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The Weatherman
The Weatherman
Checkerboard Europe
Richardson's forecast was actually a hindcast: He was "predicting" events that had taken place years before. His initial data described the state of the atmosphere over Germany and neighboring
countries at 7:00 a.m. (Greenwich time) on May 20, 1910. His goal was to model the weather in this region over a span of six hours. He chose this particular time and place not because the weather was
in any way unusual that Friday morning but rather because unusually good data were available. May 20, 1910, was one of a series of days on which weather observations were collected from coordinated
balloon ascents all over Europe. The results had been tabulated and analyzed by the Norwegian meteorologist Vilhelm Bjerknes.
The observing sites for the Bjerknes project were scattered irregularly across the map of Europe, and so Richardson had to interpolate to produce a uniform grid of data points. The grid he chose was
a checkerboard, with squares roughly 200 kilometers on a side. A pattern of 25 squares covered a diamond-shaped region extending from Denmark to Italy and from the English Channel to Poland.
Vertically, he sliced the atmosphere into five layers, with boundaries at altitudes of roughly 2, 4, 7 and 12 kilometers. (The heights were chosen so that each stratum had about the same mass of
air.) Thus the model divided the volume being studied into 125 compartments.
The pattern of squares laid out on the landscape was described as a checkerboard rather than merely a grid because different quantities were computed in the alternating black and white squares. In
one set of squares (called P cells) Richardson recorded the barometric pressure in each of the five altitude layers, and also moisture amounts and the stratospheric temperature. In the other squares
(M cells) he calculated the momentum of the atmosphere—that is, the wind speed and direction multiplied by the mass of the air.
It's not hard to see in a qualitative way how these variables would enter into a model of the atmosphere's dynamics. In particular, winds and barometric pressures have an obvious interconnection.
Pressure is the force that drives the winds; air flows from place to place in response to differences in pressure. At the same time, the convergence or divergence of winds alters the pressure in a
region, as air is either blown in or sucked out. A similar linkage connects pressure with temperature, since heat is generated when air is compressed. The model has to track all these relationships
(and others) from moment to moment. Given initial values of all the variables at time t[0], the model calculates the values at a later time t[1]; then these t[1] values form the basis of a new
calculation at time t[2], and so on.
The most important quantities to keep track of in Richardson's model turned out to be barometric pressure and three components of momentum (along north-south, east-west and up-down axes). All of
these quantities vary from place to place and from moment to moment; in other words, they are functions of latitude, longitude, height and time. There are also crucial dependencies among them. For
example, one of Richardson's equations states that the rate of change in the east-west component of momentum depends on the pressure gradient along the same axis. This relation is unsurprising; it
says that air goes where you push it. But the full atmospheric equations of motion are more complicated. In addition to the pressure-gradient term, they also includes a term representing the Coriolis
force, by which the earth's rotation twists the winds and thereby couples east-west and north-south velocities.
Vertical motions in the atmosphere are even more problematic. No vertical winds were included among the initial data. To calculate them, Richardson relied on a simple ground truth: The earth is
solid, and therefore impervious to wind. It follows that if horizontal winds are converging in some ground-level cell of the model atmosphere, then air must be flowing upward out of that cell. By the
same principle, divergent winds at ground level must be balanced by air sinking into the cell from above. | {"url":"http://www.americanscientist.org/issues/pub/the-weatherman/3","timestamp":"2014-04-19T00:11:51Z","content_type":null,"content_length":"121661","record_id":"<urn:uuid:c78dafc3-1e62-412b-a242-8332969a6a27>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00044-ip-10-147-4-33.ec2.internal.warc.gz"} |
Choosing Search Heuristics by Non-Stationary Reinforcement Learning. Metaheuristics: Computer Decision-Making
Results 1 - 10 of 27
, 2003
"... Hyperheuristics can be defined to be heuristics which choose between heuristics in order to solve a given optimisation problem. The main motivation behind the development of such approaches is
the goal of developing automated scheduling methods which are not restricted to one problem. In this paper ..."
Cited by 117 (56 self)
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Hyperheuristics can be defined to be heuristics which choose between heuristics in order to solve a given optimisation problem. The main motivation behind the development of such approaches is the
goal of developing automated scheduling methods which are not restricted to one problem. In this paper we report the investigation of a hyperheuristic approach and evaluate it on various instances of
two distinct timetabling and rostering problems. In the framework of our hyperheuristic approach, heuristics compete using rules based on the principles of reinforcement learning. A tabu list of
heuristics is also maintained which prevents certain heuristics from being chosen at certain times during the search. We demonstrate that this tabu-search hyperheuristic is an easily re-usable method
which can produce solutions of at least acceptable quality across a variety of problems and instances. In effect the proposed method is capable of producing solutions that are competitive with those
obtained using stateof -the-art problem-specific techniques for the problems studied here, but is fundamentally more general than those techniques.
- Handbook of Graph Theory, chapter 5.6 , 2004
"... Abstract Automating the neighbourhood selection process in an iterative approach that uses multiple heuristics is not a trivial task. Hyper-heuristics are search methodologies that not only aim
to provide a general framework for solving problem instances at different difficulty levels in a given dom ..."
Cited by 29 (17 self)
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Abstract Automating the neighbourhood selection process in an iterative approach that uses multiple heuristics is not a trivial task. Hyper-heuristics are search methodologies that not only aim to
provide a general framework for solving problem instances at different difficulty levels in a given domain, but a key goal is also to extend the level of generality so that different problems from
different domains can also be solved. Indeed, a major challenge is to explore how the heuristic design process might be automated. Almost all existing iterative selection hyper-heuristics performing
single point search contain two successive stages; heuristic selection and move acceptance. Different operators can be used in either of the stages. Recent studies explore ways of introducing
learning mechanisms into the search process for improving the performance of hyper-heuristics. In this study, a broad empirical analysis is performed comparing Monte Carlo based hyper-heuristics for
solving capacitated examination timetabling problems. One of these hyper-heuristics is an approach that overlaps two stages and presents them in a single algorithmic body. A learning heuristic
selection method (L) operates in harmony with a simulated annealing move acceptance method using reheating (SA) based on some shared variables. Yet, the heuristic selection and move
"... Hyper-heuristics represent a novel search methodology that is motivated by the goal of automating the process of selecting or combining simpler heuristics in order to solve hard computational
search problems. An extension of the original hyper-heuristic idea is to generate new heuristics which are n ..."
Cited by 20 (11 self)
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Hyper-heuristics represent a novel search methodology that is motivated by the goal of automating the process of selecting or combining simpler heuristics in order to solve hard computational search
problems. An extension of the original hyper-heuristic idea is to generate new heuristics which are not currently known. These approaches operate on a search space of heuristics rather than directly
on a search space of solutions to the underlying problem which is the case with most meta-heuristics implementations. In the majority of hyper-heuristic studies so far, a framework is provided with a
set of human designed heuristics, taken from the literature, and with good measures of performance in practice. A less well studied approach aims to generate new heuristics from a set of potential
heuristic components. The purpose of this chapter is to discuss this class of hyper-heuristics, in which Genetic Programming is the most widely used methodology. A detailed discussion is presented
including the steps needed to apply this technique, some representative case studies, a literature review of related work, and a discussion of relevant issues. Our aim is to convey the exciting
potential of this innovative approach for automating the heuristic design process
"... The current state of the art in hyper-heuristic research comprises a set of approaches that share the common goal of automating the design and adaptation of heuristic methods to solve hard
computational search problems. The main goal is to produce more generally applicable search methodologies. In ..."
Cited by 18 (13 self)
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The current state of the art in hyper-heuristic research comprises a set of approaches that share the common goal of automating the design and adaptation of heuristic methods to solve hard
computational search problems. The main goal is to produce more generally applicable search methodologies. In this chapter we present and overview of previous categorisations of hyper-heuristics and
provide a unified classification and definition which captures the work that is being undertaken in this field. We distinguish between two main hyper-heuristic categories: heuristic selection and
heuristic generation. Some representative examples of each category are discussed in detail. Our goal is to both clarify the main features of existing techniques and to suggest new directions for
hyper-heuristic research.
- In: Journal of Machine Learning Research
"... Grid search and manual search are the most widely used strategies for hyper-parameter optimization. This paper shows empirically and theoretically that randomly chosen trials are more efficient
for hyper-parameter optimization than trials on a grid. Empirical evidence comes from a comparison with a ..."
Cited by 17 (1 self)
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Grid search and manual search are the most widely used strategies for hyper-parameter optimization. This paper shows empirically and theoretically that randomly chosen trials are more efficient for
hyper-parameter optimization than trials on a grid. Empirical evidence comes from a comparison with a large previous study that used grid search and manual search to configure neural networks and
deep belief networks. Compared with neural networks configured by a pure grid search, we find that random search over the same domain is able to find models that are as good or better within a small
fraction of the computation time. Granting random search the same computational budget, random search finds better models by effectively searching a larger, less promising configuration space.
Compared with deep belief networks configured by a thoughtful combination of manual search and grid search, purely random search over the same 32-dimensional configuration space found statistically
equal performance on four of seven data sets, and superior performance on one of seven. A Gaussian process analysis of the function from hyper-parameters to validation set performance reveals that
for most data sets only a few of the hyper-parameters really matter, but that different hyper-parameters are important on different data sets. This phenomenon makes
- In proceedings of Congress on Evolutionary Computation(CEC2003 , 2003
"... Abstract-This paper investigates a tabu assisted genetic algorithm based hyper-heuristic (hyper-TGA) for personnel scheduling problems. We recently introduced a hyper-heuristic genetic algorithm
(hyper-GA) with an adaptive length chromosome which aims to evolve an ordering of low-level heuristics in ..."
Cited by 15 (4 self)
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Abstract-This paper investigates a tabu assisted genetic algorithm based hyper-heuristic (hyper-TGA) for personnel scheduling problems. We recently introduced a hyper-heuristic genetic algorithm
(hyper-GA) with an adaptive length chromosome which aims to evolve an ordering of low-level heuristics in order to find good quality solutions to given problems. The addition of a tabu method, the
focus of this paper, extends that work. The aim of adding a tabu list to the hyper-GA is to indicate the efficiency of each gene within the chromosome. We apply the algorithm to a geographically
distributed training staff and course scheduling problem and compare the computational results with our previous hyper-GA. 1.
, 2007
"... One of the main motivations for investigating hyper-heuristic methodologies is to provide a more general search framework than is currently available. Most of the current search techniques
represent approaches that are largely adapted for specific search problems (and, in some cases, even specific ..."
Cited by 14 (7 self)
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One of the main motivations for investigating hyper-heuristic methodologies is to provide a more general search framework than is currently available. Most of the current search techniques represent
approaches that are largely adapted for specific search problems (and, in some cases, even specific problem instances). There are many real-world scenarios where the development of such bespoke
systems is entirely appropriate. However, there are other situations where it would be beneficial to have methodologies which are more generally applicable to more problems. One of our motivating
goals is to underpin the development of more flexible search methodologies that can be easily and automatically employed on a broader range of problems than is currently possible. Almost all the
heuristics that have appeared in the literature have been designed and selected by humans. In this paper, we investigate a simulated annealing hyper-heuristic methodology which operates on a search
space of heuristics and which employs a stochastic heuristic selection strategy and a short-term memory. The generality and performance of the proposed algorithm is demonstrated over a large number
of benchmark data sets drawn from three very different and difficult (NP-hard) problems: nurse rostering, university course timetabling and onedimensional bin packing. Experimental results show that
the proposed hyper-heuristic is able to achieve significant performance improvements over a recently proposed tabu search hyper-heuristic without lowering the level of generality. We
- Proceedings of the fourth Asia-Pacific Conference on Simulated Evolution And Learning, (SEAL'02), Orchid Country Club, Singapore, 18-22 Nov 2002
"... Hyper-GA was introduced by the authors as a genetic algorithm based hyperheuristic which aims to evolve an ordering of low-level heuristics so as to find a good quality solution to a given
problem. The adaptive length chromosome hyper-GA, let's call it ALChyper-GA, is an extension of the authors pre ..."
Cited by 13 (4 self)
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Hyper-GA was introduced by the authors as a genetic algorithm based hyperheuristic which aims to evolve an ordering of low-level heuristics so as to find a good quality solution to a given problem.
The adaptive length chromosome hyper-GA, let's call it ALChyper-GA, is an extension of the authors previous work, in which the chromosome was of fixed length. The aim of a variable length chromosome
is two fold; 1) it allows dynamic removal and insertion of heuristics 2) it allows the GA to find a good chromosome length which could otherwise only be found by experimentation. We apply the
ALChyper-GA to a geographically distributed training staff and courses scheduling problem, and report that good quality solution can be found. We also present results for four versions of the
ALChyper-GA, applied to five test data sets.
- In Proc. of the Workshop on Computationally Hard Problems and Joint Inference , 2006
"... We introduce a novel decoding procedure for statistical machine translation and other ordering tasks based on a family of Very Large-Scale Neighborhoods, some of which have previously been
applied to other NP-hard permutation problems. We significantly generalize these problems by simultaneously con ..."
Cited by 12 (3 self)
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We introduce a novel decoding procedure for statistical machine translation and other ordering tasks based on a family of Very Large-Scale Neighborhoods, some of which have previously been applied to
other NP-hard permutation problems. We significantly generalize these problems by simultaneously considering three distinct sets of ordering costs. We discuss how these costs might apply to MT, and
some possibilities for training them. We show how to search and sample from exponentially large neighborhoods using efficient dynamic programming algorithms that resemble statistical parsing. We also
incorporate techniques from statistical parsing to improve the runtime of our search. Finally, we report results of preliminary experiments indicating that the approach holds promise. 1
- In Proceedings of the 10th International Conference on Principles and Practice of Constraint Programming , 2004
"... Abstract. The success of stochastic algorithms is often due to their ability to effectively amplify the performance of search heuristics. This is certainly the case with stochastic sampling
algorithms such as heuristic-biased stochastic sampling (HBSS) and value-biased stochastic sampling (VBSS), wh ..."
Cited by 12 (5 self)
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Abstract. The success of stochastic algorithms is often due to their ability to effectively amplify the performance of search heuristics. This is certainly the case with stochastic sampling
algorithms such as heuristic-biased stochastic sampling (HBSS) and value-biased stochastic sampling (VBSS), wherein a heuristic is used to bias a stochastic policy for choosing among alternative
branches in the search tree. One complication in getting the most out of algorithms like HBSS and VBSS in a given problem domain is the need to identify the most effective search heuristic. In many
domains, the relative performance of various heuristics tends to vary across different problem instances and no single heuristic dominates. In such cases, the choice of any given heuristic will be
limiting and it would be advantageous to gain the collective power of several heuristics. Toward this goal, this paper describes a framework for integrating multiple heuristics within a stochastic
sampling search algorithm. In its essence, the framework uses online-generated statistical models of the search performance of different base heuristics to select which to employ on each subsequent
iteration of the search. To estimate the solution quality distribution resulting from repeated application of a strong heuristic within a stochastic search, we propose the use of models from extreme
value theory (EVT). Our EVT-motivated approach is validated on the NP-Hard problem of resource-constrained project scheduling with time windows (RCPSP/max). Using VBSS as a base stochastic sampling
algorithm, the integrated use of a set of project scheduling heuristics is shown to be competitive with the current best known heuristic algorithm for RCPSP/max and in some cases even improves upon
best known solutions to difficult benchmark instances. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=650557","timestamp":"2014-04-18T08:15:03Z","content_type":null,"content_length":"43261","record_id":"<urn:uuid:a4faf73b-2177-43d6-8167-b5a011a64039>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00189-ip-10-147-4-33.ec2.internal.warc.gz"} |
Noncommutative geometry and the standard model with neutrino mixing
Seminar Room 1, Newton Institute
We show that allowing the metric dimension of a space to be independent of its KO-dimension and turning the finite noncommutative geometry F-- whose product with classical 4-dimensional space-time
gives the standard model coupled with gravity--into a space of KO-dimension 6 by changing the grading on the antiparticle sector into its opposite, allows to solve three problems of the previous
noncommutative geometry interpretation of the standard model of particle physics: The finite geometry F is no longer put in ``by hand" but a conceptual understanding of its structure and a
classification of its metrics is given. The fermion doubling problem in the fermionic part of the action is resolved. The action now automatically generates the full standard model coupled with
gravity with neutrino mixing and see-saw mechanism for neutrino masses. We shall also discuss three predictions of the model including a relation between boson and fermion masses at unification | {"url":"http://www.newton.ac.uk/programmes/NCG/seminars/2006090415301.html","timestamp":"2014-04-17T06:44:53Z","content_type":null,"content_length":"4833","record_id":"<urn:uuid:47686e93-685f-4cbf-806b-edf1087de7a6>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00610-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to help students understand high school geometry?
If you read the first part of this article, you can already see that the measures to take should happen before high school. The best approach involves changing how math and especially geometry is
taught BEFORE high school. Some points to consider are:
• Improve geometry teaching in the elementary and middle school so that students' van Hiele levels are brought up to at least to the level of abstract/relational.
• Include more justifications, informal proofs, and "why" questions in geometry teaching during elementary and middle school.
• In general, make students think, reason, and use their brain in different educational tasks (not just math).
This article will now concentrate only on the first point.
Understanding geometry concepts/Van Hiele levels
You can expect kids up through first grade to be in the first van Hiele level - visual. This means children recognize geometric figures based on their appearance, and not based on their properties.
On this level, children are mainly learning the names of some shapes, such as square, triangle, rectangle, and circle.
During the elementary school (grade 2, 3, 4, and on) children should investigate geometric shapes so that they will reach the second van Hiele level (descriptive/analytic). That is when they can
identify properties of figures and recognize them by their properties, instead of relying on appearance.
For example, students should come to understand that a rectangle has four right angles, and even if it is rotated on its "corner", it is a rectangle. Children should learn about parallel lines and
understand what is a parallelogram. Students should divide shapes into different shapes (such as dividing a square into two rectangles), and combine shapes to form new ones, and of course name the
new shapes. The shapes to be recognized should be rotated so they appear in different positions.
Drawing also helps. Students can learn to use a ruler, compass, and a protactor, so they can then draw squares, rectangles, parallelograms, and circles.
If all goes well, in middle school (grades 6th-8th) the student would proceed to the third Van Hiele level (abstract/relational), where he/she can understand and form abstract definitions,
distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes. And thus, the student would be prepared for the formal proofs and
deductive reasoning in high school geometry.
Experiments have shown that this is indeed possible with the right kind of geometry teaching. The key is to emphasize the geometrical concepts and providing the kids lots of hands-on activities like
drawing the figures and working with manipulatives, instead of only memorizing formulas and definitions and calculating areas, perimeters, etc. See below some examples of activities that will help
children and young people to develop their geometric thinking.
How to help students to develop understanding of a single geometry concept
• When studying a concept, show correct AND incorrect examples, and in different ways or representations (rotate the pictures upside down etc!). Students are asked to distinguish between correct
and incorrect examples. This will help prevent misconceptions.
• Aks students to draw correct and incorrect examples of a geometry concept. For example, ask them to draw parallel lines and lines that are not parallel. Tying in with this, ask them to draw a
parallelogram and a quadrilateral that is not a parallelogram.
• Tying in with the previous point, you can ask the students to provide a definition for a concept. This can get them to thinking about which properties in the definition are really necessary and
which are not. For example, ask them to define an "equilateral triangle".
• Allow the students to experiment, investigate, and play with geometrical ideas and figures. For this you could use manipulatives, lots of drawing, and computer programs (more on them below).
• Have each student make his/her own geometry concepts notebook, with examples, nonexamples, definitions and other notes or pictures.
Computers and interactive geometry
A computer can really help in geometry teaching, since it allows a dynamic, interactive manipulation of a figure. A child can move, rotate, or stretch the figure, and observe what properties stay the
For example, let's say you are teaching the concept of an isosceles triangle in 4th grade. You could simply use the Drawing Toolbar in Microsof Word, which has the AutoShape for isosceles triangle
(as well as for a right triangle and parallelogram). Let children draw one or two and then tell them to drag it from the white handles to make it bigger/smaller, and also to rotate the figure. Ask,
"What changes? What does not change? What stays the same? Can you draw this kind of thing on paper?"
There also exist dynamic geometry software that is specifically designed to teach geometry in an interactive investigational way. Such programs have been used in research experiments and in schools
with good results. After you see what can be done, it is very easy to fall in love with such a program - the idea is just great!
How can I help the student already studying high school geometry?
Perhaps your student is already studying geometry in high school and is having problems. Of course you cannot change how he/she was taught in the past. Since this is such a common problem, many
publishers have come out with textbooks that emphasize "informal" geometry and geometry concepts, instead of proofs. You could use one of those books, and simply forget about the proving.
Yet other books include proofs, but not in the same quantity or same emphasis as in previous years. These include for example Harold Jacobs Geometry: Seeing, Doing, Understanding. The link goes to my
review of this book.
And even with good preparation, high school geometry and the proofs can still be difficult. All in all, there is no quick and easy answer to the difficulties in this course. Remember that even math
teachers in schools struggle with this problem of getting students to understand and construct proofs. Maybe the explanations on Ask Dr. Math: FAQ About Proofs can be of some help.
I have reviewed several geometry books:
Geometry: Seeing, Doing, Understanding by Harold Jacobs (high school)
Geometry: A Guided Inquiry by Chakerian, Crabill, and Stein, and its supplement "Home Study Companion - Geometry" by David Chandler (high school).
Dr. Math geometry books - these are inexpensive companions to middle and high school geometry courses.
RightStart Geometry is a hands-on geometry course for middle school where much of the work is done with a drawing board, T-square, and triangles. It is more pricey, but of good quality.
These two books are my creations:
Math Mammoth Geometry 1 for grades 3-5 emphasizes hands-on drawing exercises and covers basic plane geometry topics for those grades. Price: $7.50 (download), $12.70 (softcover printed book)
Math Mammoth Geometry 2 for grades 6-7 continues the study of geometry after Math Mammoth Geometry 1, continually emphasizing conceptual understanding, besides calculation-type exercises. Price:
$5.95 download, $10.40 (softcover printed book).
Here is one high school geometry book that is "traditional" in its emphasis on proofs:
Geometry by Ray C. Jurgensen
Why is high school geometry difficult? - the first part of this article, explaining the Van Hiele levels.
Actually, I loved Geometry, but I was in the Honors course with a brilliant teacher. As a tutor and now teaching homeschooled children, I teach the same way he taught us. Mr. Kasper taught us to
flowchart the proof from either end, citing the theorem in initials beneath each step. We could flowchart some of the more difficult proofs in half the time that the two-column proofs take,
simply because we had a visual layout that easily led to the next step. We did learn to do the formal two-column proofs, but we always did them from the flowcharts, accomplishing them more
easily. I've tutored kids that do not understand the two-column proofs but catch the concept with the flowcharts quickly. On the other hand, I have had two students who needed the formal columns.
Math Lessons menu
• Place Value Ideas • Add/Subtract lessons • Multiplication
• Division • Fraction Lessons • Geometry Lessons
• Decimals Lessons • Percents Lessons • General | {"url":"http://www.homeschoolmath.net/teaching/geometry-2.php","timestamp":"2014-04-16T22:18:13Z","content_type":null,"content_length":"51003","record_id":"<urn:uuid:a54ffd4c-af90-4084-afb4-986d21c6a8be>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00246-ip-10-147-4-33.ec2.internal.warc.gz"} |
BIOL398-01/S11:Week 11
From OpenWetWare
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== Shared Journal Assignment == == Shared Journal Assignment ==
Revision as of 10:55, 29 March 2011
This journal entry is due on Tuesday, April 5 at midnight PDT (Wednesday night/Thursday morning). NOTE new due date and that the server records the time as Eastern Daylight Time (EDT). Therefore,
midnight will register as 03:00.
Individual Journal Assignment
• Store this journal entry as "username Week 11" (i.e., this is the text to place between the square brackets when you link to this page).
• Create the following set of links. (HINT: you can do all of this easily by adding them to your template and then using the template on your pages.)
□ Link to your journal entry from your user page.
□ Link back from your journal entry to your user page.
□ Link to this assignment from your journal entry.
□ Don't forget to add the "BIOL398-01/S11" category to the end of your wiki page.
Microarray Data Analysis
This is a list of steps required to analyze DNA microarray data.
1. Quantitate the fluorescence signal in each spot
2. Calculate the ratio of red/green fluorescence
3. Log transform the ratios
4. Normalize the ratios on each microarray slide
□ Steps 1-4 are performed by the GenePix Pro software.
□ You will perform the following steps:
5. Normalize the ratios for a set of slides in an experiment
6. Perform statistical analysis on the ratios
7. Compare individual genes with known data
□ Steps 5-7 are performed in Microsoft Excel
8. Pattern finding algorithms (clustering)
9. Map onto biological pathways
□ We will use software called STEM for the clustering and mapping
10. Create mathematical model of transcriptional network
Each group will analyze a different microarray dataset:
• Wild type data from the Schade et al. (2004) paper you read last week.
• Wild type data from the Dahlquist lab.
• Δgln3 data from the Dahlquist lab.
For your assignment this week, you will keep an electronic laboratory notebook on your individual wiki page that records all the manipulations you perform on the data and the answers to the questions
throughout the protocol.
You will download your assigned Excel spreadsheet from LionShare. Because the Dahlquist Lab data is unpublished, please do not post it on this public wiki. Instead, keep the file(s) on LionShare,
which is protected by a password.
Experimental Design
On the spreadsheet, each row contains the data for one gene (one spot on the microarray). The first column (labeled "MasterIndex") numbers the rows in the spreadsheet so that we can match the data
from different experiments together later. The second column (labeled "ID") contains the gene identifier from the Saccharomyces Genome Database. Each subsequent column contains the log[2] ratio of
the red/green fluorescence from each microarray hybridized in the experiment (steps 1-4 above having been done for you by the scanner software).
Each of the column headings from the data begin with the experiment name ("Schade" for Schade wild type data, "wt" for Dahlquist wild type data, and "dGLN3" for the Dahlquist Δgln3 data). "LogFC"
stands for "Log[2] Fold Change" which is the Log[2] red/green ratio. The timepoints are designated as "t" followed by a number in minutes. Replicates are numbered as "-0", "-1", "-2", etc. after the
For the Schade data, the timepoints are t0, t10, t30, t120, t12h (12 hours), and t60 (60 hours) of cold shock at 10°C.
For the Dahlquist data (both wild type and Δgln3), the timepoints are t15, t30, t60 (cold shock at 13°C) and t90 and t120 (cold shock at 13°C followed by 30 or 60 minutes of recovery at 30°C). Note
that the experimental designs are different.
1. Begin by recording in your wiki the number of replicates for each time point in your data. For the group assigned to the Schade data, compare the number of replicates with what is stated in the
Materials and Methods for the paper. Is it the same? If not, how is it different?
Between-chip Normalization
To scale and center the data (between chip normalization) perform the following operations:
• Insert a new Worksheet into your Excel file, and name it "scaled_centered".
• Go back to the "compiled_raw_data" worksheet, Select All and Copy. Go to your new "scaled_centered" worksheet, click on the upper, left-hand cell (cell A1) and Paste.
• Insert two rows in between the top row of headers and the first data row.
• In cell A2, type "Average" and in cell A3, type "StdDev".
• You will now compute the Average log ratio for each chip (each column of data). In cell C2, type the following equation:
and press "Enter". Excel is computing the average value of the cells specified in the range given inside the parentheses. Instead of typing the cell designations, you can left-click on the beginning
cell (let go of the mouse button), scroll down to the bottom of the worksheet, and shift-left-click on the ending cell.
• You will now compute the Standard Deviation of the log ratios on each chip (each column of data). In cell B3, type the following equation:
and press "Enter".
• Excel will now do some work for you. Copy these two equations (cells C2 and C3) and paste them into the empty cells in the rest of the columns. Excel will automatically change the equation to
match the cell designations for those columns.
• You have now computed the average and standard deviation of the log ratios for each chip. Now we will actually do the scaling and centering based on these values.
• Insert a new column to the right of each data column and label the top of the column as with the same name as the column to the left, but adding "_sc" for scaled and centered to the name. For
example, "wt_LogFC_t15-1_sc"
• In cell D4, type the following equation:
In this case, we want the data in cell C4 to have the average subtracted from it (cell C2) and be divided by the standard deviation (cell C3). We use the dollar sign symbols surrounding the "C" to
tell Excel to always reference that cell in the equation, even though we will paste it for the entire column. Why is this important?
• Copy and paste this equation into the entire column.
• Repeat the scaling and centering equation for each of the columns of data. Be sure that your equation is correct for the column you are calculating.
Shared Journal Assignment
• Store your journal entry in the shared Class Journal Week 11 page. If this page does not exist yet, go ahead and create it (congratulations on getting in first :) )
• Link to your journal entry from your user page.
