, where the argument x is assumed to be a real scalar.
up vote 6 All this could be avoided simply by putting the two subroutines in a module and use that module in your main program, automatically making explicit interfaces available. This way you
down vote don't have to provide the interfaces yourself, which is error prone.
As a side note, I advise the use of implicit none in ALL your code.
3 Another side note: you might want to add the tag fortran to your question. More people follow that than just fortran90. – eriktous Feb 24 '11 at 12:28
+1. @user630900, don't feel bad; this is a common and somewhat subtle issue that comes up all the time - eg, cs.rpi.edu/~szymansk/OOF90/bugs.html#8 – Jonathan Dursi Feb 24 '11 at
1 @erikous: Precisely answered! With all the subroutines in a module everything works perfectly. Thanks a lot. – user630900 Feb 25 '11 at 11:14
add comment
Not the answer you're looking for? Browse other questions tagged arrays fortran fortran90 or ask your own question. | {"url":"http://stackoverflow.com/questions/5096206/passing-assumed-shape-arrays-in-two-levels-of-subroutines-fortran-90","timestamp":"2014-04-21T04:47:36Z","content_type":null,"content_length":"68140","record_id":"<urn:uuid:abf05a60-6fe2-43d2-9558-1c000ee02bd7>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00381-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathJax Syntax Change
May 25, 2012
By jjallaire
We’ve just a made a change to the syntax for embedding MathJax equations in R Markdown documents. The change was made to eliminate some parsing ambiguities and to support future extensibility to
additional formats.
The revised syntax adds a “latex” qualifier to the $ or $$ equation begin delimiter. It looks like this:
This change was the result of a few considerations:
1. Some users encountered situations where the $<equation>$ syntax recognized standard text as an equation. There was an escape sequence (\$) to avoid this but for users not explicitly aware of
MathJax semantics this was too hard to discover.
2. The requirement to have no space between equation delimiters ($) and the equation body (intended to reduce parsing ambiguity) was also confusing for users.
3. We want to eventually support ASCIIMath and for this will require an additional qualifier to indicate the equation format.
RStudio v0.96.227 implements the new MathJax syntax and is available for download now.
for the author, please follow the link and comment on his blog:
RStudio Blog
daily e-mail updates
news and
on topics such as: visualization (
), programming (
Web Scraping
) statistics (
time series
) and more...
If you got this far, why not
subscribe for updates
from the site? Choose your flavor:
, or | {"url":"http://www.r-bloggers.com/mathjax-syntax-change/","timestamp":"2014-04-18T03:26:57Z","content_type":null,"content_length":"36990","record_id":"<urn:uuid:1f29a4fe-2b1f-4492-9fb2-524da60f7bbc>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00489-ip-10-147-4-33.ec2.internal.warc.gz"} |
Taming Serpents and Pachyderms
Vasudev Ram pointed readers to a Hacker News poll on the subject. While the raw numbers per language are interesting, I think the percentages of Like and Dislike vs. the total votes cast for a given
language are perhaps a better metric.
Thus the five most liked languages based on raw votes were:
1. Python
2. C
3. JavaScript
4. Ruby
5. SQL
And the five most disliked languages were:
1. PHP
2. Java
3. C++
4. JavaScript
5. Visual Basic
It’s rather interesting that JavaScript is on both lists (and I’m quite surprised that SQL had so many votes). I haven’t included the actual numbers since the poll is still active.
Ranking the languages by number of Like votes as percentage of total votes for that language gives a perhaps more realistic picture:
C 88%
Python 86%
Scheme 84%
Lua 84%
Lisp * 81%
Haskell 80%
Rust 79%
Clojure 78%
Erlang 76%
Go 75%
(*) Dimitri Fontaine will appreciate this.
I believe those rankings will be more stable than the raw votes. Oh, and SQL ranks about 14 according to these percentages.
For completeness, here are the five most disliked languages based on ratio of Dislike votes to total votes for the language:
Cobol 94%
ColdFusion 94%
Visual Basic 89%
Actionscript 83%
PHP 76%
As they say, YMMV.
Multisets and the Relational Model
In a comment to my previous post, David Fetter challenged me to “find a case for multisets. That we’re stuck with them doesn’t mean they’re useless.” My response was that I couldn’t help him because
multisets (or bags) are not part of the relational model (which was the point of my post) and asked David to show me an example of a multiset he’s stuck with so that we could discuss it.
While waiting for his response, I read an article titled “Toil and Trouble” by Chris Date, which was originally published in Database Programming and Design, January 1994^1, where he tackled the
issue of duplicate rows and multisets. Chris opened by stating that duplicates “are, and always were, a mistake in SQL” (and nearly 20 years later the mistake has not been corrected).
In the article, Date makes a number of points against duplicates and multisets but perhaps two of the best are the following:
1. When considering the collection (3, 6, 6, 8, 8, 8, 11) versus the set {3, 6, 8, 11} we have to distinguish between the two 6′s by saying “the first 6″ or “the second.” Date then points out that
“we have now introduced a totally new concept, one that is quite deliberately omitted from the relational model: positional addressing. … we have moved quite outside the cozy framework of
relational theory … [and] there is no guarantee whatsoever that any results that hold within that framework still apply.”
2. In response to a claim by David Beech that “mathematicians deal with such collections, called multisets or … bags” and therefore that a model with duplicate rows is at least mathematically
respectable, Date says:
“… all of the mathematical ‘bag theory’ treatments I’ve seen start off by assuming that there is a way to count duplicates! And that assumption, I contend, effectively means that bags are defined
in terms of sets—each bag element has a hidden identifying tag that distinguishes it somehow, and the bag is really a set of tag/element pairs.”
I believe that as programmers it becomes second nature to deal with duplicate items in lists and sequences. Since it is so easy to code a loop to visit each item in turn and apply some processing—in
Python you can even use built-ins or functions from itertools, that we frown on a system that, at least in theory, insists on removing duplicates and dealing only with proper (mathematical) sets.
However, we should realize that the theory, as Date says, is practical: by keeping the duplicates we lose, for example, the benefits of relational normal forms and certain optimization techniques.
In closing, Date presents the following parts and shipments database:
P pno │ pname SP sno │ pno
─────┼──────── ─────┼─────
P1 │ Screw S1 │ P1
P1 │ Screw S1 │ P1
P1 │ Screw S1 │ P2
P2 │ Screw
And considers the query “List part numbers for parts that either are screws or are supplied by supplier S1, or both.” He then presents 12 candidate SQL formulations, which someone ran for him against
SQL Server 4.2 on OS/2. I thought it would be instructive to run them against Postgres 9.3, so here they are.
SELECT pno
FROM p
WHERE pname = 'Screw'
OR pno IN
( SELECT pno
FROM sp
WHERE sno = 'S1');
Result: 3 P1, 1 P2
SELECT pno
FROM sp
WHERE sno = 'S1'
OR pno IN
( SELECT pno
FROM p
WHERE pname = 'Screw');
Result: 2 P1, 1 P2
SELECT p.pno
FROM p, sp
WHERE ( sno = 'S1' AND
p.pno = sp.pno)
OR pname = 'Screw';
Result: 9 P1, 3 P2
SELECT sp.pno
FROM p, sp
WHERE ( sno = 'S1' AND
p.pno = sp.pno)
OR pname = 'Screw';
Result: 8 P1, 4 P2
SELECT pno
FROM p
WHERE pname = 'Screw'
SELECT pno
FROM sp
WHERE sno = 'S1';
Result: 5 P1, 2 P2
SELECT DISTINCT pno
FROM p
WHERE pname = 'Screw'
SELECT pno
FROM sp
WHERE sno = 'S1';
Result: 3 P1, 2 P2
SELECT pno
FROM p
WHERE pname = 'Screw'
SELECT DISTINCT pno
FROM sp
WHERE sno = 'S1';
Result: 4 P1, 2 P2
SELECT DISTINCT pno
FROM p
WHERE pname = 'Screw'
OR pno IN
( SELECT pno
FROM sp
WHERE sno = 'S1');
Result: 1 P1, 1 P2
SELECT DISTINCT pno
FROM sp
WHERE sno = 'S1'
OR pno IN
( SELECT pno
FROM p
WHERE pname = 'Screw');
Result: 1 P1, 1 P2
SELECT pno
FROM p
GROUP BY pno, pname
HAVING pname = 'Screw'
OR pno IN
( SELECT pno
FROM sp
WHERE sno = 'S1');
Result: 1 P1, 1 P2
SELECT p.pno
FROM p, sp
GROUP BY p.pno, p.pname, sno, sp.pno
HAVING ( sno = 'S1' AND
p.pno = sp.pno)
OR pname = 'Screw';
Result: 2 P1, 2 P2
SELECT pno
FROM p
WHERE pname = 'Screw'
SELECT pno
FROM sp
WHERE sno = 'S1';
Result: 1 P1, 1 P2
As Date points out, 12 different formulations produce 9 different results! And as he further states, those are not all the possible formulations. For example, a modern revision of the third query
may be:
SELECT pno
FROM p NATURAL JOIN sp
WHERE sno = 'S1'
OR pname = 'Screw';
and the result is yet again different (6 P1 parts and 1 P2).
The bottom line is to be very, very careful when dealing with multisets in SQL.
^1 The article was republished in Relational Database Writings, 1991-1994, in Part I, “Theory Is Practical!”
Is This Relational?
This post was prompted by Hans-Juergen Schoenig’s Common mistakes: UNION vs. UNION ALL because it touches on one of my pet peeves: the claim that some feature of SQL exemplifies or conforms to the
relational model. Schoenig does not make that claim explicitly, but he does state “What [most] people in many cases really want is UNION ALL” and shows the following query and result:
test=# SELECT 1 UNION ALL SELECT 1;
(2 rows)
There are two relational faults above*. First, UNION ALL is not a relational operator. This is an area where both Ted Codd and Chris Date (and Hugh Darwen), are fully in agreement. In the “Serious
Flaws in SQL” chapter of The Relational Model for Database Management: Version 2 (1990) Codd listed duplicate rows as the first flaw and characterized “relations in which duplicate rows are permitted
as corrupted relations.” Date concurs and wrote the essay “Why Duplicate Rows Are Prohibited”(in Relational Database Writings, 1985-1989) and (with Darwen) included RM Proscription 3: No Duplicate
Tuples in their Third Manifesto, which reads:
D shall include no concept of a “relation” containing two distinct tuples t1 and t2 such that the comparison “t1 = t2” evaluates to TRUE. It follows that (as already stated in RM Proscription 2),
for every relation r expressible in D, the tuples of r shall be distinguishable by value.
Needless to say, those two “1″s are not distinguishable unless you talk about “the first 1″ and “the last 1,” i.e., ordering, which is also proscribed by the relational model because relations are
Now, the example given is synthetic so I’ll present a more realistic example. Suppose a manager asks “which employees are in department 51 or work on the Skunk Works project?” Let’s assume we have a
projects table with columns proj_no (primary key) and proj_name, an emp table with columns emp_no (primary key), last_name, first_name, and dept_no, and an assignments table with columns proj_no and
emp_no (both forming the primary key and each referencing the previous two tables, respectively). We’ll first emulate this with a CTE, so we won’t have to create or populate any tables:
WITH emp AS (SELECT 'Ben Rich'::text AS emp_name,
51 AS dept_no),
assignments AS (SELECT 'Ben Rich'::text AS emp_name,
'Skunk Works'::text AS proj_name)
SELECT emp_name
FROM emp
WHERE dept_no = 51
SELECT emp_name
FROM assignments
WHERE proj_name = 'Skunk Works';
If you run this in psql, you’ll see two rows with identical values and the manager is going to ask “Do we have two employees named Ben Rich?” However, in practice the real query will be:
SELECT first_name, last_name
FROM emp
WHERE dept_no = 51
SELECT first_name, last_name
FROM emp JOIN assignments USING (emp_no)
JOIN projects p USING (proj_no)
WHERE p.proj_name = 'Skunk Works';
Unless you change UNION ALL to UNION your result wil contain duplicate rows for employees that satisfy both predicates. However, an alternative formulation without UNION would be
SELECT first_name, last_name
FROM emp LEFT JOIN assignments USING (emp_no)
LEFT JOIN projects p USING (proj_no)
WHERE dept_no = 51
OR p.proj_name = 'Skunk Works';
This query correctly returns only one row per employee. Admittedly, the query is still somewhat synthetic. In reality, the query may include multiple other columns and several hundred rows may be
retrieved and thus the duplicate tuples and the logical error may not be so obvious.
UPDATE: Changed last query to use LEFT JOINs as correctly suggested by RobJ below.
* The second relational fault? The result column is unnamed (something Date and Darwen insist on much more than Codd).
A short detour into AngularJS and Brunch
I was planning to continue exploring database user interfaces using something completely different, as mentioned in the last post. However, I’ve taken a short detour to experiment further with the
technologies in question.
In case you’re interested in AngularJS, the Karma test runner, the Brunch application assembler, CoffeeScript and the Jade templating engine, you can follow along at GitHub where I’ve created
angular-phonecat-brunch, a derivative of the AngularJS phone catalog tutorial. And if you’re an expert on these topics, you can visit too, and let me know how to do it better!
Update: The tutorial has been completed, including changes to use the latest releases of Bower and Brunch, and also incorporates Stylus dynamic CSS stylesheets. Don’t miss the documentation README.
ANFSCD: Revisiting the Web Server
Nearly two years ago, I was considering which Python web framework to use for a user interface to Postgres: CherryPy, Flask, Werkzeug? Not entirely satisfied with the choices, I started reviewing
even more frameworks thinking I might want to write my own minimalist framework.
Several months later, somebody (through Planet Python, IIRC) referred me to a presentation by Jacob Kaplan-Moss on the history and future of Python on the web. Surprisingly, halfway through the talk
Jacob started raving about Meteor, a pure JavaScript framework, saying “we’re deluding ourselves if we think this [something like Meteor] is not the future of web applications.” This prompted me to
take a close look at Meteor and several other JS frameworks.
Tarek Ziadé’s “A new development era” essay reinforced this change in direction. Ultimately, I settled on AngularJS as the (client) framework. Two-way data binding, dependency injection and
testability are some of the features that won me over.
Angular opened the door to the Node.js world—which appears somewhat chaotic compared to Python’s (and even more to the staidness of Postgres). Like Python, Node.js has an abundance of web frameworks,
templating libraries and other tools to choose from (and master). Aside from that, are there any negatives in continuing down this path?
For one, although Angular is an open source project, unlike Python and PostgreSQL, its destiny is controlled by a behemoth. A saving grace is its large community of contributors. And perhaps some of
Angular’s innovations may eventually become part of standard HTML.
Second, in spite of Selena Deckelmann’s recent comments on JS and PG, I’m strongly partial to Python and not fond of JavaScript as an implementation language. It’s liberating not to have to use
braces (and semicolons) for code structure! To compensate, CoffeeScript appears to be the obvious alternative.
When it comes to interfacing to Postgres, although I haven’t explored it enough to do justice, node-postgres doesn’t seem to be up to par with psycopg, and I’m not about to throw away the work I’ve
done on Pyrseas, in particular the TTM-inspired interface. So Werkzeug may still play a part, as a Postgres-Python-to-JSON service, particularly now that it support Python 3. However, for contrast I
will use node-postgres in an early implementation.
Last, the Angular team’s choice for “workflow” tool (Yeoman) did not sit well with me: I don’t care for “scaffolding” and my first experience with Grunt rubbed me the wrong way. Fortunately, in the
Node.js “chaos” I found Brunch, which although not without problems, looks suitable for my purposes.
Having addressed the negatives, I’ve started work on this at GitHub, and plan to post more about it later on.
Update: Due to the change in direction, I was wondering whether I should also change the title of this blog to something like “Taming Serpents, Pachyderms and White A’s in Red Shields”, but
fortunately I discovered that at least O’Reilly uses a rhinoceros as the JavaScript mascot and rhinos are considered pachyderms. :-)
Pyrseas contributions solicited
Do you use PostgreSQL and truly believe it’s “the world’s most advanced open source database” and that its upcoming 9.3 release will make it even more awesome?
Do you also use Python and believe it’s “an easy to learn, powerful programming language” with “elegant syntax” that makes it an ideal language for developing applications and tools around
PostgreSQL, such as Pyrseas?
Then we could use your help. For starters, we want to add support for the MATERIALIZED VIEWs and EVENT TRIGGERs coming up in PG 9.3.
We have also been requested to add the capability to load and maintain “static data” (relatively small, unchanging tables) as part of yamltodb, so that it can be integrated more easily into database
version control workflows.
And for the next release, Pyrseas 0.7, we’d like to include the first version of the database augmentation tool which will support declarative implementation of business logic in the
database–starting off with audit trail columns. Some work has been done on this already, but it needs integration with the current code and tests.
Or perhaps coding is not your forte, but you’re really good at explaining and documenting technical “stuff”. Then you could give us a hand with revamping the docs, maybe writing a tutorial so that
users have a smooth ride using our tools.
Or maybe you have your own ideas as to how improve the PostgreSQL version control experience. We’d love to hear those too.
If you’d like to help, you can fork the code on GitHub, join the mailing list and introduce yourself, or leave a comment below.
Tuples in the Pythonic, TTM-inspired interface to PostgreSQL
The Third Manifesto formally describes tuple types (RM prescription 6), tuple values (prescription 9), tuple variables (prescription 12) as well as other tuple-related elements. As mentioned in the
previous post, a tuple value is a set of ordered triples each consisting of attribute name, type and value.
Class Tuple of the TTM-inspired interface to PostgreSQL models TTM tuples as Python lists of TTM Attribute objects. Lists were used rather than sets because for many practical purposes the order of
the attributes is useful (or has “meaning”), e.g., the first attribute listed is most often –even in purist relational theory presentations– the primary key or part of the primary key.
The interface stores the Tuple heading as a (Python) n-tuple of name-type tuples, in the “internal use” _heading attribute. The n-tuple was chosen over a list due to its immutability. The interface
also sets each Attribute as a Python attribute of the Tuple object. Thus, if you define a Tuple variable as follows:
film = Tuple([
Attribute('id', int, sysdefault=True),
Attribute('release_year', int)])
You can then assign or access an Attribute using simple Python syntax:
film.title = "Seven Samurai"
if film.year == 1954:
do something
The interface also stores two other internal use lists, one for nullable attributes and another for attributes that allow default values. These are to be used by upstream classes such as RelVar.
Class Tuple has a __setattr__ method tailored to deal with assignment to TTM Attributes. It disallows assignment to internal attributes after initialization, with one exception: the _tuple_version
attribute (used by RelVar for optimistic concurrency). It also doesn’t allow assignment to undefined Attributes, e.g., given the film variable above, attempting to assign to film.length will raise an
AttributeError. Finally, the assignment is “filtered” through class Attribute, so that an attempt such as film.title = 8.5 will result in a ValueError from that class.
The pyrseas.relation.tuple module defines a standalone function: tuple_values_dict. This is used to generate a dictionary of attribute values suitable for passing to Psycopg’s cursor.execute method.
For INSERT, a single currtuple argument is expected. For UPDATE, the modified Tuple is passed as a second argument and tuple_values_dict will return a dictionary solely for the attribute and values
that have changed. | {"url":"http://pyrseas.wordpress.com/","timestamp":"2014-04-17T16:50:27Z","content_type":null,"content_length":"121055","record_id":"<urn:uuid:07c60476-a445-41e0-b84b-591da4544f08>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00405-ip-10-147-4-33.ec2.internal.warc.gz"} |
An Example of Using Linear Regression of Seasonal Weather Patterns to Enhance Undergraduate Learning
Teresa Jacobson
Josh James
Neil C. Schwertman
California State University, Chico
Journal of Statistics Education Volume 17, Number 2 (2009), www.amstat.org/publications/jse/v17n2/jacobson.html
Copyright © 2009 by Teresa Jacobson, Josh James, and Neil C. Schwertman all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without
express written consent from the author and advance notification of the editor.
Key Words: Collaborative learning; Class project; Data analysis.
Group activities are an excellent way to enhance learning. When students are actively involved in a relevant project, understanding and retention are improved. The proposed activity introduces a
timely and interesting project typical of the type encountered in statistical practice. Using the computer to successfully developing an appropriate model is a valuable educational experience that
builds confidence.
I. Introduction
Whenever possible many teachers attempt to involve students in research or projects to enrich their educational environment. Such projects can serve as a capstone for highly motivated students,
bringing together the theoretical and applied aspects of their studies. The purpose of this example project is to introduce statistics undergraduates to such topics as model selection, higher order
polynomial regression and to demonstrate that basic statistics methods and computer software can be used to quite precisely define patterns in real data.
Weather data affords the opportunity to enhance the learning environment by providing interesting practical real world data for analysis. Driscoll (1988) had his meteorology students monitor the
accuracy of the TV temperature forecasts in seven U.S. cities for six months to compare the accuracy to forecasts from the National Weather Service, noting little difference in the accuracies for
temperatures. Reading (2004) developed a weather data activity for students with a focus on variations in monthly temperatures and rainfall. To encourage the use of real data in the classroom, the
National Oceanographic and Atmospheric Administration (NOAA) provides interesting data and resources. Google "NOAA Education Resources".
Weather is so popular that there is now a television channel completely devoted to weather issues. Weather can have a substantial impact on the economy, especially agriculture but other areas as
well. The 2005 hurricanes Katrina and Rita caused an extensive shutdown of oil and gas rigs in the Gulf of Mexico resulting in a significant spike in energy prices while the 2006 freeze in California
caused hundreds of millions of dollars in losses to citrus growers and wide spread unemployment. Consequently, it is not surprising there has been significant scientific efforts in studying and
predicting weather patterns, see for example, journals such as: Bulletin of the American Meteorological Society, Journal of Meteorology, Journal of Applied Meteorology and Climatology and Weather
and Forecasting.
Of course in practice the meteorological study of weather patterns use rather complex models and techniques (see for example: Brenner(1986), Brunet, et.al. (2007), Gyakum(1986), Murphy (1998),
Serra, et.al. (2001) and Stone and Weaver(2002)). A historic perspective of evolving weather prediction models initially developed for the National Meteorological Center by the Princeton University
Institute of Advanced Study in 1932 is provided by Murphy (1998) and Shuman (1989). The purpose here, however, is not to advance meteorological practice but rather increase statistics undergraduates’
appreciation of the usefulness of statistics methods and to provide practical experience using these methods and computers with real world data. A Google search of "Modeling Daily Temperatures"
lists ten references that illustrate the methods currently in use.
While most of forecasting literature focuses on short term predictions of just a few days, the long term forecasts can still be useful. For example, planning weeks in advance an outdoor activity such
as a hike, swim, or camping or perhaps even an outdoor wedding, the expected maximum (or minimum) temperatures and 95% confidence limits on the range of daily temperatures could be helpful in
determining the practicality of the activity. The minimum temperature and it’s 95% lower bound would provide valuable information for determining when to put temperature sensitive plants in an
outdoor garden.
Since weather conditions usually are quite localized, many of the studies in the scientific journals by necessity pertain to very limited locales. Like studies by Gyakum (1986), Brenner (1986) and
Mass (1987)) the example project in this paper is quite localized. Similar to Driscoll (1988) and Reading (2004), this example project for statistics undergraduates is a study of the daily
temperature patterns throughout the year based on the thirty year temperature averages for Chico, California U.S.A. This type of project is easily adaptable as a class activity even at the
elementary level.
This activity was motivated on one particularly hot day by a common keen interest among the authors of the daily temperature trends. For background, Chico, located in the Sacramento River valley and
the northern end of the California "central valley" gets very hot in the summer. Temperatures above 100F are very common but the winters are mild, rarely below 30F or -1C. To establish focus, by
consensus with the students, specific questions were addressed. The weather related questions investigated in the project are:
1. What day of the year on average has the highest maximum temperature and what is that temperature?
2. What day of the year on average has the lowest maximum temperature and what is that temperature?
3. What day of the year on average has the highest minimum temperature and what is that temperature?
4. What day of the year on average has the lowest minimum temperature and what is that temperature?
5. Is the daily trend in temperatures the same for the maximum and minimum temperatures?
6. Are the random variations over thirty years in each day’s maximum temperature roughly the same for each day throughout the year?
7. Are the random variations over thirty years in each day’s minimum temperature roughly the same for each day throughout the year?
The statistical questions addressed are:
1. Can the daily patterns in temperature be modeled adequately using polynomial regression models with day of the year as the independent variable?
2. What criteria should be used to evaluate the adequacy of the model?
To answer these questions the steps were:
1. To access the data as described in Section II.
2. To plot the data using the JMP statistical program.
3. To analyze data using the regression analysis in JMP with successive higher order polynomial models until the changes in R-square and root mean square error (RMSE) are minimal.
The two undergraduate seniors’ (first two authors) statistical education consisted of three semesters of upper division calculus based mathematical statistics and two semesters of applied statistics/
experimental design. The primary topics covered in applied statistics were analysis of variance and the general linear model with an introduction to SAS and JMP statistical programs. Other topics
included the design of experiments using blocking to decrease noise and factorial designs, orthogonal and multiple comparison procedures to enhance information, repeated measures and analysis of
covariance. While the analysis of data similar to the weather data could easily be used as project for a class, for these two students it was not a requirement or extra credit for any class. It was
purely voluntary to enrich their learning experience. They were asked if they would like to do a research project. No prodding was needed to keep them on track which indicates that the students were
enthusiastic and enjoyed doing some actual research and found it rewarding and interesting.
The students had not been introduced to time series or non-linear modeling which could be used for a more statistically appropriate analysis of the weather data. The intent was to provide students
with practice in the basic tools of research and modeling and to address the limitations of these methods.