• Link back from the journal entry to your user page.
• Sign your portion of the journal with the standard wiki signature shortcut (~~~~).
• Add the "BIOL398-01/S11" category to the end of the wiki page (if someone has not already done so). | {"url":"http://www.openwetware.org/index.php?title=BIOL398-01/S11:Week_11&diff=501032&oldid=501030","timestamp":"2014-04-19T18:50:43Z","content_type":null,"content_length":"33368","record_id":"<urn:uuid:6288f2e9-57c6-48d6-89c5-2578db02a455>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00496-ip-10-147-4-33.ec2.internal.warc.gz"} |
Quantum computers will be able to simulate particle collisions (w/video)
The latest news from academia, regulators research labs and other things of interest
Posted: Jun 01, 2012
Quantum computers will be able to simulate particle collisions (w/video)
(Nanowerk News) Quantum computers are still years away, but a trio of theorists has already figured out at least one talent they may have. According to the theorists, including one from the National
Institute of Standards and Technology (NIST), physicists might one day use quantum computers to study the inner workings of the universe in ways that are far beyond the reach of even the most
powerful conventional supercomputers (see paper in Science: "Quantum Algorithms for Quantum Field Theories").
Quantum computers require technology that may not be perfected for decades, but they hold great promise for solving complex problems. The switches in their processors will take advantage of quantum
mechanics – the laws that govern the interaction of subatomic particles. These laws allow quantum switches to exist in both on and off states simultaneously, so they will be able to consider all
possible solutions to a problem at once.
This unique talent, far beyond the capability of today's computers, could enable quantum computers to solve some currently difficult problems quickly, such as breaking complex codes. But they could
look at more challenging problems as well.
"We have this theoretical model of the quantum computer, and one of the big questions is, what physical processes that occur in nature can that model represent efficiently?" said Stephen Jordan, a
theorist in NIST's Applied and Computational Mathematics Division. "Maybe particle collisions, maybe the early universe after the Big Bang? Can we use a quantum computer to simulate them and tell us
what to expect?"
Questions like these involve tracking the interaction of many different elements, a situation that rapidly becomes too complicated for today's most powerful computers.
The team developed an algorithm – a series of instructions that can be run repeatedly – that could run on any functioning quantum computer, regardless of the specific technology that will eventually
be used to build it. The algorithm would simulate all the possible interactions between two elementary particles colliding with each other, something that currently requires years of effort and a
large accelerator to study.
Simulating these collisions is a very hard problem for today's digital computers because the quantum state of the colliding particles is very complex and, therefore, difficult to represent accurately
with a feasible number of bits. The team's algorithm, however, encodes the information that describes this quantum state far more efficiently using an array of quantum switches, making the
computation far more reasonable.
A substantial amount of the work on the algorithm was done at the California Institute of Technology, while Jordan was a postdoctoral fellow. His coauthors are fellow postdoc Keith S.M. Lee (now a
postdoc at the University of Pittsburgh) and Caltech's John Preskill, the Richard P. Feynman Professor of Theoretical Physics.
The team used the principles of quantum mechanics to prove their algorithm can sum up the effects of the interactions between colliding particles well enough to generate the sort of data that an
accelerator would provide.
"What's nice about the simulation is that you can raise the complexity of the problem by increasing the energy of the particles and collisions, but the difficulty of solving the problem does not
increase so fast that it becomes unmanageable," Preskill says. "It means a quantum computer could handle it feasibly."
Though their algorithm only addresses one specific type of collision, the team speculates that their work could be used to explore the entire theoretical foundation on which fundamental physics
"We believe this work could apply to the entire standard model of physics," Jordan says. "It could allow quantum computers to serve as a sort of wind tunnel for testing ideas that often require
accelerators today."
Subscribe to a free copy of one of our daily
Nanowerk Newsletter Email Digests
with a compilation of all of the day's news. | {"url":"http://www.nanowerk.com/news/newsid=25462.php","timestamp":"2014-04-16T13:07:49Z","content_type":null,"content_length":"40592","record_id":"<urn:uuid:9af3447b-f58c-4ea8-bf85-24cf003025e8>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00595-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Rotational Symmetries of Platonic Solids
A symmetry of a figure moves a copy of the figure to coincide with its original position. Beside the rotations shown here, the other symmetries of the Platonic solids are reflections in various
planes through the center. Symmetries are motions and form a group. If and are symmetries, is also a symmetry: move the figure with , then move the new position with . Try to understand how the
rotations shown in this Demonstration combine. | {"url":"http://demonstrations.wolfram.com/RotationalSymmetriesOfPlatonicSolids/","timestamp":"2014-04-20T08:18:30Z","content_type":null,"content_length":"43536","record_id":"<urn:uuid:a26daf2d-f5da-4e45-96e7-3a8b3160fbc9>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00516-ip-10-147-4-33.ec2.internal.warc.gz"} |
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How to help students understand high school geometry?
If you read the first part of this article, you can already see that the measures to take should happen before high school. The best approach involves changing how math and especially geometry is
taught BEFORE high school. Some points to consider are:
• Improve geometry teaching in the elementary and middle school so that students' van Hiele levels are brought up to at least to the level of abstract/relational.
• Include more justifications, informal proofs, and "why" questions in geometry teaching during elementary and middle school.
• In general, make students think, reason, and use their brain in different educational tasks (not just math).
This article will now concentrate only on the first point.
Understanding geometry concepts/Van Hiele levels
You can expect kids up through first grade to be in the first van Hiele level - visual. This means children recognize geometric figures based on their appearance, and not based on their properties.
On this level, children are mainly learning the names of some shapes, such as square, triangle, rectangle, and circle.
During the elementary school (grade 2, 3, 4, and on) children should investigate geometric shapes so that they will reach the second van Hiele level (descriptive/analytic). That is when they can
identify properties of figures and recognize them by their properties, instead of relying on appearance.
For example, students should come to understand that a rectangle has four right angles, and even if it is rotated on its "corner", it is a rectangle. Children should learn about parallel lines and
understand what is a parallelogram. Students should divide shapes into different shapes (such as dividing a square into two rectangles), and combine shapes to form new ones, and of course name the
new shapes. The shapes to be recognized should be rotated so they appear in different positions.
Drawing also helps. Students can learn to use a ruler, compass, and a protactor, so they can then draw squares, rectangles, parallelograms, and circles.
If all goes well, in middle school (grades 6th-8th) the student would proceed to the third Van Hiele level (abstract/relational), where he/she can understand and form abstract definitions,
distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes. And thus, the student would be prepared for the formal proofs and
deductive reasoning in high school geometry.
Experiments have shown that this is indeed possible with the right kind of geometry teaching. The key is to emphasize the geometrical concepts and providing the kids lots of hands-on activities like
drawing the figures and working with manipulatives, instead of only memorizing formulas and definitions and calculating areas, perimeters, etc. See below some examples of activities that will help
children and young people to develop their geometric thinking.
How to help students to develop understanding of a single geometry concept
• When studying a concept, show correct AND incorrect examples, and in different ways or representations (rotate the pictures upside down etc!). Students are asked to distinguish between correct
and incorrect examples. This will help prevent misconceptions.
• Aks students to draw correct and incorrect examples of a geometry concept. For example, ask them to draw parallel lines and lines that are not parallel. Tying in with this, ask them to draw a
parallelogram and a quadrilateral that is not a parallelogram.
• Tying in with the previous point, you can ask the students to provide a definition for a concept. This can get them to thinking about which properties in the definition are really necessary and
which are not. For example, ask them to define an "equilateral triangle".
• Allow the students to experiment, investigate, and play with geometrical ideas and figures. For this you could use manipulatives, lots of drawing, and computer programs (more on them below).
• Have each student make his/her own geometry concepts notebook, with examples, nonexamples, definitions and other notes or pictures.
Computers and interactive geometry
A computer can really help in geometry teaching, since it allows a dynamic, interactive manipulation of a figure. A child can move, rotate, or stretch the figure, and observe what properties stay the
For example, let's say you are teaching the concept of an isosceles triangle in 4th grade. You could simply use the Drawing Toolbar in Microsof Word, which has the AutoShape for isosceles triangle
(as well as for a right triangle and parallelogram). Let children draw one or two and then tell them to drag it from the white handles to make it bigger/smaller, and also to rotate the figure. Ask,
"What changes? What does not change? What stays the same? Can you draw this kind of thing on paper?"
There also exist dynamic geometry software that is specifically designed to teach geometry in an interactive investigational way. Such programs have been used in research experiments and in schools
with good results. After you see what can be done, it is very easy to fall in love with such a program - the idea is just great!
How can I help the student already studying high school geometry?
Perhaps your student is already studying geometry in high school and is having problems. Of course you cannot change how he/she was taught in the past. Since this is such a common problem, many
publishers have come out with textbooks that emphasize "informal" geometry and geometry concepts, instead of proofs. You could use one of those books, and simply forget about the proving.
Yet other books include proofs, but not in the same quantity or same emphasis as in previous years. These include for example Harold Jacobs Geometry: Seeing, Doing, Understanding. The link goes to my
review of this book.
And even with good preparation, high school geometry and the proofs can still be difficult. All in all, there is no quick and easy answer to the difficulties in this course. Remember that even math
teachers in schools struggle with this problem of getting students to understand and construct proofs. Maybe the explanations on Ask Dr. Math: FAQ About Proofs can be of some help.
I have reviewed several geometry books:
Geometry: Seeing, Doing, Understanding by Harold Jacobs (high school)
Geometry: A Guided Inquiry by Chakerian, Crabill, and Stein, and its supplement "Home Study Companion - Geometry" by David Chandler (high school).
Dr. Math geometry books - these are inexpensive companions to middle and high school geometry courses.
RightStart Geometry is a hands-on geometry course for middle school where much of the work is done with a drawing board, T-square, and triangles. It is more pricey, but of good quality.
These two books are my creations:
Math Mammoth Geometry 1 for grades 3-5 emphasizes hands-on drawing exercises and covers basic plane geometry topics for those grades. Price: $7.50 (download), $12.70 (softcover printed book)
Math Mammoth Geometry 2 for grades 6-7 continues the study of geometry after Math Mammoth Geometry 1, continually emphasizing conceptual understanding, besides calculation-type exercises. Price:
$5.95 download, $10.40 (softcover printed book).
Here is one high school geometry book that is "traditional" in its emphasis on proofs:
Geometry by Ray C. Jurgensen
Why is high school geometry difficult? - the first part of this article, explaining the Van Hiele levels.
Actually, I loved Geometry, but I was in the Honors course with a brilliant teacher. As a tutor and now teaching homeschooled children, I teach the same way he taught us. Mr. Kasper taught us to
flowchart the proof from either end, citing the theorem in initials beneath each step. We could flowchart some of the more difficult proofs in half the time that the two-column proofs take,
simply because we had a visual layout that easily led to the next step. We did learn to do the formal two-column proofs, but we always did them from the flowcharts, accomplishing them more
easily. I've tutored kids that do not understand the two-column proofs but catch the concept with the flowcharts quickly. On the other hand, I have had two students who needed the formal columns.
Math Lessons menu
• Place Value Ideas • Add/Subtract lessons • Multiplication
• Division • Fraction Lessons • Geometry Lessons
• Decimals Lessons • Percents Lessons • General | {"url":"http://www.homeschoolmath.net/teaching/geometry-2.php","timestamp":"2014-04-16T22:18:13Z","content_type":null,"content_length":"51003","record_id":"<urn:uuid:a54ffd4c-af90-4084-afb4-986d21c6a8be>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00246-ip-10-147-4-33.ec2.internal.warc.gz"} |
11.3.2 The
MySQL retrieves and displays TIME values in 'HH:MM:SS' format (or 'HHH:MM:SS' format for large hours values). TIME values may range from '-838:59:59' to '838:59:59'. The hours part may be so large
because the TIME type can be used not only to represent a time of day (which must be less than 24 hours), but also elapsed time or a time interval between two events (which may be much greater than
24 hours, or even negative).
MySQL recognizes TIME values in several formats, some of which can include a trailing fractional seconds part in up to microseconds (6 digits) precision. See Section 9.1.3, “Date and Time Literals”.
For information about fractional seconds support in MySQL, see Section 11.3.6, “Fractional Seconds in Time Values”. In particular, as of MySQL 5.6.4, any fractional part in a value inserted into a
TIME column is stored rather than discarded. With the fractional part included, the range for TIME values is '-838:59:59.000000' to '838:59:59.000000'.
Be careful about assigning abbreviated values to a TIME column. MySQL interprets abbreviated TIME values with colons as time of the day. That is, '11:12' means '11:12:00', not '00:11:12'. MySQL
interprets abbreviated values without colons using the assumption that the two rightmost digits represent seconds (that is, as elapsed time rather than as time of day). For example, you might think
of '1112' and 1112 as meaning '11:12:00' (12 minutes after 11 o'clock), but MySQL interprets them as '00:11:12' (11 minutes, 12 seconds). Similarly, '12' and 12 are interpreted as '00:00:12'.
The only delimiter recognized between a time part and a fractional seconds part is the decimal point.
By default, values that lie outside the TIME range but are otherwise valid are clipped to the closest endpoint of the range. For example, '-850:00:00' and '850:00:00' are converted to '-838:59:59'
and '838:59:59'. Invalid TIME values are converted to '00:00:00'. Note that because '00:00:00' is itself a valid TIME value, there is no way to tell, from a value of '00:00:00' stored in a table,
whether the original value was specified as '00:00:00' or whether it was invalid.
For more restrictive treatment of invalid TIME values, enable strict SQL mode to cause errors to occur. See Section 5.1.7, “Server SQL Modes”. | {"url":"http://docs.oracle.com/cd/E17952_01/refman-5.6-en/time.html","timestamp":"2014-04-17T17:45:41Z","content_type":null,"content_length":"7697","record_id":"<urn:uuid:b465a2be-d268-4bd4-9dc2-36f40a7ab7f0>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00080-ip-10-147-4-33.ec2.internal.warc.gz"} |
st: RE: AW: RE: combination foreach forvalues
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
st: RE: AW: RE: combination foreach forvalues
From "Nick Cox" <n.j.cox@durham.ac.uk>
To <statalist@hsphsun2.harvard.edu>
Subject st: RE: AW: RE: combination foreach forvalues
Date Tue, 20 Oct 2009 12:47:52 +0100
-encode- by default returns a mapping that is strictly alphabetical. As
this is what John is asking for, that is not obviously problematic.
In any case, -encode- allows you to specify a set of labels to be used.
What you may be referring to is that sometimes users want encoding in
order of first occurrence in the data. That can be tackled in various
ways, but John doesn't mention this.
However, it is easy to imagine that if some of the letters a ... z do
not occur in practice, then a straight -encode- may not be what John
wants. Consider this:
tokenize "`c(alpha)'"
forval i = 1/26 {
label def alphabetic `i' "``i''", modify
encode stringvar, gen(numvar) label(alphabetic)
More complicated alphabets e.g. with accents or diacritical marks
clearly require modified code.
Martin Weiss
"-encode- does precisely this."
I remember there being an issue with the order of the codes that
assigns, that is why I was reluctant to recommend it. Is that not an
Nick Cox
This doesn't require either -foreach- or -forvalues-. -encode- does
precisely this.
bysort stringvar : gen newvar = _n == 1
replace stringvar = sum(stringvar)
egen newvar = group(stringvar)
There is not much to explain about combining -foreach- and -forvalues-,
as you just do it if and when you need it, typically by nesting one
inside the other. But that's not the case here.
John Bunge
I have a string variable x1 with a list of values. I want to create a
numerical variable x2 in which the numbers correspond to the string
values in x1 in an ordered fashion (as a counter).
To illustrate, lets assume x1 contains all letters of the alphabet, and
I want x2 to contain a counter that corresponds to the position of the
letter in the alphabet, i.e. x1=a > x2=1, x1=b > x2=2, x1=c > x2=3,
This seems to me like a combination of foreach and forvalues, but I
cannot find information on whether and how such thing is implementable
in 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/2009-10/msg00915.html","timestamp":"2014-04-20T11:17:20Z","content_type":null,"content_length":"8406","record_id":"<urn:uuid:de0e36f1-09bc-4ec7-a574-8e452585aabf>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00227-ip-10-147-4-33.ec2.internal.warc.gz"} |
hexagonal pyramid height
February 17th 2010, 05:09 AM #1
hexagonal pyramid height
I need to calculate the height of a hexagonal pyramid given the base length sides as 2cm and the slant edges as 6cm from each vertex.
The formula?
If I understand your question correctly then you have a six sided pyramid (with a regular hexagonal base) with side lengths 2cm and corner to top-point length 6cm?
If so, a regular hexagon is made up of six regular (equilateral) triangles, so you know the distance from a corner to the centre of the hexagon is also 2cm. If you take a cross section of the
pyramid across two opposite corners you will have a 2d triangle of which you know the length of the base and hypotenuse. You should be able to work out the height using standard trig from here
The [regular] hexagon consists of six equilateral triangles, all sides of length 2 cm. A diagonal of the hexagon then has length 4. If you drop a perpendicular to the center, you'll have a right
triangle with hypotenuse 6, and one side 2. The other side, the altitude or height, can be found using Pythagoras.
February 17th 2010, 05:55 AM #2
Feb 2010
February 17th 2010, 08:52 AM #3
Feb 2010 | {"url":"http://mathhelpforum.com/geometry/129259-hexagonal-pyramid-height.html","timestamp":"2014-04-20T15:02:17Z","content_type":null,"content_length":"34502","record_id":"<urn:uuid:b9dc81a5-dfbd-4f51-a9e6-062aa76558bb>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00364-ip-10-147-4-33.ec2.internal.warc.gz"} |
Jim taught high school math for 29 years and served as math department chairman for 22 years. In addition, he taught at the Governor's Scholars Program for nine years and the Bluegrass Community and
Technical College part time for three years, worked as a Kentucky Center for Mathematics (KCM) Math Coach in a high school for two years and has worked as a Regional Teacher Partner half-time for the
last three years with the P-12 Math and Science Outreach unit. He served two non-consecutive terms as president of the Kentucky Council of Teachers of Mathematics (KCTM) and is a member of the
National Council of Teachers of Mathematics (NCTM), the Kentucky Society for Technology in Education and the Mathematical Association of America (MAA).
Jim has been a presenter numerous times at KCTM meetings and many other mathematics and teaching conferences in Kentucky and at the Annual Meeting of NCTM. Jim specializes in working with 9-12 math
teachers on content, problem solving and the use of technology to support the teaching and learning of mathematics in high schools. He is currently working with the AMSP Master Teacher Project and
with the two Special Education Math Cadres - one in Elizabethtown and one in Lexington. Jim is also a bibliophile and is currently serving as President of the Friends of the Library in Boyle County
and sings in two choirs in Danville.
Outreach Projects Current
Special Education Math Cadre (SEMC)
Number Properties and Operations (NPO)
Math Leadership Support Network (MLSN)
• Math (grades 9-12)
• Content training
• Problem solving
• Instructional technology
Degrees and Certifications
• B.S. in Mathematics, University of Kentucky
• B.A. in Teaching Secondary Mathematics, University of Kentucky
• M.S. in Teaching Secondary Mathematics, University of Kentucky
• Rank I Certification
• Math Consultant Certification
• Additional graduate work: Computer Architecture, University of Louisville; Dynamic Geometry Software, Mount Holyoke College, MA; Number Theory, Ohio State University
Awards and Associations
• Presidential Award for Excellence in Mathematics
• Tandy Technology Scholar
• Kentucky Council of Teachers of Mathematics
• Kentucky Society for Technology in Education
• National Council of Teachers of Mathematics
• Association for Supervision and Curriculum Development
• Mathematical Association of America | {"url":"http://www.uky.edu/P12MathScience/about/rtps/moore.html","timestamp":"2014-04-20T19:11:08Z","content_type":null,"content_length":"11118","record_id":"<urn:uuid:51dd2d4e-ed3f-4af4-b45f-e264bf06811f>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00065-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] Does zero based indexing drive anyone else crazy?
Webb Sprague webb.sprague at gmail.com
Sun Jul 2 18:36:14 CDT 2006
Hi Numpeans,
I have been working on a web-based scientific application for about a
year, most of which had been written in either Matlab or SPLUS/R. My
task has been to make it "driveable" through an online interface (if
anyone cares about mortality forecasting, drop me an email and we can
chat about it offline). I chose Python/Numpy for the language because
Python and Numpy are both so full featured and easy to work with
(except for one little thing...), and neither Matlab nor R could
gracefully deal with CGI programming (misguided propaganda
However.... I have spent a huge amount of my time fixing and bending
my head around off-by-one errors caused by trying to index matrices
using 0 to n-1. The problem is two-fold (threefold if you count my
limited IQ...): one, all the formulas in the literature use 1 to n
indexing except for some small exceptions. Second and more important,
it is far more natural to program if the indices are aligned with the
counts of the elements (I think there is a way to express that idea in
modern algebra but I can't recall it). This lets you say "how many
are there? Three--ok, grab the third one and do whatever to it" etc.
Or "how many? zero--ok don't do anything". With zero-based indexing,
you are always translating between counts and indices, but such
translation is never a problem in one-based indexing.
Given the long history of python and its ancestry in C (for which zero
based indexing made lots of sense since it dovetailed with thinking in
memory offsets in systems programming), there is probably nothing to
be done now. I guess I just want to vent, but also to ask if anyone
has found any way to deal with this issue in their own scientific
Or maybe I am the only with this problem, and if I were a real
programmer would translate into zero indexing without even
Anyway, thanks for listening...
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Automated temporal reasoning about reactive systems
Results 1 - 10 of 31
- JOURNAL OF THE ACM , 1998
"... Translating linear temporal logic formulas to automata has proven to be an effective approach for implementing linear-time model-checking, and for obtaining many extensions and improvements to
this verification method. On the other hand, for branching temporal logic, automata-theoretic techniques ..."
Cited by 298 (64 self)
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Translating linear temporal logic formulas to automata has proven to be an effective approach for implementing linear-time model-checking, and for obtaining many extensions and improvements to this
verification method. On the other hand, for branching temporal logic, automata-theoretic techniques have long been thought to introduce an exponential penalty, making them essentially useless for
model-checking. Recently, Bernholtz and Grumberg have shown that this exponential penalty can be avoided, though they did not match the linear complexity of non-automata-theoretic algorithms. In this
paper we show that alternating tree automata are the key to a comprehensive automata-theoretic framework for branching temporal logics. Not only, as was shown by Muller et al., can they be used to
obtain optimal decision procedures, but, as we show here, they also make it possible to derive optimal model-checking algorithms. Moreover, the simple combinatorial structure that emerges from the
, 1997
"... We apply the symbolic analysis principle to pushdown systems. We represent (possibly infinite) sets of configurations of such systems by means of finite-state automata. In order to reason in a
uniform way about analysis problems involving both existential and universal path quantification (like mode ..."
Cited by 292 (36 self)
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We apply the symbolic analysis principle to pushdown systems. We represent (possibly infinite) sets of configurations of such systems by means of finite-state automata. In order to reason in a
uniform way about analysis problems involving both existential and universal path quantification (like model-checking for branching-time logics), we consider the more general class of alternating
pushdown systems and use alternating finite-state automata as a representation structure for their sets of configurations. We give a simple and natural procedure to compute sets of predecessors for
this representation structure. We apply this procedure and the automata-theoretic approach to model-checking to define new model-checking algorithms for pushdown systems and both linear and
branching-time properties. From these results we derive upper bounds for several model-checking problems, and we also provide matching lower bounds, using reductions based on some techniques
introduced by Walukiewicz.
- Information and Computation , 1998
"... . This paper presents a prefixed tableaux calculus for Propositional Dynamic Logic with Converse based on a combination of different techniques such as prefixed tableaux for modal logics and
model checkers for ¯-calculus. We prove the correctness and completeness of the calculus and illustrate its f ..."
Cited by 56 (7 self)
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. This paper presents a prefixed tableaux calculus for Propositional Dynamic Logic with Converse based on a combination of different techniques such as prefixed tableaux for modal logics and model
checkers for ¯-calculus. We prove the correctness and completeness of the calculus and illustrate its features. We also discuss the transformation of the tableaux method (naively NEXPTIME) into an
EXPTIME algorithm. 1 Introduction Propositional Dynamic Logics (PDLs) are modal logics introduced in [10] to model the evolution of the computation process by describing the properties of states
reached by programs during their execution [15, 24, 27]. Over the years, PDLs have been proved to be a valuable formal tool in Computer Science, Logic, Computational Linguistics, and Artificial
Intelligence far beyond their original use for program verification (e.g. [4, 12, 14, 15, 24, 23]). In this paper we focus on Converse-PDL (CPDL) [10], obtained from the basic logic PDL by adding the
converse operat...
- ARTIFICIAL INTELLIGENCE , 2000
"... The last years have seen two major advances in Knowledge Representation and Reasoning. First, many interesting problems (ranging from Semi-structured Data to Linguistics) were shown to be
expressible in logics whose main deductive problems are EXPTIME-complete. Second, experiments in automated reaso ..."
Cited by 51 (3 self)
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The last years have seen two major advances in Knowledge Representation and Reasoning. First, many interesting problems (ranging from Semi-structured Data to Linguistics) were shown to be expressible
in logics whose main deductive problems are EXPTIME-complete. Second, experiments in automated reasoning have substantially broadened the meaning of “practical tractability”. Instances of realistic
size for PSPACE-complete problems are now within reach for implemented systems. Still, there is a gap between the reasoning services needed by the expressive logics mentioned above and those provided
by the current systems. Indeed, the algorithms based on tree-automata, which are used to prove EXPTIME-completeness, require exponential time and space even in simple cases. On the other hand,
current algorithms based on tableau methods can take advantage of such cases, but require double exponential time in the worst case. We propose a tableau calculus for the description logic ALC for
checking the satisfiability of a concept with respect to a TBox with general axioms, and transform it into the first simple tableaubased decision procedure working in single exponential time. To
guarantee the ease of implementation, we also discuss the effects that optimizations (propositional backjumping, simplification, semantic branching, etc.) might have on our complexity result, and
introduce a few optimizations ourselves.
- In Proc. of the 15th Nat. Conf. on Artificial Intelligence (AAAI-98 , 1998
"... The problem of modeling semi-structured data is important in many application areas such as multimedia data management, biological databases, digital libraries, and data integration. Graph
schemas (Buneman et al. 1997) have been proposed recently as a simple and elegant formalism for representing se ..."
Cited by 27 (10 self)
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The problem of modeling semi-structured data is important in many application areas such as multimedia data management, biological databases, digital libraries, and data integration. Graph schemas
(Buneman et al. 1997) have been proposed recently as a simple and elegant formalism for representing semistructured data. In this model, schemas are represented as graphs whose edges are labeled with
unary formulae of a theory, and the notions of conformance of a database to a schema and of subsumption between two schemas are defined in terms of a simulation relation. Several authors have
stressed the need of extending graph schemas with various types of constraints, such as edge existence and constraints on the number of outgoing edges. In this paper we analyze the appropriateness of
various knowledge representation formalisms for representing and reasoning about graph schemas extended with constraints. We argue that neither First Order Logic, nor Logic Programming nor
Frame-based languages are satisfactory for this purpose, and present a solution based on very expressive Description Logics. We provide techniques and complexity analysis for the problem of deciding
schema subsumption and conformance in various interesting cases, that differ by the expressive power in the specification of constraints.
- In Proc. KR-02 , 2002
"... In this paper, we study reasoning about actions and planning with incomplete information in a setting where the dynamic system is specified by adopting Linear Temporal Logic (ltl). Specifically,
we study: (i) reasoning about action effects (i.e., projection, historical queries, etc.), in such ..."
Cited by 22 (6 self)
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In this paper, we study reasoning about actions and planning with incomplete information in a setting where the dynamic system is specified by adopting Linear Temporal Logic (ltl). Specifically, we
study: (i) reasoning about action effects (i.e., projection, historical queries, etc.), in such a setting; (ii) when actions can be legally executed, assuming a non-prescriptive approach, where
executing an action is possible in a given situation unless forbidden by the system specification; (iii) the problem of finding conformant plans for temporally extended goals that consist of
arbitrary ltl formulas, thus allowing for expressing sophisticated dynamic requirements.
, 2011
"... Artifacts are entities characterized by data of interest (constituting the state of the artifact) in a given business application, and a lifecycle, which constrains the artifact’s possible
evolutions. In this paper we study relational artifacts, where data are represented by a full fledged relation ..."