The advisor on the project, suggested the topic, search the internet for the data and related literature. The students suggested and investigated all the models and wrote the initial version of the
paper. They were asked to write the paper in a form for submission to a peer reviewed journal. This was their first attempt at such a task and the initial version was more about temperature patterns
in Chico, California than about what they learned and how they benefitted from the project. Even with the limitations on statistical inference, the paper provides insight into the daily temperature
patterns throughout the year in the California central valley. The subsequent revisions required more commentary from the teacher’s prospective and the advisor became involved in the rewrite since
both students had graduated.
The graphical displays of the data and the polynomial models used to answer the questions are provided in Figures 1 to 8 and the summary of the numerical part of the statistical analyses are in
Tables I to IV. The Question 9), criteria for evaluating the models are discussed in Section III of this paper. Sections IV and V describe the analyses of the temperature and random variation data
respectively. In section VI the nine questions are answered with some concluding comments.
Accessing weather data
The weather data for the western United States (excluding Hawaii and Alaska) is available at the web address: www.wrcc.dri.edu/climsum.html. This page is: "Western Regional Climate Center". Click on
the desired state and city or location, then on "daily tabular data". There are 183 locations listed however many have no data. Other temperature data can be found for each state at the web address:
http://cdo.ncdc.noaa.gov/cgi-bin/climatenormals/climatenormals.pl. First click on "Product Selection", then click on "Daily Stations Normals". Next select the appropriate state and choose the desired
Using the daily temperatures for thirty years would be a massive data set and for just modeling the patterns, the daily averages are sufficient. Furthermore, there may be gradual climate change which
could invalidate any analysis used for prediction. The purpose of this activity, however, was to introduce statistical methodology and use of statistical programs rather than temperature prediction.
3. Criteria for evaluating the models
Three criteria were initially considered for choosing the "best" model: R-square, Root Mean Square Error (RMSE) and the p-values of the polynomial coefficients. A lively discussion of the relative
merit of each concluded that R-square and RMSE are quite similar and are best if the purpose of the model is prediction and that the p-value is better suited for testing separately each term in the
polynomial. The p-value criterion was significant at p<.05 for all parameters in all models considered for both maximum and minimum temperatures as well as all models in the analyses of the
variations or standard deviations. In fact, all the final models had all p-values <.0001. Consequently that criterion was deemed to be ineffective in discriminating the relative adequacy of the
models and only the R-square and RMSE criteria were used to measure the relative quality of the competing models.
There are other methods for determining models such as SELECT, PRESS and the stepwise, forward selection, and backward elimination regression procedures but these were considered too advanced for
4. Determining the Models for Maximum and Minimum Daily Temperatures
The data provided from the website of the Western Regional Climate Center were the mean maximum and mean minimum temperatures for each day of the year over the 30 year period 1971 to 2000. However in
some cases observations for a few years were missing in the averages. That daily temperatures on consecutive days are likely to be correlated since there are frequent "hot spells" and "cold spells"
presents a challenge. Such data often are analyzed using time series techniques. Because of the student’s background, a time series analysis was not practical. Of necessity the focus became to
introduce undergraduates to modeling using regression methods and the correlations were ignored for purposes of illustration. A challenging task of a more advanced nature would be a time series
analysis of weather data.
The average maximum daily temperatures throughout the year are displayed in Figure 1 and the individual 95% confidence limits for the range of the daily maximum temperature are displayed in Figure 2.
To find models which closely approximated the plots, simple polynomial regression models were evaluated using both SAS with the PROC REG procedure and JMP. Using the day of the year as the
independent variable, starting with just the quadratic model the order of the polynomial models was increased until an adequate fit was obtained. While SAS has many more options it is more difficult
to use. Since the analyses were identical for both SAS and JMP the simpler JMP was used for evaluating all the remaining polynomial models. The R-square and RMSE for the maximum daily temperature for
the various polynomial models are provided in Table I with the plots of the various polynomial models in Figure 3.
Table I indicates very little improvement in the model using a polynomial higher than degree 5. The next higher polynomial model only increased R-square by 0.00009 and reduced RMSE by only 0.014723.
Therefore the fifth degree polynomial model was considered to adequately fit the data with R-square of 0.998252 and RMSE of 0.594392.
While the objective of the project was to introduce polynomial regression modeling for real data, one non-linear model was investigated. One student observed the data plot in Figure 1 seemed to
follow a Beta type distribution with (day/366) on the horizontal axis. The Beta model was temperature, ^th and 210^th day of the year and x = 209.5 was used. Taking the derivative of T(x), T'(x)=
0 and solving the ratio Figure 4). While the R-square for this non-linear model is 0.989 and the root mean square error is 1.456 the polynomial linear model provided a better fit to the data.
Figure 5 is the plot of the average minimum daily temperatures with the individual 95% confidence limits on the range of the daily minimum temperatures. The average minimum daily temperature plot, as
would be expected, followed a similar pattern to the maximum temperature plot. The R-squares and RMSE for the various polynomial models are included in Table II and the plots of these models are
provided in Figure 6. The inclusion of the sixth degree term increased R-square by 0.003323 to 0.996717 and RMSE was reduced by 0.213329 to 0.511967. This model was considered to be quite adequate
fit of the data. Due to the similarity in the patterns for maximum and minimum temperatures it would be reasonable to reconsider an order six polynomial for maximum temperature but this was not done.
Since the nonlinear Beta type model was less satisfactory than the higher degree polynomial models and due to time constraints, the nonlinear model was not investigated for the average daily minimum
temperature data. The non-linear model requires the estimation of four parameters while the polynomial models requires the estimation of six or seven parameters. The small increase in the number of
parameters for the linear models is well justified by the improvement in the model’s fit as well as the simplicity of the linear models compared to the non-linear approach.
5. Modeling the Variations in Maximum and Minimum Daily Temperatures
The Western Regional Climate Center website also provides, for each day, the standard deviations of the maximum and minimum daily temperatures over the same approximate thirty year period. The
standard deviations of the maximum temperatures follow a particularly complex pattern. The "transition periods" from winter to summer and from summer to winter had the largest daily variation in
maximum daily temperatures. Throughout the entire periods from March 23 to June 23 and from September 19 to October 19, and only in these periods were the standard deviations at least 8.000. On May
19 the maximum of all the estimated standard deviation was 9.571 while a second local maximum of 8.807 occurred on September 30.
Based on the students’ knowledge of polynomials and the graphical display of the daily standard deviations in maximum temperatures it was estimated that a model of at least degree seven would be
necessary to satisfactorily describe the pattern. Figure 7 shows the fit of the polynomial models. The sixth order polynomial was used to compare the erratic pattern of the maximum standard
deviations to the corresponding pattern of the best fitting polynomial for the minimum standard deviations. Table III provides the R-squares and RMSE for the polynomial models of degrees 6, 7 and 8.
Including the eighth degree term increased R-square by 0.020975 to 0.885433 and reduced RMSE by 0.028903 to 0.33539. For such an erratic pattern this seemed quite reasonable.
The basic underlying trend in the daily variations as measured by the standard deviations of the daily minimum temperatures had a much less erratic pattern, being greatest during the winter and more
stable or consistent during the late summer to October. During the period from August 3 to October 13 the standard deviations in the minimum daily temperature were consistently less than 5.0 degrees
with 4.570 the minimum of all the estimated standard deviations occurring on August 30. From November 1 to March 23 the standard deviations in minimum daily temperatures was consistently greater than
6.0 with the largest standard deviation in the data, 7.490 occurring on January 3. For model comparison, Table IV provides the R-square and RMSE for the polynomial models of degrees two to six.
Figure 8 is the plot of the standard deviations of the daily minimum temperatures with the graphs of the polynomial models. By including the sixth degree term the R-square only increased by 0.024551
to 0.977235 but the RMSE decreased substantially by 0.057555 to 0.13090. Hence the sixth degree model was considered the best.
6. Answering the questions and concluding comments.
The study provided answers to the nine questions in the Introduction that were used to focus the project. The first four weather questions were answered immediately upon accessing the data and the
other weather questions were easily answered from the figures. Specifically, for Chico California
1. The highest average maximum temperature of 95.1F occurs on July 27 and 28.
2. The lowest average maximum temperature of 53.0F or occurs on January 2 and 3.
3. The average highest minimum temperature of 60.8F occurs during the period July 21 to 28.
4. The average lowest minimum temperature of 33.6F occurs on December 27 and 28.
5. The pattern in the daily maximum and minimum temperatures is roughly the same with the highest maximum and minimum daily occurring on the same days and the lowest maximum and minimum temperatures
only six days apart.
6. Figure 7 clearly shows that the basic pattern of random variation in maximum temperature varies greatly during the year, being much larger during the transition times, Spring and Fall.
7. Figure 8 show that the basic pattern of random variation in minimum daily temperature is much less erratic and much more consistent during the summer.
8. With R-squares greater than 0.99 and root mean square errors (RMSE) just about .5 the polynomial regression was able to model both maximum and minimum temperature patterns very well.
9. The three criteria used to evaluate the models are described in Section 3 and were in complete agreement with each other. The two used to evaluate the differences between models were R-square and
While the temperature questions are interesting the statistical questions 8) and 9) were much more important from the educational prospective. The primarily purpose of this paper was not to analyze
the temperature pattern of Chico, California but rather to demonstrate an example project and the vast analysis potential of readily available weather data. Projects such as this one, afford ample
opportunities to discuss with students many of the vexing questions and problems that can occur while modeling real data. One of the most obvious is the time scale. Fortunately for this data both the
minimum high and low daily temperatures occurs very close to the first day of the year (January 2 and December 28) and the day of the year was a natural time scale. In other data the determination of
a time scale can have significant ramifications which should be discussed. For data based on a yearly cycle it is important that the model provide values close together at the end and beginning of
the time scale. For the weather example in this paper, days one and 365 are really only one day apart and should have nearly identical average temperatures. Students could be asked how to address
this requirement. With some thought students may suggest, due to the cyclic nature of yearly data, a trigonometric model to ensure a smooth transition from the end to the beginning of the time scale.
In this project, simple trigonometric models were considered initially but not pursued due to the complexity and time constraints. Furthermore, the simpler the higher order polynomial models seemed
to adequately describe the patterns.
A second area of discussion is the importance of independence of the data. It was assumed there would be dependencies caused by hot or cold spells that last for several days. It is essential to point
out that the lack of independence in this data precludes statistical inference with the methods used. A general discussion about handling dependencies such a paired data analysis or the
Geisser-Greenhouse correction factor should be mentioned as possibilities. Neither of these could be used for this project since the thirty daily temperatures for each day of the year was not
available. It is instructive to point out to the students that more complex techniques such as time series and multivariate analysis are available and would allow statistical inference from the data.
The Editor suggested that since the maximum and minimum daily temperature patterns are closely related raises the question: Should the same degree polynomial be used for both? Specifically, students
could discuss how much should the similarity in patterns and intuition influence the modeling process? How important is it that the model make sense from a practical standpoint? It could be pointed
out that the efficacy of using the same degree polynomial for both maximum and minimum temperature models could be tested by the classical linear model technique of finding the total regression sum
of squares (RSS) for the full model (allowing different degree polynomials for each) and a restricted RSS by using the same degree polynomial for each. The difference between the two RSS’s divided by
the appropriate degrees of freedom and the MSE results in a statistic which is a pseudo F because of the dependencies. This statistic, nevertheless, provides some measure of the efficacy of using
polynomials of the same order for both maximum and minimum temperature models.
Upon graduating, statistical students, whether in graduate school or in industry, are likely to need to model and analyze data. Projects like this afford practice in using statistical programs such
as JMP and should aid in the understanding and appreciation of statistical methodology and the limitations. The students were enthusiastic and particularly enjoyed analyzing local data that had
meaning to them. The modeling of real data can enhance confidence and promote statistical maturity.
Figure 1 Daily Maximum Temperatures
Figure 2 Daily Maximum Temperatures with Confidence Limits
Tmax Quadratic Model Tmax Cubic Model
Tmax Quartic Model Tmax Quintic Model
Figure 3 Maximum Temperature Models
Figure 4 Tmax Beta Model with Confidence Limits
Figure 5 Tmin Daily Temperatures with Confidence Limits
Tmin Quadratic Model Tmin Cubic Model
Tmin Quintic Model Tmin Sextic Model
Figure 6 Minimum Temperature Models
Daily Standard Deviations Maximum Daily Standard Deviations Maximum
Temperatures Sextic Model Temperatures Septic Model
Daily Standard Deviations Maximum
Temperatures Octic Model
Figure 7 Models of Standard Deviation of Daily Maximum Temperatures
Daily Standard Deviations Minimum Daily Standard Deviations Minimum
Temperatures Quadratic Model Temperatures Cubic Model
Daily Standard Deviations Minimum Daily Standard Deviations Minimum
Temperatures Quartic Model Temperatures Sextic Model
Figure 8 Models of Standard Deviation Minimum Temperature Models
Table I
Maximum Temperature
Model R-Square Root Mean Square
Quadratic 0.88407 4.820916
Cubic 0.957602 2.919457
Quartic 0.982139 1.897503
Quintic 0.998252 0.594392
Sextic 0.998342 0.579669
Table II
Minimum Temperature
Model R-Square Root Mean Square
Quadratic 0.868455 3.223
Cubic 0.926276 2.4162
Quartic 0.975827 1.3855
Quintic 0.993394 0.7253
Sextic 0.996717 0.512
Table III
Standard Deviations of Maximum Temperatures
Model R-Square Root Mean Square
Sextic 0.690793 0.549455
Septic 0.864458 0.364293
Octic 0.885433 0.33539
Table IV
Standard Deviations of Minimum Temperatures
Model R-Square Root Mean Square
Quadratic 0.869937 0.311158
Cubic 0.931936 0.225404
Quartic 0.932711 0.224427
Quintic 0.952684 0.188456
Sextic 0.977235 0.130901
Brenner I.S. (1986). "Biases in MOS (Model Output Statistics) Forecasts of Maximum and Minimum Temperatures in Phoenix, Arizona", Weather and Forecasting, Vol. 1(3), 226-229.
Brunet M., Sigro J., Jones P.D., Saladie O., Aguilar, Moberg A., Della-Marta P.M., Lister D., Walter A. (2007). "Annual and Seasonal changes in the distribution of daily maximum and minimum
temperature data in temperature extreme indices thoughout the 1901-2005 period over mainland Spain.", Geophysical Research Abstracts 9, 07167.
Driscoll D.M. (1988). "A Comparison of Temperature and Precipitation forecasts issued by Telecasters and National Weather Service", Weather and Forecasting, Vol.3(4), 285-295.
Gyakum J.R. (1986). "Experiments in Temperature and Precipitation Forecasting in Illinois", Weather and Forecasting, Vol. 1(1), 77-88.
Mass C.F. (1987). "The ‘Banana Belt’ of Coastal Regions in Southern Oregon and Northern California", Weather Forecasting, Vol. 2(3), 253-258.
Murphy A.H. (1998). "The Early History of Probability Forecasts: Extensions and Classifications", Weather and Forecasting, Vol. 13(1), 5-15.
Reading C. (2004). "Student Description of Variation While Working with Weather Data", Statistics Education Research Journal, Vol. 3(2), 84-105.
Serra C. , Burgueno A. , Lana X. (2001). "Analysis of Maximum and Minimum daily temperatures recorded at Fabra Observatory Barcelona NE Spain in the period 1917-1998", International Journal of
Climatology, Vol. 21, 617-636.
Shuman (1989). "History of numerical weather prediction at the National Meteorological Center", Weather and Forecasting, Vol. 4, 286-296.
Stone, Weaver (2002), "Daily maximum and minimum temperature trends in a climate model", Geophysical Research Letters, Vol. 29(9), 70-71.
Teresa Jacobson
California State University, Chico
Email: kalany@gmail.com
Josh James
California State University, Chico
Email: joshjames5@hotmail.com
Neil C. Schwertman
Professor Emeritus of Statistics
California State University, Chico
Email: NSchwertman@csuchico.edu
Volume 17 (2009) | Archive | Index | Data Archive | Resources | Editorial Board | Guidelines for Authors | Guidelines for Data Contributors | Home Page | Contact JSE | ASA Publications | {"url":"http://www.amstat.org/publications/jse/v17n2/jacobson.html","timestamp":"2014-04-16T04:13:09Z","content_type":null,"content_length":"75219","record_id":"<urn:uuid:fcc1fc8a-b8bc-4188-bbb9-0a045ee40e7c>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00533-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: second-order logic lives...
Vedasystem@aol.com Vedasystem at aol.com
Wed Mar 24 07:28:07 EST 1999
In a message dated 3/23/99 8:57:55 PM Eastern Standard Time,
holmes at catseye.idbsu.edu writes:
<<In any case, no language has semantics in the absence
of a given interpretation or interpretations>>
It seems to be a standard assumption of classical mathematical logic, but
I would like to cite the following statement of Robert Kowalski:
"The assumption ... that there exists a reality composed of individuals,
functions, and relations, separate from the syntax of language, is both
unnecessary and unhelpful". (R. Kowalski. Logic without model theory. In
"What is a logical system?", ed. by D.M. Gabbay, 1994, p.38).
Holmes writes:
First-order logic has proved to be fruitful because it has helped
to solve some important long - standing mathematical problems.
What are some significant mathematical problems solved with
the help of "second-order-logic"?
Holmes writes:
<< How do we see that the definition captures our pre-formal
notion of what a natural number is? >>
Holmes tacitly assumes that everybody has the same
"pre-formal notion of what a natural number is".
But it is not the case!. Sazonov, for example, may have
a different such notion than Holmes has.
Or Holmes may have a different notion of set that Zermelo had.
A great benefit of formalization is that it allows to write down
in a formal language a formal analog of a pre-formal notion .
There are may be several such analogs.
A lot of debates go on because different people mean
under the same name a different notion.
But the formal language used must be really formal
(it means that actually it must be a formal logical system).
The first-order ZFC is a formal language in that sense.
Is "second order logic" a formal language in that sense?
Victor Makarov
Brooklyn, New York
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/1999-March/002887.html","timestamp":"2014-04-20T16:35:30Z","content_type":null,"content_length":"4174","record_id":"<urn:uuid:f3cfbc10-f540-40d5-9bf2-47f56a7ceead>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00517-ip-10-147-4-33.ec2.internal.warc.gz"} |
MCSIIP - Mathematics, Computer, and Science Instructional Improvement Programs
Certificate of Graduate Studies in Secondary Mathematics
The Certificate of Graduate Studies (COGS) in Secondary Mathematics will provide an opportunity for mathematics teachers to pursue advanced study in both mathematics and mathematics education. Goals
will include: increasing teachers’ mathematics content knowledge, increasing teachers’ pedagogical knowledge, and increasing teachers’ familiarity with current and historical research in mathematics
Admission requirements:
The applicant for the COGS in secondary mathematics will be expected to have completed a minimum of 30 semester hours at the undergraduate level of mathematics (or have a secondary mathematics
teaching certificate). Enrollment in all COGS require eligibility for graduate study (ie.: attainment of a bachelor’s degree). Matriculation must take place by the completion of 6 semester hours.
Course requirements:
Students will complete a minimum of 15 semester hours of graduate credits in classes taught both by the College of Liberal Arts & Science’s Mathematics Department (9 semester hours) and the College
of Education’s Secondary Education Department (6 semester hours).
I. Mathematics Core 6 semester hours
Select two courses from:
MATH 01.500 Foundations of Mathematics
MATH 01.522 History of Mathematics
MATH 03.550 Discrete Mathematics
MATH 01.503 Number Theory
MATH 01.502 Linear Algebra & Matrix Theory
II. Mathematics Requirement 3 semester hours
MATH 01.561 School Mathematics from an Advanced Standpoint
III. Mathematics Education 6 semester hours
MATH 01.560 Processes and Principles in School Mathematics
SMED 33.600 Problems in Math Ed I
For an application packet or for registration material, visit the graduate school web page at: http://www.rowan.edu/graduateschool/prospective_students/grad_application/index.htm
Certificate Of Graduate Study In Secondary Mathematics Education
Eric Milou, Ed.D.
Program Advisor
Department of Mathematics, Robinson Hall
(856) 256-4500, x 3876 | {"url":"http://www.rowan.edu/open/mcsiip/smecogs.html","timestamp":"2014-04-17T20:16:23Z","content_type":null,"content_length":"9154","record_id":"<urn:uuid:bd626a6b-fedb-4a68-a0d6-1ca13d1ea4cf>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00341-ip-10-147-4-33.ec2.internal.warc.gz"} |
- Current Issues in Parsing Technology , 1991
"... Introduction Many of the formalisms used to define the syntax of natural (and programming) languages may be located in a continuum that ranges from propositional Horn logic to full first order
Horn logic, possibly with non-Herbrand interpretations. This structural parenthood has been previously rem ..."
Cited by 48 (3 self)
Add to MetaCart
Introduction Many of the formalisms used to define the syntax of natural (and programming) languages may be located in a continuum that ranges from propositional Horn logic to full first order Horn
logic, possibly with non-Herbrand interpretations. This structural parenthood has been previously remarked: it lead to the development of Prolog [Col-78, Coh-88] and is analyzed in some detail in
[PerW-80] for Context-Free languages and Horn Clauses. A notable outcome is the parsing technique known as Earley deduction [PerW-83]. These formalisms play (at least) three roles: descriptive: they
give a finite and organized description of the syntactic structure of the language, analytic: they can be used to analyze sentences so as to retrieve a syntactic structure (i.e. a representation)
from which the meaning can be extracted, generative: they can also be used as the specification of the concrete representation of sentences from a more
- In Sixth Conference of the European Chapter of the Association for Computational Linguistics, Proceedings of the Conference , 1993
"... We show how techniques known from generalized LR parsing can be applied to leftcorner parsing. The esulting parsing algorithm for context-free grammars has some advantages over generalized LR
parsing: the sizes and generation times of the parsers are smaller, the produced output is more compa ..."
Cited by 23 (7 self)
Add to MetaCart
We show how techniques known from generalized LR parsing can be applied to leftcorner parsing. The esulting parsing algorithm for context-free grammars has some advantages over generalized LR
parsing: the sizes and generation times of the parsers are smaller, the produced output is more compact, and the basic parsing technique can more easily be adapted to arbitrary context-free grammars.
- In ICALP’95 , 1995
"... The critical problem of finding efficient implementations for recursive queries with bound arguments offers many open challenges of practical and theoretical import. We propose a novel approach
that solves this problem for chain queries, i.e., for queries where bindings are propagated from arguments ..."
Cited by 5 (3 self)
Add to MetaCart
The critical problem of finding efficient implementations for recursive queries with bound arguments offers many open challenges of practical and theoretical import. We propose a novel approach that
solves this problem for chain queries, i.e., for queries where bindings are propagated from arguments in the head to arguments in the tail of the rules, in a chain-like fashion. The method, called
pushdown, is based on the fact that a chain query can have associated a context-free language and a pushdown automaton recognizing this language can be emulated by rewriting the query as a particular
factorized left-linear program. The proposed method generalizes and unifies previous techniques such as the `counting' and `right-, left-, mixed-linear' methods. It also succeeds in reducing many
non-linear programs to query-equivalent linear ones.
, 1999
"... The critical problem of finding efficient implementations for recursive queries with bound arguments offers many open challenges of practical and theoretical import. In particular, we need
methods that are effective for the general case, such as non-linear programs, as well as for specialized cases, ..."
Cited by 4 (0 self)
Add to MetaCart
The critical problem of finding efficient implementations for recursive queries with bound arguments offers many open challenges of practical and theoretical import. In particular, we need methods
that are effective for the general case, such as non-linear programs, as well as for specialized cases, such as left-recursive linear programs. In this paper, we propose a novel approach that solves
this problem for chain queries, i.e., for queries where bindings are propagated from arguments in the head to arguments in the tail of the rules, in a chain-like fashion. The method, called pushdown
method, is based on the fact that each chain query can be associated with a context-free language, and that a pushdown automaton recognizing this language can be emulated by rewriting the query as a
particular factorized left-linear program. The proposed method generalizes and unifies previous techniques such as the `counting' and `right-, left-, mixed-linear' methods. It succeeds in reducing
many non-linear programs to query-equivalent linear ones.
- Proceedings of the International Conference on Logic Programming , 1994
"... The Logic Push-Down Automaton (LPDA) is introduced as an abstract operational model for the evaluation of logic programs. The LPDA can be used to describe a significant number of evaluation
strategies, ranging from the top-down OLD strategy to bottom-up strategies, with or without prediction. Two ty ..."
Cited by 4 (0 self)
Add to MetaCart
The Logic Push-Down Automaton (LPDA) is introduced as an abstract operational model for the evaluation of logic programs. The LPDA can be used to describe a significant number of evaluation
strategies, ranging from the top-down OLD strategy to bottom-up strategies, with or without prediction. Two types of dynamic programming, i.e. tabular, interpretation are defined, one being more
efficient but restricted to a subclass of LPDAs. We propose to evaluate a logic program by first compiling it into a LPDA according to some chosen evaluation strategy, and then applying a tabular
interpreter to this LPDA. This approach offers great flexibility and generalizes Magic Set transformations. It explains in a more intuitive way some known Magic Set variants and their limits, and
also suggests new developments. Keywords: logic programs, tabulation, memoing, magic-set, dynamic programming, push-down automata. 1 Introduction The recent years have seen the popularity of (at
least) two approaches to i...
- in LOLA in Ramakrishnan R.: Applications of Logic Databases, Kluwer Academic , 1995
"... In this paper we present the set-oriented bottom-up parsing system AMOS which is a major application of the deductive database system LOLA. AMOS supports the morpho-syntactical analysis of old
Hebrew and has now been operationally used by linguists for a couple of years. The system allows the declar ..."
Cited by 3 (1 self)
Add to MetaCart
In this paper we present the set-oriented bottom-up parsing system AMOS which is a major application of the deductive database system LOLA. AMOS supports the morpho-syntactical analysis of old Hebrew
and has now been operationally used by linguists for a couple of years. The system allows the declarative specification of Definite Clause Grammar rules. Due to the set-oriented bottom-up evaluation
strategy of LOLA it is particularly well suited to the analysis of language ambiguities. 1 INTRODUCTION In this paper the set-oriented bottom-up natural language parsing system AMOS, a major
application of the deductive database system LOLA [3, 4], is presented. The AMOS system serves for the morpho-syntactical analysis of old Hebrew text and is intensively used by linguists. A grammar
for old Hebrew [12] has been formalized as a Definite Clause Grammar (DCG) and represented as a LOLA program. The Definite Clause Grammar formalism and the evaluation of the corresponding logic
programs by a P...
- In Coping with Linguistic Ambiguity in Typed Feature Formalisms, Proceedings of a Workshop held at ECAI 92 , 1992
"... A parse forest is a space-efficient representation of a number of parse trees. Parse forests are produced by various context-free parsing algorithms. This paper presents an algorithm to
manipulate a parse forest according to a context-free grammar extended with parameters over a finite domain. This ..."
Cited by 1 (1 self)
Add to MetaCart
A parse forest is a space-efficient representation of a number of parse trees. Parse forests are produced by various context-free parsing algorithms. This paper presents an algorithm to manipulate a
parse forest according to a context-free grammar extended with parameters over a finite domain. This algorithm combines the disambiguation of ambiguous sentences, which is necessarily user-directed,
with the verification of context dependencies, which is done automatically. The algorithm is very time-efficient. Furthermore, our approach allows the storage of a parse forest instead of separate
parse trees to store the result of parsing an ambiguous sentence. 1 Introduction Large subsets of natural languages can be described using context-free grammars extended with some kind of parameter
mechanism, e.g. affix grammars, attribute grammars, and definite clause grammars. This paper deals with affix grammars over a finite lattice (AGFLs). The parameters in AGFLs are called affixes. AGFLs
are a ... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1516194","timestamp":"2014-04-16T12:08:27Z","content_type":null,"content_length":"32496","record_id":"<urn:uuid:bdd94425-1f86-4bf8-84b2-3a8ac6a08973>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00340-ip-10-147-4-33.ec2.internal.warc.gz"} |
MAA talk: Start your own Netflix
Comments (0)
Please log in to add your comment.
1) "Doing Data Science" with Schutt, Gattis and Crawshaw
2) "Google News Personalization: Scalable Online Collaborative Filtering" by Das, Datar, Garg, and Rajaram
Part 1:
3 Recommendation Engines
Factor analysis
Latent topic analysis
(might want to label edges)
Can think of them as items
How do we anticipate preferences?
Linear regression?
It's a start. Fix an item and find user i's rating of that item.
Good news/ Bad news
Good: closed-form solution
Bad: no sharing of information between items
Bad: too many items and missing items
Bad: this causes huge coefficients
But: you can add a prior/ penalty
But: then you'd be introducing other problems
Reducing dimensions: SVD decomposition
Interpreting SVD
U is a matrix whose rows correspond to users
V is a matrix whose rows correspond to items
S diagonal, entries <-> importance
k is rank but we can cut it down to approximate X
Computationally expensive
Start out dumber. Find U, V so that:
Q: What are minimizing?
A: Squared error (in 1st iteration):
Alternating Least Squares
Optimize how?
User by user (or item by item)
Comes down to linear regression
(keep doing this until nothing much changes)
rows of U <-> users
rows of V <-> items
columns <-> latent variables
There are d columns
You choose d
Simple problems get hard
with big data
Example: word frequency
How do you solve this?
By inspection, if there are only a few words
Or, if you have pages and pages, write an algorithm
Make sure you store 1 copy of each word (channel)
Scales to ~100 million words
Obstacle is your computer's memory
Can we squeeze out more?
Compress your words
Many cores - use them all
Can maybe do 10 trillion words
Q: What about if we have
more than 10 trillion words?
A: Use more computers!
If we have 10 computers, we can process 100 trillion words.
We might actually split up the work into parts and then send the results to a "controller" to add it all up.
What next?
As you grow your grid,
you get increasingly risky.
Say a computer fails with probability .
Then is close to 0.
But it's not equal to 0.
What is the chance that one of 1,000 computers fails?
Computers sometimes fail.
It depends on the time frame but if = 0.3%,
then chances of something failing is more than 95%.
We need to deal with robustness issues.
Hopefully separately from analytical issues.
Enter: MapReduce
MapReduce takes the robustness problems off your hands.
Copies data, deals with piping issues
Provided you can put your problem in its framework
Namely, a (series of) map step, then a reduce step.
Takes data,
performs some function,
output is of form
(key, value)
Aggregates by key.
performs function,
output is of form
(key, newvalue)
Example: word frequency
"red" ("red", 1)
(aggregates the "red"'s)
count the ("red", 1)'s
i.e. add up the 1's
Can we bring these
two ideas together?
Can we MapReduce our various Recommendation Engines?
First fix a user, so a length d column in U
Train a linear model with all known scores for U
There are not too many known scores per user
Invert a d-by-d matrix, which is small
Similarly, fix a movie.
There may be quite a few scores.
Covisitation is parallelizable
1. Factor model is parallelizable
Thank you
Latent Variables?
Example: sexiness
Certain things are associated with sexiness
Sexiness is really a combination of things
Not totally prescriptive
But we can approximate it pretty well
Is sexiness important?
Part 3:
Part 2:
Broaden our goal
Looking for X:
Picture of SVD: before
Picture of SVD: after
Stupid MapReduce
Send relevant data for U to mapper - not too large!
Have the mapper perform task and
Mapper output = (U, answer)
Fan-in does nothing, reducer does nothing
Just need each algo to be smallish and parallelizable
Start your own Netflix
Cathy O'Neil
Johnson Research Labs
Factor analysis: You like sentimental films, this film is sentimental.
Covisitation: People who like what you like also like this.
Topic analysis: This is related to stuff you like.
What is a recommendation engine?
Amazon, Netflix, Pandora, Spotify, Google News, many others.
You need LOTS of data on people and their preferences to make these work well.
With all this data we build a huge matrix X that encapsulates all the users and their preferences and attributes.
Rows are users, columns are preferences or attributes.
How do we make these computationally feasible?
1's and 0's (clicks)
User identified with history vector
Similarity of users defined by ratio of intersection and union:
Fixing u, find S(u, u') for all other u'.
Too hard! Instead, hash.
Map users to unsigned 64-bit numbers
"Similar users" more likely to collide
Thus create clusters of users.
User u reads stories S1, S2, ...
Randomize, then order story ID's
First item in common with u'?
How this works
Likelihood of collision is
S(u, u')
Basic idea behind each engine
How this really works
Use minhash to order ID's
Use more than one story
Lots of clusters per user
Remove small clusters
Latent Topic Modeling
Think about specifying a list of topics
Is this article likely to relate to this topic?
Is this person likely to enjoy this topic?
Is this the right topic?
Probabilistic Model
The model relates:
p(topic | user),
p(article | topic), and of course
p(topic | users, articles)
Iteratively computed
E-M algorithm
Maximize product of conditional likelihood over all data points
Minimize "log likelihood"
Alternate between optimizing
topics, other prob's
Mapper takes user history, makes cluster ID's
output: (userID, clusterID)'s
Fan-in collects users in same cluster
Reducer throws away small clusters, populates
a "big table" to help decide recommendations.
Latent Topic modeling is MapReducable
Choose two integers R and K and for i and j, choose user ID's i mod R and j mod K.
"Sufficient statistics" are relevant probabilities, so choose R and K large enough.
MapReduce to iterate on E-M algorithm
Clicks might not be up-votes
Some people are over-represented
People may lie
Missing information is a problem
New people are a problem
Seasonal data is a problem | {"url":"http://prezi.com/oo4_opdov8r4/maa-talk-start-your-own-netflix/","timestamp":"2014-04-18T20:47:40Z","content_type":null,"content_length":"60059","record_id":"<urn:uuid:1e1cdc1c-788d-437f-a577-36d4af99e2de>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00595-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] find_common_type broken?
Citi, Luca lciti@essex.ac...
Sun Jul 12 12:24:55 CDT 2009
> That is what I thought at first, but then what is the difference between
> array_types and scalar_types? Function signature is:
> *find_common_type(array_types, scalar_types)*
As I understand it, the difference is that in the following case:
np.choose(range(5), [np.arange(1,6), np.zeros(5, dtype=np.uint8), 1j*np.arange(5), 22, 1.5])
one should call:
find_common_type([np.int64,np.uint8,np.complex128], [int,float])
I had a look at the code and it looks like
dtype1 < dtype2 if dtype1 can safely be broadcasted to dtype2
As this is not the case, in either direction, for int32 and float32,
then neither dtype(int32) < dtype(float32) nor dtype(int32) > dtype(float32)
and this causes the problem you highlighted.
I think in this case find_common_type should return float64.
The same problem arises with:
>>> np.find_common_type([np.int8,np.uint8], [])
>>> np.find_common_type([np.uint8,np.int8], [])
here too, I think find_common_type should return e third type
which is the "smallest" to which both can be safely
broadcasted: int16.
More information about the NumPy-Discussion mailing list | {"url":"http://mail.scipy.org/pipermail/numpy-discussion/2009-July/043942.html","timestamp":"2014-04-20T01:51:56Z","content_type":null,"content_length":"3600","record_id":"<urn:uuid:b30817e0-98d1-40f0-8796-2adc826b4318>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00360-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fractional Equation? Confused...need some assurance
November 5th 2007, 06:43 PM
Fractional Equation? Confused...need some assurance
Pipes A and B can fill a storage tank in 8 hours and 12 hours, respectively. With the tank empty, pipe A was turned on at noon, and then pipe B was turned on at 1:30pm. At what time was the tank
How do I solve this??
x= ???
I got x/8 = amount of the tank filled using A
x/12 = amount of tank filled using B
maybe 1.5x/8 + x/12 = 1 ?
November 5th 2007, 07:59 PM
Pipes A and B can fill a storage tank in 8 hours and 12 hours, respectively. With the tank empty, pipe A was turned on at noon, and then pipe B was turned on at 1:30pm. At what time was the tank
How do I solve this??
x= ???
I got x/8 = amount of the tank filled using A
x/12 = amount of tank filled using B
maybe 1.5x/8 + x/12 = 1 ?
Pipe A fill the tank at a rate of 1/8 of a tank per hour
Pipe B fills the tank at a rate of 1/12 of a tank per hour.
At 1:30 pipe A has been running for 1.5 hours and so the tank is (1.5)/8
Now suppose it takes a further x hours to fill the tank, then we have an
additional x/8 of a tank added from A and x/12 of a tank from B. Hence as
the tank is now full we have:
(1.5)/8 + x/8 + x/12 = 1
Now solve this for x to find how many hours after 1:30 the tank is full, and
so the time at which it is full. | {"url":"http://mathhelpforum.com/algebra/22083-fractional-equation-confused-need-some-assurance-print.html","timestamp":"2014-04-23T20:48:47Z","content_type":null,"content_length":"5178","record_id":"<urn:uuid:d7ee52dd-9934-4cd8-a70b-83d9dce7d0be>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00595-ip-10-147-4-33.ec2.internal.warc.gz"} |
Connection Formulas
$\mathop{R_{D}\/}olimits\!\left(x,y,z\right)$ is symmetric only in $x$ and $y$, but either (nonzero) $x$ or (nonzero) $y$ can be moved to the third position by using
19.21.7 $(x-y)\mathop{R_{D}\/}olimits\!\left(y,z,x\right)+(z-y)\mathop{R_{D}\/}% olimits\!\left(x,y,z\right)=3\!\mathop{R_{F}\/}olimits\!\left(x,y,z\right)% -3\sqrt{y/(xz)},$
or the corresponding equation with $x$ and $y$ interchanged.
19.21.8 $\displaystyle\mathop{R_{D}\/}olimits\!\left(y,z,x\right)+\mathop{R_{D}\/}% olimits\!\left(z,x,y\right)+\mathop{R_{D}\/}olimits\!\ $\displaystyle=3(xyz)^{-1/2},$
19.21.9 $\displaystyle x\mathop{R_{D}\/}olimits\!\left(y,z,x\right)+y\mathop{R_{D}\/}% olimits\!\left(z,x,y\right)+z\mathop{R_{D}\/}olimits\! $\displaystyle=3\!\mathop{R_{F}\/}olimits\!\left(x,y,z\
\left(x,y,z\right)$ right).$
19.21.10 $2\!\mathop{R_{G}\/}olimits\!\left(x,y,z\right)=z\mathop{R_{F}\/}olimits\!% \left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)\mathop{R_{D}\/}olimits\!\left(x,y,% z\right)+\sqrt{xy/z},$
$zeq 0$.
Because $\mathop{R_{G}\/}olimits$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\leq 0$ if the variables are real, thereby avoiding
cancellations when $\mathop{R_{G}\/}olimits$ is calculated from $\mathop{R_{F}\/}olimits$ and $\mathop{R_{D}\/}olimits$ (see §19.36(i)).
19.21.11 $6\!\mathop{R_{G}\/}olimits\!\left(x,y,z\right)=3(x+y+z)\mathop{R_{F}\/}% olimits\!\left(x,y,z\right)-\sum x^{2}\mathop{R_{D}\/}olimits\!\left(y,z,x% \right)=\sum x(y+z)\mathop{R_{D}\/}
where both summations extend over the three cyclic permutations of $x,y,z$.
Connection formulas for $\mathop{R_{-a}\/}olimits\!\left(\mathbf{b};\mathbf{z}\right)$ are given in Carlson (1977b, pp. 99, 101, and 123–124). | {"url":"http://dlmf.nist.gov/19.21","timestamp":"2014-04-19T14:33:11Z","content_type":null,"content_length":"105405","record_id":"<urn:uuid:c101ea00-243f-484f-a689-2e549744fee4>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> Modeling intecept by T0 measure
Bill Dudley posted on Monday, December 15, 2008 - 11:04 am
I am modeling change over 6 time points with a quadratic model conditioned on type of treatment where treatment is indicated by two dummy codes. Fit is only ok, CFI = .91 and RMSEA = .07. Mod
indicies indicate that fit would be inproved by freeing i by the observation at time 0.
When I do this CFI jumps to .97 and RMSEA drops to < .05. However I am not sure how to interpret the parameters in order to graph the three curves. The I BY Time 0 path is .73. and the intercept
increases by about 50% (1.7 to 2.4).
So my understand is that the intercept is now indicated both within the growth model and directly by the time 0 observation.
Does this different specification alter how I would graph the curves (using the parameters for i s q shown under intercepts and the estimates of the effect on the two dummy codes on I S and Q?
Bengt O. Muthen posted on Monday, December 15, 2008 - 11:20 am
Freeing that parameter makes you fall outside the growth modeling framework. It indicates that you don't have a quadratic growth model, but that the first time point needs some other treatment. Often
there is a big initial drop, so that you need a 2-piece growth model.
Bill Dudley posted on Thursday, December 18, 2008 - 9:17 am
Thank you for the clarification. I can live with the only ok model fit.
However, I have one question about interpretation of one of my loadings. In one dummy coded condition the effect of the dummy code on the intercept although not significant would take the intercept
for that condition to a negative value which is meaningless.
As I graph the data I can simply fix this to zero but I wonder how I can constrain the intercept for one group to be zero in the analyses so that the the slope and quadratic terms are estimated
Bengt O. Muthen posted on Thursday, December 18, 2008 - 2:00 pm
Are you sure that a negative intercept is meaningless? It is not the same as a negative mean.
Back to top | {"url":"http://www.statmodel.com/discussion/messages/14/3814.html?1229637619","timestamp":"2014-04-19T05:44:18Z","content_type":null,"content_length":"20629","record_id":"<urn:uuid:78093135-ce44-4816-949f-815ea66f780d>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00099-ip-10-147-4-33.ec2.internal.warc.gz"} |
How dare you NOT use fraction manipulatives in 4th grade?
As the fourth grade math year runs to it end, consider this:
Were fractions at least a quarter to a third of the math you covered during the year?
Did you know that fractions: equivalence, like denominators and mixed numbers are one of 3 focus standards in the new Common Core Standards adopted by many (almost all) of the 50 states in the US?
A clear understanding and broad base of experience with fractions are crucial (CRUCIAL) for a positive and useful algebra experience later on (and yes, we 4th grade teachers are partially responsible
for the experience down the road).
It is not just "another topic" to cover: it is one of the three pillars of fourth grade mathematics.
Fractions taught in isolation without manipulatives is useless: so don't try cramming it all in numerically. Fractions is a HUGELY important CONCEPT, not simply another SKILL to check off.
Why should you use manipulatives?
Manipulatives serve four purposes:
1. They engage the senses. Multi-sensory tools have been shown to dramatically increase understanding and retention. Many students, especially those who had difficulty in school, need multisensory
tools to learn effectively.
2. They help students discover concepts. This is especially true for math concepts. It can be difficult, for example, to explain how 2/4 = 1⁄2, but with when students physically moves shapes, they
can prove it for themselves.
3. They help keep students focused. Everyone appreciates a little variety. By varying activities, you help your student to stay focused and keep learning.
4. They encourage practice. Students will often practice more with manipulatives than they will with worksheets.
1 comment:
Alexis said...
These are exactly, EXACTLY! what I am looking for/need. I don't want paper fraction manipulatives, yes they work just fine, but I want REAL ONES that don't tear or bend and that I don't have to
tediously cut out.
Now, where to buy...? | {"url":"http://offthehypotenuse.blogspot.com/2011/05/how-dare-you-not-use-fraction.html","timestamp":"2014-04-21T05:21:04Z","content_type":null,"content_length":"90140","record_id":"<urn:uuid:f1c6bd9c-eab5-4abe-9596-b1d2ee2ea43f>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00393-ip-10-147-4-33.ec2.internal.warc.gz"} |
converting decimal to binary value using recursion in java
07-24-2009, 10:25 AM
converting decimal to binary value using recursion in java
i am trying to make a recursive method that take a decimal number and convert it into its binary number. i think there will be two base cases that remainder is one or zero .can anyone help me
further with simple solution
07-24-2009, 09:13 PM
Simpler way to do the same
import java.util.Scanner;
public class toBinary {
* @author : javamadd
public static void main(String[] args) {
int num;
System.out.println("enter the decimal number");
Scanner scanboy = new Scanner(System.in);
num = scanboy.nextInt();
System.out.print("The decimal number is " + num);
String binary = Integer.toBinaryString(num);
System.out.print(" And the corresponding Binary is ");
System.out.print( ""+ binary);
07-25-2009, 03:28 AM
@OP, what you mean by recursive method? Can you explain bit more clearly. What javamadd explain is straightforward, no any recursive process going on.
07-25-2009, 01:44 PM
Reccursion method to convert decimal to binary.
hey here is the code :
class Binary
int a1,b1;
void convert(int a1)
b1 = a1 % 2;
if(a1 > 0)
System.out.println("binary bits are = "+b1); //i think the result would you get reverse of the byte so u take care about it
class DECTOBIN
public static void main(String args[])
Binary obj = new Binary();
obj.a1 = Integer.parseInt(args[0]);
Ok well. this is the code. But i didnt try it. | {"url":"http://www.java-forums.org/new-java/20072-converting-decimal-binary-value-using-recursion-java-print.html","timestamp":"2014-04-21T15:17:36Z","content_type":null,"content_length":"6310","record_id":"<urn:uuid:02a02d02-b08c-4226-b830-5cba210ad2d7>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00209-ip-10-147-4-33.ec2.internal.warc.gz"} |
Measurement Theory
Authors: Nolan L. Aljaddou
This work is an analytic establishment of the proper underlying theoretical foundation of the science of physics, in terms of its most fundamental precept - measurement - in order to formulate the
proper solution to some of its ultimate problems as well as to explain some of its greatest mysteries. It is grouped into two primary sections, the first covering the purely mathematical foundation,
with the latter the application of the mathematical tools derived in order to construct the proper physical foundation. These two groups are further divided into subsections exemplifying a dual logic
premise/implication structure of establishing necessary first principles alone, followed by the ultimate extent of the logical consequences of those principles. In the mathematical section this takes
the form of first addressing the mathematical principle common to all physical measurement and then demonstrating the extent of its logical application in deriving the various branches of mathematics
necessary for physics. In the following physical section, the nature of fundamental measures such as space and time and their exact interrelationship is established, followed by an examination of
their consequential manifest properties in producing the existence of matter and its counterpart anti-matter, as well as the nature of their mutual interaction. The origin of fundamental particle
variety is then established, following its effects through to the cosmological phenomena of black holes and the big bang, and the organization of subatomic structure. The exposition concludes in
deducing the fundamental universal principle which governs all physical phenomena in their varied forms. Broadly speaking, the categories of foundation established are listed in order as follows:
Foundations of Mathematics, Fundamental Branches of Mathematics, Quantum Physics, General Relativity, Classical Physics, Elementary Particle Physics, Cosmology, and Unification Physics - covering the
spectrum of general mathematical physics classification. Fundamental problems solved include: the identity of the most fundamental axiom of mathematics, the proper axiomatic establishment of the
calculus, the proper derivation and establishment of the correct framework of quantum physics - and clarification of quantum misunderstanding - the explanation for matter's curvature of space-time,
the reason for classical inertia and the form of the electromagnetic equations, the explanation for the divisions of fundamental particles and the fundamental forces, the mathematical proof of the
existence of Yang-Mills theory of chromodynamics and its relation to producing the big bang and the internal effects of black holes, the identity of the unification principle of physics, and several
others. Its reductionist nature establishes that it is the ultimate, unique foundation of all physical principles and by its nature the only means of solving and understanding the essential problems
addressed therein.
Comments: 12 Pages.
Download: PDF
Submission history
[v1] 2012-11-09 12:59:18
[v2] 2013-05-02 16:08:20
[v3] 2013-05-05 15:38:13
Unique-IP document downloads: 60 times
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted
as unhelpful. | {"url":"http://vixra.org/abs/1210.0175","timestamp":"2014-04-19T06:51:21Z","content_type":null,"content_length":"9625","record_id":"<urn:uuid:f9f1a370-11db-4d56-8d0f-2d1d8595c209>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00650-ip-10-147-4-33.ec2.internal.warc.gz"} |
Kevin Knight
Research interests include: artificial intelligence, natural language processing, machine translation, machine learning, automata theory, decipherment.
I think our approach to syntax in machine translation is best described in D. Barthelme's short story They called for more structure....
I recently said this out loud: "I want a four-door convertible like the one Lincoln was assassinated in."
I have been fortunate to work with great students. PhD graduates include: Steve DeNeefe (thesis on adjoining for MT), Sujith Ravi (thesis on decipherment), Jonathan May (thesis on tree automata for
NLP), Yaser Al-Onaizan (thesis on named-entity translation), Kenji Yamada (thesis on syntax-based MT), Irene Langkilde-Geary (thesis on language generation), Philipp Koehn (thesis on noun phrase
translation), and Ishwar Chander (thesis on statistical article generation). I co-advised PhD theses from Victoria Fossum, Shou-de Lin and Liang Huang, and I supervised Bryant Huang's Masters thesis
on syntax-structuring for translation.
No arrests were made in 2012. Came close. I was contacted by the police in 2013, so.
Selected Publications
(Well, OK, I went ahead and selected them all)
"Mapping between Engilsh Strings and Reentrant Semantic Graphs" (F. Braune, D. Bauer, and K. Knight), Proc. LREC, 2014.
"Dependency-Based Decipherment for Resource-Limited Machine Translation" (Q. Dou and K. Knight), Proc. EMNLP, 2013. Get paper in PDF.
"Abstract Meaning Representation for Sembanking" (L. Banarescu, C. Bonial, S. Cai, M. Georgescu, K. Griffitt, U. Hermjakob, K. Knight, P. Koehn, M. Palmer, and N. Schneider), Proc. Linguistic
Annotation Workshop (LAW VII-ID), ACL, 2013. Get paper in PDF.
"Parsing Graphs with Hyperedge Replacement Grammars," (D. Chiang, J. Andreas, D. Bauer, K. M. Hermann, B. Jones, and K. Knight), Proc. ACL, 2013. Get paper in PDF.
"Smatch: an Evaluation Metric for Semantic Feature Structures," (S. Cai and K. Knight), Proc. ACL, 2013. Get paper in PDF.
"Curating and Contextualizing Twitter Stories to Assist with Social Newsgathering," (A. Zubiaga, H. Ji, and K. Knight), Proc. IUI, 2013. Get paper in PDF.
"Semantics-Based Machine Translation with Hyperedge Replacement Grammars," (B. Jones, J. Andreas, D. Bauer, K-M. Hermann, and K. Knight), Proc. COLING, 2012. Get paper in PDF.
"Towards Probabilistic Acceptors and Transducers for Feature Structures," (D. Quernheim and K. Knight), Proc. ACL SSST Workshop, 2012. Get paper in PDF.
"DAGGER: A Toolkit for Automata on Directed Acyclic Graphs," (D. Quernheim and K. Knight), Proc. FSMNLP, 2012. Get paper in PDF.
"Large Scale Decipherment for Out-of-Domain Machine Translation," (Q. Dou and K. Knight), Proc. EMNLP, 2012. Get paper in PDF.
"Decoding Running Key Ciphers," (S. Reddy and K. Knight), Proc. ACL, Short Paper, 2012. Get paper in PDF.