Cited by 15 (10 self)
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Artifacts are entities characterized by data of interest (constituting the state of the artifact) in a given business application, and a lifecycle, which constrains the artifact’s possible
evolutions. In this paper we study relational artifacts, where data are represented by a full fledged relational database, and the lifecycle is described by a temporal/dynamic formula expressed in
µ-calculus. We then consider business processes, modeled as a set of condition/action rules, in which the execution of actions (aka tasks, or atomic services) results in new artifact states. We study
conformance of such processes wrt the artifact lifecycle as well as verification of temporal/dynamic properties expressed in µ-calculus. Notice that such systems are infinite-state in general, hence
undecidable. However, inspired by recent literature on database dependencies developed for data exchange, we present a natural restriction that makes such systems finite-state, and the above problems
- In Proc. 7th Int. Joint Conf. Theory and Practice of Software Development (TAPSOFT'97 , 1996
"... . This paper proposes an expressive extension to Propositional Linear Temporal Logic dealing with real time correctness properties and gives an automata-theoretic model checking algorithm for
the extension. The algorithm has been implemented and applied to examples. 1 Introduction In a landmark pap ..."
Cited by 10 (1 self)
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. This paper proposes an expressive extension to Propositional Linear Temporal Logic dealing with real time correctness properties and gives an automata-theoretic model checking algorithm for the
extension. The algorithm has been implemented and applied to examples. 1 Introduction In a landmark paper, [Pn77], Pnueli identified a very general and important class of computing systems now called
`reactive systems' (cf. [HP85] [Pn86]). Characterized by their ongoing behavior, reactive systems and their sub-components interact with an environment over which they have little control. Such
systems, e.g. operating systems, tend to be quite complex and they have necessitated the development of powerful tools for their verification. In [Pn77] it was argued that temporal logic is a highly
appropriate formalism for specifying and verifying the ongoing operation of reactive systems. Propositional Linear Time Logic (PLTL) [Pn77] allows the simple expression of many important system
properties at a ...
- In Description Logics , 1998
"... The problem of modeling semi-structured data is important in many application areas such as multimedia data management, biological databases, digital libraries, and data integration. In this
paper, we base our work on bdfs, which is a formal and elegant model for semistructured data [Buneman et al., ..."
Cited by 10 (0 self)
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The problem of modeling semi-structured data is important in many application areas such as multimedia data management, biological databases, digital libraries, and data integration. In this paper,
we base our work on bdfs, which is a formal and elegant model for semistructured data [Buneman et al., 1997] where schemas are graphs whose edges are labeled with formulae of a theory T. We extend
bdfs with the possibility of expressing constraints and dealing with incomplete information. In particular, we consider different types of constraints, and discuss how the expressive power of the
constraint language may influence the complexity of checking subsumption between schemas. We then set up a framework for defining bdfs schemas under the assumption that the theory T is not complete.
Finally, we propose a new semi-structured data model, which extends bdfs with both constraints and incomplete theories. We present a technique for checking subsumption in a setting where both the
constraints and the theory are expressed in a very powerful language. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=97776","timestamp":"2014-04-17T19:00:27Z","content_type":null,"content_length":"40166","record_id":"<urn:uuid:102bd2c5-bd37-40c8-b8ed-cfcc7610f0ca>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00161-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Universe of Discourse : Imaginary units
Imaginary units
Yesterday I had a phenomenally annoying discussion with the pedants on the IRC #math channel. Someone was talking about square roots, and for some reason I needed to point out that when you are
considering square roots of negative numbers, it is important not to forget that there are two square roots.
I should back up and discuss square roots in more detail. The square root of x, written √x, is defined to be the number y such that y^2 = x. Well, no, that actually contains a subtle error. The error
is in the use of the word "the". When we say "the number y such that...", we imply that there is only one. But every number (except zero) has two square roots. For example, the square roots of 16 are
4 and -4. Both of these are numbers y with the property that y^2 = 16.
In many contexts, we can forget about one of the square roots. For example, in geometry problems, all quantities are positive. (I'm using "positive" here to mean "≥ 0".) When we consider a right
triangle whose legs have lengths a and b, we say simply that the hypotenuse has length √(a^2 + b^2), and we don't have to think about the fact that there are actually two square roots, because one of
them is negative, and is nonsensical when discussing hypotenuses. In such cases we can talk about the square root function, sqrt(x), which is defined to be the positive number y such that y^2 = x.
There the use of "the" is justified, because there is only one such number. But pinning down which square root we mean has a price: the square root function applies only to positive arguments. We
cannot ask for sqrt(-1), because there is no positive number y such that y^2 = -1. For negative arguments, this simplification is not available, and we must fall back to using √ in its full
In high school algebra, we all learn about a number called i, which is defined to be the square root of -1. But again, the use of the word "the" here is misleading, because "the" square root is not
unique; -1, like every other number (except 0) has two square roots. We cannot avail ourselves of the trick of taking the positive one, because neither root is positive. And in fact there is no other
trick we can use to distinguish the two roots; they are mathematically indistinguishable.
The annoying discussion was whether it was correct to say that the two roots are mathematically indistinguishable. It was annoying because it's so obviously true. The number i is, by definition, a
number such that i^2 = -1. This is its one and only defining property. Since there is another number which shares this single defining property, it stands to reason that this other root is completely
interchangeable with i—mathematically indistinguishable from it, in other words.
This other square root is usually written "-i", which suggests that it's somehow secondary to i. But this is not the case. Every numerical property possessed by i is possessed by -i as well. For
example, i^3 = -i. But we can replace i with -i and get (-i)^3 = -(-i), which is just as true. Euler's famous formula says that e^ix = cos x + i sin x. But replacing i with -i here we get e^-ix = cos
x + -i sin x, which is also true.
Well, one of them is i, and the other is -i, so can't you distinguish them that way? No; those are only expressions that denote the numbers, not the numbers themselves. There is no way to know which
of the numbers is denoted by which expression, and, in fact, it does not even make much sense to ask which number is denoted by which expression, since the two numbers are entirely interchangeable.
One is i, and one is -i, sure, but this is just saying that one is the negative of the other. But so too is the other the negative of the one.
One of the #math people pointed out that there is a well-known Im() function, the "imaginary part" function, such that Im(i) = 1, but Im(-i) = -1, and suggested, rather forcefully, that they could be
distinguished that way. This, of course, is hopeless. Because in order to define the "imaginary part" function in the first place, you must start by making an entirely arbitrary choice of which
square root of -1 you are using as the unit, and then define Im() in terms of this choice. For example, one often defines Im(z) as !!z - \bar{z} \over 2i!!. But in order to make this definition, you
have to select one of the imaginary units and designate it as i and use it in the denominator, thus begging the question. Had you defined Im() with -i in place of i, then Im(i) would have been -1,
and vice versa.
Similarly, one #math inhabitant suggested that if one were to define the complex numbers as pairs of reals (a, b), such that (a, b) + (c, d) = (a + c, b + d), (a, b) × (c, d) = (ac - bd, ad + bc),
then i is defined as (0,1), not (0,-1). This is even more clearly begging the question, since the definition of i here is solely a traditional and conventional one; defining i as (0, -1) instead of
(0,1) works exactly as well; we still have i^2 = -1 and all the other important properties.
As IRC discussions do, this one then started to move downwards into straw man attacks. The #math folks then argued that i ≠ -i, and so the two numbers are indeed distinguishable. This would have been
a fine counterargument to the assertion that i = -i, but since I was not suggesting anything so silly, it was just stupid. When I said that the numbers were indistinguishable, I did not mean to say
that they were numerically equal. If they were, then -1 would have only one square root. Of course, it does not; it has two unequal, but entirely interchangeable, square roots.
The that the square roots of -1 are indistinguishable has real content. 1 has two square roots that are not interchangeable in this way. Suppose someone tells you that a and b are different square
roots of 1, and you have to figure out which is which. You can do that, because among the two equations a^2 = a, b^2 = b, only one will be true. If it's the former, then a=1 and b=-1; if the latter,
then it's the other way around. The point about the square roots of -1 is that there is no corresponding criterion for distinguishing the two roots. This is a theorem. But the result is completely
obvious if you just recall that i is merely defined to be a square root of -1, no more and no less, and that -1 has two square roots.
Oh well, it's IRC. There's no solution other than to just leave. [ Addenda: Part 2 Part 3 Part 4 Part 5 ]
[Other articles in category /math] permanent link | {"url":"http://blog.plover.com/math/i.html","timestamp":"2014-04-20T13:18:38Z","content_type":null,"content_length":"19872","record_id":"<urn:uuid:3806bf62-4f8b-410d-8664-bb2c873d924b>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00142-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] Newbie Question, Probability
pearu at cens.ioc.ee pearu at cens.ioc.ee
Wed Dec 27 14:39:18 CST 2006
On Wed, 27 Dec 2006, Christopher Barker wrote:
> Travis Oliphant wrote:
> > It is the
> > combination of SciPy+NumPy+Matplotlib+IPython (+ perhaps a good IDE)
> > that can succeed at being a MATLAB/IDL replacement for a lot of people.
> > What is also needed is a good "package" of it all --- like the Enthon
> > distribution. This requires quite a bit of thankless work.
> I know Robert put some serious effort into "MacEnthon" a while back, but
> is no longer maintaining that, which doesn't surprise me a bit -- that
> looked like a LOT of work.
> However, MacEnthon was much bigger than that just the packages Travis
> listed above, and I think Travis has that right -- those are the key
> ones to do. Let's "just do it!" -- first we need to solve the
> Fortran+Universal binary problems though -- that seems to be the
> technical sticking point on OS-X
Let me add a comment on the Fortran problem (which I assume to be the
(lack of) Fortran compiler problem, right?). I have been working on
f2py rewrite to support wrapping Fortran 90 types among other F90
constructs and as a result we have almost a complete Fortran parser in
Python. It is relatively easy to use this parser to automatically convert
Fortran 77 codes that we have in scipy to C codes whenever no Fortran
compiler is available. Due to lack of funding this work has been
freezed for now but I'd say that there is a hope to resolve the Fortran
compiler issues for any platform in future.
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Experimental Investigation of the Reynolds Number’S Effect on the Aerodynamic Characteristics of a Horizontal Axis Wind Turbine of the Göttingen 188 Airfoil Type
Zied Driss^1, , Sarhan Karray^1, Ali Damak^1, Mohamed Salah Abid^1
^1Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax (US), Tunisia
In this paper, a study of the effect of Reynolds number on the aerodynamic characteristics of a horizontal axis wind turbine equipped by three adjustable blades of the Göttingen 188 airfoil has been
developed. Particularly, different aerodynamic regimes defined by Reynolds numbers were investigated. To achieve this, an open wind tunnel has been used to determine the global characteristics of the
wind turbine. The obtained results consist of the recovered power, the exerted torque on the rotor in static and in dynamic modes as well as power coefficient and torque coefficient. This work has
been developed at Laboratory of Electro-Mechanic Systems (LASEM) of the National School of Engineers of Sfax (ENIS).
At a glance: Figures
Keywords: Göttingen 188 airfoil, wind turbine, horizontal axis, aerodynamic characteristics, wind tunnel
American Journal of Mechanical Engineering, 2013 1 (5), pp 143-148.
DOI: 10.12691/ajme-1-5-7
Received October 10, 2013; Revised October 23, 2013; Accepted October 24, 2013
© 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
• Driss, Zied, et al. "Experimental Investigation of the Reynolds Number’S Effect on the Aerodynamic Characteristics of a Horizontal Axis Wind Turbine of the Göttingen 188 Airfoil Type." American
Journal of Mechanical Engineering 1.5 (2013): 143-148.
• Driss, Z. , Karray, S. , Damak, A. , & Abid, M. S. (2013). Experimental Investigation of the Reynolds Number’S Effect on the Aerodynamic Characteristics of a Horizontal Axis Wind Turbine of the
Göttingen 188 Airfoil Type. American Journal of Mechanical Engineering, 1(5), 143-148.
• Driss, Zied, Sarhan Karray, Ali Damak, and Mohamed Salah Abid. "Experimental Investigation of the Reynolds Number’S Effect on the Aerodynamic Characteristics of a Horizontal Axis Wind Turbine of
the Göttingen 188 Airfoil Type." American Journal of Mechanical Engineering 1, no. 5 (2013): 143-148.
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1. Introduction
Wind energy is one such energy source that has little environmental impact, little adverse health effects, negligible security concerns, and is completely renewable. Currently wind energy is the
fastest growing source of energy. Wind energy has proven itself as a viable source. Nowadays, subsidy programs were required, to stimulate the installation of such a large wind energy capacity. As
such, there is still a lot of work needed to develop the technology, so that it is cost competitive with conventional sources. In this context, HU ^[1] developed an experimental investigation on the
properties of the near wake behind the rotor of a Horizontal-Axis Wind Turbine (HAWT) at model scale. Measurements were made with a stationary slanted hot-wire anemometer using the technique of
phase-locked averaging. The primary aim is to study the formation and development of the three-dimensional wake. Five axial locations were chosen within four chord lengths of the blades over a range
of tip speed ratios. The results show that during the downstream development of the wake, the wake centre traces a helical curve with its rotation direction opposite to that of the rotor. The
distribution of mean velocity behind the HAWT rotor reveals an expansion and a decay of the three-dimensional wake. The shapes of the mean velocity distribution are similar along the blades span at
the same downstream axial location. It is shown that the turbulence levels in the wake are higher than those in the non-wake region. The circumferential component and the radial component of the
turbulence intensity are higher than the axial component. This study offers some food of thought for better understanding of the physical features of the flow field as well as the performance of
HAWT. Grant et al. ^[2] described a wind-tunnel study of the wake dynamics of an operational, horizontal-axis wind turbine. The behaviour of the vorticity trailing from the turbine blade tips and the
effect of was interference on wake development were considered. Laser sheet visualisation (LSV) techniques were used to measure the trajectories of the trailing vorticity under various conditions of
turbine yaw and blade azimuth. Selected results obtained in the experimental study were compared with the predictions of a prescribed wake model and are being used in the further development of the
method. Barnsley and Wellicome ^[3] made surface pressure and near rotor velocity measurements, using a laser Doppler facility, at six radial positions for a 1m diameter two-bladed rotor, over the
stalling range of tip speed ratios at typical Reynolds' numbers of 300 000. Velocity measurements have been used to quantify local incidence and results illustrate clearly the development of enhanced
lift incidence due to a delay in the loss of leading edge suction peaks compared to 2D behaviour. Static hysteresis in the stall behaviour has also been identified. Power comparisons with full scale
data indicate fairly good agreement in peak power coefficient and tip speed ratio at the onset of stall but also show significant Reynolds number effects in the stalling and post stall regions. Ting
et al. ^[4] developed wind chiller in CCT Lab. Directly uses wind force to drive refrigeration system and hence reduces two times energy conversions between mechanical and electrical energies. The
wind chiller needs high wind speed for its effective work due to the large working torque is required by the compressor. For the purpose of enlarging the applied wind field by the wind machine, this
work aims to develop a dual system of wind chiller integrated with wind generator. The integrated wind generator can use the wind energy which cannot effectively drive the compressor. Therefore, the
new developed dual system can apply larger range of the wind field and further increase the total working efficiency of the wind machine. A programmable logic controller (PLC) is applied in this wind
forced dual system to select the wind chiller or the wind generator separately in terms of the rotational speed of the wind machine. In this work, the wind chiller is switched on while the
accelerated rotational speed reaches 80 rpm and off while the decelerated rotational speed reaches 60 rpm. The integrated wind generator is switched on while the decelerated rotational speed reaches
60 rpm and off while the decelerated rotational speed reaches 40 rpm. The two apparatuses in the dual system always work separately. The results show that there is ca. 18.5% increment of effective
working efficiency which is captured by the wind generator. According to these studies, we can confirm that there are several areas for further development in the design of wind turbines. There are
many opportunities to improve the mechanical, structural and electrical systems. The greatest potential for improvement, in both short and long term development, is in the field of aerodynamics. Chen
and Liou ^[5] quantitatively investigated the effects of tunnel blockage on the turbine power coefficient in wind tunnel tests of small horizontal-axis wind turbines (HAWTs). The blockage factor was
determined by measuring the tunnel velocities with and without rotors using a pitot-static tube under various test conditions. Results showed that the BF depends strongly on the rotor tip speed
ratio, the blade pitch angle, and the tunnel blockage ratio. This study also showed that the blockage correction is less than 5% for a BR of 10%, which confirms that no blockage correction for a BR
less than 10% in literatures is acceptable. Driss et al. ^[6] studied the effect of the 913 wind turbine airfoils. The numerical results obtained in the cases of the airfoils type SD2030 and BM4640
are particularly predicted and analyzed. The results, from application of the computational fluid dynamics (CFD) code "Fluent", are presented in the transversal and longitudinal planes of the
considered control volume. The Navier-Stokes equations are solved by a finite volume discretization method. The turbulence model used is the RNG k-ε. The objective is to study the effect of the
airfoil type on the aerodynamic structure flow around the horizontal-axis wind turbine. Driss and Abid ^[7] studied the aerodynamic characteristics on an open circuit tunnel. They are interested to
verify that the test vein provides a uniform out flow, a high-speed and a low-turbulence. The numerical results from the application of the CFD code are presented in the transversal and longitudinal
planes of the wind tunnel. The comparison between the numerical results and the global experimental results, conducted within a hot wire anemometry AM-4204 model, confirm the validity of the
numerical method ^[8, 9, 10].
On the basis of the previous studies, it appears that there is paucity on the study of small horizontal axis wind turbine and particularly on the Göttingen airfoil type. For this reason, an
experimental investigation is presented in this paper to study the effect of the Reynolds number on the aerodynamic characteristics of a horizontal axis wind turbine equipped by three adjustable
blades of the Göttingen 188 airfoil.
2. Geometrical Arrangement
The present work focuses on the horizontal axis wind turbine. The wind turbine is constituted of three adjustable blades of the Göttingen 188 airfoil. In this application, the airfoil is
characterised by a blade length equal to l = 100 mm and a chord length equal to C = 43 mm. The radius rotor is equal to R = 157 mm (Figure 1). Indeed, the wind turbine is equipped by a system to
change the wedging angle β, measured between the blade rotation plane and the chorde. The experimental investigation has been developed using wind tunnel. The wind turbine has been introduced through
a hole situated on the top of the test vein. Particularly, a vertical axis is used to maintain the rotor. This installation permits to study the effect of the wedging angle and the Reynolds number on
the global characteristics of the wind turbine (Figure 2).
3. Experimental Method
In this study, an experimental investigation is conducted on an open wind tunnel designed and realized in our Laboratory of Electro-Mechanic Systems at National School of Engineers of Sfax ^[7, 8, 9,
10]. To do this, the necessary equipments for the global characterization of the horizontal axis wind turbines have been installed. This involved the manufacture of an open wind tunnel and its
instrumentation. This system has been designed and realised to establish the aerodynamic characteristics of small wind turbines. It mainly consists of five compartments: a settling chamber, a
collector, a test vein, a diffuser and a drive section. A vacuum cleaner with variable speed draws the air through the test vein. The honeycomb placed at the input of this room provides a uniform
airflow. The hot wire anemometry technique has been used to justify the nature of the flow generated by the system and to ensure a uniform airflow in the test vein. The experimental device has been
used to predict the aerodynamic behaviour and investigate the conditions experienced by the wind turbines placed in the air flow. The rotor axis has been placed in the middle of the test vein having
a cross section area of 400 mm x 400 mm. By changing the rotation frequency of the vacuum cleaner SV0081C5-1F type, the wind tunnel exit-air velocity was controlled. The entire tests have been
conducted within a hot wire anemometry AM-4204 model to measure the air velocity. In the test vein, the maximum air velocity value is equal to 12.7 m/s. The rotational speed of the wind turbine rotor
was measured with a digital tachometer CA-27 model. To measure the static torque on the rotor shaft, a torque meter TQ-8800 model has been used. The dynamic torque exerted on the rotor shaft was
measured with a DC generator which transforms the torque on its axis at an electrical current. For that the generator, coupled to the dynamometer RZR-2102 model, display simultaneously the shape
speed and the dynamic torque. This dynamometer has been used to provide mechanical power to the generator which delivers an electric current in a resistive load. Torque measurement integrated into
the dynamometer, allows tracing the calibration curve that connects the electric current supplied by the generator to the dynamic torque (Figure 3). This calibration curve serves for determination of
the dynamic torque after referring to the value of the electric current supplied by the generator. This strategy offered a comprehensive understanding of aerodynamic characteristics of different
4. Experimental Results
In this study, different flow regimes defined by the Reynolds numbers equals to Re = 165093, Re = 194350, Re = 217338, Re = 242415, Re = 257044 and Re = 265403 are investigated.
4.1. Power
Figure 4 presents the variation of the recovered power depending on the revolution speed Ω of the horizontal axis-wind turbine for different Reynolds numbers equals to Re = 165093, Re = 194350, Re =
217338, Re = 242415, Re = 257044 and Re = 265403. According to these results, the presented curve shows a parabolic branch. The recovered power decreases with the increase of the revolution speed.
Indeed, it’s noted that the Reynolds number has a direct effect on the results. In these conditions, the recovered power increases with the increase of the Reynolds number. The maximal values of the
recovered power increase also with the increase of the Reynolds number. Particularly, for the Reynolds number Re = 165093, the maximal value of the recovered power is equal to P = 8.4 W for a
revolution speed equal to Ω = 885 rpm. Indeed, the minimal and maximal revolution speeds increase with the increase of the Reynolds number. In fact, for a Reynolds number Re = 165093, the revolution
speed of the wind turbine varies between Ω = 885 rpm and Ω = 1090 rpm. This interval variation increases for a Reynolds number equal to Re = 265403. In this case, the revolution speed varies between
Ω = 2010 rpm and Ω = 2100 rpm.
4.2. Power Coefficient
Figure 5 presents the variation of the power coefficient depending on the specific velocity λ of the horizontal axis-wind turbine for different Reynolds numbers equal to Re = 165093, Re = 194350, Re
= 217338, Re = 242415, Re = 257044 and Re = 265403. According to these results, the presented curve shows a parabolic branch. Indeed, it’s noted that the Reynolds number has a direct effect on the
results. Particularly, the power coefficient increases with the increase of the Reynolds number at the same specific velocity λ. In these conditions, for the Reynolds number Re = 165093, the maximal
value of the power coefficient is equal to C[p] = 0.21 for a specific velocity equal to λ = 2. Indeed, the minimal and maximal specific velocity values increase with the increase of the Reynolds
number. In fact, for a Reynolds number Re = 165093, the specific velocity of the wind turbine varies between λ = 2 and λ = 2.3. However, with a Reynolds number Re = 265403, the specific velocity
varies between λ = 2.63 and λ = 2.72.
4.3. Dynamic Torque
Figure 6 presents the variation of the dynamic torque depending on the revolution speed Ω of the horizontal axis-wind turbine for different Reynolds numbers equal to Re = 165093, Re = 194350, Re =
217338, Re = 242415, Re = 257044 and Re = 265403. According to these results, the dynamic torque value decreases with the increase of the revolution speed Ω. Also, it’s clear that the Reynolds number
has a direct effect on the cartographies presentation. Particularly, the dynamic torque value increases with the increase of the Reynolds number at the same revolution speed Ω. In fact, the maximal
value of the dynamic torque reaches M[d] = 0,072 N.m for the Reynolds number equal to Re = 165093 and a revolution speed equal to Ω = 925 rpm. However, the maximal value of the dynamic torque reaches
M[d] = 0,088 N.m for the Reynolds number equal to Re = 265403 and a revolution speed equal to Ω = 2002 rpm. This confirms that the maximal value increases with the increase of the Reynolds number.
Indeed, it’s clear that the minimal value of the dynamic torque increases with the increase of the Reynolds number. For example, the minimal value of the dynamic torque reaches M[d] = 0,008 N.m for
the Reynolds number equal to Re = 265403 and a revolution speed equal to Ω = 2140 rpm.
4.4. Dynamic Torque Coefficient
Figure 7 presents the variation of the dynamic torque coefficient depending on the specific velocity λ of the horizontal axis-wind turbine for different Reynolds numbers equal to Re = 165093, Re =
194350, Re = 217338, Re = 242415, Re = 257044 and Re = 265403. According to these results, it’s noted that the presented curve shows a parabolic branch. In these conditions, the dynamic torque
coefficient decreases with the increase of the specific velocity λ. Indeed, it’s clear that the Reynolds number has a direct effect on the cartographies presentation. The maximal values of the
dynamic torque coefficient decrease also with the increase of the Reynolds number. In fact, for the Reynolds number Re = 165093, the maximal value of the dynamic torque coefficient is equal to C[Md]
= 0.108 for a specific velocity equal to λ = 1.94. However, for the Reynolds number Re = 265403, the maximal value of the dynamic torque coefficient is equal to C[Md] = 0.059 for a specific velocity
equal to λ = 2.52.
4.5. Static Torque
Figure 8 presents the variation of the static torque depending on the wedging angle β of the horizontal axis-wind turbine for different Reynolds numbers equals to Re = 165093, Re = 194350, Re =
217338, Re = 242415, Re = 257044 and Re = 265403. According to these results, the presented curve shows a parabolic branch. Indeed, it’s noted that the Reynolds number has a direct effect on the
cartographies presentation.
In these conditions, the static torque increases with the increase of the Reynolds number. Particularly, the maximal values of the static torque increase with the increase of the Reynolds number. In
fact, for the Reynolds number Re = 165093, the maximal value of the static torque is equal to M[s] = 2.8 N.m for a wedging angle equal to β = 50°.
4.6. Static Torque Coefficient
Figure 9 presents the variation of the static torque coefficient depending on the wedging angle β of the horizontal axis-wind turbine for different Reynolds numbers equal to Re = 165093, Re = 194350,
Re = 217338, Re = 242415, Re = 257044 and Re = 265403. According to these results, it’s noted that the presented curve shows a parabolic branch. Indeed, it’s clear that the Reynolds number hasn’t an
effect on the cartographies presentation. In these conditions, the maximal value of the static torque coefficient value is equal to C[Ms] = 0.06 for a wedging angle equal to β = 50°. Indeed, the
static torque coefficient value becomes null for a wedging angle equal to β = 0°.
5. Conclusion
In this paper, an experimental investigation has been developed to study the Reynolds numbers effect on the global characteristics of the horizontal axis wind turbine equipped by three adjustable
blades of the Göttingen 188 airfoil. Evaluation of the rotor performance based on the power and torque produced is reported to optimize and to improve the experimental conditions of the wind turbine.
According to the experimental results, it’s noted that the Reynolds number has a direct effect on the global characteristics, except the static torque coefficient.
In the future, we intend to change the blade profiles to optimize the output of the wind turbines. Therefore, we propose to develop an experimental investigation within a particle image velocimetry
laser (PIV) system for a finer survey of the local out-flow features. Also, it is interested to find the necessary material for the manufacture of the wind turbines.
A: Swept area of the rotor (m^2)
C: chord length (m)
D: diameter of the rotor (m)
l: length of the blade (m)
M: Torque (N.m)
P: power (W)
R: radius of the rotor (m)
V: Speed of air (m/s)
Greek Letters
β: wedging angle (°)
μ: dynamic viscosity of the fluide (Pa.s)
Ω: revolution speed of the rotor (rpm)
ρ: density of fluid (kg.m^-3)
Adimensionnels numbers
[ ]: Specific velocity
[ ]Coefficient of torque
s: static
d: dynamic
[1] D. HU, Near wake of a model horizontal-axis wind turbine, Journal of Hydrodynamics 21 (2) (2009) 285-291. doi: 10.1016/S1001-6058(08)60147-X
[2] I. Grant, M. Mo, X. Pan, P. Parkin, J. Powell, H. Reinecke, K. Shuang, F. Coton, D. Lee, An experimental and numerical study of the vortex laments in the wake of an operational, horizontal-axis,
wind turbine, Journal of Wind Engineering and Industrial Aerodynamics 85 (2000) 177-189. doi: 10.1016/S0167-6105(99)00139-7
[3] M.J. Barnsley, J.F. Wellicome, Wind tunnel investigation of stall aerodynamics for a 1.0 m horizontal axis rotor, Journal of Wind Engineering and Industrial Aerodynamics 39 (1992) 11-21. doi:
[4] C.C. Ting, C.W. Lai, C.B. Huang, Developing the dual system of wind chiller integrated with wind generator, Applied Energy 88 (2011) 741-747. doi: 10.1016/j.apenergy.2010.09.002
[5] Z. Driss, W. Triki, M. S. Abid, Numerical investigation of the Rutland 913 wind turbine airfoils effect on the aerodynamic structure flow, Science Academy Transactions on Renewable Energy Systems
Engineering and Technology 1 (4) (2011) 116-123.
[6] T.Y. Chen, L.R. Liou, Blockage corrections in wind tunnel tests of small horizontal-axis wind turbines, Experimental Thermal and Fluid Science 35 (2011) 565-569. doi: 10.1016/
[7] Z. Driss, M. S. Abid, Numerical and experimental study of an open circuit tunnel: aerodynamic characteristics, Science Academy Transactions on Renewable Energy Systems Engineering and Technology
2 (1) (2012) 116-123.