"The Secrets of the Copiale Cipher," (K. Knight, B. Megyesi, and C. Schaefer), Journal of Research into Freemasonry and Fraternalism, 2(2), 2012.
"Deciphering Foreign Language," (S. Ravi and K. Knight), Proc. ACL, 2011. Get paper in PDF.
"Bayesian Inference for Zodiac and Other Homophonic Ciphers," (S. Ravi and K. Knight), Proc. ACL, 2011. Get paper in PDF.
"Unsupervised Discovery of Rhyme Schemes," (S. Reddy and K. Knight), Proc. ACL, 2011. Get paper in PDF.
"What We Know About the Voynich Manuscript," (S. Reddy and K. Knight), Proc. ACL Workshop on Language Technology for Cultural Heritage, Social Sciences, and Humanities (LaTeCH), 2011. Get paper in
"The Copiale Cipher," (K. Knight, B. Megyesi, and C. Schaefer), part of invited talk at ACL Workshop on Building and Using Comparable Corpora (BUCC), 2011. Get paper in PDF.
"Automatic Analysis of Rhythmic Poetry with Applications to Generation and Translation", (E. Greene, T. Bodrumlu, and K. Knight), Proc. EMNLP, 2010. Get paper in PDF.
"Re-Structuring, Re-Labeling, and Re-Aligning for Syntax-Based Statistical Machine Translation", (W. Wang, J. May, K. Knight, and D. Marcu), Computational Linguistics, 36(2), 2010. Get paper in PDF.
"A Decoder for Probabilistic Synchronous Tree Insertion Grammars", (S. DeNeefe, K. Knight, and H. Vogler), Proc. ACL Workshop on Applications of Tree Automata in Natural Language Processing, 2010.
Get paper in PDF.
"Fast, Greedy Model Minimization for Unsupervised Tagging", (S. Ravi, A. Vaswani, K. Knight, and D. Chiang), Proc. COLING, 2010. Get paper in PDF.
"Does GIZA++ Make Search Errors?", (S. Ravi and K. Knight), Computational Linguistics, Squibs & Discussion, 36(3), 2010. Get paper in PDF.
"Unsupervised Syntactic Alignment with Inversion Transduction Grammars", (A. Pauls, D. Klein, D. Chiang, and K. Knight), Proc. NAACL, 2010. Get paper in PDF.
"Bayesian Inference for Finite-State Transducers", (D. Chiang, J. Graehl, K. Knight, A. Pauls, and S. Ravi), Proc. NAACL, 2010. Get paper in PDF.
"Minimized Models and Grammar-Informed Initialization for Supertagging with Highly Ambiguous Lexicons", (S. Ravi, J. Baldridge, and K. Knight), Proc. ACL, 2010. Get paper in PDF.
"Efficient Inference through Cascades of Weighted Tree Transducers", (J. May, K. Knight, and H. Vogler), Proc. ACL, 2010. Get paper in PDF.
"A Statistical Model for Lost Language Decipherment", (B. Snyder, R. Barzilay, and K. Knight), Proc. ACL, 2010. Get paper in PDF.
"Bayesian Inference with Tears", Get tutorial workbook in PDF.
"Binarization of Synchronous Context-Free Grammars", (L. Huang, H. Zhang, D. Gildea, and K. Knight), Computational Linguistics, 35(4), 2009. Get paper in PDF.
"Synchronous Tree Adjoining Machine Translation", (S. DeNeefe and K. Knight), Proc. EMNLP, 2009. Get paper in PDF.
"Minimized Models for Unsupervised Part-of-Speech Tagging", (S. Ravi and K. Knight), Proc. ACL, 2009. Get paper in PDF.
"Fast Consensus Decoding over Translation Forests", (J. DeNero, D. Chiang, and K. Knight), Proc. ACL, 2009. Get paper in PDF.
"Learning Phoneme Mappings for Transliteration without Parallel Data", (S. Ravi and K. Knight), Proc. NAACL, 2009. Get paper in PDF.
"11,001 New Features for Statistical Machine Translation", (D. Chiang, K. Knight, and W. Wang), Proc. NAACL, 2009. Best Paper Award. Get paper in PDF.
"Combining Constituent Parsers", (V. Fossum and K. Knight), Proc. NAACL (short paper), 2009. Get paper in PDF.
"Faster MT Decoding through Pervasive Laziness", (M. Pust and K. Knight), Proc. NAACL (short paper), 2009. Get paper in PDF.
"A New Objective Function for Word Alignment", (T. Bodrumlu, K. Knight, and S. Ravi), Proc. NAACL Workshop on Integer Linear Programming for NLP, 2009. Get paper in PDF.
"The Power of Extended Top-Down Tree Transducers", (A. Maletti, J. Graehl, M. Hopkins, and K. Knight), SIAM J. Comput. 39(2), pp. 410-430, 2009. Get paper in PDF.
"Probabilistic Methods for a Japanese Syllable Cipher", (S. Ravi and K. Knight), Proc. International Conference on the Computer Processing of Oriental Languages, 2009. Get paper in PDF.
"Applications of Weighted Automata in Natural Language Processing", (K. Knight and J. May), Handbook of Weighted Automata (M. Droste, W. Kuich, H. Vogler, eds.), 2009.
"Attacking Decipherment Problems Optimally with Low-Order N-gram Models", (S. Ravi and K. Knight), Cryptologia, 33(4), pp. 321-334, 2009. Get paper in PDF.
"Training Tree Transducers", (J. Graehl, K. Knight, and J. May), Computational Linguistics, 34(3), 2008. Get paper in PDF.
"Using Syntax to Improve Word Alignment Precision for Syntax-Based Machine Translation", (V. Fossum, K. Knight, and S. Abney), Proc. Workshop on Statistical MT, ACL, 2008. Get paper in PDF.
"Name Translation in Statistical Machine Translation: Learning When to Transliterate", (U. Hermjakob, K. Knight, and H. Daume III), Proc. ACL, 2008. Get paper in PDF.
"Attacking Decipherment Problems Optimally with Low-Order N-gram Models", (S. Ravi and K. Knight), Proc. EMNLP, 2008. Get paper in PDF.
"Automatic Prediction of Parser Accuracy", (S. Ravi, K. Knight, and R. Soricut), Proc. EMNLP, 2008. Get paper in PDF.
"Overcoming Vocabulary Sparsity in MT Using Lattices", (S. DeNeefe, U. Hermjakob, and K. Knight), Proc. AMTA, 2008. Get paper in PDF.
"Using Bilingual Chinese-English Word Alignments to Resolve PP-Attachment Ambiguity in English", (V. Fossum and K. Knight), Proc. AMTA (student session), 2008. Get paper in PDF.
"Capturing Practical Natural Language Transformations", (K. Knight), Machine Translation, 21(2), 2007. Get draft in PDF. The complete publication is available at www.springerlink.com.
"Syntactic Re-Alignment Models for Machine Translation", (J. May and K. Knight), Proc. EMNLP-CoNLL, 2007. Get paper in PDF.
"What Can Syntax-Based MT Learn from Phrase-Based MT?", (S. DeNeefe, K. Knight, W. Wang, D. Marcu), Proc. EMNLP-CoNLL, 2007. Get paper in PDF.
"Binarizing Syntax Trees to Improve Syntax-Based Machine Translation Accuracy", (W. Wang, K. Knight and D. Marcu), Proc. EMNLP-CoNLL, 2007. Get paper in PDF.
"Statistical Syntax-Directed Translation with Extended Domain of Locality", (L. Huang, K. Knight, A. Joshi), Proc. AMTA (poster), 2006. Get paper in PDF.
"Tiburon: A Weighted Tree Automata Toolkit", (J. May and K. Knight), Proc. International Conference on Implementation and Application of Automata (CIAA), Lecture Notes in Computer Science (Springer),
v. 4094/2006, 2006. Get paper in PDF.
"Building an English-Iraqi Arabic Machine Translation System for Spoken Utterances with Limited Resources", (J. Riesa, B. Mohit, K. Knight, D. Marcu), Proc. Interspeech, 2006. Get paper in PDF.
"Unsupervised Analysis for Decipherment Problems", (K. Knight, A. Nair, N. Rathod, and K. Yamada), Proc. ACL-COLING (poster), 2006. Get paper in PDF.
"Scalable Inference and Training of Context-Rich Syntactic Models", (M. Galley, J. Graehl, K. Knight, D. Marcu, S. DeNeefe, W. Wang, and I. Thayer), Proc. ACL-COLING, 2006. Get paper in PDF.
"Discovering the Linear Writing Order of a Two-Dimensional Ancient Hieroglyphic Script", (Shou-de Lin and Kevin Knight), Artificial Intelligence, volume 170(4-5), 2006. Get draft in PDF.
"A Better N-Best List: Practical Determinization of Weighted Finite Tree Automata", (J. May and K. Knight), Proc. NAACL-HLT, 2006. Get paper in PDF.
"Capitalizing Machine Translation", (W. Wang, K. Knight, D. Marcu), Proc. NAACL-HLT, 2006. Get paper in PDF.
"Synchronous Binarization for Machine Translation", (H. Zhang, L. Huang, D. Gildea, K. Knight), Proc. NAACL-HLT, 2006. Get paper in PDF.
"Relabeling Syntax Trees to Improve Syntax-Based Machine Translation Quality", (B. Huang and K. Knight), Proc. NAACL-HLT, 2006. Get paper in PDF.
"Transonics: A Practical Speech-to-Speech Translator for English-Farsi Medical Dialogs", (Robert Belvin, Emil Ettelaie, Sudeep Gandhe, Panayiotis Georgiou, Kevin Knight, Daniel Marcu, Scott Millward,
Shrikanth Narayanan, Howard Neely, David Traum), Proc. ACL poster/demo, 2005.
"Interactively Exploring a Machine Translation Model", (Steve DeNeefe, Kevin Knight, and Hayward H. Chan), Proc. ACL poster/demo, 2005. Get paper in PDF.
"Machine Translation in the Year 2004", (K. Knight, D. Marcu), Proc. ICASSP, 2005. Get paper in PDF.
"An Overview of Probabilistic Tree Transducers for Natural Language Processing", (K. Knight, J. Graehl), Proc. of the Sixth International Conference on Intelligent Text Processing and Computational
Linguistics (CICLing), Lecture Notes in Computer Science, copyright Springer Verlag, 2005. Get paper in PDF.
"Text Simplification for Information Seeking Applications", (B. Beigman Klebanov, K. Knight, D. Marcu), In: On the Move to Meaningful Internet Systems, eds. R. Meersman and Z. Tari, Lecture Notes in
Computer Science (3290), copyright Springer Verlag, 2004.
"What's in a Translation Rule?", (M. Galley, M. Hopkins, K. Knight, D. Marcu), Proc. NAACL-HLT, 2004. Get paper in PDF.
"Training Tree Transducers", (J. Graehl, K. Knight), Proc. NAACL-HLT, 2004. Get paper in PDF.
"The Transonics Spoken Dialogue Translator: An Aid for English-Persian Doctor-Patient Interviews", (S. Narayanan, S. Ananthakrishnan, R. Belvin, E. Ettaile, S. Gandhe, S. Ganjavi, P. G. Georgiou, C.
M. Hein, S. Kadambe, K. Knight, D. Marcu, H. E. Neely, N. Srinivasamurthy, D. Traum, and D. Wang), AAAI Fall Symposium, 2004.
"Syntax-based Language Models for Machine Translation", (E. Charniak, K. Knight, and K. Yamada), Proc. MT Summit IX, 2003. Get paper in PDF.
"Feature-Rich Statistical Translation of Noun Phrases", (P. Koehn and K. Knight), Proc. ACL, 2003. Get paper in PDF.
"Syntax-based Alignment of Multiple Translations: Extracting Paraphrases and Generating New Sentences", (B. Pang, K. Knight, and D. Marcu), Proc. NAACL-HLT, 2003. Get paper in PDF.
"Transonics: A Speech to Speech System for English-Persian Interactions", (S. Narayanan, S. Ananthakrishnan, R. Belvin, E. Ettaile, S. Ganjavi, P. Georgiou, C. Hein, S. Kadambe, K. Knight, D. Marcu,
H. Neely, N. Srinivasamurthy, D. Traum, and D. Wang), Proc. IEEE ASRU, 2003.
"Finding the WRITE Stuff: Automatic Identification of Discourse Structure in Student Essays", (J. Burstein, D. Marcu, and K. Knight), IEEE Intelligent Systems, Jan/Feb, 2003.
"Translation with Scarce Bilingual Resources", (Y. Al-Onaizan, U. Germann, U. Hermjakob, K. Knight, P. Koehn, D. Marcu, K. Yamada), Machine Translation, 2003.
"Fast Decoding and Optimal Decoding for Machine Translation", (U. Germann, M. Jahr, K. Knight, D. Marcu, and K. Yamada), Artificial Intelligence, 2003.
"Teaching Statistical Machine Translation", (K. Knight), Proc. MT Summit IX Workshop on Teaching Machine Translation, 2003. Get paper in PDF.
"Cognates Can Improve Statistical Translation Models", (G. Kondrak, D. Marcu, and K. Knight), Proc. NAACL-HLT, 2003. Get paper in PDF.
"Desperately Seeking Cebuano", (Douglas W. Oard, David Doermann, Bonnie Dorr, Daqing He, Philip Resnik, Amy Weinberg, William Byrne, Sanjeev Khudanpur, David Yarowsky, Anton Leuski, Philipp Koehn,
and Kevin Knight), Proc. NAACL-HLT, 2003.
"Using a Large Monolingual Corpus to Improve Translation Accuracy", (R. Soricut, K. Knight, and D. Marcu), Proceedings of the 6th AMTA Conference, 2002.
"Learning a Translation Lexicon from Monolingual Corpora", (P. Koehn and K. Knight), Proc. of ACL Workshop on Unsupervised Lexical Acquisition, 2002. Get paper in PDF.
"Summarization Beyond Sentence Extraction: A Probabilistic Approach to Sentence Compression", (K. Knight and D. Marcu), Artificial Intelligence, 139(1), 2002.
"Named Entity Translation: Extended Abstract", (Y. Al-Onaizan and K. Knight), Proc. HLT, 2002. Get paper in Postscript.
"Translating Named Entities Using Monolingual and Bilingual Resources", (Y. Al-Onaizan and K. Knight), Proc. of the Conference of the Association for Computational Linguistics (ACL), 2002. Get paper
in PDF.
"Machine Transliteration of Names in Arabic Text", (Y. Al-Onaizan and K. Knight), Proc. of ACL Workshop on Computational Approaches to Semitic Languages, 2002. Get paper in PDF.
"A Decoder for Syntax-Based Statistical MT", (K. Yamada and K. Knight), Proc. of the Conference of the Association for Computational Linguistics (ACL), 2002. Get paper in PDF.
"A Syntax-Based Statistical Translation Model", (K. Yamada and K. Knight), Proc. of the Conference of the Association for Computational Linguistics (ACL), 2001. Get paper in PDF.
"Fast Decoding and Optimal Decoding for Machine Translation", (U. Germann, M. Jahr, K. Knight, D. Marcu, and K. Yamada), Proc. of the Conference of the Association for Computational Linguistics
(ACL), 2001. Best Paper Award. Get paper in PDF.
"Knowledge Sources for Word-Level Translation Models", (P. Koehn and K. Knight), Empirical Methods in Natural Language Processing conference (EMNLP), 2001. Get paper in PDF.
"Translating with Scarce Resources", (Y. Al-Onaizan, U. Germann, U. Hermjakob, K. Knight, P. Koehn, D. Marcu, K. Yamada), National Conference on Artificial Intelligence (AAAI), 2000. Get paper in PDF
"Preserving Ambiguities in Generation via Automata Intersection", (K. Knight and I. Langkilde), National Conference on Artificial Intelligence (AAAI), 2000. Get paper in PDF.
"Estimating Word Translation Probabilities from Unrelated Monolingual Corpora Using the EM Algorithm", (P. Koehn and K. Knight), National Conference on Artificial Intelligence (AAAI), 2000. Get paper
in PDF.
"Statistics-Based Summarization --- Step One: Sentence Compression", (K. Knight and D. Marcu), National Conference on Artificial Intelligence (AAAI), 2000. Outstanding Paper Award. Get paper in PDF.
"Decoding Complexity in Word-Replacement Translation Models", Computational Linguistics, Squibs & Discussion, 25(4), 1999. Get paper in PDF.
"Mining Online Text", Communications of the ACM, 42(11), November 1999.
"A Statistical MT Tutorial Workbook," unpublished, August 1999. Get paper in PDF. Get pretty version from Richard Wicentowski in PDF. Get paper in Word.
"A Computational Approach to Deciphering Unknown Scripts", (K. Knight and K. Yamada), Proceedings of the ACL Workshop on Unsupervised Learning in Natural Language Processing, 1999. Get paper in PDF.
"Machine Transliteration", (K. Knight and J. Graehl), Computational Linguistics, 24(4), 1998. Get paper in PDF.
"Translation with Finite-State Devices," (K. Knight and Y. Al-Onaizan), Proceedings of the 4th AMTA Conference, 1998. Get paper in PostScript.
"Generation that Exploits Corpus-based Statistical Knowledge", (I. Langkilde and K. Knight), Proc. of the Conference of the Association for Computational Linguistics (COLING/ACL), 1998. Get paper in
"Translating Names and Technical Terms in Arabic Text", (B. Stalls and K. Knight), Proc of the COLING/ACL Workshop on Computational Approaches to Semitic Languages, 1998. Get paper in PDF.
"The Practical Value of N-Grams in Generation", (I. Langkilde and K. Knight), Proc. of the International Natural Language Generation Workshop, 1998. Get paper in PDF.
"Automating Knowledge Acquisition for Machine Translation", AI Magazine 18(4), 1997. Get paper in PDF.
"Machine Transliteration", (K. Knight and J. Graehl), Proc. of the Conference of the Association for Computational Linguistics (ACL), 1997. Get paper in PDF.
"It's Time for Your Evaluation", special section, What Makes a Compelling Empirical Evaluation in AI, IEEE Expert, 11(5), 1996.
"Learning Word Meanings by Instruction", Proc. of the National Conference on Artificial Intelligence (AAAI), 1996. Get paper in PDF.
"Two-Level, Many-Paths Generation", (K. Knight and V. Hatzivassiloglou), Proc. of the Conference of the Association for Computational Linguistics (ACL), 1995. Get paper in PDF.
"Filling Knowledge Gaps in a Broad-Coverage MT System", (K. Knight, I. Chander, M. Haines, V. Hatzivassiloglou, E. Hovy, M. Iida, S. K. Luk, R. Whitney, and K. Yamada), Proc. of the International
Joint Conference on Artificial Intelligence (IJCAI), 1995. Get paper in PostScript.
"Unification-Based Glossing", (V. Hatzivassiloglou and K. Knight), Proc. of the International Joint Conference on Artificial Intelligence (IJCAI), 1995. Get paper in PostScript.
"Integrating Knowledge Bases and Statistics in MT", (K. Knight, I. Chander, M. Haines, V. Hatzivassiloglou, E. Hovy, M. Iida, S. Luk, A. Okumura, R. Whitney, K. Yamada), Proc. of the Conference of
the Association for Machine Translation in the Americas (AMTA), 1994. Get paper in PostScript.
"Building a Large-Scale Knowledge Base for Machine Translation", (K. Knight and S. Luk), Proc. of the National Conference on Artificial Intelligence (AAAI), 1994. Get paper in PDF.
"Automated Postediting of Documents", (K. Knight and I. Chander), Proc. of the National Conference on Artificial Intelligence (AAAI), 1994. Get paper in PDF.
"Are Many Reactive Agents Better Than a Few Deliberative Ones?" Proc. of the International Joint Conference on Artificial Intelligence (IJCAI), 1993.
"Unification", The Encyclopedia of Artificial Intelligence, Second Edition, John Wiley and Sons, 1992.
"Integrating Knowledge Acquisition and Language Acquisition", Applied Intelligence, 1(1), 1992.
Artificial Intelligence, Second Edition (E. Rich and K. Knight), McGraw-Hill Book Company, 1991.
"Connectionist Ideas and Algorithms", Communications of the ACM, 33(11), November, 1990. Get paper in PDF.
"Knowledge and Natural Language Processing", (J. Barnett, K. Knight, I. Mani, and E. Rich), Communications of the ACM, 33(8), August, 1990.
"Unification: A Multidisciplinary Survey,", ACM Computing Surveys, 21(1), 1989. Get paper in PDF.
Been doing some painting: | {"url":"http://www.isi.edu/~knight/","timestamp":"2014-04-20T00:37:16Z","content_type":null,"content_length":"35656","record_id":"<urn:uuid:afbc907d-0a05-491a-8745-047d84615568>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00532-ip-10-147-4-33.ec2.internal.warc.gz"} |
Harrison, NY Math Tutor
Find a Harrison, NY Math Tutor
...My primary area of expertise is chemistry. I hold undergraduate degrees in chemistry, biology and education. My graduate degree is also in chemistry, with a concentration in physical
chemistry, and I will complete my doctorate in chemistry in the fall of 2014 from the City University of New York.
4 Subjects: including prealgebra, ACT Math, chemistry, elementary math
...I believe practice makes perfect. As to my own personal knowledge, I am well traveled and I have a Masters. I also read as much as I can and keep an open mind.
20 Subjects: including algebra 1, prealgebra, reading, English
...Therefore, I am fluent in Mandarin, Japanese and English. I am also proficient to teach math to children up to 7th grade because I have tutored my own nieces so far and they both have the best
math scores in their class (level 1).I can teach Mandarin at business level and Japanese at conversational level. I am easygoing and get along with children at all ages very well.
6 Subjects: including calculus, algebra 1, prealgebra, precalculus
Currently, I am a math teacher in a Northern Westchester HS. I have been teaching/tutoring for over 15 years. I have taught most levels of math including MS through College. Please feel free to
contact me, if needed. I look forward to working with you.
11 Subjects: including SAT math, algebra 1, algebra 2, Microsoft Excel
My name is Qian, I recently graduated from Rutgers University majored in Mechanical Engineering. I am very good with math and enjoy working with kids. I am tutoring my brother with SAT math, and
he is been doing well. I am very patient and comfortable teaching all levels of math.
8 Subjects: including calculus, Chinese, algebra 1, algebra 2
Related Harrison, NY Tutors
Harrison, NY Accounting Tutors
Harrison, NY ACT Tutors
Harrison, NY Algebra Tutors
Harrison, NY Algebra 2 Tutors
Harrison, NY Calculus Tutors
Harrison, NY Geometry Tutors
Harrison, NY Math Tutors
Harrison, NY Prealgebra Tutors
Harrison, NY Precalculus Tutors
Harrison, NY SAT Tutors
Harrison, NY SAT Math Tutors
Harrison, NY Science Tutors
Harrison, NY Statistics Tutors
Harrison, NY Trigonometry Tutors | {"url":"http://www.purplemath.com/harrison_ny_math_tutors.php","timestamp":"2014-04-18T08:29:46Z","content_type":null,"content_length":"23755","record_id":"<urn:uuid:ff5dd323-b209-43fe-90c1-43f8d9dc02cb>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00487-ip-10-147-4-33.ec2.internal.warc.gz"} |
Excel shortcuts
How To Use A Spreadsheet
Excel® for the Mac and PC-Windows
by John D. Winter
Most good spreadsheets have very similar capabilities, but the syntax of the commands differs slightly.
I will use the keyboard command and mouse syntax of Excel® by Microsoft for this example. I am
assuming you have a mouse. In what follows, what you enter on the keyboard will be in bold. Special
keys, like the key labeled “Enter” will be written as: <Enter>, and menu options will be bold-italic.
Let’s suppose you have a number of data points such as data on a series of cylinders. You want to
perform some statistical analysis, perhaps to find the sum, mean and standard deviation of the various
data sets. The first step is to set up the organization of the rows and/or columns. Perhaps you decide to
list the rows as the separate measurements, and the columns as your measurements on each as follows:
Row 1 contains the titles of the columns as text. Each box in which you enter something is called a
“cell”. Excel recognizes the data in a cell as you type it in as either text or a number by the first
character. So we begin by moving the cursor (either with the mouse or the keyboard arrow keys) to the
cell A1 (column A row 1). When the cursor is in a cell, that cell appears to have a dark border. Typing
the first “C” of “Cylinder” alerts Excel that the cell will contain text, and not a number. Excel is quite
good at figuring out your input. It can recognize numbers, text, even a variety of date formats. For now
we type Cylinder in cell A1. Notice as you type, the input is shown at the top, in the “formula bar”, as
well as in the cell itself. You can backspace, delete, etc. in order to get your input correct. When it’s
OK hit the <Return> or <Enter> key to place the word in the cell. If it’s incorrect after you have hit
<Enter>, you can still correct it by simply typing it again in any highlighted cell. If the entry is a long
one, you can highlight the cell, move the mouse pointer to the incorrect spot in the formula bar, and
correct it there, and <Enter> it again.
Next we move the highlight to cell B1 and type: Diameter. There is a short-cut at this point. Rather
than hit <Enter> we can simply move the highlight to C1 and the enter will be automatic. Try it. Then
type: Length in C1. Notice that your titles are a bit crammed together. Wouldn’t it be nice if columns
B and C were a bit wider? Let’s do that first. Begin by moving the mouse cursor to the top row (with
the column letters in it: A, B, C, etc.). Go to column A and set the cursor exactly on the line that
separates the A and B columns. Notice that the cursor has changed shape to a vertical line with double
arrows across it. Press the left mouse button, and while holding it down, drag the column width to a
size that will contain the title and leave a bit of space on the sides. Release the mouse button when
you’re satisfied. Do the same for columns B and C. Excel also has a nice feature that makes column
widths fit automatically, after you’ve entered all the data: Format/Column/AutoFit Selection. The
preceeding syntax means to choose the menu items at the top of the spreadsheet in sequence. Click the
left mouse button first on the word File at the top left, then choose the Column option, and finally
choose AutoFit Selection.