[8] Z. Driss, A. Damak, H. Kchaou, M.S. Abid, Experimental investigation on wind tunnel, Tunisian Japanese Symposium on Science, Society and (2011) 1-4.
[9] A. Damak, Z. Driss, H. Kchaou, M.S. Abid, Conception et réalisation d’une soufflerie à aspiration, 4^ème Congrès International Conception et Modélisation des Systèmes Mécaniques (2011) 1-7.
[10] A. Damak, Z. Driss, M.S. Abid, Experimental investigation of helical Savonius rotor with a twist of 180°, Renewable Energy 52 (2013) 136-142. doi: 10.1016/j.renene.2012.10.043 | {"url":"http://pubs.sciepub.com/ajme/1/5/7/index.html","timestamp":"2014-04-16T16:00:18Z","content_type":null,"content_length":"91245","record_id":"<urn:uuid:b0c7e042-9729-4a44-977f-3b7b7091bb4e>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00355-ip-10-147-4-33.ec2.internal.warc.gz"} |
Al-Khwarizmi - Islamic Mathematics - The Story of Mathematics
Muhammad Al-Khwarizmi (c.780-850 AD)
One of the first Directors of the House of Wisdom in Bagdad in the early 9th Century was an outstanding Persian mathematician called Muhammad Al-Khwarizmi. He oversaw the translation of the major
Greek and Indian mathematical and astronomy works (including those of Brahmagupta) into Arabic, and produced original work which had a lasting influence on the advance of Muslim and (after his works
spread to Europe through Latin translations in the 12th Century) later European mathematics.
The word “algorithm” is derived from the Latinization of his name, and the word "algebra" is derived from the Latinization of "al-jabr", part of the title of his most famous book, in which he
introduced the fundamental algebraic methods and techniques for solving equations.
Perhaps his most important contribution to mathematics was his strong advocacy of the Hindu numerical system, which Al-Khwarizmi recognized as having the power and efficiency needed to revolutionize
Islamic and Western mathematics. The Hindu numerals 1 - 9 and 0 - which have since become known as Hindu-Arabic numerals - were soon adopted by the entire Islamic world. Later, with translations of
Al-Khwarizmi’s work into Latin by Adelard of Bath and others in the 12th Century, and with the influence of Fibonacci’s “Liber Abaci” they would be adopted throughout Europe as well.
An example of Al-Khwarizmi’s “completing the square” method for solving quadratic equations
Al-Khwarizmi’s other important contribution was algebra, a word derived from the title of a mathematical text he published in about 830 called “Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala”
(“The Compendious Book on Calculation by Completion and Balancing”). Al-Khwarizmi wanted to go from the specific problems considered by the Indians and Chinese to a more general way of analyzing
problems, and in doing so he created an abstract mathematical language which is used across the world today.
His book is considered the foundational text of modern algebra, although he did not employ the kind of algebraic notation used today (he used words to explain the problem, and diagrams to solve it).
But the book provided an exhaustive account of solving polynomial equations up to the second degree, and introduced for the first time the fundamental algebraic methods of “reduction” (rewriting an
expression in a simpler form), “completion” (moving a negative quantity from one side of the equation to the other side and changing its sign) and “balancing” (subtraction of the same quantity from
both sides of an equation, and the cancellation of like terms on opposite sides).
In particular, Al-Khwarizmi developed a formula for systematically solving quadratic equations (equations involving unknown numbers to the power of 2, or x^2) by using the methods of completion and
balancing to reduce any equation to one of six standard forms, which were then solvable. He described the standard forms in terms of "squares" (what would today be "x^2"), "roots" (what would today
be "x") and "numbers" (regular constants, like 42), and identified the six types as: squares equal roots (ax^2 = bx), squares equal number (ax^2 = c), roots equal number (bx = c), squares and roots
equal number (ax^2 + bx = c), squares and number equal roots (ax^2 + c = bx), and roots and number equal squares (bx + c = ax^2).
Al-Khwarizmi is usually credited with the development of lattice (or sieve) multiplication method of multiplying large numbers, a method algorithmically equivalent to long multiplication. His lattice
method was later introduced into Europe by Fibonacci.
In addition to his work in mathematics, Al-Khwarizmi made important contributions to astronomy, also largely based on methods from India, and he developed the first quadrant (an instrument used to
determine time by observations of the Sun or stars), the second most widely used astronomical instrument during the Middle Ages after the astrolabe. He also produced a revised and completed version
of Ptolemy's “Geography”, consisting of a list of 2,402 coordinates of cities throughout the known world.
<< Back to Islamic Mathematics Forward to Medieval European Mathematics >>
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Home | The Story of Mathematics | List of Important Mathematicians | Glossary of Mathematical Terms | Sources | Contact
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Memoirs on Differential Equations and Mathematical Physics
Table of Contents: Volume 41, 2007
Nikolay Viktorovich Azbelev
Mem. Differential Equations Math. Phys. 41 (2007), pp. 1-3.
Mikhail Drahlin
Mem. Differential Equations Math. Phys. 41 (2007), pp. 5-10.
E. I. Bravyi
On the Solvability of the Cauchy Problem for Systems of Two Linear Functional Differential Equations
Mem. Differential Equations Math. Phys. 41 (2007), pp. 11-26.
download pdf file.
R. Hakl and S. Mukhigulashvili
On a Periodic Boundary Value Problem for Third Order Linear Functional Differential Equations
Mem. Differential Equations Math. Phys. 41 (2007), pp. 27-42.
download pdf ile.
I. Kiguradze and Z. Sokhadze
A Priori Estimates of Solutions of Systems of Functional Differential Inequalities, and Some of Their Applications
Mem. Differential Equations Math. Phys. 41 (2007), pp. 43-67.
download pdf file.
A. Lomtatidze, Z. Opluštil, and J. Šremr
On a Nonlocal Boundary Value Problem for First Order Linear Functional Differential Equations
Mem. Differential Equations Math. Phys. 41 (2007), pp. 69-85.
download pdf file.
V. V. Malygina
Positiveness of the Cauchy Function and Stability of a Linear Differential Equation with Distributed Delay
Mem. Differential Equations Math. Phys. 41 (2007), pp. 87-96.
download pdf file.
J. Rebenda
Asymptotic Properties of Solutions of Real Two-Dimensional Differential Systems with a Finite Number of Constant Delays
Mem. Differential Equations Math. Phys. 41 (2007), pp. 97-114.
download pdf file.
A. Rontó and A. Samoilenko
Unique Solvability of Some Two-Point Boundary Value Problems for Linear Functional Differential Equations with Singularities
Mem. Differential Equations Math. Phys. 41 (2007), pp. 115-136.
download pdf file.
V. Z. Tsalyuk
Two Versions of the W-Method for Quadratic Variational Problems with Many Linear Constraints
Mem. Differential Equations Math. Phys. 41 (2007), pp. 137-149.
download pdf file.
Malkhaz Ashordia
On a Priori Estimates of Bounded Solutions of Systems of Linear Generalized Ordinary Differential Inequalities
Mem. Differential Equations Math. Phys. 41 (2007), pp. 151-156.
download pdf file.
N. A. Izobov, S. E. Karpovich, L. G. Krasnevsky, and A. V. Lipnitsky
Quasi-Integrals of Three-Dimensional Linear Differential Systems with Skew-Symmetric Coefficient Matrices
Mem. Differential Equations Math. Phys. 41 (2007), pp. 157-162.
download pdf file.
Sulkhan Mukhigulashvili
On a Priori Estimates of Solutions of Nonlinear Functional Differential Inequalities of Higher Order with Boundary Conditions of Periodic Type
Mem. Differential Equations Math. Phys. 41 (2007), pp. 163-165.
download pdf file.
Vakhtang Badagadze
Shalva Gelashvili
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Foundations of Computer Science - Tick Errata
Latest version of ML Tick 6
• The submission date should read Wednesday 14th January 2009
• When scaling the co-ordinate system a single value is used to specify both the width and the height of the square. This means that unless your output image is square the rendering will have an
aspect ratio distortion. You may rectify this if you wish but it is acceptable to complete the tick as written. (Thanks to Sebastian).
• In part 7 you are given 256 as the multiplicative factor - it should be 255 (thus giving a total of 256 different values). Again, it is acceptable to fix this in your submission or to complete
the tick as written. (Thanks to Rhodri). | {"url":"http://www.cl.cam.ac.uk/teaching/0809/FoundsCS/ERRATA/","timestamp":"2014-04-18T00:16:28Z","content_type":null,"content_length":"3280","record_id":"<urn:uuid:e87e04d6-af8a-4eb0-94cd-2bb60c238956>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00429-ip-10-147-4-33.ec2.internal.warc.gz"} |
Properties of permutations with unknown pattern avoidance descriptions
up vote 7 down vote favorite
Many properties of permutations can be stated in terms of classical patterns. For example:
• a permutation is stack-sortable if and only if it avoids 231 (Knuth 1975)
• a permutation corresponds to a smooth Schubert variety if and only if it avoids 1324 and 2143 (Lakshmibai and Sandhya 1990)
For other properties we need a stronger notion of a pattern, e.g., the mesh patterns introduced by Brändén and Claesson (2011). For example:
• a permutation corresponds to a factorial Schubert variety if and only if it avoids 1324 and (2143,{(2,2)}) (These are the so-called forest-like permutations, Bousquet-Mélou and Butler 2007)
• a permutation is sortable in two passes through a stack if and only if it avoids 2341 and (3241,{(1,4)}) (These are the so-called West-2-stack-sortable permutations, West 1990)
There are also properties which have not been translated into patterns (to my knowledge):
The Question
What permutation properties do you know that have not been described by the avoidance of patterns
I recently wrote an algorithm that given a finite set of permutations outputs the mesh patterns that the permutations avoid. This algorithm is called BiSC (derived from the last names of three people
that inspired me to write the algorithm) and can conjecture the descriptions given in the first two lists above. It is available at http://staff.ru.is/henningu/programs/bisc/bisc.html and described
in the paper http://arxiv.org/abs/1211.7110.
This is a community wiki question since it there is obviously not a single best answer
What means permutation avoids patern? – Alexander Chervov Dec 18 '12 at 19:50
add comment
4 Answers
active oldest votes
Here's one idea. For every permutation $\pi$ of length $n$, there are $n^2+1$ permutation of length $n+1$ containing $\pi$. However, once you look at permutations of length $n+2$, this
quantity depends on $\pi$. Ray and West gave a proof that for $\pi$ of length $n$ the number of permutations of length $n+2$ containing $\pi$ is $$ (n^4+2n^3+n^2+4n+4-2j)/2, $$ where
$0\le j\le k-1$ depends on $\pi$. Perhaps you could give a description of this statistic in terms of patterns of $\pi$?
up vote 3 down
vote accepted References and a bit more discussion can be found in this paper: http://www.math.ufl.edu/~vatter/publications/pp2007-problems/
add comment
Derangements. More generally, properties that allow superexponentially many permutations.
up vote 1 down vote
add comment
I hope I understood the question correctly. I have a feeling that questions on permutations of algebraic as opposed to combinatorial nature, could be candidates.
Lakshmibai and Sandhya's theorem is a geometric question and it is a significant theorem because it reduces geometry to combinatorics. With this understanding of your question let me
attempt to give four examples:
(1) A permutation being of specific order $m$ .
Suppose we attempt pattern avoidance like: for any $k$ relatively prime to $m$ it should not have a length $k$ cycle. A permutation of order, for example $m^2$, will also satisfy
up vote 0 down that criterion and will be accepted wrongly.
(2) Permutation being even. (avoidance criterion may not work: because presence of an even number of cycles of any particular length, as opposed odd number of them, will be ok)
(3) Some irreducible character vanishing in it. This is conjugacy class question. Can be argued similarly
(4) Commuting with another specific permutation.
add comment
A source of interesting examples may come from infinite groups with finite presentation, possibly extending your methods to words instead of just permutations (i.e. allowing repetitions).
Given a set of generators $\{x,y,\dots,z\}$ of the group $G$, which words in the alphabets of $\{x,\, x^{-1},y,\, y^{-1},\dots z,\, z^{-1}\}$ correspond to minimal length presentations of
up vote elements of $G$? In this generality, of course, the problem is intractable, but in principle one optimal answer could be given (and actually is, in some concrete cases) precisely in terms of
0 down avoidance of a list of patterns (starting, of course, from avoiding $xx^{-1}$). Clearly, an algorithm as yours may prove very useful to formulate conjecture about patterns.
add comment
Not the answer you're looking for? Browse other questions tagged permutations pattern-avoidance co.combinatorics or ask your own question. | {"url":"https://mathoverflow.net/questions/115636/properties-of-permutations-with-unknown-pattern-avoidance-descriptions","timestamp":"2014-04-17T07:02:23Z","content_type":null,"content_length":"64731","record_id":"<urn:uuid:4c2d6b70-f4b0-4d8e-b601-e97078db43df>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00334-ip-10-147-4-33.ec2.internal.warc.gz"} |
Subtracting Without Borrowing
Subtraction without borrowing is slightly harder than simply subtracting one digit numbers, as the previous section showed, but it is similar in the sense that you are still going to be subtracting
one digit numbers, you'll just be doing it more than once for the same problem. The problems will be lined up in columns, so you would subtract the column on the right first and then work your way to
the left. Examples are shown below:
Subtracting Two Digit Minus One Digit Numbers
As you get better with subtraction, you’ll be asked to subtract one digit numbers from two digit numbers. For example, a two digit minus one digit number subtraction problem may look like this:
One of the most important things to remember about subtraction with more than one digit is that you have to keep your place values lined up. In our problem above, the ones’ columns are aligned at the
right. This is very important. After you make sure the columns are lined up, all you have to do is subtract by column, from right to left. When you’re ready to start subtraction, make sure you start
with the ones’ column, or the column furthest to the right. We have this circled in red in our example below:
When we subtract the ones column, 6 – 4, we get 2 as our answer, so we write it in the ones’ column beneath the answer bar. Next, we have to subtract the tens’ column. In this example, there is only
one digit in the tens’ column, so we simply bring it down into our answer, as shown by the red arrow in the example below.
Thus, 12 is our final answer.
Let’s try another example. This time, we’ll let you try working it out, and then you can compare your answer with ours. Here’s the problem:
Now, try it on your own. When you’re done, compare it to our answer.
Does your answer agree with ours? If not, go back and re-subtract the ones column. Then make sure you brought down the tens’ column, since there was not another number to subtract. If you did get 42,
congrats! You did it correctly!
Subtracting Two Digit Minus Two Digit Numbers
The next step in subtraction is taking two digit numbers and subtracting other two digit numbers. A two digit minus two digit number subtraction problem would look like this:
The most important thing you need to remember in doing a problem like this is that you always start with the ones’ column (or the column furthest to the right) and work your way to the left of the
problem. This may seem wrong, because normally we do things from left to right, not from right to left. However, when doing subtraction, we always want to start at the right and work towards the
The first part of the problem, starting with the ones’ column, looks like this:
Notice the ones’ column is circled, because we started with it. Now, let’s look at the second step in solving the problem:
Thus, our final answer is 55.
Let’s try that one more time to make sure you understand. Your new problem is:
Now, following the same pattern as above, see if you can work through this one on your own, and then check your answer with ours.
Thus, our final answer is 53.
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to access more Math resources like . WyzAnt Resources features blogs, videos, lessons, and more about Math and over 250 other subjects. Stop struggling and start learning today with thousands of free | {"url":"http://www.wyzant.com/resources/lessons/math/elementary_math/subtraction/subtracting_without_borrowing","timestamp":"2014-04-16T13:35:17Z","content_type":null,"content_length":"74468","record_id":"<urn:uuid:b9394c71-cb28-4ac5-a739-8179e1728906>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00490-ip-10-147-4-33.ec2.internal.warc.gz"} |
intermediate algebra
Posted by Lueshelle on Friday, April 13, 2012 at 6:59pm.
Find the quotient...Show work.....y^4+y+1/y^2-9
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Bayesian classi cation (autoclass): Theory and results
Results 1 - 10 of 20
- Proceedings of the 14 th Conference on Uncertainty in Artificial Intelligence , 1998
"... 1 ..."
- LEARNING IN GRAPHICAL MODELS , 1995
"... ..."
- Machine Learning , 1999
"... . This paper shows that the accuracy of learned text classifiers can be improved by augmenting a small number of labeled training documents with a large pool of unlabeled documents. This is
important because in many text classification problems obtaining training labels is expensive, while large qua ..."
Cited by 803 (17 self)
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. This paper shows that the accuracy of learned text classifiers can be improved by augmenting a small number of labeled training documents with a large pool of unlabeled documents. This is important
because in many text classification problems obtaining training labels is expensive, while large quantities of unlabeled documents are readily available. We introduce an algorithm for learning from
labeled and unlabeled documents based on the combination of Expectation-Maximization (EM) and a naive Bayes classifier. The algorithm first trains a classifier using the available labeled documents,
and probabilistically labels the unlabeled documents. It then trains a new classifier using the labels for all the documents, and iterates to convergence. This basic EM procedure works well when the
data conform to the generative assumptions of the model. However these assumptions are often violated in practice, and poor performance can result. We present two extensions to the algorithm that
improve ...
- Data Mining and Knowledge Discovery , 2005
"... Data mining applications place special requirements on clustering algorithms including: the ability to find clusters embedded in subspaces of high dimensional data, scalability, end-user
comprehensibility of the results, non-presumption of any canonical data distribution, and insensitivity to the or ..."
Cited by 561 (12 self)
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Data mining applications place special requirements on clustering algorithms including: the ability to find clusters embedded in subspaces of high dimensional data, scalability, end-user
comprehensibility of the results, non-presumption of any canonical data distribution, and insensitivity to the order of input records. We present CLIQUE, a clustering algorithm that satisfies each of
these requirements. CLIQUE identifies dense clusters in subspaces of maximum dimensionality. It generates cluster descriptions in the form of DNF expressions that are minimized for ease of
comprehension. It produces identical results irrespective of the order in which input records are presented and does not presume any specific mathematical form for data distribution. Through
experiments, we show that CLIQUE efficiently finds accurate clusters in large high dimensional datasets.
- TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING , 2007
"... Often, in the real world, entities have two or more representations in databases. Duplicate records do not share a common key and/or they contain errors that make duplicate matching a dif cult
task. Errors are introduced as the result of transcription errors, incomplete information, lack of standard ..."
Cited by 256 (7 self)
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Often, in the real world, entities have two or more representations in databases. Duplicate records do not share a common key and/or they contain errors that make duplicate matching a dif cult task.
Errors are introduced as the result of transcription errors, incomplete information, lack of standard formats or any combination of these factors. In this article, we present a thorough analysis of
the literature on duplicate record detection. We cover similarity metrics that are commonly used to detect similar eld entries, and we present an extensive set of duplicate detection algorithms that
can detect approximately duplicate records in a database. We also cover multiple techniques for improving the ef ciency and scalability of approximate duplicate detection algorithms. We conclude with
a coverage of existing tools and with a brief discussion of the big open problems in the area.
"... Clustering in data mining is a discovery process that groups a set of data such that the intracluster similarity is maximized and the intercluster similarity is minimized. These discovered
clusters are used to explain the characteristics of the data distribution. In this paper we propose a new metho ..."
Cited by 88 (16 self)
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Clustering in data mining is a discovery process that groups a set of data such that the intracluster similarity is maximized and the intercluster similarity is minimized. These discovered clusters
are used to explain the characteristics of the data distribution. In this paper we propose a new methodology for clustering related items using association rules, and clustering related transactions
using clusters of items. Our approach is linearly scalable with respect to the number of transactions. The frequent item-sets used to derive association rules are also used to group items into a
hypergraph edge, and a hypergraph partitioning algorithm is used to find the clusters. Our experiments indicate that clustering using association rule hypergraphs holds great promise in several
application domains. Our experiments with stock-market data and congressional voting data show that this clustering scheme is able to successfully group items that belong to the same group.
Clustering of items can ...
, 1998
"... We examine methods for clustering in high dimensions. In the first part of the paper, we perform an experimental comparison between three batch clustering algorithms: the
Expectation–Maximization (EM) algorithm, a “winner take all ” version of the EM algorithm reminiscent of the K-means algorithm, a ..."
Cited by 78 (0 self)
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We examine methods for clustering in high dimensions. In the first part of the paper, we perform an experimental comparison between three batch clustering algorithms: the Expectation–Maximization
(EM) algorithm, a “winner take all ” version of the EM algorithm reminiscent of the K-means algorithm, and model-based hierarchical agglomerative clustering. We learn naive-Bayes models with a hidden
root node, using high-dimensional discrete-variable data sets (both real and synthetic). We find that the EM algorithm significantly outperforms the other methods, and proceed to investigate the
effect of various initialization schemes on the final solution produced by the EM algorithm. The initializations that we consider are (1) parameters sampled from an uninformative prior, (2) random
perturbations of the marginal distribution of the data, and (3) the output of hierarchical agglomerative clustering. Although the methods are substantially different, they lead to learned models that
are strikingly similar in quality. 1
"... In traditional data clustering, similarity of a cluster of objects is measured by pairwise similarity of objects in that cluster. We argue that such measures are not appropriate for transactions
that are sets of items. We propose the notion of large items, i.e., items contained in some minimum fract ..."
Cited by 63 (5 self)
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In traditional data clustering, similarity of a cluster of objects is measured by pairwise similarity of objects in that cluster. We argue that such measures are not appropriate for transactions that
are sets of items. We propose the notion of large items, i.e., items contained in some minimum fraction of transactions in a cluster, to measure the similarity of a cluster of transactions. The
intuition of our clustering criterion is that there should be many large items within a cluster and little overlapping of such items across clusters. We discuss the rationale behind our approach and
its implication on providing a better solution to the clustering problem. We present a clustering algorithm based on the new clustering criterion and evaluate its effectiveness.
- Data Mining and Knowledge Discovery , 2003
"... We present a new methodology for exploring and analyzing navigation patterns on a web site. The patterns that can be analyzed consist of sequences of URL categories traversed by users. In our
approach, we rst partition site users into clusters such that users with similar navigation paths through th ..."
Cited by 53 (0 self)
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We present a new methodology for exploring and analyzing navigation patterns on a web site. The patterns that can be analyzed consist of sequences of URL categories traversed by users. In our
approach, we rst partition site users into clusters such that users with similar navigation paths through the site are placed into the same cluster. Then, for each cluster, we display these paths for
users within that cluster. The clustering approach weemployis model-based (as opposed to distance-based) and partitions users according to the order in which they request web pages. In particular, we
cluster users by learning a mixture of rst-order Markov models using the Expectation-Maximization algorithm. The runtime of our algorithm scales linearly with the number of clusters and with the size
of the data � and our implementation easily handles hundreds of thousands of user sessions in memory. In the paper, we describe the details of our method and a visualization tool based on it called
WebCANVAS. We illustrate the use of our approach on user-tra c data from msnbc.com. Keywords: Model-based clustering, sequence clustering, data visualization, Internet, web 1
- Machine Learning , 2001
"... The EM algorithm is a popular method for parameter estimation in a variety of problems involving missing data. However, the EM algorithm often requires signi cant computational resources and has
been dismissed as impractical for large databases. We presenttwo approaches that signi cantly reduce the ..."
Cited by 35 (1 self)
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The EM algorithm is a popular method for parameter estimation in a variety of problems involving missing data. However, the EM algorithm often requires signi cant computational resources and has been
dismissed as impractical for large databases. We presenttwo approaches that signi cantly reduce the computational cost of applying the EM algorithm to databases with a large number of cases,
including databases with large dimensionality. Both approaches are based on partial E-steps for which we can use the results of Neal and Hinton (1998) to obtain the standard convergence guarantees of
EM. The rst approach is a version of the incremental EM, described in Neal and Hinton (1998), which cycles through data cases in blocks. The number of cases in each block dramatically e ects the e
ciency of the algorithm. We provide a method for selecting a near optimal block size. The second approach, which we call lazy EM, will, at scheduled iterations, evaluate the signi cance of each data
case and then proceed for several iterations actively using only the signi cant cases. We demonstrate that both methods can signi cantly reduce computational costs through their application to
high-dimensional real-world and synthetic mixture modeling problems for large databases. Keywords: Expectation Maximization Algorithm, incremental EM, lazy EM, online EM, data blocking, mixture
models, clustering. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=149668","timestamp":"2014-04-17T01:54:16Z","content_type":null,"content_length":"38197","record_id":"<urn:uuid:c87ad45b-a2d5-46b5-8884-8c8e067211d4>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00593-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Total # Posts: 68,190
What does each electron outside of a nucleus occupy?
Write multiplication equation that has a solution of 9/14
Write a multiplication equation that has a solution of -14.8
help i have no idea!!!!!!
1.It took Jack 4 hours to drive from his house to the beach at a speed of 45km/h. If he takes 5 hours to make the return trip home,what will his speed be?
When I print them from my printer they aren't like glossy. That's how we need them. I asked Walgreen's and they said they can't do that because it's photo-shopping or something..
I'm printing my pictures out from Walgreens so like how would I get them there.
Can I then print it out? Like the real pictures?
Her moms like never home, she usually gets home really late.
I have a question about like life... it relates to school as well. Like I have a huge project due THIS Thursday and I helped take care of my friends dogs and I have to take pictures of them and my
friend says I can take them now and when I ask her she always says she can't...
For carbon to move from an animal to a car, what processes need to happen? Photosynthesis, combustion, and respiration? Respiration, fossilization, and photosynthesis? Fossilization, decomposition,
and respiration? Or respiration, photosynthesis, and decomposition?
Is this equation balanced? Does it show that no atoms are lost or gained in chemical reactions? 8Fe + S8 --->8FeS
A 5.0-nC charge is at (0, 0) and a -2.0-nC charge is at (3.0 m, 0). If the potential is taken to be zero at infinity, what is the electric potential energy of a 1.0-nC charge at point (0, 4.0 m)?
Answer 1.5 × 10-8 J 3.6 × 10-9 J 1.1 × 10-8 J 7.7 × 10-9 J
What is the minimal resistance of a 100 W light bulb designed to be used in a 120 V circuit? Answer 12.0 Ω 144 Ω 1.2 Ω 0.83 Ω
Statistic Help please
1. no;no 2. 20%
Algebra I
Solve the linear equation by elimination 10x-9y=46 -2x+3y=10
science quiz pls help!!!
i agree with al of them but 3 and 7
Some water molecules are split into hydrogen and oxygen when going from: A. Atmosphere to clouds B. Ocean to atmosphere C. Ground water to surface water D. Surface water to biosphere EXPLAIN!
A 3.0-kg object moving 8.0 m/s in the positive x direction has a one-dimensional elastic collision with an object (mass = M) initially at rest. After the collision the object of unknown mass has a
velocity of 6.0 m/s in the positive x direction. What is M?
STATISTIC ****HELP****
your a hoe bag
Settlers reached the organ territory in 1840's. Which is the verb of the sentence? a)1840's b)reached c)oregon territory d)settlers
How many gallons of gasoline were sold in 2013
You deposit $7,900 in a money-market account that pays an annual interest rate of 4.3%. The interest is compounded quarterly. How much money will you have after 3 years?
Environmental Concern Speech
Can someone give their opinion about the beginning of my speech? "A few short days ago, my mother and I left our house to journey to the local grocery store. While my mother veered onto the highway,
I quickly glanced over at the side of the road, only for my attention to ...
Square on top 3x+12 sides 5x-3 and 7 on bottom 18
a 15 foot ladder is placed against a vertical wall of a building with the bottom of the ladder stand on level ground 12 feet from the base of the building. How high up the wall does the ladder reach?
Science asap
Which organisms are capable of converting gaseous nitrogen in the air into a form that other living organisms can use? nitrogen-fixing bacteria denitrifying bacteria decomposers producers Which step
in the nitrogen cycle is accelerated at the beginning of the eutrophication pr...
Complete the following nuclear reaction: ^16,v8 O + ^4, v2 He → ________ + ^19, v10 Ne 1p 1n 2H 3H
The tail of neil's dog is 5 1/4 inches long. This length is between which two inch marks on a ruler?
If 0.75 g of a gas dissolves in 1.0 L of water at 20.0 kPa of pressure, how much will dissolve at 105.4 kPa of pressure?
Grade 10 English
Based on the movie, The Red Violin what are 3 ways it differs from most Hollywood blockbusters?
state the A value of each quadratic equation 1. Y=2x^2-4x+1 Please help
Personal Improvement
I'd like some additional examples.
Personal Improvement
Like humbleness, generosity, of that type.
Personal Improvement
What are some examples of values that one could stand up to?
Us or we? "Our characteristics make us/we and friends go to the movies."
Social studies***help***
what makes mining in cerro rico mountain so hard? A.distance from the city of potosi B.isolation from other workers C.dangerous working conditions and low pay D.lack of job security i pick B.