Now move the highlight to cell A3. Type the numbers 1-9 for the cylinders you measure down the
column A to complete the organization. If you make any mistakes, simply type the new data over the
old. Your spreadsheet should look like the figure above. Next enter the data so that the spreadsheet
looks like this:
Notice a couple of things at this point. Text aligns to the left margin of the cells (“left-justified”) while
numbers are right-justified. This makes things look a little messy and we will fix these cosmetic things
later. Also, typing 4.0 in B11 results in a “4”. Excel takes only the number of “significant” digits that it
thinks you intend. We can treat this later as well. First let’s get some statistics on our data. Move the
highlight to cell B13. Let’s determine the sum of column B. Operations like sum, mean, etc are called
“functions”. Functions are listed in the manual for Excel, but can also be found using the Help
command in the upper right part of the menu bar. We want SUM, so we type: =SUM(B3:B11) in cell
B13 (case doesn’t matter). You must enter the “=“ sign first, which signals Excel that you are about to
enter a formula, and not a name or number. The colon means that you are specifying a range of cells,
in this case the sum of rows 3 through 11 in column B. Hit <Enter> and you have it. Some fun, huh?
Move the highlight to B14 and type: =average(B3:B11) to get the mean. Do this in lower case. Watch
the entry in the formula bar at the top of the spreadsheet. When you hit the <Enter> key, the formula in
the bar at the top changes “average” to upper case. This means that Excel recognized your entry as an
Excel function. It left “Cylinder” as it was entered before, since that isn’t a function. This is a nice
feature, and I always enter my functions in lower case, letting Excel tell me if I entered them correctly
when it changes to upper case.
Let’s do a standard deviation too, but in a different manner. Type only: =stdevp( in cell B15 for the
standard deviation, but do not yet hit <Enter>. Now move the mouse cursor to cell B3, hold down the
left button, and “drag” the highlight to cell B11. Note that it now says “=stdevp(B3:B11” in the bar at
the top. Just move the cursor to the upper function bar and add the right parenthesis and hit <Enter>.
Excel will then create the standard deviation for the column of data in cell B11.
In order to know what your values are, you should type: Sum in cell A13, Mean in A14, and S. Dev.
in A15. The sheet will now look like:
Explore the spreadsheet for a moment here. Move the highlight around from cell to cell, and notice that
the cell contents are always shown in the formula bar at the top of the spreadsheet. It tells you what’s
in the highlighted cell. When you highlight cell B11, you see a “4” in this line. That’s a relief, since
there’s a 4 in the cell. Now move to B13. While there’s a 51.9 in the cell, the formula bar tells us that
there is a function “=SUM(B3:B11)” there. This is a good way to check and correct errors in the
spreadsheet. Now note the number of decimals in rows 14 and 15. Excel likes to be rigorous, I guess.
However, it’s entirely misleading. Your measurements do not imply this level of accuracy. We can fix
this up quite easily. In fact let’s format a whole block for just one decimal place. Formatting works just
like in a word processor. First highlight the cell, or range of cells, and then format it. So begin by
highlighting the range to format. Using the mouse, move the highlight to cell B3. But we want a whole
block, not just a cell, so press the left button and drag the highlight to cell D15. That’s right, D15. I
have some plans for column D and you can format empty cells in advance for future use. You could
also move the cursor to the “D” at the top of the column, to highlight the whole column if you prefer.
Now, we want to format the numbers to one decimal place, so choose Format up in the menu area.
Click it with the mouse, or hit Alt-F. The underlined F in Format is the key that, if hit while working
in the menu area, will execute the selection. The <Alt> key gets you into the menus via the keyboard.
Next select Cells. Once again, you can click on it or choose “e” on the keyboard (e because C is for
Column...note that the e is underlined) You don’t need to press Alt this time, since you are already in
menus when you hit Alt the first time. Excel brings up a new menu with a number of formatting
choices. Choose Number for the formatting options that pertains to numbers. You can either select
from the offerings, or type in your own. We’ll do the latter. Move the cursor down to where it says
“General” (the default choice) by using the mouse or Tab key and type: 0.0. Excel interprets this as
any number with a single decimal. It also adds this to the list for future selection. Hit <Enter> or click
“OK” and everything will be formatted to one decimal place. Shortcut: use the “.00 →.0” icon in the
menu bar to reduce the number of decimal places in a highlighted cell by one.
Next let’s calculate the volume of our cylinder. The formula is :
Volume = area x length = πr2 x length
Thus each cell in the volume column = ½ the cell in the same row of the diameter column, squared,
times π times the cell in the length column. Move to cell D1 and type: Volume. Then move to D3 and
type: = to initiate a formula, and type your formula, which is: =(b3/2)^2*pi()*c3. The math functions
are like in most programming languages:
+ add * multiply
- subtract ^ exponent
/ divide
Be careful with “priorities” when you create formulas. Exponents have priority over multiplication and
division, which, in turn, have priority over addition and subtraction. Use parentheses to change this, or
simply when in doubt. For example, 3 + 4/2^5 is the same as 3 + 4/25 which is 3.00002. ((3+4)/2)^5,
on the other hand, is (3.5)5, or 525.22. And (3+4/2)^5 is 55, or 3125. Note again, that typing in lower
case lets you know that Excel recognizes your input by changing it to upper case. “PI()” is an Excel
function for the value 3.14159... Thus in the formula above you told Excel to take the value in cell B3,
divide that by 2, and square the quotient, multiply the result by π, and then multiply the whole thing by
the value in cell C3. I repeat, the parentheses are important here, since exponents have a priority over
division. We can retype this formula for all of the cylinders, but here’s a shortcut that really makes
spreadsheets fast. We can copy cell D3 to cells D4 through D11 and the cell references in the formula
will change relative to the proper row!! To see how this works, be sure D3 is still highlighted. Now
select Edit in the menu area, and then Copy. (Excel’s menu tells you that there is a keyboard short-cut
for this function. You could simply hold down the <Ctrl> key while hitting the “C” key to copy a
highlighted cell.) An even easier way to copy is to click the mouse on the little double-sheet-of-pater
icon in the menu area (it’s just to the right of the little scissors icon, which means “cut”). When you
have copied, the cell D3 has a weird margin, indicating a process in transition. Now look down at the
left corner of the very bottom line. This is called the “Status bar”. It usually says “Ready” (a nice
status), but now it says “Select destination and press ENTER or choose Paste”. So, we do what it says:
highlight the cells D4 through D11 (remember how? Move to D4, press the left mouse button, and drag
to D11). Now either simply hit <Enter> (which “pastes” the copied cell into the destination cell (or
range of cells in the present case) and column D is now filled with the volumes for all 9 cylinders!
Move the cursor down column D and note the formulas in the formula bar. Although we copied the
formula for row 3 to the other rows, the references to B3 and C3 were automatically adjusted to B4
and C4 for row 4, B5 and C5 for row 5, etc. These are called “relative references”, meaning that the
cell references are determined relative to the cell to which you copy them. If, on the other hand, you
don’t want the reference to shift, you must use an “absolute reference”. Suppose that you want to use a
constant, such as the gravitational constant, or a fixed length, or π, that you have stored in a specific
cell, like A2. If you copy this reference in the method above, when the formula is copied down a row,
the next row will think p is in cell A3, and not A2. To avoid this, type in the absolute reference by
using the $ sign. $A$2 is the absolute reference for cell A2. It will not change as you copy. For
example, if you had copied the formula: =(B3/2)^2*PI()*$C$3 above instead of the formula that you
did, copying it down a row would result in: =(B4/2)^2*PI()*$C$3. The relative reference to B3
changed to B4, but cell C3, referenced as $C$3, stayed the same. You can use the $ independently in
the reference: $C3 makes only the column absolute, while C$3 makes only the row absolute.
Forgetting to “anchor” absolute references is one of the most common errors made with spreadsheets.
As a final step, we will compute the sum, average, and standard deviation for columns C and D. We
could do this by typing in the functions for the data as we did for the lengths, but this is a waste of
time. Try to do this all at once using the Copy-Paste shortcut. If you have trouble, here’s the method in
1) Highlight B13:B15.
2) Copy (either Edit-Copy on the menu, Control-C, or the copy icon).
3) Highlight C13:D15.
4) <Enter>.
This copied the sum, average, and standard deviation for the diameter data to both the length and
volume data at the same time. The spreadsheet will look like this:
In Pat’s Paleo class, he asks students to calculate individual deviations from the mean. This requires a
new column after each measurement column. This can be done at this point by inserting columns after
columns B and C. To insert a column after B, move the cursor to the cell labelled “C” at the very top
of the C column. This will highlight all of the column. Then select Insert/Columns. This procedure
inserts columns to the left of the column containing the highlight, so be sure you highlight column C to
insert a column between B and C. The new column becomes column C, and all columns to the left of it
change labels to D, E, F, etc, but the formulas shift too, so the calculations remain the same. We
won’t add any columns in this exercise.
Let’s save the file at this point. Use File-Save from the menu (or the little diskette icon in the menu
area) and make up a name for the file, like “Cyl”. Excel adds an extension “.xls”.
As a further exercise, let’s create two graphs of our data. The first is a histogram of the volume data.
We’d like to know the “frequency distribution” of calculated volumes that we have in various ranges.
Excel fails miserably at creating histograms, but it has a very powerful “macro” language that is really
a programming language, like Basic. It’s quite opaque to learn (the manual is nearly worthless for
this). With some experience you can write macros to do almost anything. I’ve written a macro to do
histograms. If you are on the Geology Department machine, it’s already there. If you have your own
computer and want a copy, feel free to copy it. Excel requires that the macro is loaded with the
spreadsheet in order to run. I find this a nuisance, but I suppose it was designed to keep down the
clutter for big businesses with lots of macros. However, one can save a macro with a spreadsheet so
they will always be used together by saving them as a “worksheet”. But this is a digression. Let’s load
the macro.
Select File/Open. Then look for a macro called “HIST.XLM”. The extension “xlm” means it’s a
macro. The macro covers the spreadsheet, and we don’t want that, so select Window then CYL.XLS to
get it back. Excel can hold several spreadsheets, macros, and graphs at once. You can switch from one
to another by using the Window menu.
First we must choose the ranges that we want to use for the histogram. Look at the data in column D.
The values range from 39 to 301. From this data we might choose the intervals 0-50, 50-100, 100-150,
150-200, 200-250, 250-300, and 300-350. Begin the histogram by highlighting the data to plot, in this
case the range D3:D11. Next execute the macro by clicking Macro:Run or use the “hot-key” assigned
to the macro: Ctrl-h (hold down the <Ctrl> key and hit “h”, for you Mac folks you must hold down
both the “option” and keys while hitting “h”) The hist macro should be on the list if you’ve loaded it
correctly above, so choose it by double-clicking the left mouse button on it, or select it and hit OK.
The macro copies the block and then prompts you for the low range (not 0). This means the first
increment above 0, and for us this is 50. So type: 50 and <Enter>. Next it prompts for the high range,
so enter 350. Finally it asks for the increment, so we enter 50. The spreadsheet will then look like this:
Note that the data in column G says there are 2 values below 50, 2 between 50-100, etc.
Next we graph the data. Excel is rather “intuitive” about graphs. It attempts to pre-empt your choices,
and it usually does a good job. When it doesn’t, you may have some trouble getting what you want at
first. Excel was designed as a business tool, so thinks bar charts are the norm. In the sciences we prefer
Cartesian (x-y) graphs, but this is a bar graph, so we have an easy start. Excel even calls them
“Charts”, true to its business origins. Making charts can become quite complex, and you can
experiment around with multiple data sets, formatting, labels, etc. for a long time. I’ll just supply you
with the quickest way to get a graph created, and you can explore the options at a later date. The
simplest method is to use the nifty “Chart Wizard”. Begin the graph by highlighting the data to plot, in
this case columns cells F3:G10. Next click the Chart Wizard icon in the menu area. It looks like a bar
graph with one of the bars crashing and burning (I think Microsoft wanted it to look like a magic
wand). Then just choose from the options offered. First choose the chart type, in this case Column.
Next choose the sub-type: I chose #1 (the default value). Select Next, and you are presented with
options to change the data choice information, most of which is applicable to multiple data sets. It is all
correct in this case, so proceed to the next menu. Choose No Legend, and ander Titles, pick labels,
such as “Frequency Distribution of Cylinder Volumes” as the chart title, “Number of Cylinders” for
the Y-axis caption, and “Volume Range” for the X-axis caption. Choose Finish, and you have your
chart. If the chart does not look as you might like, experiment with resizing it (drag the little dark
square “handles” on the margins), or double-click the chart itself and format parts. You do this by
double-clicking the object you want to reformat, and choose among the options. You may also use the
Format option in the menu area. For example, let’s reformat the y-axis. Double-click the chart, then
double-click any y-axis label. You should have a menu area pop up onto the screen. Choose the Scale
label. Then click on and overtype the Maximum to make it 5, the Major tick is better as 1 also. Click
OK, and it is reformatted. The bar labels are better if they are printed vertically. Double click them and
choose Alignment to adjust them. Then your histogram is done.
Let’s do one more graph, a comparison of diameters and lengths to see if there is a correlation of
length and width in the samples. That means graphing C3:C11 vs. B3:B11. Begin by highlighting
(dragging) rows 3 through 11 of columns B and C with the mouse. Next click the Wizard, and progress
through the menus as before, Except, this time choose the XY Scatter chart type, and subtype 1. Add
an appropriate title and axis captions. There is our chart with the Diameter as X and the Length as Y.
On your own adjust the scales for X as 4 to 8 and for Y as 3 to 7. The graph is done.
You might wonder how to graph two non-adjacent columns. Like perhaps Diameter vs. Volume. To do
this you highlight one, like Diameter. Then hold down the Control key as you highlight the other, and
proceed as usual. Note that Excel naturally chooses the first column in your highlight as the x-axis.
How to reverse the axes so that you can plot the second column as the x axis in Excel is not easy, and
invovles reformatting the data choices in the chart once you have created it. It is much better if you
plan ahead. In your future spreadsheets, if you plan to make x-y graphs, always attempt to have the
data for the x-axis in the left-most data column.
We can see that there is a good correlation between length and diameter, in other words the cylinders
have very similar shapes, regardless of their size. We can test the linearity of the fit, and even get the
equation of the line of best fit by performing a linear regression. The equation we seek is a simple
line, with the familiar equation
y = mx + b
or for us: Length = m(Diameter) + b
Excel has a Function that calculates a linear regression. The name of the function is LINEST. The
format is LINEST(y-range, x-range, C,S), where C is a switch to force the y-intercept (b in the
equation above) to zero. If C =“False”, the line is forced through the origin (b = 0), if C is “True” b is
calculated. If S is “False” only m and b are calculated. If S is “True”, more regression statistics are
calculated. An added complexity, is that the regression function returns more than one number. The
return data is called an array by Excel. We rarely want all of the statistics in the array. Suppose we
want only m, b, and the r2 value. r2 is called the “coefficient of determination”, and is an estimate of
the degree to which the data fits a linear relationship. These three values are in the first three rows in
the 2 columns of the array. To get all there we must highlight a range of three rows by 2 columns
where we want the results. Lets choose F12:G14.
Highlight this range. Next type the formula “=linest(c3:c11,b3:b11, true,true)”. But this is to be an
array, not a simple formula. To enter it as an array, hold down both the <Ctrl> and <Shift> keys as you
hit <Enter>. Your spreadsheet will look like this:
The first value returned (cell F12) is the slope, m, and equals 0.967. The intercept, b, is in cell G12 and
equals -0.997. If you want only the slope and intercept you could define two individual cells for that
data only and avoid the array of statistics, etc. You could type: =linest(c3:c11,b3:b11)” in cell E12 for
the slope and “=index(linest(c3:c11,b3:b11),2)” in cell E13 for the intercept. Excel 5.0 permits
another easier and more straightforward option: the slope now is slope(c3:c11,b3:b11) and the
intercept is intercept(c3:c11,b3:b11). The r2 value is rsq(c3:c11,b3:b11). However you do it, the
resulting equation is:
Length = 0.967 x Diameter - 0.997
Excel version 6 and later have a snappy new way to do regressions if you don’t want all the statistical
information. Click and highlight the chart. Then choose Chart/Add Trendline. Then click on the
Linear box (for a linear regression) and Options. Put a check mark in the Display equation on chart
and the Display R-squared value on chart options. Then click OK. You get the regression line drawn
on your chart and the formula for the regression printed as well. You can drag the formula to some
more convenient position if you like.
Notice that the regression values are the same as in our previous method.
Let’s take a diameter value and calculate a Length from it using this formula. I took row 7 and plugged
7.1 in for the diameter and got 5.8 for the Length, as compared to 6.2 as the measured length. I
repeated the regression forcing b to zero and got: Length = 0.802 x Diameter + 0 and calculated a
length of 5.7 from the 7.1 diameter. The r2 value is a measure of how good a linear fit we have, and
would be 1.0 for a perfect fit. The value of 0.96 isn’t too bad. With b = 0, r2 was 0.93, suggesting a
relatively good fit as well (could you imagine a cylinder with zero length and any diameter other than
I mentioned that the spreadsheet might look better if the names weren’t left-justified and the numbers
right-justified. You can fix up this cosmetic touch by highlighting the cells and selecting
Format/Cells/Alignment/Center (or clicking the center icon in the menu area). However, this will
lose the nice decimal point alignment and make column D look worse.
The most important part of a spreadsheet is the initial organization. A good spreadsheet is properly
labeled. Begin with a shreadsheet title in row 1. Excel will let long titles extend into neighboring rows
to the right if they are empty. Then leave a blank row. Label each data column column, and any
constants that you plan to use (π, the gas constant, the gravitational constant, etc.). The label and the
constant must be in two adjacent cells, since Excel can only recognize a number in a cell if it is alone.
Place the constants in the beginning, and then create your columns and rows of data below that. Use
whatever formulas you need to manipulate the data, and be sure to refer to your constants with
absolute references ($). If you plan to graph the results, be sure that your x-axis data is in the left-most
column of data. A little time spent at the beginning visualizing the overall organization can save you
lots of time and work later.
Excel is full of other goodies, which you can explore in the future. This was only an introduction, but it
covers most of what we’ll do in class. Save the file if you wish, on your own diskette (use the
File/Save command or the diskette icon). If you want to print the spreadsheet, choose File/Print from
the menu (or use the printer icon). Most spreadsheets and graphs look best when printed in landscape
mode (rows parallel to the long dimension of the paper) which can be done by
File/Print/Options/Page Setup. | {"url":"http://www.docstoc.com/docs/125723804/Excel-shortcuts","timestamp":"2014-04-21T09:11:03Z","content_type":null,"content_length":"78514","record_id":"<urn:uuid:b2e010ac-dfcd-4467-8bcc-dbb54add11b9>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00651-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ramanujan's most beautiful identity
G. H. Hardy called the following equation Ramanujan’s “most beautiful identity.” For |q| < 1,
If I understood it, I might say it’s beautiful, but for now I can only say it’s mysterious. Still, I explain what I can.
The function p on the left side is the partition function. For a positive integer argument n, p(n) is the number of ways one can write n as the sum of a non-decreasing sequence of positive integers.
The right side of the equation is an example of a q-series. Strictly speaking it’s a product, not a series, but it’s the kind of thing that goes under the general heading of q-series.
I hardly know anything about q-series, and they don’t seem very motivated. However, I keep running into them in unexpected places. They seem to be a common thread running through several things I’m
vaguely familiar with and would like to understand better.
As mysterious as Ramanujan’s identity is, it’s not entirely unprecedented. In the eighteenth century, Euler proved that the generating function for partition numbers is a q-product:
So in discovering his most beautiful identity (and others) Ramanujan followed in Euler’s footsteps.
Reference: An Invitation to q-series
Hei-Chi Chan, was a former professor of mine and he discovered a generalization of Ramanujan’s “most beautiful identity.” Here is a link to his paper: https://edocs.uis.edu/hchan1/www/Chan=
For a very readable proof of this identity, see:
Given his unorthodox background, I wonder if Ramanujan knew of Euler’s work. In any case, I am constantly amazed of these things that “real” mathematicians dream up!
According to <a href="http://en.wikipedia.org/wiki/Srinivasa_Ramanujan" Wikipedia (citing <a href="http://www.amazon.com/Ramanujan-Essays-Surveys-History-Mathematics/dp/0821826247" Berndt &
Rankin, Ramanujan independently rediscovered Euler’s identity.
Ramanujan wasn’t entirely self-taught. Hardy mentions some books as ones Ramanujan had definitely read, others he was unsure about.
It’s plausible that Ramanujan rediscovered Euler’s identity. He was interested in partitions, and it’s natural to want to find the generating function of a sequence you’re interested in.
Tagged with: Math, Ramanujan
Posted in Math | {"url":"http://www.johndcook.com/blog/2012/10/27/ramanujans-most-beautiful-identity/","timestamp":"2014-04-20T03:11:41Z","content_type":null,"content_length":"33513","record_id":"<urn:uuid:28d02103-ef93-4bbc-8f01-42b9ea9ba085>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00290-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rectangle is Inscribed (Did I do it right?)
March 28th 2010, 11:41 AM #1
Rectangle is Inscribed (Did I do it right?)
Hi I am working on this Optimization Question but like usual, I am never sure if I am doing it right.
The question is:
A rectangle is inscribed in a semi circle of radius 25cm. Find the largest area of such a rectangle.
Here is what i have done so far:
Can someone tell me what steps I have done wrong? And what should I do next?
Hi I am working on this Optimization Question but like usual, I am never sure if I am doing it right.
The question is:
A rectangle is inscribed in a semi circle of radius 25cm. Find the largest area of such a rectangle.
Here is what i have done so far:
Can someone tell me what steps I have done wrong? And what should I do next?
1. The labeling in the drawing should have 2x as length of the rectangle. The 2nd line should read: $x^2+y^2=25^2$
2. All calculations are OK. Calculate now y and A.
To find y,
I plugged my x into y = sqrt(625 - x^2) and got 17.68
when i plugged it into my area formula i got 625
Would that be the correct answer?
March 28th 2010, 12:41 PM #2
March 28th 2010, 01:26 PM #3
March 29th 2010, 12:03 PM #4 | {"url":"http://mathhelpforum.com/pre-calculus/136097-rectangle-inscribed-did-i-do-right.html","timestamp":"2014-04-19T07:14:48Z","content_type":null,"content_length":"41142","record_id":"<urn:uuid:98292803-1106-45cd-b5fb-68282bc9dba2>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00442-ip-10-147-4-33.ec2.internal.warc.gz"} |
Got Homework?
Connect with other students for help. It's a free community.
• across
MIT Grad Student
Online now
• laura*
Helped 1,000 students
Online now
• Hero
College Math Guru
Online now
Here's the question you clicked on:
why heat capacity integrals required for accurate enthalpy measurements??????
• one year ago
• one year ago
Best Response
You've already chosen the best response.
No clue what you're talking about.
Best Response
You've already chosen the best response.
How are you going to calculate heat from changes in temperature? Surely something like this:\[Q = \int d{\rm T} \thinspace m \thinspace C_p({\rm T})\]
Best Response
You've already chosen the best response.
i am talking about enthalpy measurement using the relation\[dH=Cp*dT\] and then integrating it over the range of temperature, well acually enthalpy is supposed to be a function of temp. and
pressure. at constant pressure if i want to calculate the enthalpy at constant pressure ............. well never mind..!! this was a question in physical chemistry by castellan. i was not able to
figure out what might be the answer but now i know the ans.. so u may forget this question.!! the ans is ridiculously simple.. the heat capacity at constant pressure of a substance is not
independent of temp. but is a function of temp. so we dont take it as a constant quantity and thus for ACCURATE measurements we use heat capacity integrals.(not from the formula u mentioned carl)
but from this formula\[Cp=a+bT+cT^2+dT^3+.......\] i hope its a bit clear now// apologies for writing such a long answer!!
Best Response
You've already chosen the best response.
did you get the help you needed?
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
• Problem Solving 19 Hero
• Engagement 19 Mad Hatter
• You have blocked this person.
• ✔ You're a fan Checking fan status...
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.
You haven't written a testimonial for Owlfred. | {"url":"http://openstudy.com/updates/50c33a09e4b0c673d53d7a1e","timestamp":"2014-04-19T15:25:08Z","content_type":null,"content_length":"35620","record_id":"<urn:uuid:51429490-a365-4043-819b-1685ef624810>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00105-ip-10-147-4-33.ec2.internal.warc.gz"} |
problems on sufficiency, completeness and ancillarity
October 9th 2007, 12:52 PM
dave chen
problems on sufficiency, completeness and ancillarity
Let, X_1, X_2...X_N. be i.i.d from the uniform distribution on (0,1), and let M=max{X_1, X_2….X_N}, show that X_1/M and M are independent
I know probably I need to show X_1/M is minimal sufficient and M is ancillary, and use Basu theorem but I don't know how to do it.
Let (X_1,Y_1),(X_2,Y_2)… (X_N,Y_N) be i.i.d. and absolutely continuous with common density
f(x,y)= 2/ theta^2 , x>0, y>0, x+y<theta
f(x,y)= 0 , otherwise
(This is the density for a uniform distribution on the region inside a triangle in R^2 )
a)find a minimal sufficient statistic for the family of joint distributions.
b)Find the density for your minimal sufficient statistic
c)Is the minimal sufficient statistic complete.
My answer is T=max{(X_1,Y_1),(X_2,Y_2)… (X_N,Y_N)} is minimal sufficient but I don't know if it is correct, also the rest .
Let , X_1,X_2….X_N be i.i.d from a discrete distribution Q on {1,2,3}. Let p_i=Q({i})=P(X_j=i), i=1,….3. and assume we know that p_1=1/3, but have no additional knowledge of Q. Define
N_i= # { j <=n: X_j=i }
a)show that T =(N_1, N_2 ) is sufficient
b)Is T minimal sufficient ? If so, explain why. If not , find a minimal sufficient statistic.
I don't know how to work part b)
http://www.mathhelpforum.com/math-he...c/progress.gif Thanks a bunch!
October 9th 2007, 01:57 PM
Please don't double post. See rule #1 here. | {"url":"http://mathhelpforum.com/advanced-statistics/20261-problems-sufficiency-completeness-ancillarity-print.html","timestamp":"2014-04-18T19:17:01Z","content_type":null,"content_length":"5204","record_id":"<urn:uuid:c60ca3c2-af37-46f0-ab48-50082dbc71af>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00198-ip-10-147-4-33.ec2.internal.warc.gz"} |
Automatic resize of Math formulas
Currently OOo does not seem to offer any off-the-shelf solution for automatically changing the size of all Math formulas in a document or a selection. This would be a pretty useful tool, in my
While waiting for a cleaner and brighter solution, a macro could be operated to this goal, according to the following logic:
1. create the collection of the embedded objects (in the document, or selection)
2. loop on the collection
3. if the current object is a formula,
3.1 apply the new size
3.2 and refresh
Ok, and I append below my current implementation, that, on the other hand, has two problems:
A. when the collection contains more than 20 (yes, exactly 20 is the threshold) formulas, the debugger is activated, with the error
"BASIC runtime error. - Object variable not set"
set by the line 3 above (the line
if oTR3.EmbeddedObject.supportsService("com.sun.star.formula.FormulaProperties") then
in the implementation below);
B. the logic for refreshing the resized formulas (line 3.2 above) does not work, so that the size change must be "manually notified " by double clicking one by one all the formulas.
Any suggestions to solve these issues A and B? Thank a lot everybody.
... and here is the code:
sub Prova
newSize = InputBox("New formula font size:", "BaseFontHeight", 10)
if isNull(newSize) or newSize = "" then exit sub
oDoc = ThisComponent
oSel = oDoc.CurrentController.Selection
if oSel.supportsService("com.sun.star.text.TextEmbeddedObject") then
if oSel.EmbeddedObject.supportsService("com.sun.star.formula.FormulaProperties") then
SetFormulaSize(oSel, newSize)
msgbox "ok..."
exit sub
if oSel.supportsService("com.sun.star.text.TextRanges") then
oTR = oSel.getByIndex(0)
oEnum = oTR.createEnumeration()
while oEnum.hasMoreElements()
oTR1 = oEnum.nextElement()
oEnum1 = oTR1.createEnumeration()
while oEnum1.hasMoreElements()
oTR2 = oEnum1.nextElement()
if oTR2.TextPortionType = "Frame" then
oEnum2 = oTR2.createContentEnumeration("com.sun.star.text.TextEmbeddedObject")
while oEnum2.hasMoreElements()
oTR3 = oEnum2.nextElement()
if oTR3.EmbeddedObject.supportsService("com.sun.star.formula.FormulaProperties") then
SetFormulaSize(oTR3, newSize) 'msgbox "Formula"
msgbox "ok..."
exit sub
end sub
sub SetFormulaSize(obj, newSize)
oFormula = obj.EmbeddedObject
oFormula.BaseFontHeight = newSize
temp = oFormula.Formula ' some tricks to force OO.o redraw formulas '*** THESE DO NOT WORK!***
oFormula.Formula = ""
oFormula.Formula = temp
end sub
Re: Automatic resize of Math formulas
A further note on the post (sorry for this missing point...):
I tested the macro on OOo 2.3 and 2.4, on both Linux (OpenSuse 10.3) and WinXp, and the mentioned behavior is exactly the same.
Re: Automatic resize of Math formulas
try the following. At least it solves your second problem ...
It may also be possible to iterate the code over all objects of a drawpage. Then, you should not run into the max 20 problem
Good luck,
Sub Main
oShape = ThisComponent.Drawpage.getByIndex(0)
if oShape.supportsService("com.sun.star.text.TextEmbeddedObject") then
oEO = oShape.EmbeddedObject
if oEO.supportsService("com.sun.star.formula.FormulaProperties") then
oXEO = oShape.ExtendedControlOverEmbeddedObject
iCurrentState = oXEO.currentState
if oEO.BaseFontHeight = 100 then
oEO.BaseFontHeight = 30
oEO.BaseFontHeight = 100
dispatcher = createUnoService("com.sun.star.frame.DispatchHelper")
dispatcher.executeDispatch(oEO.CurrentController.Frame, ".uno:Draw", "", 0, Array())
End Sub
Re: Automatic resize of Math formulas
Thank you so much.
I tried your solution, so that I discovered that the reason for which the formulas do not refresh after being resized by the macro seems to be related to their "activation": only after the first
double click on it, a formula "becomes alive" and it is correctly refreshed by the macro.
If, for example, I close a document and then reopen it, for having its formulas correctly visualized after their automatic resize by the macro, I must double click on each of them.
Hence, I suspect that what is missing in the macro is some code simulating the double click (the "activation"...) of each formula to be resized.
Any idea on this issue?
Thanks again.
Re: Automatic resize of Math formulas
Hi Luca,
does the code work or not ? It should do all resizing without the need for any user interaction ...
In my code,
is the equivalent of doubleclicking on the formula. Normally, this should be sufficient to also update the view. Here, you have to add the draw dispatch to actually update the view.
Good luck,
Re: Automatic resize of Math formulas
Dear ms777,
I suppose that you are right: for the first formula in the document, obtained by:
oShape = ThisComponent.Drawpage.getByIndex(0)
your code actually works.
But may you show me how to get all formulas in the current selection, or in the current document, so to apply your code to them?
Even in the simple case of a single selected formula, I tried to get it by:
oShape = ThisComponent.CurrentController.Selection
but I got an exception (com.sun.star.embed.WrongStateException) on the line:
Thank you again | {"url":"https://forum.openoffice.org/en/forum/viewtopic.php?f=45&t=5579","timestamp":"2014-04-19T17:46:28Z","content_type":null,"content_length":"34305","record_id":"<urn:uuid:7916d488-4fff-4cda-a1b8-c5065aba0ae1>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00656-ip-10-147-4-33.ec2.internal.warc.gz"} |
Super Pokeno Bonus
Contributed by Jason Berger
This game requires:
• a standard 52-card deck of cards, two 6-sided dice
• a set of Pokeno boards
• a supply of chips or counters
• pen and paper, to record the players' point scores.
Pokeno sets normally come with 12 boards, each printed with a 5x5 array of playing-cards, in which each row or column representing a poker hand. Chips are used to cover the cards depicted on the
boards as they are called.
The number of players can be up to 12, or more if you have more pokeeno boards (more than one set). The Pokeno boards are distributed, one to each player
Players take turns to be caller: the caller rolls the dice and then draws and calls cards. The procedure is as follows.
1. Roll 2 dice to establish a base for the number of cards to be drawn:
□ 2, 3, 4 : base is 20
□ 5, 6, 7 : base is 30
□ 8, 9, 10 : base is 40
□ 11 : the player loses 10 points on the scoresheet, and the dice pass to the next player clockwise to roll and draw.
□ 12 : the player wins 10 points on the scoresheet and rolls again
2. After a base is established, roll both dice again to determine what number to add to the base
□ 2 to 9 : face value
□ 10 : zero
□ 11 : 1
□ 12 : the player wins 10 points on the scoresheet and rolls again
3. Roll 1 die to set the bonus hand for that round
□ 1 = pair
□ 2 = two pair
□ 3 = 3 of a kind
□ 4 = straight
□ 5 = flush
□ 6 = full house
4. Roll 1 die to set the multiplier for the bonus hand score
The basic scores for each hand type are:
• pair : 5 points
• 2 pairs : 10 points
• 3 of a kind : 15 points
• straight : 20 points
• flush : 25 points
• full house : 30 points
• 4 of a kind : 35 points
• straight flush : 40 points
• 1st roll total is 4; 2nd roll total is 11; 3rd roll 5: 4th roll 3. Draw 21 cards; bonus hand is flush, which will score 75 instead of 25.
• 1st roll is 12, repeat 1st roll 7; 2nd roll is 9; 3rd roll 1, 4th roll 4. The caller scores 10 points and is going to draw 39 cards (30 + 9), the bonus hand is a pair, scoring 20 instead of 5.
• 1st roll 12, repeat 12, repeat 11. The player scores 10 points (10 + 10 - 10) and passes the dice to the left, for the next player to roll.
Having rolled, the caller draws cards, one at a time, from the top of the face down shuffled deck, and calls out the rank and suit of each card drawn. Every player who has the called card on his or
her pokeno board covers it with a chip.
When the indicated number of cards have been drawn all players calculate their scores, scoring for any hand formed by the covered cards in each row and column of their board. Note that on a pokeno
board there is at least one example of each hand type (pair though straight flush) but anything higher than a full house cannot be the bonus hand.
Incomplete rows and columns can be scored, according to the cards covered. Note that:
• Straight, flush, full house and straight flush are 5-card combinations, so can only be scored if the whole row is covered
• It is possible to score just a pair or 3 of a kind on a full house or 4 of a kind row if only those cards are covered
• Each row or column is only scored once, and only for the highest ranking poker hand in it. For example a complete full house must be scored as a full house, not a pair or 3 of a kind. If you
complete a straight flush you score 40 - you cannot count it as an ordinary straight instead, even if straight is the bonus hand and would score more.
When the scores have been calculated and recorded, the dice are passed to the next player clockwise.
Five rounds of rolling, calling and scoring are played (not counting rounds where the caller loses the turn to call by throwing an 11). If there are more than 5 players, you may agree to play
additional rounds if necessary, continuing until each player has had an opportunity to roll the dice. | {"url":"http://www.pagat.com/invented/dice-pokeno.html","timestamp":"2014-04-17T10:11:28Z","content_type":null,"content_length":"8800","record_id":"<urn:uuid:547d4e62-6943-40a2-ad7f-501bc5c2fd3f>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00532-ip-10-147-4-33.ec2.internal.warc.gz"} |
Enumeration of groups of order 720
Posted by: Alexandre Borovik | October 6, 2010
Enumeration of groups of order 720
I discovered this quotation in some research policy document:
One of the first areas of pure mathematics to be computerised was group theory, as predicted in1945 by Alan Turing who wrote of his design for the NPL ACE computer, that
There will be positively no internal alterations to be made even if we wish suddenly to switch from calculating the energy levels of the neon atom to the enumeration of groups of order 720.
Why there was a need for a computer to enumerate groups of order 720? Even in Turing’s times, published papers already contained all the ingredients for (admittedly, long and tedious) enumeration of
groups of order 720 by hand. Although one ingredient that was missing was perhaps the Hall-Higman Theorem (not even the theorem itself, but the ideology of using linear algebra on “internal modules”
in groups.
Posted in Uncategorized | {"url":"http://micromath.wordpress.com/2010/10/06/enumeration-of-groups-of-order-720/?like=1&source=post_flair&_wpnonce=ad7ead3064","timestamp":"2014-04-21T02:09:00Z","content_type":null,"content_length":"54384","record_id":"<urn:uuid:b024123a-3e0e-4a4c-bc7e-8846edb37a66>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00017-ip-10-147-4-33.ec2.internal.warc.gz"} |
Algebra II - Math Learning Guides
Rational Functions
Sometimes counting on all your fingers just won't cut it. Figuring out arithmetic calculations on your smartphone won't do the trick, either. When you've had it with figuring out simple stuff, like
slicing and dicing a pizza into enough pieces to feed your army of deadbeat friends, check out rational functions.
Some right-brain types would argue that this stuff is neither rational nor functional, but what do word nerds know?
You can do cool stuff with rational expressions. Say you're really into energy drinks. Sadly, you know they're oozing with all kinds of caloric nightmares—sugar, more sugar, and tons of sugar. How
do you figure out how much your BMI (Body Mass Index) will increase if you pack in the sugar? You set up a rational function of the ratio between your height measurement and your weight. But we're
getting ahead of ourselves. Let's get back to basics.
As a reminder, a rational number is a ratio of two integers:
"Poly . . . huh?" It sounds like something a parrot would say.
A polynomial is a numerical stew that contains variables, constants, and positive whole number exponents. A polynomial could be something like this: 6x^2 + 3x – 6 (call it polynomial "A"). Another
example of a polynomial is 5x (call it polynomial "B"). If we take the first polynomial and divide it by the second polynomial, we now have a numerator and a denominator. Now we have a rational
expression that looks something like this:
Generalizing the concept of rational expressions to its nuts and bolts, we can say that a rational expression can be written like this:
where A and B are both polynomials.
We can do great things with all this polynomial magnificence, except for one little thing. We can never, ever have a polynomial in the denominator that equals zero. No, the equation won't blow up,
melt, burst into flames, or otherwise self-destruct if the denominator gets into the zero zone. But your work won't work. Trust us. In our little equation, 5x can never be zero. Just don't do it. | {"url":"http://www.shmoop.com/rational-functions/","timestamp":"2014-04-19T22:12:44Z","content_type":null,"content_length":"27445","record_id":"<urn:uuid:fa802382-a775-4400-9ca5-34c8e829567d>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00623-ip-10-147-4-33.ec2.internal.warc.gz"} |
Luigi C. Berselli
Luigi C. Berselli received a degree in mathematics from the Scuola Normale Superiore and the University of Pisa in 1995 and the Ph.D. degree in mathematics from the University of Pisa in 2000. He
joined the Department of Applied Mathematics of the Faculty of Engineering of the University of Pisa in 2000, and he is an Associate Professor of mathematical analysis since 2006. He is a Member of
the executive committee for the Ph.D. School in mathematics of the University of Pisa and of the American Mathematical Society. His main scientific interests are the mathematical analysis of the
Navier-Stokes equations and the development of efficient large-eddy simulation models for the simulation of turbulent flows.
Biography Updated on 16 January 2011
Scholarly Contributions [Data Provided by | {"url":"http://www.hindawi.com/76954206/","timestamp":"2014-04-17T05:18:08Z","content_type":null,"content_length":"8970","record_id":"<urn:uuid:52f5450e-360a-4e43-866d-293c4f365e46>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00512-ip-10-147-4-33.ec2.internal.warc.gz"} |
Help with an economics question?
September 5th 2011, 08:52 PM #1
Aug 2011
Help with an economics question?
Hi there,
I missed my last Microeconomics class due to travel plans and have no idea how to set up this problem. Could you possibly help me get started on this or send me an explanation of a similar
Here it is:
Maria can read either 20 pages of economics or 50 pages of sociology in an hour. She spends 5 hours per day studying.
a. Graph her production possibilities frontier per day for reading these subjects.
b. What is Maria’s opportunity cost for reading 100 pages of sociology? Explain.
Kindest thanks,
Re: Help with an economics question?
(a) wants a graph of how she could spend the 5 hours studying. one axis would have the pages of econ, the other would have sociology. on one end of the graph, she would spend the whole 5 hours
reading econ(100 pages), and 0 hours reading sociology(0 pages). on the other end of the graph, she'd spend 0 hours reading econ (0 pages), and 5 reading sociology (250 pages). you'd draw a line
from one point to the other.
here is an image of the problem: http://cdis.missouri.edu/exec/data/c.../UFig02_03.gif
(b) is asking : how long does it take for Maria to read 100 pages of sociology? if she can read 50 pages in one hour, then that means it would take her 2 hours to read 100 pages.
now, in that same 2 hours , how many pages of economics could she have read? well, according to the problem, at 20 pages per hour, she could have read 40 pages of economics in that 2 hours.
so the opportunity cost is 40 pages of economics.
Re: Help with an economics question?
(a) wants a graph of how she could spend the 5 hours studying. one axis would have the pages of econ, the other would have sociology. on one end of the graph, she would spend the whole 5 hours
reading econ(100 pages), and 0 hours reading sociology(0 pages). on the other end of the graph, she'd spend 0 hours reading econ (0 pages), and 5 reading sociology (250 pages). you'd draw a line
from one point to the other.
here is an image of the problem: http://cdis.missouri.edu/exec/data/c.../UFig02_03.gif
(b) is asking : how long does it take for Maria to read 100 pages of sociology? if she can read 50 pages in one hour, then that means it would take her 2 hours to read 100 pages.
now, in that same 2 hours , how many pages of economics could she have read? well, according to the problem, at 20 pages per hour, she could have read 40 pages of economics in that 2 hours.
so the opportunity cost is 40 pages of economics.
Yay! Thank you. This makes so much more sense now.
September 5th 2011, 09:08 PM #2
Aug 2011
September 6th 2011, 09:15 AM #3
Aug 2011 | {"url":"http://mathhelpforum.com/business-math/187363-help-economics-question.html","timestamp":"2014-04-16T16:49:48Z","content_type":null,"content_length":"36304","record_id":"<urn:uuid:fd4e4db4-2b53-40ac-a5e2-eda411cded8c>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00462-ip-10-147-4-33.ec2.internal.warc.gz"} |
Explicit metrics
up vote 39 down vote favorite
Every surface admits metrics of constant curvature, but there is usually a disconnect between these metrics, the shapes of ordinary objects, and typical mathematical drawings of surfaces.
Can anyone give an explicit and intuitively meaningful formulas for negatively curved metrics that are related to an embedding of a surface in space?
There is an easy way to do this for an open subset of the plane. If the metric of the plane is scaled by a function that is $\exp$ of a harmonic function, the scaling factor is at least locally the
norm of the derivative of a complex analytic function, so the resulting metric is still flat; the converse is true as well. Therefore, the sign of curvature of a conformally modified metric $\exp(g)$
depends only on the sign of the laplacian of the $g$. If the value of $g$ at a point is less than the average value in a disk centered at that point, then the metric $\exp(g) ds_E$ is negatively
curved, where $ds_E = \sqrt(dx^2 + dy^2)$ is Euclidean arc length.
For example, in a region $R$, if we impose a limit that speed is not to exceed the distance to the complement of $R$, this defines a non-positively curved metric. (The metric is 1/(distance to
boundary)$ds_E$). In this metric, geodesics bend around corners: it doesn't pay to cut too close, it's better to stay closer to the middle. If the domain is simply-connected, you see one and only one
image of everything, no matter where you are.
There are a number of other ways to write down explicit formulas for negatively curved or non-positively curved metrics for a subset of the plane, but that's not the question: what about for closed
surfaces in space? Any closed surface $M^2$ has at least a total of $4 \pi$ positive curvature, where the surface intersects its convex hull. If $M$ is a double torus, how can this be modified to
make it negative? It would be interesting to see even one good example of a negatively curved metric defined in terms of Euclidean geometry rather than an indirect construction. (In particular: it
can be done by solving PDE's, but I wwant something more direct than that.)
mg.metric-geometry dg.differential-geometry gt.geometric-topology
This is a great question, and it is indeed somewhat surprising that nobody has figured anything like this out yet. I find it rather difficult to envision a negatively curved metric on the double
1 torus, given what it looks like when embedded in $R^3$. We know that it is possible to change the "visual" metric into a negatively curved metric via a conformal factor, but it is highly unclear
how to do this without using heavy analytic machinery. I imagine that Bill and other visually oriented mathematicians have already tried to do this, so I doubt it's an easy question to answer. –
Deane Yang Sep 23 '10 at 20:40
2 Although it doesn't answer your question, I've been looking at a related thing recently. Given a finite group $G$ acting on a surface $\Sigma$, try to find an embedding of $\Sigma$ in $S^3$ such
that the action of $G$ on $\Sigma$ extends to an action of $G$ on $S^3$. So there's all kinds of restrictions on when this is possible, but looking at examples can be quite pleasant. – Ryan Budney
Sep 23 '10 at 20:46
1 @Ryan: Yes, that's a fun problem. There are many possible subgroups of $O(3)$, and many possible constructions. An interesting extra geometric condition: when are they equivariant minimal
surfaces? Sometimes they're not equivariantly compressible, which I believe means they can be made into minimal surfaces. – Bill Thurston Sep 23 '10 at 21:38
1 A slightly cheezy modification of Deane Yang's idea would be to remove a large disc from a hyperbolic surface, so large that the complement is a thin regular neighbourhood of a graph in the
surface. Embed that regular neighbourhood in Euclidean space much like how one constructs zero Gauss curvature embeddings of cylinders and Moebius bands in Euclidean space. – Ryan Budney Sep 24
'10 at 0:13
Dear Bill, I like very much the picture and the idea of this non-positively curved metric (though, funny enough in the beginning I thought, that these circles are wholes in the paddle), and I have
1 a question. Does this type of construction works for domains in higher-dimensional spaces? For some domains in R^n (maybe convex)? Or. for example, suppose we want to prove that a complement to
some hyperplane arrangement in $C^n$ is $K(\pi,1)$, is there a chance that by some similar kind of method we could find a complete, non-positively curved metric on its complement? (this would do
the job). – Dmitri Sep 24 '10 at 22:15
show 13 more comments
3 Answers
active oldest votes
Here's an answer to an analogous question, not Bill's original question, but also a question about how to specify (in a simple way) a non-positively curved metric on a compact Riemann surface
$C$ of genus $g>1$, in this instance, one that has been specified as an algebraic curve somewhere (as opposed to being given it as a surface in $3$-space).
The construction is easy: If the curve has been specified as an algebraic curve, then, more-or-less by algorithmic means, one can write down a basis for the holomorphic differentials on $C$
(which is a complex vector space of dimension $g$). Now select two of these differentials, say, $\omega$ and $\eta$, that have no common zeroes on $C$. (Again, this can be tested
algebraically). Now consider the metric $g = \omega\circ\bar\omega + \eta\circ\bar\eta$. This $g$ will have non-positive curvature. In fact, the curvature will vanish at only a finite number
up vote of points and will otherwise be strictly negative. (Of course, you can add more terms. If you take a basis $\omega_1,\ldots,\omega_g$ of the holomorphic differentials on $C$, then the metric
11 down $g = \omega_1\circ\bar{\omega_1} +\cdots + \omega_g\circ\bar{\omega_g}$ will have strictly negative curvature except when $C$ is hyperelliptic, in which case, the curvature will vanish at the
vote Weierstrass points of $C$.)
For example, if you take a hyperelliptic curve, say $y^2 = (x-\lambda_1)(x-\lambda_2)\cdots(x-\lambda_{2g+2})$ (with the $\lambda_i$ being distinct and, say, nonzero), then a basis for the
holomorphic differentials will be given by $\omega_i = x^{i-1}dx/y$ for $i = 1,\ldots, g$. Moreover, $\omega_1$ and $\omega_g$ (for example) have no common zeros. Thus, the smooth metric $g =
(1 + |x|^{2(g-1)})|dx|^2/|y|^2$ has negative curvature on this curve except at a finite number of points.
add comment
This is not a complete answer either -- in particular, I can't write down any formulas yet, but I wanted to share some pictures I made to help build my intuition. Zachary Treisman's
construction may be related.
As Bill Thurston's illustration of the (1/d)-metric taught me, by drawing disks living on the surface which represent how far I can get in a certain constant time, I can recover a great deal
of tactile intuition for a metric on a surface, even if my eyes are showing me a different one.
So let me try that out in a really simple and special case: flattening a torus, that is visualizing a flat metric on an embedded torus.
The embedding I considered is parametrized by coordinates (u,v) which each live in [0,2π]:
where the radius of a meridian circle is 1 and c is the distance from the center of a meridian circle to the center of the "outer" hole of the torus. If c>1, then the torus will not
I first placed disks in a triangular lattice in a rectangle with aspect ratio c. Here c=2. These are the disks of constant speed:
I rescaled the rectangle so that it had dimensions [0,2π]x[0,2π] and plotted the disks on the surface of the torus. Here are the results (the disks seem to be peeling off the torus because I
shrunk the torus so that it wouldn't intersect the disks and then didn't do enough fiddling to make it perfect):
up vote c=2
6 down
vote c=5
I found it useful to imagine the disks moving on the embedded torus by isometries (which are just translations in the rectangle picture). You can see how the disks get sheared on the
embedding if we take v to v+c so that the meridians of the torus all rotate -- this is an effect of the differing principal curvatures of the embedding. The outer disks move much faster than
the inner ones if we take u to u+c; this rotates the embedding about a central vertical axis.
How hard would it be to make these pictures for negatively curved surfaces? I don't know a nice parametrization for an embedded double torus, let alone one that plays nicely with a
negatively curved metric. I also haven't been able to pick out the right features to focus on in these pictures in order to imagine what happens in other cases, so any guidance or comments
would be appreciated!