Social studies***help***
2.what makes mining in cerro rico mountain so hard? A.distance from the city of potosi B.isolation from other workers C.dangerous working conditions and low pay D.lack of job security my answer is D.
A B B B
Environmental Concern Speech
I am struggling to prepare a environmental concern speech that must consist of 2-minutes' worth of writing. In fact, I am torn and conflicted about which concern to chose: deforestation, pollution,
or neglect of nature. What do you think? Which do you think will be the gre...
Social studies***help***
what makes mining in cerro rico mountain so hard? A.distance from the city of potosi B.isolation from other workers C.dangerous working conditions and low pay D.lack of job security
How many grams of CL2 are needed to produce 7.20 moles of chloroform?
How many grams of CL2 are needed to produce 7.20 moles of chloroform?
B. Mechanical digestion.
Some animals, but not humans, have an organ called a gizzard located near the top of their digestive tract. This organ is small and can contain small rocks. It also has thick, muscular walls. Based
on this physical description, which of the following is most likely to be the f...
Please help!
Perfect Thank you!
Please help!
If you can please help, I would greatly appreciate it! Determine the equation of g(x) that results from translating the function f(x) = (x + 10)^2 to the right 12 units. Thank you!
Three point charges of 8 C, 3 C, and -5 C, are locate at the top, bottom left, and bottom right corners respectively of an equilateral triangle of side 1Å. Find the magnitude and direction of the net
force on the 3 C charge.
Math 117
Thank you so much, greatly appreciated!
Math 117
Awesome. Thank you so much!
Find all angles, 0≤A<360, that satisfy the equation below, to the nearest 10th of a degree. 2tanA+6=tanA+3
how would you would model 5 times 2/3
No because a product is the answer to a multiplication or division problem and 6 can't be the multiple in a problem if its the product or answer.
the most prominent line in the spectrum of an element is 267.2 in what region of the electromagnetic spectrum is this line found
Management for Organizations
he most important difference between managing and leadership can best be described as follows
A 2.2 C charge is located on the x-axis at x = -1.5 m. A 5.4 C charge is located on the x-axis at x = 2.0m. A 3.5 C charge is at the origin. Find the net force acting on the 3.5 C charge.
A 2.2 C charge is located on the x-axis at x = -1.5 m. A 5.4 C charge is located on the x-axis at x = 2.0m. A 3.5 C charge is at the origin. Find the net force acting on the 3.5 C charge.
Please help me
Math 2 questions
A rectangular prism has a width of 92 ft and a volume of 240 ft3. Find the volume of a similar prism with a width of 23 ft. Round to the nearest tenth, if necessary. 3.8 ft3 60 ft3 15 ft3 10.4 ft3 A
pyramid has a height of 5 in. and a surface area of 90 in2. Find the surface a...
A 2.2 uC charge is located on the x-axis at x = -1.5 m. A 5.4 uC charge is located on the x-axis at x = 2.0m. A 3.5 uC charge is at the origin. Find the net force acting on the 3.5 C charge.
thank you Ms. Sue
list three sets of rhyming words from this poem. One set has four words. Summer. When summer blues the skies and thrushes sing for hours, and gold and orange butterflies float by like flying flowers.
Although I squint my eyes the way a thinker does, somehow, I just can't r...
Math plz help me!!! URGENT!!
Thank u soooo much,Damon! :D
Math plz help me!!! URGENT!!
Find the volume of a cone with a radius of 10 mm and a height of 6 mm. 628 mm3 600 mm3 1,884 mm3 1,254 mm3*** Find the lateral area of a cone with a radius of 7 ft. and a slant height of 13 ft. Use
3.14 for π and round to the nearest whole number. 439 ft2 324 ft2 572 ft2 ...
statistic help please
The sense of smell is closely linked to: A. proper functioning of the auditory canal. B. perceptual constancy. C. memories and emotions. D. the physiology of the taste buds.
What are the exact solutions within the interval [0,2pi)? sin2x=cosx
City Cellular purchased $28,900 in cell phones on April 25. The terms of sale were 4/20, 3/30, n/60. Freight terms were F.O.B. destination. Returned goods amounted to $650. . If the manufacturer
charges a 4 1/2 % late fee, how much would City Cellular owe if it did not pay th...
Biology help please!
What is your opinion of the balance of nature hypothesis? Would the deer on the island be better off, worse off, or about the same without the wolves. Defend your position.
thanks Ms. Sue
Someone can help me this Marking the rhythm please. Mark the strong beats in each line: Mary had a little lamb, its fleece was white as snow, and everywhere that Mary went, the lamb was sure to go.
your acquisition program expects to award a vehicle production contract in April 2005. The first item from this contract is expected to be delivered in January 2006. The estimated unit cost for each
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The specific heat of ethanol is 0.59 cal/g Celsius If one adds 450 cal of heat to 37 g of ethanol at 20 celsius what would the final temp be
When a teapot boils, we see a white cloud rising from the spout. The white cloud is composed of: A. water in the liquid phase B. water in the gaseous phase c. water molecules coming apart to from H+
and OH- ions D. H+ and OH- ions coming together to form water molecules. Explain!
What is a mutation? How does this change the information in a DNA molecule?
"That is what I did to make my difference." That's what I put for my body closing sentence and my teacher considers it lame. How could I improve it?
It was A.
What is a trend line on a scatter plot? A. a visual representation of the relationship between two variables. B. a line connecting only the highest and lowest values on a graph. C. A line connecting
all the data points on a scatter plot* D. Any line of data on a graph
When the rate of the reaction 2NO+O2=2NO2 was studied, the rate was found to double when the O2 concentration alone was doubled but to quadruple when the NO concentration alone was doubled. Which of
the following mechanisms accounts for these observations? a.) Step 1: NO + O2=...
Determine the rate constant for the following second-order reaction: A=C+2D given that the initial concentration of A is .3 M and that the concentration of C increase to .01 M in 200 seconds. Express
the result in terms of the rate law for the loss of A.
A food manufacturer samples 7 bags of pretzels off the assembly line and weighs their contents. If the sample mean is 14.2 oz and the sample standard deviation is 0.60 oz., find the 95% confidence
interval of the true mean.
MATH 104
Last semester, a certain professor gave 24 As out of 244 grades. If one of the professor's students from last semester were selected randomly, what is the probability that student received an A?
(Assume that each student receives a grade.)
can u explain? 1 question
Really? Thank you.
can u explain? 1 question
What is the formula for volume of a pyramid?
i need yur help with this..
Thank you.
i need yur help with this..
Sorry. my answer for #2 was originally 105. wasn't paying attention.
i need yur help with this..
Find the surface area of a square pyramid with a base length of 11 cm and a slant height of 15 cm. 472 cm2 451 cm2*** 781 cm2 330 cm2 Find the volume of a rectangular prism with the following
dimensions: Length = 5 mm Base = 7 mm Height = 3 mm 142 mm3 *** 105 mm3 126 mm3 130 mm3
I think it is told from the second person point of view
"You should just go somewhere. You ruined our relationship, and I don't want to talk to you anymore," Bertha said to Mitchell as he begged her for forgiveness, "But, Bertha, you are the best girl in
the world. You make my dreams come true." Bertha ignor...
"I don't want to put my shoes on," cried Timmy. Timmy had been fussy ever since snack time. He didn't get a cheese stick because Michael ate the last one. Then he didn't get to sleep at all during
nap time. Now he was really worked up. "I want to go ...
First, you will need to wash your hands and gather all of your materials. Once you've done that, follow all of the directions in your cookbook. Put your crispy treats in the oven and cook for 30-35.
Once the treats are cooled, you and your friends can enjoy. a)first person...
Timmy Turner was rushing to get to school because he was going on a field trip. Timmy felt so happy and excited that he was going on his first field trip of the year. Timmy thought that everything
was going to go good that day. a)first person b) second person c) third person I...
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Posts by
Total # Posts: 1,243
i don't know how to solve this. looked through all my note but i'm not sure what kind of formula i am to use or if i even need one! please help me to understand this: "calculate the work need to make
room for the products in the combustion of S8(s) to SO2(g) at 1 ...
please help me? :) topic: standard enthalpies of formation "Calculate ΔHf° of octane, C8H18(l), given the entalpy of combustion of octane to CO2(g) and H2O(l) is -5471kJ/mol. The standard enthalpies
of formation ofCO2 and H2O are given: CO2(g)ΔHf°=-393.5...
oh! i get it now. q mug + q coffee = 0 after typing many numbers on my calculator i got the final temperature to equal 84.65 degrees C. I'm sure this is right, thank you very much :)
topic: thermochemistry i'm not really sure how to go about his question. i'm not sure with formula to use. (perhaps Cspht= q/(m x deltaT)?)can you please help me step by step or give me a hint and
i'll post my answer, pretty please? "250g of hot coffee at 95 d...
chem- acids and bases
you're the best, thank you so much!
chem- acids and bases
correction for typo: "- how do i find the concentration of HNO2, is it 0.20M? " i meant NO2^- instead of HNO2
chem- acids and bases
sorry to take up your time again, acids and bases just really confuse me, i'd really appreciate your help again. "What is the pH at the end point for the titration of 0.20M HNO2 by 0.20M NaOH? Ka
nitrous acid= 4.5 x 10^-4." this is what i do know: - end point, so...
thanks again!
what will the pH at the end point of 0.0812M Ba(OH)2 be when titrated with HCl? a) 9.0 b) 8.0 c) 12.2 d) 7.0 e) 6.0 for this question, can i just assume the pH will be 7 since it's s titration with a
strong base and a strong acid? (or do i have to do calculations?) I'm...
oh! i get it now, thank you very much =)
what is the pH change after the addition of 10mL of 1.0M sodium hydroxide to 90mL of a 1.0M NH3/1.0M NH4^+ buffer? Kb ammonia= 1.8 x 10^-5 i found the pH for the original buffer to be 9.255, but i'm
not sure how to go about the rest of the question, please, someone help me.
Mr. Forrester has a swimming pool that measures 3 1/3 yards by 8 yards. If the deck around the pool is 2 2/3 yards wide, what is the outside perimeter of the deck?
College Physics
nvm I got this one. but thank you! :]
College Physics
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hang on I dont know why it keeps doing that. how much heat would be required to warm Earths ocean by 1.0 degrees C assuming that the volume is 137*10^7 km ^3 and the density of sea water is 1.03 g/cm
3. also assume that the heat capacity of seawater is the same as that of water
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How much heat would be required to warm Earth¡¯s oceans by 1.0 ¡ãC? Assume that the volume of Earth¡¯s oceans is 137 x ¡¼10¡½^7 ¡¼km¡½^3 and that the density of sea water is 1.03 g/cm3. Also...
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Matches for: Author/Editor=(Goerss_Paul)
Contemporary Mathematics
2004; 507 pp; softcover
Volume: 346
ISBN-10: 0-8218-3285-9
ISBN-13: 978-0-8218-3285-1
List Price: US$131
Member Price: US$104.80
Order Code: CONM/346
As part of its series of Emphasis Years in Mathematics, Northwestern University hosted an International Conference on Algebraic Topology. The purpose of the conference was to develop new connections
between homotopy theory and other areas of mathematics.
This proceedings volume grew out of that event. Topics discussed include algebraic geometry, cohomology of groups, algebraic \(K\)-theory, and \(\mathbb{A}^1\) homotopy theory. Among the contributors
to the volume were Alejandro Adem, Ralph L. Cohen, Jean-Louis Loday, and many others.
The book is suitable for graduate students and research mathematicians interested in homotopy theory and its relationship to other areas of mathematics.
Graduate students and research mathematicians interested in homotopy theory and its relationship to other areas of mathematics.
• A. Adem -- Constructing and deconstructing group actions
• M. Behrens and S. Pemmaraju -- On the existence of the self map \(v_2^9\) on the Smith-Toda complex \(V(1)\) at the prime 3
• C. Broto, R. Levi, and B. Oliver -- The theory of \(p\)-local groups: a survey
• R. L. Cohen and A. Stacey -- Fourier decompositions of loop bundles
• D. Dugger and D. C. Isaksen -- Weak equivalences of simplicial presheaves
• B. Fresse -- Koszul duality of operads and homology of partitions posets
• W. Gajda -- On \(K_{\ast}(\mathbb{Z})\) and classical conjectures in the arithmetic of cyclotomic fields
• G. Gutman -- Finite group actions in elliptic cohomology
• L. Hesselholt -- Topological Hochschild homology and the de Rham-Witt complex for \(\mathbb{Z}_{(p)}\)-algebras
• M. Hovey -- Homotopy theory of comodules over a Hopf algebroid
• J. F. Jardine -- Bousfield's \(E_{2}\) model theory for simplicial objects
• Y. Kamiya and K. Shimomura -- A relation between the Picard group of the \(E(n)\)-local homotopy category and \(E(n)\)-based Adams spectral sequence
• A. Libman -- Homotopy limits of monad algebras
• J.-L. Loday and M. Ronco -- Trialgebras and families of polytopes
• M. A. Mandell -- Equivariant symmetric spectra
• B. Richter and A. Robinson -- Gamma homology of group algebras and of polynomial algebras
• L. Scull -- Formality and \(S^1\)-equivariant algebraic models
• B. Shipley -- A convenient model category for commutative ring spectra
• P. Symonds -- The Tate-Farrell cohomology of the Morava stabilizer group \(S_{p-1}\) with coefficients in \(E_{p-1}\)
• J. M. Turner -- Characterizing simplicial commutative algebras with vanishing André-Quillen homology | {"url":"http://www.ams.org/cgi-bin/bookstore/booksearch?fn=100&pg1=CN&s1=Goerss_Paul&arg9=Paul_Goerss","timestamp":"2014-04-24T17:01:21Z","content_type":null,"content_length":"16680","record_id":"<urn:uuid:241d7800-6f08-4323-82c3-5695f99e1019>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00258-ip-10-147-4-33.ec2.internal.warc.gz"} |
Meeting Details
For more information about this meeting, contact Mary Anne Raymond.
Title: The Campbell-Baker-Hausdorff Formula
Seminar: Slow Pitch Seminar
Speaker: Nigel Higson, Penn State
It is fundamental to nearly everything that if a and b are numbers, then exp ( a ) exp ( b ) = exp ( a + b ) ; that is, the exponential function converts addition into multiplication. If A and B are
square matrices, then the corresponding formula is false, which is not surprising since multiplication of matrices is not commutative. In its place there is the Campbell-Baker-Hausdorff formula exp (
A ) exp ( B ) = exp ( C ) , where the matrix C is the sum of a series that begins C = A + B + 1/2[A,B] + 1/12[A,[A,B]] + 1/12[B,[B,A]] + ... where the brackets denote commutator: [X,Y] = XY - YX . I
shall prove the formula and try to explain the foundational role it plays in Lie group theory.
Room Reservation Information
Room Number: MB106
Date: 02 / 03 / 2009
Time: 05:00pm - 06:00pm | {"url":"http://www.math.psu.edu/calendars/meeting.php?id=4366","timestamp":"2014-04-19T00:29:36Z","content_type":null,"content_length":"3723","record_id":"<urn:uuid:b6db3325-c995-4843-8c4b-e9f4cbd7a006>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00234-ip-10-147-4-33.ec2.internal.warc.gz"} |
Transfer Function of ODE using Laplace Transform
December 11th 2008, 02:47 AM #1
Super Member
Dec 2008
Transfer Function of ODE using Laplace Transform
Hi. I'm wondering if anyone can help me with the process of finding the Transfer functions of some differential equations.
I just don't get it tbh! My lecturer gave us the following very ambiguous example:
$a_{2} \frac{d^{2}q_0}{dt^2} + a_{1}\frac{dq_{0}}{dt} + a_{0}q_{0} = b_{0} q_{i}$
$\frac{d^{2}q_0}{dt^2} + \frac{a_{1}}{a_{2}}\frac{dq_{0}}{dt} + \frac{a_{0}}{a_{2}}q_{0} = \frac{b_{0}}{a_{2}} q_{i}$
$\mathcal{L}[\frac{d^{2}q_0}{dt^2} + \frac{a_{1}}{a_{2}}\frac{dq_{0}}{dt} + \frac{a_{0}}{a_{2}}q_{0}] = \mathcal{L}[\frac{b_{0}}{a_{2}} q_{i}]$
$[s^{2} Q_{0}(s)-sq_{0}(0)-q_{0}'(0)] + \frac{a_{1}}{a_{2}}[sQ_{0}(s)-q_{0}(0)] + \frac{a_0}{a_2}Q_{0}(s) = \frac{b_0}{a_2}Q_{i}(s)$
Assuming 0 initial conditions:
$s^{2} Q_{0}(s) + \frac{a_{1}}{a_{2}}sQ_{0}(s) + \frac{a_0}{a_2}Q_{0}(s) = \frac{b_0}{a_2}Q_{i}(s)$
$[s^2 + \frac{a_{1}}{a_{2}}s + \frac{a_0}{a_2}]Q_{0}(s) = \frac{b_0}{a_2}Q_{i}(s)$
$Q_{0}(s) = \frac{b_0}{a_2}Q_{i}(s) \times \frac{1}{s^2 + \frac{a_{1}}{a_{2}}s + \frac{a_0}{a_2}}$
And apparently the transfer function is given by: $\frac{Q_{0}(s)}{Q_{i}(s)}$
Apparently, this is a representation of a time domain dynamical system, with forcing function $q_i$, and response function $q_0$.
Which is fair enough!
But how on earth does that apply to finding the transfer function of, for example, these two:
$\frac{d^{2}x}{dt^{2}} + 2 \sigma \omega_{n} \frac{dx}{dt} + \omega_{n}^{2}x = y sin(wt)$
$\frac{d^{2}x}{dt^{2}} + 2 \sigma \omega_{n} \frac{dx}{dt} + \omega_{n}^{2}x = K \omega_{n}^{2}$
I'm told that sigma and omega are constants in this case...
Hi. I'm wondering if anyone can help me with the process of finding the Transfer functions of some differential equations.
I just don't get it tbh! My lecturer gave us the following very ambiguous example:
$a_{2} \frac{d^{2}q_0}{dt^2} + a_{1}\frac{dq_{0}}{dt} + a_{0}q_{0} = b_{0} q_{i}$
$\frac{d^{2}q_0}{dt^2} + \frac{a_{1}}{a_{2}}\frac{dq_{0}}{dt} + \frac{a_{0}}{a_{2}}q_{0} = \frac{b_{0}}{a_{2}} q_{i}$
$\mathcal{L}[\frac{d^{2}q_0}{dt^2} + \frac{a_{1}}{a_{2}}\frac{dq_{0}}{dt} + \frac{a_{0}}{a_{2}}q_{0}] = \mathcal{L}[\frac{b_{0}}{a_{2}} q_{i}]$
$[s^{2} Q_{0}(s)-sq_{0}(0)-q_{0}'(0)] + \frac{a_{1}}{a_{2}}[sQ_{0}(s)-q_{0}(0)] + \frac{a_0}{a_2}Q_{0}(s) = \frac{b_0}{a_2}Q_{i}(s)$
Assuming 0 initial conditions:
$s^{2} Q_{0}(s) + \frac{a_{1}}{a_{2}}sQ_{0}(s) + \frac{a_0}{a_2}Q_{0}(s) = \frac{b_0}{a_2}Q_{i}(s)$
$[s^2 + \frac{a_{1}}{a_{2}}s + \frac{a_0}{a_2}]Q_{0}(s) = \frac{b_0}{a_2}Q_{i}(s)$
$Q_{0}(s) = \frac{b_0}{a_2}Q_{i}(s) \times \frac{1}{s^2 + \frac{a_{1}}{a_{2}}s + \frac{a_0}{a_2}}$
And apparently the transfer function is given by: $\frac{Q_{0}(s)}{Q_{i}(s)}$
Apparently, this is a representation of a time domain dynamical system, with forcing function $q_i$, and response function $q_0$.
Which is fair enough!
But how on earth does that apply to finding the transfer function of, for example, these two:
$\frac{d^{2}x}{dt^{2}} + 2 \sigma \omega_{n} \frac{dx}{dt} + \omega_{n}^{2}x = y sin(wt)$
$\frac{d^{2}x}{dt^{2}} + 2 \sigma \omega_{n} \frac{dx}{dt} + \omega_{n}^{2}x = K \omega_{n}^{2}$
I'm told that sigma and omega are constants in this case...
It does not, you have a specific input here rather than an arbitary input. The idea of a transfer function does not apply to these, you can find the output corresponding to these inputs by
multiplying the Laplace transform of the input by the transfer function of the appropriate system and then taking the inverse LT.
Oh-yes, what is $y$?
It does not, you have a specific input here rather than an arbitary input. The idea of a transfer function does not apply to these, you can find the output corresponding to these inputs by
multiplying the Laplace transform of the input by the transfer function of the appropriate system and then taking the inverse LT.
Oh-yes, what is $y$?
I think it's an arbitrary function of t.
$\mathcal L (y(t)) = Y(s)$
Quoting straight from the exam paper that these questions were taken from:
"For the time domain differential equation :
$\frac{d^2x}{dt^2}+2\sigma \omega_{n} \frac{dx}{dt}+\omega^2_{n}x=y.sin({\omega t})$
and assuming zero initial conditions, find the transfer function:
$F(s) = \frac{Y(s)}{X(s)}$"
And also:
"For the Differential Equation:
$\ddot{x} + 2\sigma\omega_n\dot{x} + \omega_n^2 x = K\omega_n^2$
find the transfer function when: $x(0) = \dot{x}(0)=0$"
I'm still not sure how to proceed! Especially in the 2nd question as there seems to be NO function of y.
December 11th 2008, 03:55 AM #2
Grand Panjandrum
Nov 2005
December 11th 2008, 04:08 AM #3
Super Member
Dec 2008
December 11th 2008, 06:08 AM #4
Super Member
Dec 2008 | {"url":"http://mathhelpforum.com/differential-equations/64492-transfer-function-ode-using-laplace-transform.html","timestamp":"2014-04-16T06:36:32Z","content_type":null,"content_length":"50568","record_id":"<urn:uuid:a4c78346-07b5-42dd-be7c-6c9638fb3bfa>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00060-ip-10-147-4-33.ec2.internal.warc.gz"} |
Academic Senate
California State University, Long Beach Policy Statement
December 6, 2004
Master of Science in Mathematics, Option in Mathematics Education
for Secondary School Teachers (code MATHMS04)
This new option was recommended by the Academic Senate on October 28, 2004
and approved by the President on November 30, 2004.
Option in Mathematics Education for Secondary School Teachers is designed for people holding a California Single Subject teaching credential in mathematics and teaching in middle, junior high, or
high schools. This option will give students greater expertise in mathematics and mathematics education (curriculum, teaching, learning, assessment and research). The program includes a blend of
courses from pure and applied mathematics as well as mathematics education, and also may include coursework from the College of Education.
1. A bachelor's degree in mathematics or mathematics education, or a bachelor's degree with at least 24 upper division units in mathematics from an accredited college or university.
2. A California Single Subject Credential in mathematics
3. Coursework in mathematics should include MATH 247, 310, 341, 355, 380, and 361A or 364A or equivalent with a grade of "C" or better.
Advancement to Candidacy
In addition to University requirements stated elsewhere in this catalog, the student must have completed the prerequisites above and must have satisfied the Graduate Writing Assessment Requirement
(GWAR). The Student must file for Advancement to Candidacy after completion of at least 6 units (and recommend filing before completing 9 units) of the Program of Study, with at least a 3.0 grade
point average. The Student's Program of Study must be approved by the Mathematics Education Graduate Advisor, Mathematics & Statistics Department Chair, Associate Dean for Graduate Accountability in
the College of Natural Sciences and Mathematics and Dean of Graduate Studies.
Requirements for the Option in Mathematics Education for Secondary School Teachers
1. A minimum of 30 units of graduate level or approved upper division coursework which includes the following:
A. A minimum of 9 graduate or approved upper division units of mathematics (those marked by * in the CSULB Catalog), including at least one 500 level mathematics course. If not previously taken
for BS or credential, this course of study this course of study must include MATH 410 and 444 as a part of these requirements.
B. A minimum of 15 graduate units of mathematics education including:
1) MTED 511 and 512; and
2) At least 9 units in mathematics education chosen in consultation with the Mathematics Education Graduate Advisor from the following courses: MTED 540, 550, 560, 580, 590, 695
C. A minimum of 6 units of approved upper division or graduate electives from mathematics, mathematics education or approved College of Education courses, chosen in consultation with the
Mathematics Education Graduate Advisor. If the student plans to teach at the Community College level she/he must take at least 18 units of graduate or approved upper division mathematics for
parts A and C. If the student intends to do a thesis or project he/she must take EDP 520.
2. Complete one of the following three options:
A. Pass a comprehensive written examination in mathematics education and in one area of pure or applied mathematics;
B. Subject to the approval of the Mathematics Education Committee of the Department of Mathematics and Statistics, write a thesis in mathematics education and defend it orally (MTED 698);
C. Pass one comprehensive written examination in mathematics education or in one area of pure and applied mathematics and complete a project in mathematics education (MTED 697), subject to the
approval of the Mathematics Education Committee of the Department of Mathematics & Statistics.
Mathematics Education Thesis
Students choosing the thesis option must consult with the Mathematics Education Graduate Advisor to select a thesis advisor. A thesis is then written in consultation with the mathematics education
faculty advisor who will guide the student in choosing the thesis topic and supervise the process of writing the thesis. After a thesis topic is chosen it must be approved by the Mathematics
Education Committee. A thesis committee of three faculty members, including the thesis advisor is then chosen to approve the final work. During the writing of the thesis, students must enroll in MTED
Mathematics Education Project
Students choosing the research project option must seek out a Faculty Advisor in mathematics education. A project is then written in consultation with the mathematics education faculty advisor who
will guide the student in choosing a topic and supervise the process of writing the project report. After a project topic is chosen it must be approved by the Mathematics Education Committee. During
the writing of the project report, students must enroll in MTED 697.
EFFECTIVE: Spring 2005
Code: MATHMS04
College: 65
Career: GR
IPEDS (Major) ERSS: 17011
IPEDS (Degree) ERSD: 17011 | {"url":"http://csulb.edu/divisions/aa/grad_undergrad/senate/documents/policy/2004/12/","timestamp":"2014-04-16T13:03:59Z","content_type":null,"content_length":"31258","record_id":"<urn:uuid:0a927ff6-51c5-4ca9-910f-9cbf4240135b>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00239-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math: Integers
Number of results: 214,795
Two sides of a triangle measure 14 and 10. If A is the set of possible lengths for the third side, how many elements of A are integers? Is the correct answer 19 elements of A are integers?
Monday, October 1, 2012 at 5:21pm by Mark
Two sides of a triangle measure 14 and 10. If A is the set of possible lengths for the third side, how many elements of A are integers? Is the correct answer 19 elements of A are integers?
Monday, October 1, 2012 at 8:58pm by Mark
Math 115 #20
Label each statement as true or false a. All integers are real numbers. b. All negative integers are whole numbers. c. Zero is neither positive nor negative. d. All integers are whole numbers. I
think all answera are TRUE am I correct
Friday, January 9, 2009 at 5:36pm by Deb
Three consecutive integers (x-1), x, (x+1) (could have said x,x+1,x+2 or x+10,x+11,x+12 etc) "the square of the largest one equals the sum of the squares of the two other" ----> (x+1)^2 = x^2 + (x-1)
^2 x^2 + 2x + 1 = x^2 + x^2 - 2x + 1 0 = x^2 - 4x x(x-4) = 0 x = 0 or x = 4...
Wednesday, March 24, 2010 at 8:46pm by Reiny
The sum of 11 consecutive positive integers is 2002. What is the greatest of these 11 integers? *************************************** I truly appreciate the time and effort that you all give
answering our questions and taking the time to explain how to get the answers when ...
Wednesday, April 4, 2012 at 11:50pm by Please Help w/ Math
The greater of two consecutive even integers is six less than twice the smaller. Find the integers. Let the smaller be x. That means that the larger is x+2. From your statement x+2=2x-6 Solve for x.
I hope this helps. Thanks for asking.
Sunday, June 3, 2007 at 1:09pm by Jaime
I just wanted to check my answers with anyone willing to take the time. Identify all sets that -3/4 belongs to: a.whole #s, integers, rational #s b.rational #s c.integers, rational #s d.odd #s, whole
#s, integers, rational #s Ithought it was C How many centimeters are equal to...
Thursday, December 14, 2006 at 7:15pm by Katie
heeeeeeeeeeeeeelp maths
How many integers 1≤N≤1000 can be written both as the sum of 26 consecutive integers and as the sum of 13 consecutive integers?
Monday, May 27, 2013 at 6:09am by Anubhav
Math ( Number Theory )
How many numbers from 1 to 1000 inclusive can be expressed as the sum of k >= 2 consecutive positive integers for some value of k ? Sorry to post question from Brilliant, I got 997 integers, but it's
wrong. If you think you can answer ...