^u3[r_, [Theta], u0] := r Cos[[Theta]] + u0
^v3[r_, [Theta], v0] := r Sin[[Theta]] + v0
^cent[u0_, v0_] := Module[{rad = c + Cos[v0]}, {rad Cos[u0], rad Sin[u0], Sin[v0]}]
^c = 2; xx = 16; yy = 8; max = (.8 [Pi])/xx;
^lattice = Flatten[2 [Pi]/ xx Mod[Table[{m + 1/2. n, Sqrt[3]/2. n}, {m, 1, xx}, {n, 0, yy - 1}], xx], 1];
^toruspic = ParametricPlot3D[Module[{rad = 1.01 c + .96 Cos[v]} (* perturb so that disks won't intersect torus *), {rad Cos[u], rad Sin[u], .96 Sin[v]}], {u, 0, 2 [Pi]}, {v, 0, 2 [Pi]},
PlotStyle -> {LightBlue, Specularity[1, 20], Lighting -> Automatic}, Mesh -> None, Boxed -> False, Axes -> False]
^Show[toruspic, Show[Table[u0 = lattice[[i, 1]]; v0 = lattice[[i, 2]]; ParametricPlot3D[ Module[{rad = c + Cos[1/(Sqrt[3]/2. yy/xx)*v3[r, [Theta], v0]]}, {rad Cos[ u3[r, [Theta], u0]], rad
Sin[u3[r, [Theta], u0]], Sin[1/(Sqrt[3]/2. yy/xx)*v3[r, [Theta], v0]]}], {r, 0, max}, {[Theta], 0, 2 [Pi]}, PlotStyle -> {Specularity[White, 40], Blue, Opacity[.6]}, Mesh -> {2, 0},
MeshStyle -> Opacity[.4]], {i, Length[lattice]}]], PlotRange -> All, Boxed -> False, Axes -> False]
@jc: Those are nice pictures! good ingenuity in finding a way to show a flat metric. Ideas: (a) Putting patterns on surfaces is called "texture mapping", implemented in hardware and
1 well-suported in software on modern computers. It should be easy to draw an image of a photograph (say) wrapped around the torus. I don't know what tools do this conveniently. In Mma,
check MeshShading and Image Processing. There are other options with separate software. (b) For conformal toruses: you can use stereographi projection from $S^3$ (or work out formula from
derivatives). $\to $ next comment – Bill Thurston Sep 25 '10 at 10:40
@jc, cont'd: For hyperbolic surfaces: the universal covers of $(n m p)$ orbifold is a hypergeometric function (for the special case $22\infty$ it's $\cos$), and many finite-sheeted covers
of these orbifolds have simple desriptions as algebraic curves. These are also related to congruence subgroups and modular forms. There are well-developed ways to describe minimal
surfaces in $\mathbb R^3$ using complex analytic data of this sort. Willmore surfaces are nice. There are very large bodies of mathematics lurking nearby --- the problem is how to get
just a little that you want. – Bill Thurston Sep 25 '10 at 10:59
@Bill Thurston: Thanks for the comments and directions for further exploration. I've confused myself a few times trying to think through a more basic question: is there a way of
understanding what happens to small geodesic disks under such a texture mapping from just comparing the Gaussian curvature of the metric we'd like to impose to the principal curvatures of
the embedding? The Jacobian of the mapping seems to enter as well. – j.c. Sep 25 '10 at 16:41
add comment
I'm thinking that the trouble with the metric in the positively curved parts of the surface comes from the fact that when building these things in $R^3$ out of a polygon in a Euclidean or
hyperbolic plane, we need to do a bit of stretching, because we run out of dimensions for just rolling - in $R^4$ a torus can be flat, yes? So how about if we fix a particular curvature for a
curve on the surface, say the one corresponding to the shortest (in the embedded, Euclidean sense) generator of the fundamental group for a basepoint in the middle of the picture (again, in
up vote the embedded, Euclidean sense). Then we can scale tangents by the inverse of the curvature in the direction they specify, so that the directions that were the most stretched if we think of
3 down the surface as coming from a glued polygon are the easiest to move along. This could be done so that the osculating circles, once scaled, all end up the same size as our particular circle.
vote This would have the effect, for example, of "tightening the belt" around the outside of the positively curved region, so that Deane Yang's flipping of the cylinder would happen.
1 !octagon I've thought a little more about this, but still not enough to give an explicit answer as was originally asked for. But I thought I'd share this picture I drew of geometry being
put on a genus two surface. From this picture, my idea is to choose a cycle in homology that looks nice on the given surface in space corresponding to each of the heavy dashed lines, which
are lines in the Poincare model shown. Then use the embedding to find nearby cycles for the thin equidistant curves, and arrange – Zachary Treisman Oct 27 '10 at 21:38
(continued) the tangents so that the time it takes to traverse the cycles for the equidistant curves matches the time it takes to traverse the originally chosen cycle. I feel like this
would give a metric at least on much of the surface - at some point the equidistant curves start bumping into each other... – Zachary Treisman Oct 27 '10 at 21:42
add comment
Not the answer you're looking for? Browse other questions tagged mg.metric-geometry dg.differential-geometry gt.geometric-topology or ask your own question. | {"url":"https://mathoverflow.net/questions/39791/explicit-metrics","timestamp":"2014-04-16T13:58:00Z","content_type":null,"content_length":"79267","record_id":"<urn:uuid:1d1a2070-1c88-4e51-af03-30b28dcf81ea>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00112-ip-10-147-4-33.ec2.internal.warc.gz"} |
AW: st: AW: AW: Something wrong with the -count- option in the -egen-?
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
AW: st: AW: AW: Something wrong with the -count- option in the -egen-?
From "Martin Weiss" <martin.weiss1@gmx.de>
To <statalist@hsphsun2.harvard.edu>
Subject AW: st: AW: AW: Something wrong with the -count- option in the -egen-?
Date Thu, 13 Aug 2009 11:26:43 +0200
I would say that the "mistake" that you made is an obvious hazard, as the
-sum- -total- and -count- functions for -egen- are not easily separated in
users` heads, and the added complication of an -expression- being allowed as
the argument compounds the problems.
Admittedly, I had to read the definition in the -h egen, count()- to solve
your puzzle. Still, you can combine the -count()- function with the -by-
prefix, and it can be handy then to see how many non-missing entries there
are for a certain variable...
set obs 5
gen id=_n
expand 10
set seed 43987
gen x=rnormal()
//25 % artificially missing
replace x=. if /*
*/ runiform()<0.25
//How many non-missing per group?
bys id: egen mycount=/*
*/ count(x)
list, noobs sepby(id)
-----Ursprüngliche Nachricht-----
Von: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] Im Auftrag von gjhxmu@sina.com
Gesendet: Donnerstag, 13. August 2009 11:16
An: statalist
Betreff: Re: st: AW: AW: Something wrong with the -count- option in the
Thank you for your help, and now I know the inner rationale for -count- in
the -egen-.
The -total- option is helpful.
BTW, what is the most value for -count- option in the -egen- ?
I can get r(N) quickly after stand alone -count- .
Best regards,
----- Original Message -----
From: Martin Weiss <martin.weiss1@gmx.de>
To: <statalist@hsphsun2.harvard.edu>
Subject: st: AW: AW: Something wrong with the -count- option in the -egen-?
Date: 2009-8-13 17:02:02
BTW, the standalone -count- command leaves "r(N)" behind, so you can process
the number further. If you want it in your dataset, you can use -egen,
input x
egen yyy=/*
*/ total(x==2)
list, noobs
-----Urspr?gliche Nachricht-----
Von: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] Im Auftrag von Martin Weiss
Gesendet: Donnerstag, 13. August 2009 10:49
An: statalist@hsphsun2.harvard.edu
Betreff: st: AW: Something wrong with the -count- option in the -egen-?
Nothing is wrong, luckily... As -h egen- says about the -count- option: "
creates a constant ... containing the number of nonmissing observations of
So the surprising result of your last line is easily explained: The
-expression- fed to -count- evaluates to nonmissing -either 0 or 1- for
every one of those five rows of your dataset, and -count- picks up this
number. All it cares about is the fact that the expression is non-missing,
not its content. If you want to -count-, use the standalone -count-...
input x
egen y=count(x)
egen yy=count(x>2)
egen yyy=count(x==2)
//another xmpl,
//20 is not even in the list of values
egen yyyy=count(x==20)
//the following dummy is counted
//it has 5 non-missing entries...
gen byte dummy=x==20
//stand alone -count-
cou if x==2
-----Urspr?gliche Nachricht-----
Von: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] Im Auftrag von gjhxmu@sina.com
Gesendet: Donnerstag, 13. August 2009 10:37
An: statalist
Betreff: st: Something wrong with the -count- option in the -egen-?
I typed the following in the stata and found -count- option didn't work
Anything wrong?
input x
egen y=count(x)
egen yy=count(x>2)
egen yyy=count(x==2)
| x y yy yyy |
| 1 4 5 5 |
| 2 4 5 5 |
| 3 4 5 5 |
| 5 4 5 5 |
| . 4 5 5 |
Thank you for any help!
Best regards,
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2009-08/msg00597.html","timestamp":"2014-04-20T14:09:26Z","content_type":null,"content_length":"11203","record_id":"<urn:uuid:8245f3b9-89a4-42cf-bcce-27340f57c345>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00382-ip-10-147-4-33.ec2.internal.warc.gz"} |
1. Precisely
Below are a number of statements:
1. Precisely one of these statements is untrue.
2. Precisely two of these statements are untrue. 3. Precisely three of these statements are untrue. 4. Precisely four of these statements are untrue. 5. Precisely five of these statements are untrue.
6. Precisely six of these statements are untrue. 7. Precisely seven of these statements are untrue. 8. Precisely eight of these statements are untrue. 9. Precisely nine of these statements are
untrue. 10. Precisely ten of these statements are untrue. Which of these statements is true?
The ten statements all contradict each other. So there can be at most one statement true. Now suppose there is no statement true. That would mean that statement 10 indeed would be true, which results
in a contradiction. This means that exactly nine statements must be untrue, and thus only statement 9 is true..
Download Logical Interview Questions And Answers PDF
Previous Question Next Question
Below is an equation that is not correct yet. By adding a number of plus signs and minus A banana plantation is located next to a desert. The plantation owner has 3000 bananas that he wants to
signs between the ciphers on the left side (without changes the order of the ciphers), transport to the market by camel, across a 1000 kilometre stretch of desert. The owner has only one camel,
the equation can be made correct. 123456789 = 100 How many different ways are there to which carries a maximum of 1000 bananas at any moment in time, and eats one banana every kilometre it
make the equation correct? travels. What is the largest number of bananas that can be delivered at the market? | {"url":"http://www.globalguideline.com/interview_questions/Answer.php?a=Below_are_a_number_of_statements_examin_these&page=11","timestamp":"2014-04-18T16:15:51Z","content_type":null,"content_length":"25438","record_id":"<urn:uuid:2c24fcc9-eba4-42ae-9968-c9dac2213a4e>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00070-ip-10-147-4-33.ec2.internal.warc.gz"} |
A gas which obeys the gas laws and the gas equation PV = nRT strictly at all temperatures and pressures is said to be an ideal gas. The molecules of ideal gases are assumed to be volume less points with no attractive forces between one another. But no real gas strictly obeys the gas equation at all temperatures and pressures. Deviations from ideal behaviour are observed particularly at high pressures or low temperatures. The deviation from ideal behaviour is expressed by introducing a factor Z known as compressibility factor in the ideal gas equation. Z may be expressed as Z = PV / nRT
• In case of ideal gas, PV = nRT ∴ Z = 1
• In case of real gas, PV ≠ nRt ∴ Z ≠ 1
Thus in case of real gases Z can be < 1 or > 1
(i) When Z < 1, it is a negative deviation. It shows that the gas is more compressible than expected from ideal behaviour.
(ii) When Z > 1, it is a positive deviation. It shows that the gas is less compressible than expected from ideal behaviour.
Causes of deviation from ideal behaviour
The causes of deviations from ideal behaviour may be due to the following two assumptions of kinetic theory of gases. There are
• The volume occupied by gas molecules is negligibly small as compared to the volume occupied by the gas.
• The forces of attraction between gas molecules are negligible.
The first assumption is valid only at low pressures and high temperature, when the volume occupied by the gas molecules is negligible as compared to the total volume of the gas. But at low temperature or at high pressure, the molecules being in compressible the volumes of molecules are no more negligible as compared to the total volume of the gas.
The second assumption is not valid when the pressure is high and temperature is low. But at high pressure or low temperature when the total volume of gas is small, the forces of attraction become appreciable and cannot be ignored.
Van Der Waal’s Equation
The general gas equation PV = nRT is valid for ideal gases only Van der Waal is 1873 modified the gas equation by introducing two correction terms, are for volume and the other for pressure to make the equation applicable to real gases as well.
Volume correction
Let the correction term be v
∴ Ideal volume v[i] = (V – v)
Now v ∝ n or v = nb
[n = no. of moles of real gas; b = constant of proportionality called Van der Waal’s constant]
∴V[i] = V – nb
b = 4 × volume of a single molecule.
Pressure Correction
Let the correction term be P
∴ Ideal pressure P[i] = (P + p)
Now, = P ∝ (n/V)^2 = an^2 / V^2
Where a is constant of proportionality called another Van der Waal’s constant.
Hence ideal pressure
P[i] = (P + an^2 / V^2)
Here, n = Number of moles of real gas
V = Volume of the gas
a = A constant whose value depends upon the nature of the gas
Substituting the values of ideal volume and ideal pressure, the modified equation is obtained as
(P + an^2 / V^2) (V–nb) = nRT
Illustration 14. 1 mole of SO[2] occupies a volume of 350 ml at 300K and 50 atm pressure. Calculate the compressibility factor of the gas.
Solution: P = 50 atm V = 350 ml = 0.350 litre
n = 1 mole
T = 300L Z = PV / nRT
∴ Z = 50 × 0.350 / 1 × 0.082 × 300 = 0.711
Thus SO[2 ]is more compressible than expected from ideal behaviour.
Exercise 15.
Out of NH[3] and N[2] which will have
(a) Larger value of a
(b) Larger value of b
Exercise 16.
2 moles of NH[3] occupied a volume of 5 litres at 27°C. Calculate the pressure if the gas obeyed Vander Waals equation. Given a = 4.17 litre^2 atm mole^–2, b = 0.037 litre/mole.
Exercise 17.
Calculate the percentage of free volume available in 1 mole gaseous water at 1 atm and Density of liquid is 0.958g / mL.
Vander Waals equation, different forms
At low pressures: ‘V’ is large and ‘b’ is negligible in comparison with V. The Vander Waals equation reduces to: Deviation of gases from ideal behaviour with pressure.
(P + a / V^2) V = RT; PV + a/ V = RT
PV = RT – a/V or PV < RT
This accounts for the dip in PV vs P isotherm at low pressures.
At fairly high pressures The plot of Z vs P for N[2] gas at different temperature is shown here.
a/V^2 may be neglected in comparison with P. The Vander Waals equation becomes
P (V – b) = RT
PV – Pb = RT
PV = RT + Pb or PV > RT
This accounts for the rising parts of the PV vs P isotherm at high pressures.
At very low pressures: V becomes so large that both b and a/V^2 become negligible and the Vander Waals equation reduces to PV = RT. This shows why gases approach ideal behaviour at very low
Hydrogen and Helium: These are two lightest gases known. Their molecules have very small masses. The attractive forces between such molecules will be extensively small. So a/V^2 is negligible even at
ordinary temperatures. Thus PV > RT. Thus Vander Waals equation explains quantitatively the observed behaviour of real gases and so is an improvement over the ideal gas equation.
Vander Waals equation accounts for the behaviour of real gases. At low pressures, the gas equation can be written as,
(P + a/v^2[m]) (V[m]) = RT or Z = V[m] / RT = 1 – a/V[m]RT
Where Z is known as compressibility factor. Its value at low pressure is less than 1 and it decreases with increase of P. For a given value of V[m], Z has more value at higher temperature.
At high pressures, the gas equation can be written as
P (V[m] – b) = RT
Z = PV[m] / RT = 1 + Pb / RT
Here, the compressibility factor increases with increase of pressure at constant temperature and it decreases with increase of temperature at constant pressure. For the gases H[2] and He, the above
behaviour is observed even at low pressures, since for these gases, the value of ‘a’ is extremely small.
Illustration 15. One litre of a gas at 300 atm and 473 K is compressed to a pressure of 600 atm and 273 K. The compressibility factors found are 1.072 & 1.375 respectively at initial and final
states. Calculate the final volume.
Solution: P[1]V[1] = Z[1]nRT[1] and P[2]V[2] = Z[2]nRT[2]
P[2]V[2] / T[2 ]× T[1] / P[1]V[1] = Z[2] / Z[1]
or V[2] = Z[2] / Z[1] × T[1] / T[1] × P[1]V[1] / P[2] = 1.375 / 1.072 × 273 / 473 × 300 × 1 / 600 = 370.1 ml
Illustration 16. The behaviour of a real gas is usually depicted by plotting compressibility factor Z versus P at a constant temperature. At high temperature and high pressure, Z is usually more than
one. This fact can be explained by van der Waal’s equation when
(A) the constant ‘a’ is negligible and not ‘b’
(B) the constant ‘b’ is negligible and not ‘a’
(C) both constants ‘a’ and ‘b’ are negligible
(D) both the constants ‘a’ and ‘b’ are not negligible.
Solution: (P + n^2a / V^2) (V – nb) = nRT
At low pressures, ‘b’ can be ignored as the volume of the gas is very high. At high temperatures ‘a’ can be ignored as the pressure of the gas is high.
∴ P (V–b) = RT
PV – Pb = RT => PV = RT + Pb
PV / RT = Z = 1 + Pb / RT
Hence, (A) is correct.
Exercise 18. The compressibility factor for CO[2] at 273K and 100atm pressure is 0.2005. Calculate the volume occupied by 0.2 mole of CO[2] gas at 100 atm assuming ideal behaviour
Some other important definitions
Relative Humidity (RH)
At a given temperature it is given by equation
RH = partial pressure of water in air / vapour pressure of water
Boyle’s Temperature (T[b])
Temperature at which real gas obeys the gas laws over a wide range of pressure is called Boyle’s Temperature. Gases which are easily liquefied have a high Boyle’s temperature [T[b](O[2])] = 46 K]
whereas the gases which are difficult to liquefy have a low Boyle’s temperature [T[b](He) = 26K].
Boyle’s temperature T[b] = a / Rb = 1/2 T[1]
where T[i] is called Inversion Temperature and a, b are called van der Waals constant.
Critical Constants
• Critical Temperature (T[c]): It (T[c]) is the maximum temperature at which a gas can be liquefied i.e. the temperature above which a gas can’t exist as liquid.
• Critical Pressure (P[c]): It is the minimum pressure required to cause liquefaction at T[c]
P[c] = a/27b^2
• Critical Volume: It is the volume occupied by one mol of a gas at T[c] and P[c]
V[c] = 3b
Molar heat capacity of ideal gases: Specific heat c, of a substance is defined as the amount of heat required to raise the temperature of is defined as the amount of heat required to raise the
temperature of 1 g of substance through 1^0C, the unit of specific heat is calorie g^-1 K^-1. (1 cal is defined as the amount of heat required to raise the temperature of 1 g of water through 1^0C)
Molar heat capacity C, is defined as the amount of heat required to raise the temperature of 1 mole of a gas trough 1^0C. Thus,
Molar heat capacity = Sp. Heat molecular wt. Of the gas
For gases there are two values of molar heats, i.e., molar heat at constant pressure and molar heat at constant molar heat at constant volume respectively denoted by C[p] and C[v]. C[p] is greater
than C[v] and C[p]-R = 2 cal mol^-1 K^-1.
From the ratio of C[p] and C[v], we get the idea of atomicity of gas.
For monatomic gas C[p] = 5 cal and C[v] =3 cal
∴ λ = 4/3 = 1.67 (γ is poisson's ratio = C[p] / C[v])
for diatomic gas C[p] = 7 cal and C[v] = 5 cal
For polyatomic gas C[p] = 8 cal and C[v]= cal
γ = 8/6 = 1.33
also C[p] = C[p]m,
where, C[p] and C[v] are specific heat and m, is molecular weight.
Illustration 17. Calculate Vander Waals constants for ethylene
T[C] = 282.8 k; P[C] = 50 atm
Solution: b = 1/8 RT[c] / P[c] = 1/8 × 0.082 × 282.8 / 50 = 0.057 litres/mole
a = 27 / 64 R^2 × T^2[C] / P[c] = 27/64 × (0.082)^2 × (282.8)^2 × 1/50 = 4.47 lit^2 atm mole^–2 | {"url":"http://www.askiitians.com/iit-jee-states-of-matter/deviation-from-ideal-gas-behavior/","timestamp":"2014-04-19T04:29:49Z","content_type":null,"content_length":"157717","record_id":"<urn:uuid:98e27fd2-0e70-425d-a3b9-4651788d10f9>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00429-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: HOMEWORK 2, MAT 568, FALL 2010
Due: Thursday, Oct 28.
1. Let (M, g) be a Riemannian manifold and f : M M a diffeomor-
phism. Let fg be the Levi-Civita connection for the metric fg and g
the Levi-Civita connection for the metric g. Prove that
= f g
X Y = g
From this, deduce that the (3, 1) Riemann curvature tensor transforms
naturally under pullback:
= f
2. Let g = 2g, where is a positive constant, so that g is a rescaling of
g. Show that | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/025/1321289.html","timestamp":"2014-04-17T13:07:27Z","content_type":null,"content_length":"7541","record_id":"<urn:uuid:6da8eb91-5871-4b05-8e58-858554739050>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00642-ip-10-147-4-33.ec2.internal.warc.gz"} |
the first resource for mathematics
Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. 2nd ed.
(English) Zbl 1094.65125
Springer Series in Computational Mathematics 31. Berlin: Springer (ISBN 3-540-30663-3/hbk). xvii, 644 p. EUR 84.95 (2006).
[For the 1st edition (2002) see Zbl 0994.65135.]
The second revised edition of the monograph is a fine work organized in fifteen chapters, updated and extended. Thus, Chapters VII and XIII become respectively: “Non-canonical Hamiltonian systems”
and “Oscillatory Differential Equations with Constant High Frequencies” and Ch. XIV is a new one, entitled: “Oscillatory differential equations with varying high frequencies”. The bibliography is
also enriched with some old and new titles including those of the authors.
All in all the second edition is larger than the previous one with more than 130 pages. In the Preface to this new edition the authors provide a detailed list of major additions and changes. It
contains 16 issues. As a general remark, fairly sophisticated aspects of numerical algorithms coexist with more applicative aspects such as long-time energy conservation, or round-off error analysis.
Although the book has a genuine bias toward numerical methods to solve conservative (Hamiltonian) systems, readers far removed from this “thin” subset of the set of smooth dynamical systems given by
differential equations, would fairly profit.
In fact the authors are concerned with two large and quite different classes of numerical schemes. The first class contains methods which preserve the structure of the flow, i.e., symmetric and
symplectic methods. The second one is the class of methods which conserve some first integrals of the systems. One-step methods as well as multi step methods are considered.
The material of the book is organized in sections which are rather self-contained, so that one can dip into the book to learn a particular topic without having to read the rest of the book or even
the rest of the chapter. A person interested in geometric numerical integration will find this book extremely useful. However, the theory of the numerical methods that preserve some particular
properties of the flow of a dynamical system has come to maturity. The authors provide an exhaustive narrative of this story.
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65Lxx Numerical methods for ODE
34Cxx Qualitative theory of solutions of ODE
37Cxx Smooth dynamical systems: general theory
37M15 Symplectic integrators (dynamical systems)
37Jxx Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
70Fxx Dynamics of a system of particles, including celestial mechanics
65-02 Research monographs (numerical analysis) | {"url":"http://zbmath.org/?q=an:1094.65125&format=complete","timestamp":"2014-04-19T19:48:07Z","content_type":null,"content_length":"24671","record_id":"<urn:uuid:7523a872-6518-472b-bc9b-9b5b2b2c2b55>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00268-ip-10-147-4-33.ec2.internal.warc.gz"} |
Braingle: 'Dots on a Line' Brain Teaser
Dots on a Line
Math brain teasers require computations to solve.
Puzzle ID: #1172
Category: Math
Submitted By: Ichabod Clay
How many points can you place on a single line?
Show Answer
What Next?
contagion Even if it were a finite length line the answer would still be infinite. The more critical factor is that you can make infinite subdivisions of a line with a point at each
Oct 22, 2001 subdivision.
dark_shadow25200 ok if you make a line and put two points on it you have a segment not a line anymore so if you put 10 dots on it you will have 9 segments and nomore line
Oct 23, 2001
piffle The person said points, not dots, though.
May 30, 2002
Sane Man what a childish questions...
Feb 11, 2005
paul726 double yawn.
Mar 04, 2006
coltonr1 points not dots gosh
Mar 15, 2006
calmsavior whats with the calculator?
Apr 07, 2006
stil On line segments-- Marking C on line segment AB gives NEW line segments AC and CB while AB remains. If you mark 10 points, you create 45 line segments.
Apr 27, 2006
grungy49 I can't even see how the difficulty on this is 1.5. It should be more like 0.01. Sorry, but this question is infinitely easy.
Oct 25, 2006
LeafFan4life LOL why is there a calculator here is this supposed to be trick question??
Apr 02, 2007
horse_luver Umm... riiiiight...
May 08, 2007
flowergirl1219 1st one I got wrong tonight
Jun 27, 2007
P.S. don't forget to send me a message saying what subject you want me to make a quiz on!!!!
javaguru Yeah, stupidly simple.
Feb 05, 2009
I was going to make the observation that there are an infinite number of points on an infintesimally small line segment, but I see that my point has more or less already been made,
although it's been made far fewer than an infinite number of times.
Jimbo This is simply the statement of the first proposition of Euclidian Geometry although it all seems like Greek to some people!
Apr 03, 2009 | {"url":"http://www.braingle.com/brainteasers/teaser.php?id=1172&op=0&comm=1","timestamp":"2014-04-20T00:47:18Z","content_type":null,"content_length":"33619","record_id":"<urn:uuid:650537c4-4b72-4265-b1a3-2ef28ffcaaa2>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00500-ip-10-147-4-33.ec2.internal.warc.gz"} |
Logistic Regression
Video - How To Do Logistic Regression in Excel
Simple Explanation of How Logistic Regression Analysis Is Performed
Logistic regression is normally used to calculate the probability of an event occurring. Logistic regression analysis is performed by fitting data to a logit regression function logistic curve. The
input variables (the predictor variables) can be numerical or categorical (dummy input variables).