Tuesday, April 9, 2013 at 10:37am by Lucy
Give examples of numbers that are integers and numbers that are not integers
Wednesday, April 29, 2009 at 10:38pm by Alisha
The average of a set of five different positive integers is 360. The two smallest integers in the set are 99 and 102. What is the largest possible integer in this set?
Friday, January 29, 2010 at 1:20pm by Mark
there are 8 consecutive integers that add up to 31. Only two of the integers are equal. what is the greatest number for any one integer? I assume that you are saying that of the 8 integers, only one
is repeated. If you include 8 as an integer, the sum will be greater than 31, ...
Thursday, August 16, 2007 at 7:55pm by jigar
Identify all sets to which the number 3 belongs A. Whole numbers, integers, rational numbers B. Rational numbers C. Integers, rational numbers D. Even numbers, whole numbers, integers, rational
numbers I THINK IT'S A
Wednesday, December 12, 2012 at 4:04pm by Frances
The product of two consecutive integers is two hundred forty. Find the two integers.
Sunday, April 10, 2011 at 11:16pm by Karina
thnx and if it says the sum of the squares of three consecutive positive integers is 194. Find the integers. is it x=1st x+3= 2nd x+5= 3rd? and would adding all these give me a result of 194?
Sunday, July 13, 2008 at 9:26pm by Chin Po
Math 115 #20
a. All integers are real numbers. TRUE b. All negative integers are whole numbers. FALSE, whole numbers are the positive integers and zero c. Zero is neither positive nor negative. TRUE d. All
integers are whole numbers. FALSE, e.g. -5 is not a whole number
Friday, January 9, 2009 at 5:36pm by Reiny
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?
Tuesday, October 14, 2008 at 7:33pm by Isobelle
Paulo withdraws the same amount from his bank account each week to pay for lunch. Over the past four weeks, he withdrew one hundred twenty dollars. Which rule best applies to determine the change in
his account each week? 1. The product of two positive integers is a positive ...
Wednesday, September 11, 2013 at 12:10pm by Bilbo
don't understand help please When one half of the second of two consecutive integers is subtracted from the first of the two consecutive integers, the result is 2.
Thursday, December 6, 2007 at 11:57pm by keauanna
consecutive even integers can be represented by x,x+2,x+4 and so on. write an equation and solve to find 3 consecutive even integers whose sum is 54.
Monday, May 24, 2010 at 7:59pm by Sara!
how do i solve the sum of the first n even positive integers is h. the cum of the first n odd positive integers is k. determine the value of h-k, in terms of n
Monday, May 31, 2010 at 7:41pm by alicia
It says that the sides are "consecutive odd integers." That means that each side is a different length. What 3 consecutive odd integers include 65?
Monday, April 19, 2010 at 5:00pm by Ms. Sue
The difference of the cubes of two consecutive odd positive integers is 400 more than the sum of their squares. Find the sum of the two integers.
Sunday, March 10, 2013 at 3:38am by nice
An eighth-grade student claims she can prove that subtraction of integers is commutative. She points out that if a and b are integers, then a-b = a+ -b. Since addition is commutative, so is
subtraction. What is your response?
Friday, August 24, 2012 at 9:51am by jonny
Math-Algebra 1
You know the rule that integers that do not divide evenly by two are odd integers. 51 is not divisible by 2. You conclude that 51 is odd.
Sunday, June 12, 2011 at 7:30pm by kenny
The greater of two consecutive even integers is six less than twice the smaller. Find the integers. The greater of two consecutive even integers is six less than twice the smaller. Find the integers
How about x + 2 = 2x - 6 almost .. it's x + (x+2)=2x-2
Sunday, June 3, 2007 at 1:10pm by Jaime
Integers are positive or negative whole numbers, so they are all integers.
Thursday, April 3, 2008 at 2:09pm by msp
How many positive integers less than 1000 are odd but not a multiple of 5
Monday, April 1, 2013 at 5:05am by please help me out!!
Yes. The exponents are also integers. All integers are rational.
Thursday, July 10, 2008 at 9:38pm by drwls
Algebra Please Help
The sum of three consecutive odd integers is 51. What are the integers?
Monday, October 6, 2008 at 2:34pm by Tina
The product of 2 positive integers is 1000. What is the smallest possible sum of these 2 integers?
Thursday, April 18, 2013 at 4:08am by HELP ME...uRGENT!
the greater of two consecutive integers is 20 more than twice the smaller. what are the integers
Thursday, June 30, 2011 at 3:52pm by chris
Algebra 1
The larger of two consecutive integers is 7 greater than twice the smaller. Find the integers. A) 4,5 B) -8,-9 C) -5,-6
Wednesday, October 24, 2012 at 2:11pm by Monica
Give your own example of a function using a set of at least 4 ordered pairs. The DOMAIN will be any four integers between 0 and +10. The RANGE will be any four integers between -12 and 5.
Sunday, November 15, 2009 at 9:53pm by sm
Give your own example of a function using a set of at least 4 ordered pairs. The DOMAIN will be any four integers between 0 and +10. The RANGE will be any four integers between -12 and 5
Saturday, September 11, 2010 at 12:39am by tim
Let S(n) denote the sum of digits of the integer n. Over all positive integers, the minimum and maximum values of S(n)/S(5n) are X and Y, respectively. The value of X+Y can be written as a/b , where
a and b are coprime positive integers. What is the value of a+b?
Tuesday, February 19, 2013 at 3:41am by John
If x and y are positive integers and x/5 + y/8 = 17/20, what is the value of x + y?
Monday, September 16, 2013 at 4:15pm by Alice
The four values x, y, x−y and x+y are all positive prime integers. What is the sum of all the four integers?
Wednesday, April 24, 2013 at 12:14am by ayan
Math/ Algebra
Give an example of a function using a set of at least 4 ordered pairs. The domain will be any four integers between 0 and +10. The range will be any four integers between -12 and 5. Explain why your
example models a function. Give an example of at least four ordered pairs that...
Wednesday, March 7, 2012 at 10:50pm by Dawn
quad. eq.
find 3 consecutive integers such that the product of the second and third integer is 20 Take three integers x, y, and z. The for xyz, we want y*z = 20 The factors of 20 are 20*1 10*2 5*4. 20*1 are
not consecutive. 10*2 are not consecutive. But 5 and 4 are consecutive; ...
Wednesday, January 31, 2007 at 7:15pm by chrisw
The greater of two consecutive integers is 15 more than twice the smaller. Find the integers.
Saturday, October 16, 2010 at 8:12pm by Krysta
The larger of two consecutive integers is 10 more than 4 times the smaller. Find the integers.
Tuesday, October 18, 2011 at 11:17pm by Anonymous
The larger of two consecutive integers is 10 more than 4 times the smaller. Find the integers.
Tuesday, October 18, 2011 at 11:25pm by Anonymous
The product of two consecutive positive integers is 11 more than their sum. What the integers? Please help.
Thursday, December 8, 2011 at 10:26pm by Anonymous
Math: Integers
Hey guys! I need help with this problem..it tells me to write a number line to tell the answer..the question is : What is the smallest integer greater than -2.1? Is it to -2 right? I have no idea. I
forgot all this back in elementary T_T. Yes. Negative integers are also ...
Friday, June 16, 2006 at 11:14pm by kim
3^2 - 1^2 = 8 = 4*2 5^2 - 3^2 = 25 - 9 = 4*4 7^2 - 3^2 = 40 = 4*10 For any integer n, 2n+1 is odd. Consider a pair of integers m and n. The difference of the squares of odd integers can be written
(2n+1)^2 - (2m+1)^2 = 4n^2 - 4m^2 - 4(n+m) = 4(n^2 - m^2 + n - m) which is ...
Thursday, December 16, 2010 at 10:11pm by drwls
consecutive integers --- x, x+1 3x + 4(x+1) = 39 3x + 4x + 4 = 39 7x = 35 x = 5 integers are 5 and 6
Wednesday, October 20, 2010 at 4:13pm by Reiny
Need a formula; the product of two consecutive integers is 41 more than their sum. Find the integers.
Monday, December 7, 2009 at 3:07pm by Nelson
Find the number of positive integers <1000 that can be expressed as 2^k−2^m, where k and m are non-negative integers.
Friday, April 12, 2013 at 5:43am by rohit
Find the number of positive integers <1000 that can be expressed as 2^k−2^m, where k and m are non-negative integers.
Friday, April 12, 2013 at 9:47pm by Ian
Algebra II
let the smallest of the odd integers be x then the next 3 consecutive integers are x+2,x+4, and x+6 5(x + x+2) = 7(x+2 + x+6) - 10 10x + 10 = 14x + 46 -4x = 36 x = -9 so the 4 integers are -9, -7,
-5, and -3 check: 5times(sum of first two) = 5(-9 -7) = -80 7times(sum of 2nd ...
Tuesday, April 13, 2010 at 6:52pm by Reiny
Fast way: Half of 545 = 272 must fall between two perfect sequres, namely 256 and 289, which means that the integers are 16 and 17. Standard way: Let x be one of the integers, then x+1 is the other.
So that: x^2+(x+1)^2=545 2x^2+2x-544=0 x^2+x-272=0 (x+17)(x-16)=0 So x=16 or x...
Tuesday, June 5, 2012 at 9:34pm by MathMate
The lesser of two consecutive even integers is 10 more than one-half the greater. Find the integers.
Monday, October 1, 2007 at 7:42pm by Sara
The lesser of two consecutive even integers is 10 more than one-half the greater. Find the integers.
Monday, October 1, 2007 at 8:04pm by Sara
The sum of the squares of two consecutive odd integers decreased by the product of the integers is the same as sixty-seven.
Wednesday, May 4, 2011 at 7:22pm by zainab
Give your own example of at least four ordered pairs that does not model a function. The domain will be any four integers between 0 and +10. The range will be any four integers between -12 and +5.
Monday, August 1, 2011 at 9:54am by tamika
Find the sum of all integers c, such that for some integers a and b satisfying a<b<c, a(b−c)^4+b(c−a)^4+c(a−b)^4=836
Tuesday, April 30, 2013 at 11:15am by Pete
how do you solve integers? There are about a thousand different types of problems with integers, so it depends on what type of problem. Do you have something specific in mind? In general, modulo
arithmetic is used, but it is taught in very few schools these days. Sometimes ...
Tuesday, October 10, 2006 at 11:05am by Anonymous
Twice the greater of two consecutive odd integers is 13 less than three times the lesser. Find the integers.
Tuesday, October 9, 2007 at 8:13pm by mike
There are three consecutive integers the square of the largest one equals the sum of the squares of the two other.Find the integers
Wednesday, March 24, 2010 at 8:46pm by Tom
MATH combinatorics HELP!!!!!
Determine the least positive integer n for which the following condition holds: No matter how the elements of the set of the first n positive integers, i.e. {1,2,…n}, are colored in red or blue,
there are (not necessarily distinct) integers x,y,z, and w in a set of the same ...
Saturday, May 11, 2013 at 9:49am by Bob K.
math plllls heelp
The sum of 26 consecutive integers starting at n is n+(n+1)+...(n+25) = 26n + 1+...+25 = 26n + 25(26)/2 = 26n + 325 The sum of 13 consecutive integers starting at m is 13m + 78 We want 26n+325 =
13m+78 where n <= 25 and m <= 70 That means that m=2n+19 Starting with n=1, ...
Wednesday, May 29, 2013 at 1:21am by Steve
what are 3 consecutive positive even integers that the product of the second and third integers is twebnty more than ten times the first integer?
Sunday, November 28, 2010 at 7:35pm by naicha
i need a website that will help me with my integers
Thursday, April 17, 2008 at 5:32pm by dylan
Part 1 In your own words, define the word “function.” Give your own example of a function using a set of at least 4 ordered pairs. The domain will be any four integers between 0 and +10. The range
will be any four integers between -12 and 5.Explain why your example models a ...
Wednesday, November 30, 2011 at 11:03pm by king
Math - Linear algebra
These problems belong to a class called diophantine equations. If these are what you are doing in class, they will be solved in a different way. For linear algebra, I will solve it similar to the
previous problem, as follows. Let integers p=number of pennies d=number of dimes ...
Sunday, September 11, 2011 at 4:36pm by MathMate
math, algebra
2a+2ab+2b I need a lot of help in this one. it says find two consecutive positive integers such that the sum of their square is 85. how would i do this one i have no clue i know what are positive
integers.but i don't know how to figure this out. Let n be a postive integer, ...
Friday, January 26, 2007 at 4:14am by sana
Write TWO EQUATIONS that have your two integers as solutions. The two integers are between -12 and +12
Sunday, November 15, 2009 at 11:38pm by denise
11th grade mths
Please check urgently .I have to submit the assignment A give an example of a function whose domain equals the set of real numbers and whose range equals the set? the set {-1,0,1} BGive an example of
a function whose domain equals (0,1)and whose range equals [0,1] C.Give n ...
Tuesday, September 18, 2012 at 2:13am by Fatima
discrete math
Let A= {for all m that's an element of the integers | m=3k+7 for some k that's an element of positive integers}. Prove that A is countably infiite. Note: you must define a function from Z+ to A, and
then prove that the function you definied is a bijection
Saturday, December 4, 2010 at 3:23pm by Samantha
Math History
The reason your research hasn't produced a specific person is that if there was a certain person who first devised integers some 5,000 years ago, his name was not recorded. What is far more likely is
that integers were developed by many people over a long period of time.
Tuesday, March 11, 2008 at 7:32pm by Ms. Sue
not quite, the set of "non-negative integers would include zero" whereas the set of positive integers does not include zero.
Sunday, October 21, 2012 at 8:48am by Reiny
Let S be the number of perfect squares among the integers from 1 to 20136. Let Q be the number of perfect cubes among the same integers. What is the relationship between S and Q?
Friday, December 13, 2013 at 7:49pm by Mary
the smallest 3 consecutive integers is added to twice the largest , the result so obtained is fifteen less than four times the middle integer. find the integers
Thursday, March 8, 2012 at 3:02am by simon
The sum of the squares of the largest and smallest of three consecutive odd integers is 353 less than 3 times the square of the middle one. Find the integers.
Tuesday, January 29, 2013 at 5:14pm by Karen
since you only want an explanation and not a proof, we could argue as follows for any two consecutive integers, one has to be even and one has to be odd, so the product of any two has to be divisible
by 2 for any 3 consecutive integers, one has to be divisible by 3, (just like...
Monday, August 17, 2009 at 7:50pm by Reiny
3)How mnay subsets of 6 integers taken from the numbers 1,2,3...,20 are there such that there are no consecutive integers in any subset (e.g. if 5 is in the subset then 4 and 6 cannot be in it)? This
is a fairly challenging problem, what have you tried so far? There are 20 ...
Sunday, October 15, 2006 at 10:42am by Ali
The sum of the integers from 40 to 60, inclusive, is 1050. what is the sum of the integers from 60 to 80 inclusive? Please help me find the pattern and how to use it instead of adding
60+61+62+63..... etc.
Wednesday, January 9, 2013 at 1:45am by Sarah
The sum of the integers from 40 to 60, inclusive, is 1050. what is the sum of the integers from 60 to 80 inclusive? Please help me find the pattern and how to use it instead of adding
60+61+62+63..... etc.
Wednesday, January 9, 2013 at 1:45am by Sarah
college math question
Find the GCD of 24 and 49 in the integers of Q[sqrt(3)], assuming that the GCD is defined. (Note: you need not decompose 24 or 49 into primes in Q[sqrt(3)]. Please teach me . Thank you very much. The
only integer divisor of both 24 and 49 is 1. I don't know what you mean by Q[...
Thursday, November 16, 2006 at 10:55am by student
discrete math
which positive integers less than 12 are relatively prime to 13 Since 13 has no factors, then all integers 2, 3, 4, 5, 6....11 are relatively prime to 13
Monday, November 6, 2006 at 9:41pm by thisha
Write a java application that finds the smallest of several integers. Prompt the user for the number of integers that will be input, then prompt the user for that number of integers. Evaluate the
integers to determine the smallest value. Run the program to test for 3 numbers...
Sunday, October 21, 2012 at 11:15pm by saleh
The sum of two positive integers is 60 and their positive difference is 26. What is the positive difference between the squares of the two integers? Can anyone explain this one to me?
Thursday, September 17, 2009 at 8:57pm by Michael
Suppose f(x) is a degree 8 polynomial such that f(2^i)=1/2^i for all integers 0≤i≤8. If f(0)=a/b, where a and b are coprime positive integers, what is the value of a+b?
Sunday, April 21, 2013 at 11:56pm by Sambaran G.
algebra 1
The sum of four consecutive integers is decreased by 30, the result is the fourth integer. Find the four integers.
Tuesday, October 1, 2013 at 9:41pm by Tracy
8th grade Maths
The product of three consecutive integers is -1716. What is the greatest of the three integers?
Sunday, September 21, 2008 at 2:56pm by seera
the sum of the two squares of two consecutive even integers is 1,648. what are the integers?
Tuesday, November 18, 2008 at 7:45pm by Allie
Find the sum of all positive integers c such that for some positive integers a and b {a!⋅b!=c!} {a+c+3}
Sunday, September 8, 2013 at 2:30pm by sa
Write two equations that have your two integers as solutions. Show how you built the equations using your integers, numbers used 5 and 8. Solve your system of equations by the addition/subtraction
method. Make sure you show the necessary 5 steps.
Wednesday, September 28, 2011 at 10:54am by Klint
write a polynomial that represents the sum off an odd integers 2n+1 and the next two consecutive odd integers
Tuesday, April 9, 2013 at 5:51pm by Anonymous
the sum of four consecutive integers is -72. write an equation to model this situation and find the values of the four integers
Sunday, August 4, 2013 at 8:23pm by lucindatamakloe
just a little stuck can you help me out? 1.)Sales (in millions) of a company are modeled by the expression for x items produced and sold (in millions). If 4 (million)items are produced and sold, what
would the sales be? 2.)The sum of four consecutive odd integers is -336. Set ...
Thursday, July 14, 2005 at 5:44pm by Angel
U is the set of positive integers less than or equal to 30. A is the set of natural numbers that are multiples of 5 in U. B is the subset of all of the even integers in U. a) Find n(A U B) b) Find n
(A intersect B) c) Find n(A U B' ) d Find n(B intersect A') Thank you so much!!!!!
Friday, January 24, 2014 at 7:00am by Lucas
U is the set of positive integers less than or equal to 30. A is the set of natural numbers that are multiples of 5 in U. B is the subset of all of the even integers in U. a) Find n(A U B) b) Find n
(A intersect B) c) Find n(A U B' ) d Find n(B intersect A') Thank you so much!!!!!
Saturday, January 25, 2014 at 12:42am by Lucas
How many ways can the integers 1,1,2,3,4,5,6 be arranged in a row, so that no integer is immediately adjacent to two strictly larger integers? NOTE: The sequence 3,1,2,4,5,6,1 is not valid as 1 is
adjacent to 2 and 3, both of which are strictly larger than 1. The sequence 3,1,...
Thursday, June 27, 2013 at 1:17am by Help
Show that there are no positive integers n for which n4 + 2n3 + 2n2 + 2n + 1 is a perfect square. Are there any positive integers n for which n4 +n3 +n2 +n+1 is a perfect square? If so, find all such
Friday, April 2, 2010 at 12:44pm by Diamond
there are three consecutive positive integers such that the sum of the squares of the smallest two is 221. write and equation to find the three consecutive positive integers let x= the smallest
Tuesday, October 30, 2012 at 1:15am by eddy
The sum of the squares of two consecutive positive even integers is one hundred sixty-four. Find the two integers.
Monday, February 4, 2013 at 1:23am by LaShawn
4 distinct integers p, q, r and s are chosen from the set {1,2,3,…,16,17}. The minimum possible value of (p/q)+(r/s) can be written as a/b, where a and b are positive, coprime integers. What is the
value of a+b?
Tuesday, April 23, 2013 at 5:32am by OIan
4 distinct integers p, q, r and s are chosen from the set {1,2,3,…,16,17}. The minimum possible value of p/q+r/s can be written as a/b, where a and b are positive, coprime integers. What is the value
of a+b?
Tuesday, April 23, 2013 at 5:35am by OIan
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st: RE: minimum of "all other variables"
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st: RE: minimum of "all other variables"
From "Nick Cox" <n.j.cox@durham.ac.uk>
To <statalist@hsphsun2.harvard.edu>
Subject st: RE: minimum of "all other variables"
Date Tue, 17 Oct 2006 08:59:41 +0100
unab myvars : ab-AB
foreach v of local myvars {
local others : list myvars - v
egen max_`v' = rowmax(`others')
Steve Vaisey
> I have been wrestling with a problem for a while, and I've finally
> decided to bring it to you for what will probably be an embarassingly
> quick solution. Here's the problem. Thanks to Nick, I have a program
> (-myboolean-) that turns fuzzy sets into multiset fuzzy
> intersections.
> So, for example, with fuzzy sets A and B, running -myboolean A B-
> yields four new variables: ab Ab aB AB (where lower = absence
> [1-set],
> upper = presence [set], and -- for example -- AB = min(A,B)). Enough
> background I hope.
> What I'd like to do now is to create new variables that contain the
> maximum value of all OTHER configurations for each configuration
> produced by -myboolean-. Manually, this is easy to do: otherAB =
> max(Ab,aB,ab). But with 32 or 64 configuration variables,
> this is very
> time consuming. Is there a way to automate this so that:
> foreach var of varlist ab - AB {
> gen other`var' = max(***all other vars in varlist but `var'***)
> }
> One final note: in the crisp set case (where set membership is always
> 0/1) other`var' would always = 1-`var'. But this is not true in the
> fuzzy case. Hence the difficulty.
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* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2006-10/msg00601.html","timestamp":"2014-04-20T00:45:20Z","content_type":null,"content_length":"6886","record_id":"<urn:uuid:6002edf9-76f2-474f-8b78-06ba68b7931c>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00516-ip-10-147-4-33.ec2.internal.warc.gz"} |
Dirac matrices
From Encyclopedia of Mathematics
Four Hermitian matrices
where Dirac equation in a form which is covariant with respect to the Lorentz group of transformations. The matrices
where Pauli matrices while Klein–Gordon equation:
where d'Alembert operator.
Introduced by P. Dirac in 1928 in the derivation of the Dirac equation.
For references see – of Dirac equation.
How to Cite This Entry:
Dirac matrices. V.D. Kukin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dirac_matrices&oldid=11569
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098 | {"url":"http://www.encyclopediaofmath.org/index.php/Dirac_matrices","timestamp":"2014-04-18T18:12:08Z","content_type":null,"content_length":"19441","record_id":"<urn:uuid:3c041558-69f9-425d-a25f-f5649df4de75>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00567-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: memorization & teaching styles
Replies: 0
memorization & teaching styles
Posted: Apr 23, 2001 9:25 AM
I am a junior in an undergraduate math/secondary education program. I
am trying to learn what teaching styles work the best to help students
learn and understand the material being taught. I have gained some
valuable input from current teachers through this discussion.
However, I also have an opinion on this topic. I have experience
working with students from lower level elementary through college
level math.
Some students really understand the concepts behind doing mathematical
procedures. These are usually the same ones who have no trouble
memorizing basic formulas, such as area formulas, pythagorean theorem,
etc. However, there are many students who memorize formulas long
enough to be tested on them and then forget them. This type of
learning must be prevented if students are going to have any usable
knowledge when they are done with school.
For example, I tutored a junior in high school just last week for the
SATs. Her parents were afraid that she would have trouble with the
math part even through her grades in school are good. When I heard
that she had a 97 average in Pre-Calculus, I assumed that her parents
worrying was unwarranted. However, when I began to work with her, I
realized what the problem was. She had crammed for every math test
for some time and really knew very little about the basic concepts.
In fact on one problem on which she became stuck, I questioned her
about what was wrong and she replied that she didn't have her
calculator and needed to know what twelve minus five was. I tried not
to act shocked, but you can imagine what was going through my head.
The point is that she will study her review book and probably do okay
on SATs, but will anything she "learns" really help her in the long
The other thing that I found interesting was that she told me that the
reason that she is bad at math and that she hates math is that she has
had boring teachers every year for quite some time. I really don't
think that this is just an excuse. While I know that putting variety
into teaching math is often quite difficult, I firmly believe that the
students level of understanding after such teaching is well worth the | {"url":"http://mathforum.org/kb/thread.jspa?threadID=390094","timestamp":"2014-04-17T12:36:59Z","content_type":null,"content_length":"15724","record_id":"<urn:uuid:895a7f12-911e-4b60-a95b-0d4408fc37e9>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00457-ip-10-147-4-33.ec2.internal.warc.gz"} |
11-dimensional supergravity
gravity, supergravity
Spacetime configurations
Quantum theory
String theory
Critical string models
Extended objects
Topological strings
$N=1$supergravity in $d = 11$.
for the moment see the respective section at D'Auria-Fre formulation of supergravity
The action functional
Kinetic terms
The higher Chern-Simons term
under construction
$\int_X \left( \frac{1}{6} \left( C \wedge G \wedge G - C \wedge \frac{1}{8} \left( p_2 + (\frac{1}{2}p_1)^2 \right) \right) \right)$
where $p_i$ is the $i$th Pontryagin class.
$\lambda := \frac{1}{2}p_1 \,.$
Concerning the integrality of
$I_8 := \frac{1}{48}(p_2 + (\lambda)^2)$
on a spin manifold $X$. (Witten96, p.9)
First, the index of a Dirac operator on $X$ is
$I = \frac{1}{1440}(7 (\frac{1}{2}p_1)^2 - p_2) \in \mathbb{Z} \,.$
Notice that $1440 = 6 x 8 x 30$. So
$p_2 - (\frac{1}{2}p_2)^2 = 6 ( (\frac{1}{2}p_1)^2 - 30 x 8 I)$
is divisble by 6.
Assume that $(\frac{1}{2}p_1)$ is further divisble by 2 (see the relevant discussion at M5-brane).
$(\frac{1}{2}p_1) = 2 x \,.$
Then the above becomes
$p_2 - (\frac{1}{2}p_2)^2 = 24 ( x^2 - 30 x 2 I)$
and hence then $p_2 + (\frac{1}{2}p_1)^2$ is divisible at least by 24.
But moreover, on a Spin manifold the first fractional Pontryagin class $\frac{1}{2}p_1$ is the Wu class $u_4$ (see there). By definition this means that
$x^2 = x (\frac{1}{2}p_1) \; mod \; 2$
and so when $(\frac{1}{2}p_1)^2$ is further divisible by 2 we have that $p_2 - (\frac{1}{2}p_1)^2$ is divisible by 48. Hence $I_8$ is integral.
There is in fact a hidden 1-parameter deformation of the Lagrangian of 11d sugra. Mathematically this was maybe first noticed in (D’Auria-Fre 82) around equation (4.25). This shows that there is a
topological term which may be expressed as
$\propto \int_{X_11} G_4 \wedge G_7$
where $G_4$ is the curvature 3-form of the supergravity C-field and $G_7$ that of the magnetically dual C6-field. However, (D’Auria-Fre 82) consider only topologically trivial (trivial instanton
sector) configurations of the supergravity C-field, and since on them this term is a total derivative, the authors “drop” it.
The term then re-appears in the literatur in (Bandos-Berkovits-Sorokin 97, equation (4.13)). And it seems that this is the same term later also redicovered around equation (4.2) in (Tsimpis 04).
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
brane in supergravity charged under gauge field has worldvolume theory
black brane supergravity higher gauge field SCFT
D-brane type II RR-field super Yang-Mills theory
$(D = 2n)$ type IIA $\,$ $\,$
D0-brane $\,$ $\,$ BFSS matrix model
D2-brane $\,$ $\,$ $\,$
D4-brane $\,$ $\,$ D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane $\,$ $\,$
D8-brane $\,$ $\,$
$(D = 2n+1)$ type IIB $\,$ $\,$
D1-brane $\,$ $\,$ 2d CFT with BH entropy
D3-brane $\,$ $\,$ N=4 D=4 super Yang-Mills theory
D5-brane $\,$ $\,$ $\,$
D7-brane $\,$ $\,$ $\,$
D9-brane $\,$ $\,$ $\,$
(p,q)-string $\,$ $\,$ $\,$
(D25-brane) (bosonic string theory)
NS-brane type I, II, heterotic circle n-connection $\,$
string $\,$ B2-field 2d SCFT
NS5-brane $\,$ B6-field little string theory
M-brane 11D SuGra/M-theory circle n-connection $\,$
M2-brane $\,$ C3-field ABJM theory, BLG model
M5-brane $\,$ C6-field 6d (2,0)-superconformal QFT
M9-brane/O9-plane heterotic string theory
topological M2-brane topological M-theory C3-field on G2-manifold
topological M5-brane $\,$ C6-field on G2-manifold
solitons on M5-brane 6d (2,0)-superconformal QFT
self-dual string self-dual B-field
3-brane in 6d
11d supergravity was originally found in
The description of 11d supergravity in terms of the D'Auria-Fre formulation of supergravity originates in
of which a textbook account is in
The topological deformation (almost) noticed in equation (4.25) of D’Auria-Fre 82 later reappears in (4.13) of
and around (4.2) of
Classical solutions
Bosonic solutions of eleven-dimensional supergravity were studied in the 1980s in the context of Kaluza-Klein supergravity. The topic received renewed attention in the mid-to-late 1990s as a result
of the branes and duality paradigm and the AdS/CFT correspondence.