Logistic regression is often called logit regression, the logistic model, or logit regression.
Using Logistic Regression
Logistic regression is used in social and medical sciences. For example, one medical use of logistic regression might be used to predict whether a person will have a stroke based upon the person's
height, weight, and age. Marketers often use logistic regression to calculate the probability of whether or not a prospect will purchase.
Here is how the calculation is done (without wasting much time on theory):
The probability of the event occurring is given as follows:
P(X) = e**L/ (1+e**L)
The only variable in the above equation is L. L is called the Logit. The formula for L depends on the input variables. As a logistic regression example, if we were trying to predict the probability
of a new prospect buying based upon the prospect's age and gender, then the equation for the Logit (L) would be the following:
L is the Logit and L = Constant + A*Age + B*Gender
We need to solve for Constant, A, and B. Once we have solved for these, we have solved for L. L can then be plugged into the probability equation P(X) above and we have the probability of the
prospect purchasing.
So, the question is: How do we solve for Constant, A, and B?
We go back to our original customer and prospect data. We have recorded the age, gender, and whether each prospect purchased for all of our hundreds of previous prospects. For each of our previous
prospect, we construct the following equation:
P(X)**Y * [ 1 - P(X) ]**(1-Y)
Y = 1 if the prospect purchased and Y = 0 if the prospect did not purchase.
P(X) is the probability equation and P(X) = e**L/ (1+e**L)
L is the Logit and L = Constant + A*Age + B*Gender
The equation P(X)**Y * [ 1 - P(X) ]**(1-Y) is maximized when P(X) approaches 1 (100%) when Y=1 and when P(X) approaches 0 when Y = 0.
Ultimately what we are doing is determining the Constant, A, and B that will maximize the sum of all P(X)**Y * [ 1 - P(X) ]**(1-Y) equations that we have calculated for each previous prospect.
This would be difficult to do by hand. It is best to use a tool like the Excel Solver. In fact you can consider Excel to be your logistic regression software. The attached video shows this being
When you have found the ideal combination of (Constant, A, and B) that makes P(X) its most accurate for as many previous prospects as possible, the sum of all [ P(X)**Y * [ 1 - P(X) ]**(1-Y) ]
equations will be maximized.
Once you have found that Constant, A, and B that maximizes that sum, you can then plug the Constant, A, and B into the Logit equation:
L = Constant + A*Age + B*Gender
After this, you have the correct Logit (L), which can then be plugged into the probability equation: P(X) = e**L/ (1+e**L) and you have the most accurate probability of whether your new prospect will
The attached video provides a logistic regression tutorial of how the calculation is performed using the Excel Solver. You can consider Excel to your logistic regression software.
Copyright 2013 | {"url":"http://www.excelmasterseries.com/Excel_Statistical_Master/Excel-Logistic-Regression.php","timestamp":"2014-04-21T07:08:51Z","content_type":null,"content_length":"39197","record_id":"<urn:uuid:357fef3f-f19c-4442-ab89-06f517c610cc>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00197-ip-10-147-4-33.ec2.internal.warc.gz"} |
Vol. 130, No. 1, 2005 · Contents
Neil A. Watson:
A Nevanlinna theorem for superharmonic functions on Dirichlet regular Greenian sets
Anna Andruch-Sobilo, Malgorzata Migda:
On the oscillation of solutions of third order linear difference equations of neutral type
Gary Chartrand, Donald W. VanderJagt, Ping Zhang:
Homogeneously embedding stratified graphs in stratified graphs
V. Marraffa:
A scalar Volterra derivative for the $PoU$-integral
Tin-Lam Toh, Tuan-Seng Chew:
On belated differentiation and a characterization of Henstock-Kurzweil-Ito integrable processes
Vinayak V. Joshi, B. N. Waphare:
Characterizations of 0-distributive posets
Vladmir Samodivkin:
Minimal acyclic dominating sets and cut-vertices
Alfonz Haviar, Gabriela Monoszova:
Constructions of cell algebras
Dragan Stevanovic:
All graphs in which each pair of distinct vertices has exactly two common neighbors
J. Jezek, V. Slavik:
Compact elements in the lattice of varieties
[Journals Homepage]
[Journal Home] [All Journals] [ELibM Home] [EMIS Home]
Publication date for the electronic files: 28 Feb 2005. Last modified: 20 January 2010.
© 2005–2010 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition | {"url":"http://www.emis.de/journals/MB/130.1/index.html","timestamp":"2014-04-19T22:27:40Z","content_type":null,"content_length":"2565","record_id":"<urn:uuid:f783e0fb-df02-4954-a717-7f8878de0c56>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00254-ip-10-147-4-33.ec2.internal.warc.gz"} |
Is this true?
April 22nd 2010, 05:47 PM #1
Junior Member
Apr 2009
Is this true?
$\lim_{x\rightarrow\infty}f(x) = e^{\lim_{x\rightarrow\infty}ln(f(x))}$
I think I've seen or used this before but it might've just been something similar, I don't have my calculus book handy.
If the limit exists, yes. The quick way using only continuity is that $\lim_{x\to\infty}f(x)=\lim_{z\to 0}f\left(\frac{1}{z}\right)=\lim_{z\to 0}\text{exp}\left(\ln\left(f\left(\frac{1}{z}\righ
t)\right)\right)=\text{exp}\left(\lim_{z\to 0}\ln\left(f\left(\frac{1}{z}\right)\right)\right)$ and work backwards.
April 22nd 2010, 05:52 PM #2 | {"url":"http://mathhelpforum.com/differential-geometry/140816-true.html","timestamp":"2014-04-19T18:43:41Z","content_type":null,"content_length":"34001","record_id":"<urn:uuid:37497bb4-85c6-4e88-ac94-14b403e974b1>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00454-ip-10-147-4-33.ec2.internal.warc.gz"} |
Advogato: Blog for oubiwann
The Lambda Calculus: A Brief History
Over this past weekend I took a lovely journey into the heart of the lambda calculus, and it was quite amazing. My explorations were made within the context of
. Needless to say, this was a romp of pure delight. In fact, it was
much fun and helped to clarify for me so many nooks and crannies of something that I had simply not explored very thoroughly in the past, that I
to share :-)
The work done over the past few days is on its way to becoming part of the
documentation for LFE
. However, this is also an excellent opportunity to share some clarity with a wider audience. As such, I will be writing a series of blog posts on λ-calculus from a very hands-on (almost practical!)
perspective. There will be some overlap with the LFE documentation, but the medium is different and as such, the delivery will vary (sometimes considerably).
This series of posts will cover the following topics:
1. A Brief History
2. A Primer for λ-Calculus
3. Reduction Explained
4. Church Numerals
5. Arithmetic
6. Logic
7. Pairs and Lists
8. Combinators
The point of these posts is not to expound upon that which has already been written about endlessly. Rather, the hope is to give a very clear demonstration of what the lambda calculus
is, and to do so with clear examples and concise prose. When the gentle reader is able see the lambda calculus in action, with lines of code that clearly show what is occurring, the mystery will
disappear and an intuition for the subject matter will quite naturally begin to arise. This post is the first in the series; I hope you enjoy them as much as I did rediscovering λ-calculus :-)
Let us start at the beginning...
A Brief History
The roots of functional programming languages such as Lisp, ML, Erlang, Haskell and others, can be traced to the concept of recursion in general and λ-calculus in particular. In previous posts, I
touched upon
how we ended up with the lambda
as a symbol for the anonymous function as well as
how recursion came to be a going concern
in modern mathematics and then computer science.
In both of those posts we saw
Alonzo Church
play a major role, but we didn't really spend time on what is quite probably considered his greatest contribution to computer science, if not mathematics itself: λ-calculus. Keep in mind that the
Peano axioms
made use of recursion, that
Giuseppe Peano
played a key role in
Bertrand Russell
’s development of the
, that Alonzo Church sought to make improvements on the Principia, and λ-calculus eventually arose from these efforts.
Invented in 1928, Alonzo didn't publish λ-calculus until 1932. When an inconsistency was discovered, he revised it in 1933 and republished. Furthermore, in this second paper, Church introduced a
means of representing positive integers using lambda notation, now known as Church numerals. With Church and Turing both publishing papers on computability in 1936 (based respectively upon λ-calculus
and the concept of
Turing machines
), they proposed solutions to the
. Though
preferred Turing's approach,
suggested that they were equivalent definitions in 1939. A few years later,
proposed the
Church Thesis
(1943) and then later formally demonstrated the equivalence between his teacher's and Turing's approaches giving the combination the name of the
Church-Turing Thesis
(1952, in his
Introduction to Metamathematics
). Within eight years,
John McCarthy
published his now-famous paper describing the work that he had started in 1958: "Recursive Functions of Symbolic Expressions and Their Computation by Machine". In this paper, McCarthy outlined his
new programming language Lisp, citing Church's 77-page book (1941,
Calculi of Lambda Conversion
), sending the world off in a whole new direction.
Since that time, there has been on-going research into λ-calculus. Indisputably, λ-calculus has had a tremendous impact on research into computability as well as the practical applications of
programming languages. As programmers and software engineers, we feel its impact -- directly and indirectly -- on a regular, almost daily basis.
Syndicated 2013-04-09 05:22:00 (Updated 2013-04-09 05:22:48) from Duncan McGreggor | {"url":"http://www.advogato.org/person/oubiwann/diary/295.html","timestamp":"2014-04-21T12:19:48Z","content_type":null,"content_length":"10559","record_id":"<urn:uuid:7959dc5a-a202-4404-ba41-1c832312c495>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00141-ip-10-147-4-33.ec2.internal.warc.gz"} |
Van Nuys Calculus Tutor
Find a Van Nuys Calculus Tutor
Although I enjoyed the challenge of teaching math and physics in several private high schools, I always felt I was at my best in one on one situations, which in a classroom setting are all too
brief. Therefore, once I retired from teaching in Los Angeles in 2005, I immediately kept myself busy as a...
31 Subjects: including calculus, English, chemistry, writing
...I also cover test taking skills. We can remember things that make sense to us and things that relate to other material that we've learned. Reviewing for these tests not only helps a student
score better, but more importantly, it gives the student an opportunity to put all the material they've learned into an integrated whole.
24 Subjects: including calculus, chemistry, English, geometry
...I specialize in helping students construct essays for the written portion of the exam. I've often been touted a "computer whiz," but more than anything I am humble, knowledgeable, and patient
when it comes to technology and sharing computer knowledge with others (my first "student" was my mother...
60 Subjects: including calculus, chemistry, reading, Spanish
...I went to Europe twice and took multiple guided tours of historical locations, prompting my desire to major in history at Duke. I excelled in geometry in high school, and am comfortable
applying the principles of geometry to higher levels of math. I started tutoring pre-calculus for the 2012 academic year and have had great results with my students thus far.
18 Subjects: including calculus, chemistry, geometry, Spanish
...Obtained from my extensive experience in teaching, one of my strongest teaching capabilities is to provide real-life examples for the subjects. After linking interesting scientific applications
to abstract concepts in math, physics and chemistry, my students are able to learn, to enjoy, and to master the subjects. As a teacher, I feel that this is my greatest reward.
10 Subjects: including calculus, chemistry, statistics, algebra 1 | {"url":"http://www.purplemath.com/van_nuys_ca_calculus_tutors.php","timestamp":"2014-04-19T12:11:13Z","content_type":null,"content_length":"24114","record_id":"<urn:uuid:cadf4efd-84f3-4666-a53d-ce382a39e218>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00141-ip-10-147-4-33.ec2.internal.warc.gz"} |
Topological Quantum Field Theories”, Publ
Results 1 - 10 of 39
, 2004
"... We describe the basic assumptions and key results of loop quantum gravity, which is a background independent approach to quantum gravity. The emphasis is on the basic physical principles and how
one deduces predictions from them, at a level suitable for physicists in other areas such as string theor ..."
Cited by 274 (9 self)
Add to MetaCart
We describe the basic assumptions and key results of loop quantum gravity, which is a background independent approach to quantum gravity. The emphasis is on the basic physical principles and how one
deduces predictions from them, at a level suitable for physicists in other areas such as string theory, cosmology, particle physics, astrophysics and condensed matter physics. No details are given,
but references are provided to guide the interested reader to the literature. The present state of knowledge is summarized in a list of 35 key results on topics including the hamiltonian and path
integral quantizations, coupling to matter, extensions to supergravity and higher dimensional theories, as well as applications to black holes, cosmology and Plank scale phenomenology. We describe
the near term prospects for observational tests of quantum theories of gravity and the expectations that loop quantum gravity may provide predictions for their outcomes. Finally, we provide answers
to frequently asked questions and a list of key open problems.
- Comm.Math.Phys.227
"... Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H
for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has ..."
Cited by 80 (12 self)
Add to MetaCart
Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a
time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models ” having a finite dimensional internal state space with no natural
tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be
efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a
radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm. 1.
, 1995
"... smolin@phys.psu.edu y permanent address I. Introduction In the last years significant progress has been made towards the construction of a quantum theory of gravity in several different
directions. Three of these directions, in particular, have involved the use of new ideas and mathematical stru ..."
Cited by 52 (25 self)
Add to MetaCart
smolin@phys.psu.edu y permanent address I. Introduction In the last years significant progress has been made towards the construction of a quantum theory of gravity in several different directions.
Three of these directions, in particular, have involved the use of new ideas and mathematical structures that seem, in different ways, well suited to the problem of describing the geometry of
spacetime quantum mechanically. These are string theory[1], topological quantum field theory[2, 3, 4, 5, 6, 7], and non-perturbative quantum gravity, based on the loop representation [8, 9, 10, 11,
12, 13, 14]. Furthermore, despite genuine differences, there are a number of concepts shared by these approaches, which suggests the possibility of a deeper relation between them[15, 54]. These
include the common use of one dimensional rather than pointlike excitations, as well as the appearance of structures associated with knot theory, spin networks and duality. There are also senses in
which each deve...
, 2002
"... A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum
gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, dis ..."
Cited by 51 (9 self)
Add to MetaCart
A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity
provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a
semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory
on deSitter spacetime. Furthermore, one may derive directly from the Wheeler-deWitt equation corrections to the energy-momentum relations for matter fields of the form E 2 = p 2 +m 2 +αlPlE 3 +...
where α is a computable dimensionless constant. This may lead in the next few years to experimental tests of the theory. To study the excitations of the Kodama state exactly requires the use of the
spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a
boundary Chern-Simons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation. The paper is written as an introduction to loop quantum
gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout.
- Asian J. Math , 1999
"... 1. Introduction. The Witten-Reshetikhin-Turaev (WRT) invariant of a compact connected oriented 3-manifold M may be formally defined by [16] ..."
- J. Knot Theory Ram , 1997
"... Abstract: We provide, with proofs, a complete description of the authors ’ construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple
tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery ..."
Cited by 30 (6 self)
Add to MetaCart
Abstract: We provide, with proofs, a complete description of the authors ’ construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile
category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of Rep
(Uq(sl2)) (q a principal 4r th root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by
Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. 1 1
, 2000
"... I sketch the main lines of development of the research in quantum gravity, from the first explorations in the early thirties to nowadays. ..."
Cited by 21 (0 self)
Add to MetaCart
I sketch the main lines of development of the research in quantum gravity, from the first explorations in the early thirties to nowadays.
"... this paper a new approach to the problem of constructing a quantum theory of gravity in the cosmological context is proposed. It is founded on results from four separate directions of
investigation, which are: 1) A new point of view towards the interpretation problem in quantum cosmology[1, 2, 3, 4] ..."
Cited by 19 (12 self)
Add to MetaCart
this paper a new approach to the problem of constructing a quantum theory of gravity in the cosmological context is proposed. It is founded on results from four separate directions of investigation,
which are: 1) A new point of view towards the interpretation problem in quantum cosmology[1, 2, 3, 4], which rejects the idea that a single quantum state, or a single Hilbert space, can provide a
complete description of a closed system like the universe. Instead, the idea is to accept Bohr's original proposal that the quantum state requires for its interpretation a context in which we
distinguish two subsystems of the universe-the quantum system and observer. However, we seek to relativize this split, so that the boundary between the part of the universe that is considered the
system and that which might be considered the observer may be chosen arbitrarily. The idea is then that a quantum theory of cosmology is specified by giving an assignment of a Hilbert space and
algebra of observables to every possible boundary that can be considered to split the universe into two such subsystems. A quantum state of the universe is then an assignment of a statistical state
to every one of these Hilbert spaces, subject to certain conditions of consistency. Each of these states is interpreted to contain the information that an observer on one side of each boundary might
have about the system of the other side. This formulation then accepts the idea that each observer can only have incomplete information about the universe, so that the most complete description
possible of the universe is given by the whole collection of incomplete, but mutually compatible quantum state descriptions of all the possible observers. At the same time, the information of
different observers is, to some extent, ...
, 1993
"... : The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy
braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techn ..."
Cited by 15 (3 self)
Add to MetaCart
: The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy
braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques gives the dimension and an explicit basis for the vector space of the topological
quantum field theory associated to any Riemann surface with arbitrary coloured punctures. * Supported by DFG, SFB 288 "Differentialgeometrie und Quantenphysik" 1 e-mail:
karowski@vax1.physik.fu-berlin.dbp.de 2 e-mail: schrader@vax1.physik.fu-berlin.dbp.de 1 1. Introduction Since the early days of topological quantum field theories there was the question whether such
field theories have a lattice formulation analogous to lattice gauge theory. The reason is that one would like to work in a context with mathematically well defined quantities instead of more or less
formal functional integrals. This qu...
, 1996
"... We describe a deformation of the observable algebra of quantum gravity in which the loop algebra is extended to framed loops. This allows an alternative nonperturbative quantization which is
suitable for describing a phase of quantum gravity characterized by states which are normalizable in the meas ..."
Cited by 14 (2 self)
Add to MetaCart
We describe a deformation of the observable algebra of quantum gravity in which the loop algebra is extended to framed loops. This allows an alternative nonperturbative quantization which is suitable
for describing a phase of quantum gravity characterized by states which are normalizable in the measure of Chern-Simons theory. The deformation parameter, q, is e i¯h 2 G 2 =6 , where is the
cosmological constant. The Mandelstam identities are extended to a set of relations which are governed by the Kauffman bracket so that the spin network basis is deformed to a basis of SU(2) q spin
networks. Corrections to the actions of operators in non-perturbative quantum gravity may be readily computed using recoupling theory; the example of the area observable is treated here. Finally,
eigenstates of the q-deformed Wilson loops are constructed, which may make possible the construction of a q-deformed connection representation through an inverse transform. internet addresses: | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=222972","timestamp":"2014-04-17T13:20:35Z","content_type":null,"content_length":"38176","record_id":"<urn:uuid:c245deaa-5f24-47a3-83e1-827015ffe262>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00473-ip-10-147-4-33.ec2.internal.warc.gz"} |
Another Chicken or Egg: Sequence or Series
up vote 3 down vote favorite
This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" quantities; on the other hand, decimal expansions -- especially infinite -- are
more likely to be series).
Secondly, is it natural for sequences to be placed prior to series in a calculus course?
So, which one is more original, a sequence or a series?
After-dinner edit. We define a sequence to be... a function mapping the positive numbers to a set(?). We define a series to be... a formal infinite sum $\sum_{n=1}^\infty a_n$(?). Tell me what is
your way to "define" these two guys, I do not believe they are very related.
There are no doubts that it is easier to define convergence of series via convergence of sequences, but it does not imply their "primogeniture". The notion of Cauchy sequence is an elegant way to
build the apparatus of not only sequences but also of real numbers; as such it can serve as a definition of series: a series is a formal infinite sum $\sum_{n=1}^\infty a_n$, and it is called a
convergent series if for any $\epsilon>0$ there exists an $N=N(\epsilon)$ such that for any $m>n>N$ the sum $|a_n+\dots+a_m|<\epsilon$. The real numbers then are nothing but representatives of
equivalence classes of convergent series. (I have no desire here to expand all the details.) A sequence $b_n$ is convergent when the corresponding series $\sum_{n=1}^\infty a_n$ where $a_1=b_1$ and
$a_n=b_n-b_{n-1}$ for $n\ge2$ converges. It would be honest to say that, besides the trivialities like "algebra of limits", the techniques for investigating convergence of series are quite
independent from that of sequences. And it does not sound impossible to do series prior to sequences.
Historically, all these convergence/divergence issues were purely intuitive for both sequences and series, and they both were on the market for many centuries. I ask whether their exists an
overwhelming historical support to the notion of sequence to lead.
ho.history-overview sequences-and-series teaching gm.general-mathematics
10 We define series using sequences. How would you define sequences using series? – Harun Šiljak Aug 26 '12 at 10:19
"Tell me what is your way to "define" these two guys, I do not believe they are very related." In my first calculus course (still not sure whether I should refer to it as a calculus, or a real
1 analysis course, though) we used the following definition of series: Let there be a sequence $(a_n)$ in a normed space $X$ and let $s_k=\sum_{n=1}^k a_n$ for $n\in\mathbb{N}$. Series in $X$ is an
ordered pair $(a_n,s_k)$ ($a_n,s_k\in X$) which is consisted of two sequences $(a_n)$ and $(s_k)$, former being the terms, and latter being partial sums of series. – Harun Šiljak Aug 26 '12 at
should be CW – Steven Gubkin Aug 26 '12 at 18:22
add comment
4 Answers
active oldest votes
This is not a precise answer, mainly some thoughts.
Historically both ideas seem very old (2000+ years), and sometimes it is hard to tell what point of view is predominant.
Say if one looks at the Method of Exhaustion or things around Zeno's paradoxes what does arise 'sequences' or 'series'? For some of the constructions, the one seems more natural for others
the other. In any case, in some form series already arose then.
up vote 8 One more point in favor of the fact that series where around early on: while in today's courses differentiation comes before integeration, in an intuitive sense I think integration is
down vote rather the easier or more natural idea, and historically early forms of integeration are very old (cf. Method of Exhaustion, which in my opinion counts as some sort of integeration). And
integration and series sort of go together.
For your second question, for a course today, I would however start with sequence.
If one wants to talk about series in a somewhat rigorous form just having the notion/word sequence at ones disposal already seems like a big plus.
add comment
I would argue that it is natural to introduce series first. Why are sequences interesting? The sequence $1$, $3/2$, $7/4$, etc. converges to $2$. Who cares?
I think the most natural answer to ``who cares'' is series. Write $e = 1 + 1 + 1/2 + 1/6 + \cdots$ on the blackboard, and I expect that students will know what is meant, and think it's
cool. We write $1/3 = .3333\dots$ in precalculus courses without first discussing convergence, and this isn't really all that different.
up vote 4
down vote Having introduced series, one can continue and write things like $1 - 1 + 1 - 1 + \cdots$ or whatever on the blackboard, and perhaps scare the students a little bit and explain that it is
possible to write down formulas which are complete nonsense. (Or maybe only almost-complete nonsense, Ramanujan argued in cold blood that $1 + 2 + 3 + 4 + \cdots = -1/12$.) This motivates
a more cautious approach to the subject, i.e. discussing convergence of sequences.
really? series are just situations where elements of a sequence get added. we could be doing other stuff like multiplication, exponentiation, transformation, etc. etc., with elements of
a sequence---so I don't really agree with it is "natural" to introduce series first...what if the elements of our sequence do not come from a space where addition is defined? we can
still have sequences.... – Suvrit Aug 26 '12 at 14:51
1 This is a calculus class. Addition is defined. – Frank Thorne Aug 26 '12 at 15:06
(I know, I was just being snide because of the "natural" in there) – Suvrit Aug 26 '12 at 15:10
I definitely like your point, Frank. Your example with $e$ could be acomplished with the limit of $(1+1/n)^n$ as $n\to\infty$, which is extremely useful in showing many other limits but
at the same time impractical for actual computation of $e$. The series, on other hand, can be successfully used not only to compute the number but also to demonstrate its irrationality
to a first year undergrad. – Wadim Zudilin Aug 27 '12 at 9:36
1 I am pretty sure that we should give the credit of that "almost-complete nonsense" you have mentioned to Euler :) – Amir Asghari May 5 '13 at 19:34
show 1 more comment
I have a rather radical idea. I start with Maclaurin Series! Let's see how it works. You first see graphically and "globally" that you get closer and closer to the function and when adding
infinite terms you get the function. Then you have a point-wise look. For example, consider the Maclaurin series of Exp(x), you ask what happens at, say, x=1 (see the corresponding
y-coordinates). Alongside "the convergent graphs", you have a numerical series. Playing with different functions, you get some interesting numerical series that without having any definition
up vote at hand it is not possible to decide whether they are convergent or not, ex. 1-1+1-1+1... It leads students to a definition of convergent numerical series and back again, convergent
1 down functional series! On the middle, we touch sequences.
It seems strange, but usually I have a big picture for each of my courses and this idea works well within the picture I have for Single Variable Calculus.
add comment
Most calculus students will see limits of sequences first, because definite integrals are limits of sequences of Riemann sums.
up vote 0 down vote
+1 for your CW comment: I really forgot about this natural option. – Wadim Zudilin Aug 27 '12 at 9:28
add comment
Not the answer you're looking for? Browse other questions tagged ho.history-overview sequences-and-series teaching gm.general-mathematics or ask your own question. | {"url":"http://mathoverflow.net/questions/105509/another-chicken-or-egg-sequence-or-series/105541","timestamp":"2014-04-19T15:12:55Z","content_type":null,"content_length":"77777","record_id":"<urn:uuid:2324e39f-1942-4277-abd2-35ae46c21ad7>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00038-ip-10-147-4-33.ec2.internal.warc.gz"} |