One of the earliest solutions of eleven-dimensional supergravity is the maximally supersymmetric Freund-Rubin background with geometry $AdS_4 \times S^7$ and 4-form flux proportional to the volume
form on $AdS_4$.
• Peter Freund, Mark Rubin, Dynamics of Dimensional Reduction Phys.Lett. B97 (1980) 233-235 (inSpire)
The radii of curvatures of the two factors are furthermore in a ratio of 1:2. The modern avatar of this solution is as the near-horizon limit of coincident M2-branes.
• Mike Duff, Kai Stelle?, Multimembrane solutions of D = 11 supergravity , Phys.Lett. B253 (1991) 113-118 (web)
Shortly after the original Freund-Rubin solution was discovered, Englert discovered a deformation of this solution where one could turn on flux on the $S^7$; namely, singling out one of the Killing
spinors of the solution, a suitable multiple of the 4-form one constructs by squaring the spinor can be added to the volume form in $AdS_4$ and the resulting 4-form still obeys the supergravity field
equations, albeit with a different relation between the radii of curvature of the two factors. The flux breaks the SO(8) symmetry of the sphere to an $SO(7)$ subgroup.
• Francois Englert, Spontaneous Compactification of Eleven-Dimensional Supergravity Phys.Lett. B119 (1982) 339 (inSPIRE)
Some of the above is taken from this TP.SE thread.
A classification of symmetric solutions is discussed in
Truncations and compactifications
Topology and anomaly cancellation
Discussin of quantum anomaly cancellation and Green-Schwarz mechanism in 11D supergravity includes the following articles.
See also the relevant references at M5-brane.
Description by exceptional generalized geometry
• Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials (arXiv:0804.1362) | {"url":"http://ncatlab.org/nlab/show/11-dimensional%20supergravity","timestamp":"2014-04-21T02:01:05Z","content_type":null,"content_length":"98849","record_id":"<urn:uuid:b5803521-5b68-4195-87c8-afba7b509abf>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00307-ip-10-147-4-33.ec2.internal.warc.gz"} |
Function proof
February 14th 2010, 08:40 PM
Function proof
Let g be defined on a set containing the range of a function f. If f is continuous at z0 and g is continuous at f(z0), then g(f(z)) is continuous at z0.
February 14th 2010, 08:59 PM
February 15th 2010, 12:15 PM
The definition of continuity we're unsing involves limits and is NOT the epsilon delta method.
I havn' really gotten anywhere because I havn't a clue how to do this..
February 15th 2010, 02:05 PM
Does it involve sequences? I mean (given f:X->R, a in the set of limit points of X), did you define limits like :
$\lim_{x\to a}{f(x)}=L$
if, and only if, for all sequence $(x_n)_{n\in\mathbb{N}}$ such that $x_n\in X\backslash{\{a\}}\forall n\in\mathbb{N}$ and $\lim_{n\to \infty}x_n=a$ we have $\lim_{n\to\infty}f(x_n)=L$?
February 15th 2010, 06:42 PM
Maybe his definition of continuity is
"f is continuous in a iff $\lim_{x\to a}{f(x)}=f(a)$ "
February 15th 2010, 06:53 PM
This is a classic and easy to find on the internet theorem...but let met give you an outline of the proof.
Let $f:A\mapsto B$ and let $g:B\mapsto C$ where $f,g$ are both continuous. (we actually only have to have $g':D\mapsto C$ where $g'|B=g$ but that is neither here nor there). Then $gf:A\mapsto C$
is also continuous.
Proof: Let $B_{\varepsilon}(g(f(c)))$ be arbitrary. Since $f(c)\in B$ and $g$ is continuous there exists some $B_{\delta}(f(c))$ such that $g\left(B_{\delta}(f(c)\right)\subseteq B_{\varepsilon}
(g(f(c)))$. And since $f$ is continuous there exists some $B_{\sigma}(c)$ such that $f\left(B_{\sigma}(c)\right)\subseteq B_{\delta}(f(c))$ and so $g\left(f\left(B_{\sigma}(c)\right)\right)\
subseteq g\left(B_{\delta}(f(c))\right)\subseteq B_{\varepsilon}(g(f(c)))$
Or much nicer, if you are using the more topological definition (which is equivalent to the regular in metric spaces) that $k:X\mapsto Y$ is continuous iff $k^{-1}(O)$ is open in $X$ whenever $O$
is open in $Y$ merely note that $(gf)^{-1}(O)=f^{-1}(g^{-1}(O))$ and $g^{-1}(O)$ is ope in $B$ and so $f^{-1}(g^{-1}(O))$ is open in $A$. Done. | {"url":"http://mathhelpforum.com/differential-geometry/128857-function-proof-print.html","timestamp":"2014-04-20T19:47:43Z","content_type":null,"content_length":"12986","record_id":"<urn:uuid:01449bad-2b3f-4494-851b-175fce49ae4d>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00296-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wave Propagation along a Transmission Line
General Instructions
This spectral simulation is an interactive Java applet. You can change parameters by clicking on the vertical arrow keys. The five control buttons at the lower right are used to start (triangle) and
pause (square) the simulation, to skip forward or back one section at a time (double triangles), and to change speed (+ and -).
After the simulation is complete, the start button takes you back to the beginning of the simulation. You may experience a delay at this point.
Wave Propagation along a Transmission Line
When a sine wave from an RF signal generator is placed on a transmission line, the signal propagates toward the load. This signal, shown here in yellow, appears as a set of rotating vectors, one at
each point on the transmission line.
In our example, the transmission line has a characteristic impedance of 50 ohms. If we choose a load of 50 ohms, then the amplitude of the signal will not vary with position along the line. Only the
phase will vary along the line, as shown by the rotating vectors in yellow.
If the load impedance does not perfectly match the characteristic impedance of the line, there will be a reflected signal that propagates toward the source. At any point along the transmission line,
that signal also appears to be a constant voltage whose phase is dependent upon physical position along the line.
The voltage seen at one particular point on the line will be the vector sum of the transmitted and reflected sinusoids. We can demonstrate this by looking at two examples.
Example 1: Perfect Match: 50 Ohms
Set the terminating resistor to 50 ohms by using the "down arrow" dialog box. Notice there is no reflection. We have a perfect match. Each rotating vector has a normalized amplitude of 1. If we were
to observe the waveform at any point with a perfect measuring instrument, we would see equal sine wave amplitudes anywhere along the transmission line. The signal amplitudes are indicated by the
green line.
Example 2: Mismatched Load: 200 Ohms
Now let's intentionally create a mismatched load. Set the terminating resistor to 200 ohms by using the down arrow. Hit the PLAY button and notice the change in the reflected waveform. If it were
possible to measure just the reflected wave, we would see that its amplitude does not vary with position along the line. The only difference between the reflected (blue) signal, say at point "z6" and
point "z4", is the phase.
But the amplitude of the resultant waveform, indicated by the standing wave (green), is not constant along the entire line because the transmitted and reflected signals (yellow and blue) combine.
Since the phase between the transmitted and reflected signals varies with position along the line, the vector sums will be different, creating what's called a "standing wave".
With the load impedance at 200 ohms, a measuring device placed at point z6 would show a sine wave of constant amplitude. The sine wave at point z4 would also be of constant amplitude, but its
amplitude would differ from that of the signal at point z6. And the two would be out of phase with each other. Again, the difference is shown by the green line, which indicates the amplitude at that
point on the transmission line.
The impedance along the line also changes, as shown by the points labeled z1 through z7.
The VSWR, or Voltage Standing Wave Ratio, is the ratio of the highest amplitude signal to the lowest amplitude signal, as measured along the transmission line. A "perfect" VSWR is 1. | {"url":"http://www.home.agilent.com/upload/cmc_upload/All/Wave_Propagation_along_a_Transmission_Line.htm?cc=US&lc=eng","timestamp":"2014-04-16T16:20:05Z","content_type":null,"content_length":"7390","record_id":"<urn:uuid:c02e4bb2-2969-46e0-a1cc-6d5dc86cc8cc>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00224-ip-10-147-4-33.ec2.internal.warc.gz"} |
Problem regarding relativistic momentum/force/velocity
1. The problem statement, all variables and given/known data
Consider a particle of rest mass m0 subjected to a constant force F. The particle starts at rest; then at time t>0, the constant force F is applied. Calculate the particle's velocity v as a function
of time t for times t>0. HINT: first calculate the particle's momentum as a function of time (easy!), then figure out its velocity v(t).
2. Relevant equations
3. The attempt at a solution
I used F=d(Y*m0*v)/dt and tried to solve it for v, but ended up doing a lot of algebra (which wasn't supposed to happen, apparently) and my equations lacked a variable, t. | {"url":"http://www.physicsforums.com/showthread.php?t=263978","timestamp":"2014-04-16T13:54:52Z","content_type":null,"content_length":"20247","record_id":"<urn:uuid:74642cf4-8a69-42a6-bd35-6d618742d267>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00287-ip-10-147-4-33.ec2.internal.warc.gz"} |
Moving Charges and Magnetic Moments
hence the "donut" shape of the magnetic field
I still don't understand this... the field is in the shape of a cylinder with the wire as it's axis.
Since it has two endpoints, (which, by the way, i'm assuming implies magnetic dipoles)...
Even if I take it to be a donut, where do you find endpoints in a 'donut'?
... it is a constant vector pointing from the south pole to the north pole of the magnetic field.
That's what I pointed out earlier... If the direction of the field keep on changing with position, how can there a fixed north pole for the magnetic moment to point at!!!
I am sorry if my answers are not good, but honestly, I can't visualize what you are trying to say! I just can't see the two poles at two distinct places throughout the spread of the field and hence i
can't see any sign of the magnetic moment.
I am really sorry that after almost a week, I couldn't provide you with a satisfactory answer! :( | {"url":"http://www.physicsforums.com/showthread.php?p=4267749","timestamp":"2014-04-20T08:40:54Z","content_type":null,"content_length":"72703","record_id":"<urn:uuid:1d349eb0-26cb-4169-a125-bca3428aac45>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00533-ip-10-147-4-33.ec2.internal.warc.gz"} |
Proposed Revisions to LP
Logic Programming is a 10-point level 9 course in our third-year undergraduate programme. At present students have 20 hours of lectures, 8 one-hour tutorials with exercises, and two written
assignments which are assessed with marks and feedback, but do not contribute to final grades. There are two final examinations, one a written theory exam and one a programming practical, weighted
50% each.
We propose that the two existing coursework assignments should each contribute 10% to the final grade, reducing the exam weighting to 40% for each exam. This coursework will contribute to the
assessment of learning outcomes "To be able to construct well crafted Prolog programs of moderate size and sophistication" and "To understand the principles of declarative specification".
This proposal includes two sample exercises: we estimate a student would spend about 12 hours working on these together.
The corresponding revisions in the course descriptor would be from this:
Summary of Intended Learning Outcomes
1 - To understand the principles of declarative specification.
2 - To be able to construct well crafted Prolog programs of moderate size and sophistication.
3 - To be able to interpret problems in a style that suits logic programming.
Assessment Weightings
Written Examination 100
Assessed Assignments 0
Oral Presentations 0
Assessment Information
There is no assessed coursework for this course. Attendance at timetabled programming laboratories is mandatory, however, and 50% of the final mark is obtained from a programming exam taken in a
computing laboratory.
Study Pattern
Lectures 10
Tutorials 8
Timetabled Laboratories 20
Non-timetabled assessed assignments 0
Private Study/Other 62
Total 100
to this:
Assessment Weightings
Written Examination 80
Assessed Assignments 20
Oral Presentations 0
Assessment Information
Two pieces of assessed coursework each contribute 10% to the final course grade, and are particularly directed at learning outcomes 1 and 2. There are two examinations, one a written theory exam
and one a programming practical, each contributing 40% to the final course grade
Study Pattern
Lectures 20
Tutorials 8
Timetabled Laboratories 0
Non-timetabled assessed assignments 12
Private Study/Other 60
Total 100
Alex Simpson, Alan Smaill
15 August 2012 | {"url":"http://www.inf.ed.ac.uk/student-services/committees/board-of-studies/meetings/2012-08-22/proposal-lp","timestamp":"2014-04-19T22:07:20Z","content_type":null,"content_length":"21247","record_id":"<urn:uuid:fe83b373-bfb9-4525-92ab-fd526c8841f1>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00120-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sort of a Book Blurb + Sort of Math = Weird Post
March critique giveaway open through March 28! Scroll down one post.
I usually post about MG books here, or books that ride the line between MG and YA. But that doesn't mean I don't read YA. One of my favorite authors is, in fact, the YA writer John Green. Not because
I like all of his books equally well (I don't, actually), but because as far as the sheer intelligence of his writing, and his ability to convey things I thought only I thought, I can't remember when
I've encountered his like. I'll at least pick up anything he writes; that is for sure.
I recently read his newest novel, The Fault in Our Stars. And it. Is. Brilliant. This year, his (next) Printz may very well come.
However. The book says something that makes what's left of my mathematical mind (which has languished over three or so decades) shudder: That some infinities are bigger than others. That, for
example, there are more numbers (not speaking solely of integers, but of all possible rationals and irrationals) between, say, zero and a million than between zero and one.
No, there aren't.
And here I'd really like to diverge from any direct comment on the novel. This is really no longer about the novel. It's just the compulsion that still arises within me every now and then to speak my
mathematical piece.
No, some infinities are not bigger than others. Such a notion doesn't make sense. Between zero and one lies an infinite number of fractions and decimals, almost all of them irrational, or
non-repeating decimals. 0.5039285715790432... and on and on; you get the picture. No pattern to them. This means there's always another one. And another one. And one more. And yet one more.
For every such number, for any number at all, that lies between zero and a million, you can find one that lies between zero and one to pair with it in a one-to-one correspondence. You will never run
out of numbers between zero and one, any more than you'll run out of numbers between zero and a trillion bazillion.
And this is just one of the things that make math, and creation, amazingly cool.
15 comments:
Oh man. I get what you're saying. I think. :-) My husband is a math teacher and would like this post very much.
I'm going out to get this book for DS, the math-person in this house.
Interesting. I'm (slowly) working on a post about numbers, but they actually make my head hurt.
Barbara and Mirka -- It's cool that each of your houses has a math person. :) I do think math people would tend to like John Green.
Crack You Whip -- I hope your head won't hurt TOO much. :) Actually, I recently found quite a collection of picture books that are math related. I hope they'll help kids make friends with math at
a young age.
Okay, I'll admit it, this post is way over my head =/
I've never read any of his books, actually--I'll have to check this one out (despite the math inaccuracies ;).
I wanted to thank you, too, for all the great book recommendations you make here. I just finished Secrets at Sea (loved it!) and just started The Aviary (love it so far!).
Ugh, my head hurts. Math and I do not get along. We never have and I doubt we ever will. But I'm a huge fan of John Green and can hardly wait to read his latest.
Ruth -- A couple of times I came out of my college math classes feeling like I'd just visited a different planet, or like there was so much more to REASONING than I ever thought. It was when math
made me feel transported that I found the connection between math and story.
Faith -- I do have a feeling we'd like many of the same books. :) My favorite John Green books are An Abundance of Katherines and now his newest. I haven't liked all. But I do have some good book
recs coming up, I think. I've been finding more books lately that I really enjoy.
Cynthia -- I think he's outdone himself this time. The Fault in our Stars will be up for many an award, I suspect.
That is really an interesting thought. I'm so bad about numbers but they are facinating!
Christina -- I think that's my attraction for numbers -- the fascination. The MYSTERY. :)
Maybe you should have been around last night to help my son with his math homework. :)
In high school I asked that very question of my math teacher. He got flustered. My father, the math wizard, tried to explain it to me, the math idiot, but I still can't picture it.
Susan -- LOL.
Bish -- The teacher got flustered? Booooo. :)
I thought this book was really well-written, too, and the characters were smart (maybe almost too smart; I couldn't help wondering if in John-Green world, having cancer automatically turns you
into a genius). That is a really cool fact about numbers, though. Ever since that book, I've been noticing things about creation that are just amazing (and marveling at how even very smart people
can miss it)!
"I couldn't help wondering if in John-Green world, having cancer automatically turns you into a genius)."
LOL, Amy. That's actually a good point. Some writers love to write about smart characters (mea culpa) but that can seem unreal at times and that's good to remember. You might like SUGAR AND ICE
by Kate Messner; there's quite a lot about the Fibonacci sequence and how it's all over nature. | {"url":"http://marciahoehne.blogspot.com/2012/03/sort-of-book-blurb-sort-of-math-weird.html?showComment=1332723536034","timestamp":"2014-04-20T01:02:19Z","content_type":null,"content_length":"127815","record_id":"<urn:uuid:161d6f18-5652-48f7-a3c3-f2116f0d5907>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00446-ip-10-147-4-33.ec2.internal.warc.gz"} |
Reflections in the Why
I took this photo last summer.
Didn’t know what to do with it. Still don’t. Not enough there for a rich task. A warm-up?
My first question: Suppose Tim Horton’s offers the next size. How much should they charge?
First, students will identify a geometric sequence in the number of Timbit. The common ratio, r, is 2. The next size is an 80 pack.
Students will also need to think about unit prices. And ignore the price-ending-in-nine nonsense. The unit prices are 20¢, 18¢, 16¢. An arithmetic sequence! The common difference, d, is 2¢. The next
unit price is 14¢.
Students will solve a problem that involves both — both! — a geometric and an arithmetic sequence. Rare in the textbook, rarer still in the real-world. Okay, this may excite math teachers more than
their students.
My follow-up question: Suppose Tim Horton’s continues this pricing. How many Timbits should you get for free?
Math Picture Book Post #6: Fika
For fans of arrays (and those with OCD), there’s much to like about Fika, the Ikea cookbook. Each recipe spans two pages: the ingredients on the first, the finished product on the second.
A sample:
My daughters and I have been talking skip counting, equal grouping, repeated addition, arrays, multiplication, etc. “How many? How do you know?”
We got in on the act:
Our “family recipe”
Pythagorean Exploration
I don’t love this textbook task.
Too many substeps before students return to the question: what’s the relationship between the length of the sides of a right triangle?
“For each right triangle, write an addition statement…”? C’mon!
But I’m hesitant to join the down with textbooks revolution; I don’t want to associate myself with the back to basics movement. So in conversations where the suggested alternative is more worked
examples, I soften my criticism.
Besides, it gives me something to modify. Instead of completing the table, I could challenge students to find right triangles and ask “What do you notice?”
One problem: this requires “attend to precision” to do some heavy lifting.
This leads to some truly awkward feedback: “Are you sure it’s a right triangle? You might want to measure again.”
GeoGebra may provide a solution.
Pythagorean Mistakes
Consider the math mistakes below. Not real samples of student work (for that, go here), but real mistakes. I’ve seen each one. I think you’ll recognize them.
Answer questions 1 and 2.
1. What math mistake did each student make?
2. What are some implications for our work?
Good. Now answer questions 3 and 4.
3. What role did memorization of the times table play?
4. What are some implications for the conversations we could be having?
[Misleading Graph] Peyton Manning vs. Russell Wilson
Peyton Manning has more than 10 times as many pass attempts as Russell Wilson http://t.co/lGEilsvaTT—
ESPN Stats & Info (@ESPNStatsInfo) January 27, 2014
Does the graph create the impression that Peyton Manning has about 10 times as many pass attempts as Russell Wilson?
One approach would be to show students the graph and ask how this visual representation could be misleading. Point to the sizes of the circles.
A different approach could be to remove information (and add perplexity). Show them this:
Have students estimate Peyton Manning’s career pass attempts. I’m anticating many students will compare the sizes of the circles. They’ll think about how many green circles could fit in the orange
circle. They may not think 100, but I’m confident they’ll think much more than 10. They may have other strategies. Have students share them.
Give students rulers (and the formula A = πr² if they ask for it). Ask them if they’d like to revise their estimate.
Reveal this:
Were students misled? I’m anticipating some will compare the diameters. Take advantage of that. If not, challenge them to find out why the circles are the sizes they are.
Given Manning’s circle, have students draw Wilson’s circle to the correct size. Again, have students share strategies.
(I’ve created this applet in GeoGebra. Not sure what, if anything, it gets me.)
Allowing students to possibly be misled by a misleading graph… should’ve thought of that earlier.
I don’t think @ESPNStatsInfo is trying to suggest a much wider experience gap. Seahawks fans may disagree, but the tweet backs me up. This is accidental: the result of focussing on graphic, not info,
in infographic.
World’s Worst Person In Sports
Last week, Keith Olbermann named the Canucks’ Tom Sestito “World’s Worst Person In Sports.” In a game against the Kings, Sestito racked up 27 penalty minutes. His total ice time for the night? One
27:00 to 0:01 is an impressive stat. It’s hard to imagine this being surpassed. Sure, twenty-seven minutes can be topped. Randy Holt holds the NHL record for most penalty minutes in one game (67).
The NHL record for most penalties in one game (10) belongs to Chris Nilan. But to do so in one second?! Inconceivable.
“I’d describe [Sestito] as a hockey player except he’s not,” Olbermann says. To make this point, he goes on to compare Sestito to Gretzky. That’s right: “The Great One” is his hockey player/”boxing
hobo on skates” referent. In 101 games, Sestito had scored 9 goals, 885 shy of Gretzky’s record. Olbermann notes that Sestito would have to play about 10 000 games, or 123 seasons, to break the NHL
record. Well, yeah, assuming he can keep up this pace.
I considered giving this the three-act treatment and bleeping Olbermann. But “When will Sestito break Gretzky’s record?” is not the first question that comes to your mind, is it? A more natural
question re: Sestito might be “How many seasons would Sestito have to play to break Dave “Tiger” Williams’ record of 3966 career PIMs?” Apples to apples.
Olbermann, 54, followed this up by feuding with Tom Sestito’s sister, 13, on Twitter. Nice use of a unit rate by the kid:
@KeithOlbermann $7,268 for 1 second of work. That's probably just a little bit more than your liberal left wing ass makes. #tomsestito—
Victoria Sestito (@vsestito32) January 15, 2014
Math Picture Book Post #5: 100 Snowmen
I’m not usually a fan of equations in math picture books. But I like 100 Snowmen by Jennifer Arena and Stephen Gilpin. On each page, students can use the mental math strategy of adding one to a
double to determine basic addition facts to 19. Each number is represented as both a number to be doubled and one more than a number to be doubled. Take five. Here, students can double five and add
one more to determine five plus six.
Here, five is not doubled, but one more than four, which is doubled.
Dot cards can be used to draw attention to the doubles plus one strategy. Ask “How many do you see? How do you see them?”
To practice this strategy, students can play a game.
Taking turns:
• Roll a ten-sided die
• Build the number
• Build one more than the number
• Cover the sum with a transparent counter
The first player to cover all of the sums wins.
On the last page, every single snowmen is added.
This suggests a different mental math strategy: making tens.
Previous Math Picture Book Posts: 1 2 3 4 | {"url":"http://reflectionsinthewhy.wordpress.com/","timestamp":"2014-04-18T11:24:45Z","content_type":null,"content_length":"83458","record_id":"<urn:uuid:801b4373-6f8b-45f7-b493-b5f7b21f0250>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00134-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Christoffel symbol as tensor
Here's an interpretation I have held for while (since I never had the chance to sort it out myself... but this might be my opportunity!)... if it's wrong, please correct me.
For simplicity, take an affine space ("a vector space, but forget the origin").
Given a point x on that space, its location is represented by a tuple of numbers that depends on the choice of origin.
Given two points x and y on that space, the displacement from x to y (i.e., y-x) is a vector, independent of any choice of origin.
Is something like this happening with the Christoffel symbols and the tensor C
(If so, is this why the Christoffel symbols are sometimes called
affine connections
? Or does "affine" refer to the "affine parameter" in the geodesic equation? Based on
, I might have to go find the references to Cartan.) | {"url":"http://www.physicsforums.com/showpost.php?p=294111&postcount=11","timestamp":"2014-04-19T04:37:48Z","content_type":null,"content_length":"8278","record_id":"<urn:uuid:a5348142-a638-4b3a-ad5e-e70d0834cf96>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00318-ip-10-147-4-33.ec2.internal.warc.gz"} |
Computing "first" in LR(1) parsing
remove.haberg@matematik.su.se (Hans Aberg)
26 Nov 2001 21:57:11 -0500
From comp.compilers
| List of all articles for this month |
From: remove.haberg@matematik.su.se (Hans Aberg)
Newsgroups: comp.compilers
Date: 26 Nov 2001 21:57:11 -0500
Organization: Mathematics
Keywords: LR(1)
Posted-Date: 26 Nov 2001 21:57:11 EST
I want to compute the function "first" of LR(1) parsing: If x is a
string of grammar symbols, first(x) is the set of terminals that begin
the strings derivable from x with respect to the given grammar, plus
the empty string, if derivable as well. In Aho et al, "Compilers", sec
4.4, p.189, 3, it says that if x is a nonterminal and there is a
production x -> y1 ... yk, then to first(x) one should add first(yi)
if the empty string is part of first(y1), ..., first(y{i-1}).
But it does not say what happens if this procedure recurses, that is,
if say when computing first(y1), one recurses back via some other
relations to first(x). If that happens, is that not a LR(1) grammar,
or how should this situation be handled?
Hans Aberg * Anti-spam: remove "remove." from email address.
* Email: Hans Aberg <remove.haberg@member.ams.org>
* Home Page: <http://www.matematik.su.se/~haberg/>
* AMS member listing: <http://www.ams.org/cml/>
Post a followup to this message
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Row products of random matrices
Seminar Room 1, Newton Institute
We define the row product of K matrices of size d by n as a matrix of size d^K by n, whose rows are entry-wise products of rows of these matrices. This construction arises in certain computer science
problems. We study to what extent the spectral and geometric properties of the row product of independent random matrices resemble those properties for a d^K by n matrix with independent random
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible. | {"url":"http://www.newton.ac.uk/programmes/DAN/seminars/2011022314001.html","timestamp":"2014-04-17T04:23:19Z","content_type":null,"content_length":"5885","record_id":"<urn:uuid:55d4eaa5-9cc1-488e-a28b-ca25a944d2ec>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00419-ip-10-147-4-33.ec2.internal.warc.gz"} |
Loyola University Chicago - Mathematics and Statistics
MATH 118
• Math 118 Common Syllabus
Review. Prerequisite Material from MATH 117 [1.5 Weeks]
Quick review of algebra, lines, circles, quadratic expressions, and functions, followed by a more comprehensive review of the definitions and properties of exponential functions and logarithms.
Exponential growth modeling can be covered lightly.
Chapter 8. Sequences, Series, and Limits [2 Weeks]
8.1 – Sequences: definition of sequence, arithmetic/geometric sequences, recursively defined sequences.
8.2 – Series: sums of sequences, definition of series, arithmetic/geometric series. Emphasize: summation notation. Binomial theorem is optional.
8.3 – Limits: introduction to limits, infinite series, decimals as series, special series.
Chapter 9. Trigonometric Functions [3 Weeks]
9.1 – The unit circle: equation of unit circle, angles, negative angles, angles greater than 360 degrees, arc length, special points on unit circle.
9.2 – Radians: motivation of radians, radius corresponding to an angle, arc length revisited, area of slices, special points on unit circle revisited.
9.3 – Cosine and sine: definition of cosine and sine, signs of cosine and sine, pythagorean identity, graphs of cosine and sine.
9.4 – More trigonometric functions: tangent, sign of tangent, connections between cosine, sine, and tangent, graph of tangent, definitions of cotangent, secant, cosecant.
9.5 – Trigonometry in right triangles: definition of trigonometric functions via right triangles, two sides of a right triangle, one side and one angle of a right triangle
9.6 – Trigonometric identities: relationship between cosine, sine, tangent, identities for negative angles, identities involving pi/2, identities involving multiples of pi
Chapter 10. Trigonometric Algebra and Geometry [3 Weeks]
10.1 – Inverse trigonometric functions: arccosine, arcsine, and arctangent functions.
10.2 (Optional) – Inverse trigonometric identities, graphical and algebraic approach to evaluation at –t
10.3 – Using trigonometry to compute area: area of triangle/parallelogram via trigonometry, ambiguous angles, areas of polygons, trigonometric approximations.
10.4 – Law of Sines and Law of Cosines: statement and uses of laws of sines/cosines, when to use which law.
10.5 – Double-Angle and Half-Angle Formulas: sine/cosine double-angle and half-angle formulas. The corresponding formulas for tangent are optional.
10.6 – Addition and subtraction formulas: sine/cosine/ sum and difference formulas. The corresponding formulas for tangent are optional.
Chapter 11. Applications of Trigonometry [2.5 Weeks]
Suggestion: If time is short, 11.1 is optional. Focus on 11.2, 11.3, and 11.5.
Suggestion: Quickly review Chapter 3, Section 2 before covering 11.2
11.1 (Optional) – Parametric curves: curves in the plane, inverse functions as parametric curves, shifts/flips of parametric curves. Stretches of parametric curves is optional.
11.2 – Transformations of trigonometric functions: amplitude, period, phase shift, modeling periodic phenomena, modeling with data.
11.3 Polar Coordinates: Definition of polar coordinates, conversion between polar/rectangular coordinates, graphs of circles and rays. Other polar graphs are optional.
11.4 (Optional) – Algebraic and geometric introduction to vectors, addition and subtraction, scalar multiplication, dot product.
11.5 – The complex plane: complex numbers as points in the plane, geometric interpretation of multiplication/division of complex numbers, De Moivre's theorem, finding complex roots.
• Textbook Information
There is one required book for this course:
Axler, Sheldon. Algebra and Trigonometry (packaged with WileyPLUS). 1st ed. ISBN-13: 9781118088418. Hoboken, NJ: Wiley, 2012.
Many instructors do require WileyPLUS. If your section is using WileyPLUS, it is important that you buy a bundle that includes a WileyPLUS access code since you will need it in order to submit
homework. A WileyPLUS access code includes an online electronic version of the textbook. You can obtain an access code in one of three ways:
1. Buy a new hard copy of the text bundled with WileyPLUS access code from the bookstore.
2. Buy a College Algebra and Trigonometry 1E WileyPLUS access code directly from the publisher’s website.
3. Buy a College Algebra and Trigonometry 1E WileyPLUS access code directly from the bookstore.
Note on WileyPLUS access: If you purchased a MATH 118 access code over the past year, you will not need to purchase another one. Your access code is valid for one year.
If you go for option (2) or (3) above, then you will only have access to the book electronically through WileyPLUS. Whether you have purchased an access code or not, all students have immediate
and free access to register for our class WileyPLUS page. Free trial access expires two weeks after the semester starts.
• Available Tutoring Resources
Center for Tutoring and Academic Excellence
The Center for Tutoring & Academic Excellence offers free collaborative learning opportunities that include small group tutoring and tutor-led study halls to Loyola students. To learn more or
request tutoring services, visit the Center for Tutoring & Academic Excellence online at http://www.luc.edu/tutoring.
Loyola Math Club Tutoring
The Loyola Math Club offers free tutoring to students in Math 118 (and other courses). Further details regarding the important information of where and when will be posted once announced.
• WileyPLUS
WileyPLUS is an online, interactive environment for teaching and learning. As part of the required text for MATH 118, you are asked to purchase an access code for WileyPLUS. This access code is
active for up to one year. Thus, if you used WileyPLUS in MATH 117 in Fall 2012, your access code from last semester will allow you to enroll in your MATH 118 section this semester at no extra
cost. Many sections of MATH 118 will use WileyPLUS as a required homework component of the final course grade.
Follow the steps below to register for your section in WileyPLUS:
1. Go to the WileyPLUS homepage.
2. In the Students section of the homepage, click on the Get started button.
3. Enter LOYOLA UNIV - CHICAGO as the school name and click Find.
4. Click the plus sign next to PRECALCULUS/MATH 118 to expand the section list.
5. Click on your section.
6. Create a WileyPLUS account or log in using an existing account.
7. If you have purchased a hard copy of the text with the access code, enter your code to register for the course. Otherwise you can purchase a code (for $55.00).
Note: All students may register for WileyPLUS immediately, even if you have not received your copy from the bookstore at the start of the semester. If you have not received an access code by
January 28, you will no longer have access to WileyPLUS.
• Free Mathematica Install
Mathematica is a powerful computing environment that is designed for use in engineering, mathematics, finance, physics, chemistry, biology, and a wide range of other fields. Loyola students and
faculty can download and install the latest copy of Mathematica for free. You must be logged on to Loyola VPN, and then visit the following ITS webpage, https://myits.luc.edu/mathematica.
• Sample Math 118 Final Exams
Spring Semester 2011
Spring Semester 2012
Other Useful Loyola Sites | {"url":"http://luc.edu/math/courseresources/math118/","timestamp":"2014-04-19T17:07:58Z","content_type":null,"content_length":"31667","record_id":"<urn:uuid:e78fc074-043c-4c3a-ba26-bf7817131804>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00453-ip-10-147-4-33.ec2.internal.warc.gz"} |
Older CS252 Announcements
The quiz review session is Sunday, April 19 in 310 Soda from 2-3 pm.
Homework 4 is available. It is due Thursday, April 16 in 283 Soda by 5PM.
Questionnaire 3 is available. It is due Monday, April 13 by 5PM. It is also available in latex form.
Homework 3 Solutions are available.
Homework 3 is available. It is due Thursday, April 2 in 283 Soda by 5PM.
Project questionnaire 2 is available as postscript and latex. It is due Wednesday, March 18 by 5 pm.
Quiz 1 Statistics are available. Quiz 1 and Quiz 1 Solutions are available.
Please fill out this course feedback form and bring it to the quiz. (.tex also available.)
Homework 2 Solutions are available.
Past quizzes and solutions are available.
Project meetings are Thursday, February 26. The signup sheet is near Joe's office (464 Soda). I'll also bring it to class on Wednesday.
Project Questionnaire 1 is available. It is due on Wednesday, February 25 by 5:00 in the 635 doorbox. (Or just bring it to class.)
Homework 2 is available. It is due on Thursday, February 26 by 5:00 PM in 283 Soda.
Homework 1 Solutions are available.
An optional postscript scoreboard, xfig .fig scoreboard, postscript Tomasulo status, and xfig .fig Tomasulo status are available if you would like to use them. (The .figs are included in case you
need to modify or improve the figure.)
An official requirement for HW1 is that we have your picture (either you send us a URL, or have us take your picture). I'll be in 464 Soda on Tuesday from 2-3PM and Thursday from 3-4PM with the
camera, and will be available before and after class on Wednesday.
HW1 is now available. It is due on Thursday, February 12 by 5:00 in 283 Soda.
The Prerequisite Quiz and Prerequisite Quiz Solutions are now available.
The handouts from cs152 are now available (and should be working).
The Handouts section of the CS152 homepage from Fall 1997 includes the midterms from this semester as well as pointers to past exams. Solutions are included.
There was a screwup in the department, so the textbook was not ordered. ASUC said today that the books would be in in less than 1 week. They can also be found in local book stores (Cody's and a few
in Barnes and Noble), as well as at WWW bookstores.
Prepare for the Prerequisite Quiz before class starts.
Grads from other departments and undergrads interested in taking CS 252 in Spring, read this before class starts.
The first class meeting will be on Wednesday January 21.
The errata sheet for H&P is available. | {"url":"http://www.cs.berkeley.edu/~pattrsn/252S98/old-announcements.html","timestamp":"2014-04-21T14:41:46Z","content_type":null,"content_length":"4266","record_id":"<urn:uuid:86bd37a8-b119-4d1e-8b2d-c18fbf6a0f5e>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00120-ip-10-147-4-33.ec2.internal.warc.gz"} |
Are Different Proofs of a Theorem Really the Same?
Date: 07/05/2006 at 20:42:28
From: Eric
Subject: Are all mathematical proofs basically the same
When I was in Junior High School I remember being surprised to find
out that there are many different proofs of the Pythagorean Theorem.
I remember wondering if all of those proofs are in some way really the
A few nights ago it occurred to me that there is a better way to ask
this question. If you have a mathematical system with several axioms
(call them A, B, C, D, E and F), is it possible to have two proofs of
a theorem in this system where one proof uses only axioms A, B and C
and the other proof uses only axioms D, E and F? In other words is
it possible for two proofs to use no common axioms?
Date: 07/05/2006 at 22:43:58
From: Doctor Tom
Subject: Re: Are all mathematical proofs basically the same
Hi Eric -
Very nice question! The study of this sort of thing is called
"metamathematics", or "foundations of mathematics", and is incredibly
deep and interesting.
Let's take a simple example. Suppose we have a system that consists
of objects and an operator * on those objects, and contains the
following two axioms (and perhaps others):
A) There exists an identity element e in the system such that if a
is any object in the system, e*a = a*e = a.
B) The system contains at least two objects.
Here's my "theorem":
There is at least one object in the system.
Clearly this can be proven from A alone or from B alone, and clearly
axioms A and B are independent.
Now this seems trivial, but it satisfies your conditions, and it's
possible to construct more interesting axiom systems and theorems that
also satisfy your requirements.
So the answer is no--not all proofs are essentially equivalent. A
much more common situation is that a theorem can be proved starting
from different subsets (possibly overlapping) of the axioms, but if
the axioms are independent, meaning that none of them is a consequence
of the fact that some subset of others forces it to be true, then the
two proofs are obviously not equivalent.
- Doctor Tom, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/68519.html","timestamp":"2014-04-17T08:24:56Z","content_type":null,"content_length":"7428","record_id":"<urn:uuid:fb8e1801-3b85-473f-8f58-8a39ea6c4344>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00228-ip-10-147-4-33.ec2.internal.warc.gz"} |
MASS - MASS 2000 (Course Outline)
MATH 497A: FINITE GROUPS, SYMMETRY, AND ELEMENTS OF GROUP REPRESENTATIONS (4:3:1)
TIME: MWRF 11:15 am - 12:05 pm
TEXT: “Representation Theory: A First Course”, W. Fulton and J. Harris, Springer-Verlag, Graduate Texts in Mathematics, No. 129, 3rd edition, 1991
Starting with very concrete examples of symmetry groups, the course will develop, in an interactive manner based on examples, the theory of group representations and its relations to different areas
of mathematics and physics.
Representation theory studies the ways in which a group acts on vector spaces. The course will emphasize the way in which the set of such representations has itself a group-like structure. There is a
deep connection between representation theory and the structure of 3-dimensional space. Structure coefficients coming from representation theory can be used to define topological invariants of knots
and 3-dimensional manifolds. These invariants are related to current attempts to model quantum gravity in mathematical physics.
The course will start from the representation theory of the dihedral group, which is the symmetry group of a regular n-gon. It will continue with the study of a continuous group, U(1), for which the
representation theory is connected to the Fourier transform.
The main nonabelian example will be SU(2), related to 3-dimensional rotations, and its subgroups, among which are the symmetry groups of the Platonic solids.
It is widely thought that, in the same way in which quantum mechanics used noncommutative mathematics, the long sought unification between gravity and quantum mechanics, called quantum field theory,
will require new types of mathematical structures such as the ones described above.
An introduction to this subject was presented in a mini-course in REU 1999.
MATH 497B: PROJECTIVE AND NON-EUCLIDEAN GEOMETRY (4:3:1)
TIME: MWRF 10:10 am - 11:00 am
TEXT: I am planning to use the book “Introduction to Geometry” by H.S.M. Coxeter, 2nd edition, 1989, John Wiley & Sons, and possibly the notes by Svetlana Katok “Arithmetic and Geometry of the
Unimodular Group” (MASS 1997)
The main topics are: Isometries and similarities in the Euclidean space. Review of coordinates, complex numbers, and matrices. Affine, projective, and hyperbolic geometries. Axioms and geometric
transformations in these geometries. The theorems of Desargues and Pappus. Straight lines and circles as geodesics in the euclidean and hyperbolic geometries.
To get a better hold of these concepts, we shall usually study in detail the case of the two dimensional geometries (affine, projective, or hyperbolic plane). The geometric transformations of these
planes can be encoded in a two-by-two matrix. Usually, these matrices are elements of the unimodular group SL(2,R). The properties of this group and of its elements will play an important role in
this course and will be studied in detail.
MATH 497C: REAL AND P-ADIC ANALYSIS (4:3:1)
TIME: MWRF 1:25 pm - 2:15 pm
TEXT: Recommended books: “p-adic Numbers” by F.Q. Gouvea, 2nd edition, Springer-Verlag, 1997 and “A Course in p-adic Analysis” by A. Robert, Springer-Verlag, Graduate Texts in Mathematics, No. 198,
The real numbers are obtained from rationals Q with the usual Euclidean distance by a procedure called completion which can be applied to any metric space. The Euclidean distance, however, is not the
only way to measure “closeness” between rational numbers. Using other (p-adic) distances, we obtain completions of rationals, called p-adic numbers and denoted by Q[p]. The p-adic distance satisfies
the “strong triangle inequality” |x + y| ≤ max(|x|, |y|) which causes surprising properties of p-adic numbers and leads to interesting deviations from the classical real analysis, for instance, a
series ∑a[n] in Q[p] converges if and only if lim[n→∞]a[n] = 0—a calculus student's dream come true!
In this course we shall develop a theory of p-adic numbers and study p-adic analysis vs. real analysis, in other words, examine what form the basic ideas of calculus take in the p-adic context.
Particular topics will include: sequences andseries, elementary functions defined by means of power series, continuity and differentiability, antidirivation and integration.
MATH 497D: MASS SEMINAR (3:3:0)
TIME: T 9:05 am - 11:00 am
This seminar is designed to focus on selected interdisciplinary topics in algebra, geometry and analysis. These areas will be related to the other MASS courses. Seminar sessions may include
presentations from student homework solutions. | {"url":"http://www.math.psu.edu/mass/mass/2000/outline.php","timestamp":"2014-04-21T02:57:26Z","content_type":null,"content_length":"9912","record_id":"<urn:uuid:a4c8bbb1-fd1d-4fa4-aba6-165d3787699c>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00046-ip-10-147-4-33.ec2.internal.warc.gz"} |
Digital Logic Design
CS302 - Digital Logic & Design
Lesson No. 01
Analogue versus Digital
Most of the quantities in nature that can be measured are continuous. Examples include
· Intensity of light during the day: The intensity of light gradually increases as the sun
rises in the morning; it remains constant throughout the day and then gradually decreases
as the sun sets until it becomes completely dark. The change in the light throughout the
day is gradual and continuous. Even with a sudden change in weather when the sun is
obscured by a cloud the fall in the light intensity although very sharp however is still
continuous and is not abrupt.
· Rise and fall in temperature during a 24-hour period: The temperature also rises and
falls with the passage of time during the day and in the night. The change in temperature is
never abrupt but gradual and continuous.
· Velocity of a car travelling from A to B: The velocity of a car travelling from one city to
another varies in a continuous manner. Even if it abruptly accelerates or stops suddenly,
the change in velocity seemingly very sudden and abrupt is never abrupt in reality. This
can be confirmed by measuring the velocity in short time intervals of few milliseconds.
The measurable values generally change over a continuous range having a minimum and
maximum value. The temperature values in a summer month change between 23 0C to 45 0C.
A car can travel at any velocity between 0 to 120 mph.
Digital representing of quantities
Digital quantities unlike Analogue quantities are not continuous but represent quantities
measured at discrete intervals. Consider the continuous signal as shown in the figure 1.1.
To represent this signal digitally the signal is sampled at fixed and equal intervals. The
continuous signal is sampled at 15 fixed and equal intervals. Figure 1.2. The set of values (1,
2, 4, 7, 18, 34, 25, 23, 35, 37, 29, 42, 41, 25 and 22) measured at the sampling points
represent the continuous signal. The 15 samples do not exactly represent the original signal
but only approximate the original continuous signal. This can be confirmed by plotting the 15
sample points. Figure 1.3. The reconstructed signal from the 15 samples has sharp corners
and edges in contrast to the original signal that has smooth curves.
If the number of samples that are collected is reduced by half, the reconstructed signal will
be very different from the original. The reconstructed signal using 7 samples have missing
peak and dip at 34 0C and 23 0C respectively. Figure 1.4. The reason for the difference
between the original and the reconstructed signal is due to under-sampling. A more accurate
representation of the continuous signal is possible if the number of samples and sampling
intervals are increased. If the sampling is increased to infinity the number of values would still
be discrete but they would be very close and closely match the actual signal.
CS302 - Digital Logic & Design
Figure 1.1 Continuous signal showing temperature varying with time
Figure 1.2 Sampling the Continuous Signal at 15 equal intervals
CS302 - Digital Logic & Design
Figure 1.3 Reconstructed Signal by plotting 15 sampled values
Figure 1.4 Reconstructed Signal by plotting 7 sampled values
Electronic Processing of Continuous and Digital Quantities
Electronic Processing of the continuous quantities or their Digital representation requires
that the continuous signals or the discrete values be converted and represented in terms of
voltages. There are basically two types of Electronic Processing Systems.
· Analogue Electronic Systems: These systems accept and process continuous signals
represented in the form continuous voltage or current signals. The continuous quantities
are converted into continuous voltage or current signals by transducers. The block diagram
describes the processing by an Analogue Electronic System. Figure 1.5.
CS302 - Digital Logic & Design
Digital Electronic Systems: These systems accept and process discrete samples
representing the actual continuous signal. Analogue to Digital Converters are used to
sample the continuous voltage signals representing the original signal.
Do the Digital Electronic Systems use voltages to represent the discrete samples of the
continuous signal? This question can be answered by considering a very simple example of a
calculator which is a Digital Electronic System. Assume that a calculator is calibrated to
represents the number 1 by 1 millivolt (mV). Thus the number 39 is represented by the
calculator in terms of voltage as 39 mV. Calculators can also represent large numbers such as
6.25 x 1018 (as in 1 Coulomb = 6.25 x 1018 electrons). The value in terms of volts is 6.25 x 1015
volts! This voltage value can not be practically represented by any electronic circuit. Thus
Digital Systems do not use discrete samples represented as voltage values.
Figure 1.5
Analogue Electronic System processing continuous quantities
Digital Systems and Digital Values
Digital systems are designed to work with two voltage values. A +5 volts represents a logic
high state or logic 1 state and 0 volts represents a logic low state or logic 0 state. The Digital
Systems which are based on two voltage values or two states can easily represent any two
values. For example,
· The numbers `0' and `1'
· The state of a switch `on' or `off'
· The colour `black' and `white'
· The temperature `hot' and `cold'
· An object `moving' or `stationary'
Representing two values or two states is not very practical, as many naturally occurring
phenomenons have values or state that are more than two. For example, numbers have
widely varying ranges, a colour palette might have 64 different shades of the colour red, the
temperature of boiling water at room temperature varies from 30 0C to 100 0C. Digital Systems
are based on the Binary Number system which allows more than two or multiple values to be
represented very conveniently.
Binary Number System
CS302 - Digital Logic & Design
The Binary Number System unlike the Decimal number system is based on two values.
Each digit or bit in Binary Number system can represent only two values, a `0' and a `1'. A
single digit of the Decimal Number system represents 10 values, 0, 1, 2 to 9. The Binary
Number System can be used to represent more than two values by combining binary digits or
bits. In a Decimal Number System a single digit can represent 10 different values (0 to 9),
representing more than 10 values requires a combination of two digits which allows up to 100
values to be represented (0 to 99). A Combination of Binary Numbers is used to represent
different quantities.
Represent Colours: A palette of four colours red, blue, green and yellow can be
represented by a combination of two digital values 00, 01, 10 and 11 respectively.
Representing Temperature: An analogue value such as 39oC can be represented in a
digital format by a combination of 0s and 1s. Thus 39 is 100111 in digital form.
Any quantity such as the intensity of light, temperature, velocity, colour etc. can be
represented through digital values. The number of digits (0s and 1s) that represents a quantity
is proportional to the range of values that are to be represented. For example, to represent a
palette of eight colours a combination of three digits is used. Representing a temperature
range of 00 C to 1000 C requires a combination of up to seven digits.
Digital Systems uses the Binary Number System to represent two or multiple values,
stores and processes the binary values in terms of 5 volts and 0 volts. Thus the number 39
represented in binary as 100111 is stored electronically in as +5 v, 0v, 0v, +5 v, +5 v and +5 v.
Advantages of working in the Digital Domain
Handling information digitally offers several advantages. Some of the merits of a digital
system are spelled out. Details of some these aspects will be discussed and studied in the
Digital Logic Design course. Other aspects will be covered in several other courses.
· Storing and processing data in the digital domain is more efficient: Computers are
very efficient in processing massive amounts of information and data. Computers process
information that is represented digitally in the form of Binary Numbers. A Digital CD stores
large number of video and audio clips. Sam number of audio and video clips if stored in
analogue form will require a number of video and audio cassettes.
· Transmission of data in the digital form is more efficient and reliable: Modern
information transmission techniques are relying more on digital transmission due to its
reliability as it is less prone to errors. Even if errors occur during the transmission methods
exist which allow for quick detection and correction of errors.
· Detecting and Correcting errors in digital data is easier: Coding Theory is an area
which deals with implementing digital codes that allow for detection and correction of multi-
bit errors. In the Digital Logic Design course a simple method to detect single bit errors
using the Parity bit will be considered.
· Data can be easily and precisely reproduced: The picture quality and the sound quality
of digital videos are far more superior to those of analogue videos. The reason being that
the digital video stored as digital numbers can be exactly reproduced where as analogue
video is stored as a continuous signal can not be reproduced with exact precision.
· Digital systems are easy to design and implement: Digital Systems are based on two-
state Binary Number System. Consequently the Digital Circuitry is based on the two-
voltage states, performing very simple operations. Complex Microprocessors are
implemented using simple digital circuits. Several simple Digital Systems will be discussed
in the Digital Logic course.
CS302 - Digital Logic & Design
Digital circuits occupy small space: Digital circuits are based on two logical states.
Electronic circuitry that implements the two states is very simple. Due to the simplicity of
the circuitry it can be easily implemented in a very small area. The PC motherboard having
an area of approximately 1 sq.ft has most of the circuitry of a powerful computer. A
memory chip small enough to be held in the palm of a hand is able to store an entire
collection of books.
Information Processing by a Digital System
A Digital system such as a computer not only handles numbers but all kinds of information.
· Numbers: A computer is able to store and process all types of numbers, integers, fractions
etc. and is able to perform different kinds of arithmetic operations on the numbers.
· Text: A computer in a news reporting room is used to write and edit news reports. A
Mathematician uses a computer to write mathematical equations explaining the dissipation
of heat by a heat sink. The computer is able to store and process text and symbols.
· Drawings, Diagrams and Pictures: A computer can store very conveniently complex
engineering drawings and diagrams. It allows real life still pictures or videos to be
processed and edited.
· Music and Sound: Musicians and Composers uses\ a computer to work on a new
compositions. Computers understand spoken commands.
A Digital System (computer) is capable of handling different types of information, which is
represented in the form of Binary Numbers. The different types of information use different
standards and binary formats. For example, computers use the Binary number system to
represent numbers. Characters used in writing text are represented through yet another
standard known as ASCII which allows alphabets, punctuation marks and numbers to be
represented through a combination of 0s and 1s.
Digital Components and their internal working
Digital system process binary information electronically through specialized circuits
designed for handling digital information. These circuits as mentioned earlier operate with two
voltage values of +5 volts and 0 volts. These specialized electronic circuits are known as Logic
Gates and are considered to be the Basic Building Blocks of any Digital circuit.
The commonly used Logic Gates are the AND gate, the OR gate and the Inverter or NOT
Gate. Other gates that are frequently used include NOR, NAND, XOR and XNOR. Each of
these gates is designed to perform a unique operation on the input information which is known
as a logical or Boolean operation.
Large and complex digital system such as a computer is built using combinations of these
basic Logic Gates. These basic building blocks are available in the form of Integrated Circuit or
ICs. These gates are implemented using standard CMOS and TTL technologies that
determine the operational characteristics of the gates such as the power dissipation,
operational voltages (3.3v or 5 v), frequency response etc.
CS302 - Digital Logic & Design
Figure 1.6 Symbolic representations of logic gates.
Combinational Logic Circuits and Functional Devices
The logic gates which form the basic building blocks of a digital system are designed to
perform simple logic operations. A single logic gate is not of much use unless it is connected
with other gates to collectively act upon the input data. Different gates are combined to build a
circuit that is capable of performing some useful operation like adding three numbers. Such
circuits are known as Combinational Logic Circuits or Combinational Circuits. An Adder
Combinational Circuit that is able to add two single bit binary numbers and give a single bit
Sum and Carry output is shown. Figure 1.7.
Implementing large digital system by connecting together logic gates is very tedious and
time consuming; the circuit implemented occupies large space, are power hungry, slow and
are difficult to troubleshoot.
Figure 1.7 1-bit Full-Adder Combinational Circuit
Digital circuits to perform specific functions are available as Integrated Circuits for use.
Implementing a Digital system in terms of these dedicated functional units makes the system
more economical and reliable. Thus an adder circuit does not have to be implemented by
connecting various gates, a standard Adder IC is available that can be readily used. Other
CS302 - Digital Logic & Design
commonly used combinational functional devices are Comparators, Decoders, Encoders,
Multiplexers and Demultiplexers.
Sequential logic and implementation
Digital systems are used in vast variety of industrial applications and house hold electronic
gadgets. Many of these digital circuits generate an output that is not only dependent on the
current input but also some previously saved information which is used by the digital circuit.
Consider the example of a digital counter which is used by many digital applications where a
count value or the time of the day has to be displayed. The digital counter which counts
downwards from 10 to 0 is initialized to the value 10. When the counter receives an external
signal in the form of a pulse the counter decrements the count value to 9. On receiving
successive pulses the counter decrements the currently stored count value by one, until the
counter has been decremented to 0. On reaching the count value zero, the counter could
switch off a washing machine, a microwave oven or switch on an air-conditioning system.
The counter stores or remembers the previous count value. The new count value is
determined by the previously stored count value and the new input which it receives in the
form of a pulse (a binary 1). The diagram of the counter circuit is shown in the figure. Figure
Digital circuits that generate a new output on the basis of some previously stored
information and the new input are known as Sequential circuits. Sequential circuits are a
combination of Combinational circuits and a memory element which is able to store some
previous information. Sequential circuits are a very important part of digital systems. Most
digital systems have sequential logic in addition to the combinational logic. An example of
sequential circuits is counters such as the down-counter which generates a new decremented
output value based on the previous stored value and an external input. The storage element or
the memory element which is an essential part of a sequential circuit is implemented a flip-flop
using a very simple digital circuit known as a flip-flop.
Figure 1.8
A Counter Sequential Circuit
Programmable Logic Devices (PLDs)
The modern trend in implementing specialized dedicated digital systems is through
configurable hardware; hardware which can be programmed by the end user. A digital
CS302 - Digital Logic & Design
controller for a washing machine can be implemented by connecting together pieces of
combinational and sequential functional units. These implementations are reliable however
they occupy considerable space. The implementation time also increases. A general purpose
circuit that can be programmed to perform a certain task like controlling a washing machine
reduces the implementation cost and time.
Cost is incurred on implementing a digital controller for a washing machine which requires
that an inventory of all its components such as its logic circuits, functional devices and the
counter circuits be maintained. The implementation time is significantly high as all the circuit
components have to be placed on a circuit board and connected together. If there is a change
in the controller circuit the entire circuit board has to be redesigned. A PLD based washing
machine controller does not require a large inventory of components to be maintained. Most of
the functionality of the controller circuit is implemented within a single PLD integrated circuit
thereby considerably reducing the circuit size. Changes in the controller design can be readily
implemented by programming the PLD.
Programmable Logic Devices can be used to implement Combinational and Sequential
Digital Circuits.
Memory plays a very important role in Digital systems. A research article being edited by a
scientist on a computer is stored electronically in the digital memory whilst changes are being
made to the document. Once the document has be finalized and stored on some media for
subsequent printing the memory can be reused to work on some other document. Memory
also needs to store information permanently even when the electrical power is turned off.
Permanent memories usually contain essential information required for operating the digital
system. This important information is provided by the manufacturer of a digital system.
Memory is organized to allow large amounts of data storage and quick access. Memory
(ROM) which permanently stores data allows data to be read only. The Memory does not allow
writing of data. Volatile memory (RAM) does not store information permanently. If the power
supplied to the RAM circuitry is turned off, the contents of the RAM are permanently lost and
can not be recovered when power is restored. RAM allows reading and writing of data. Both
RAM and ROM are an essential part of a digital system.
Analogue to Digital and Digital to Analogue conversion and Interfacing
Real-world quantities as mention earlier are continuous in nature and have widely varying
ranges. Processing of real-world information can be efficiently and reliably done in the digital
domain. This requires real-world quantities to be read and converted into equivalent digital
values which can be processed by a digital system. In most cases the processed output needs
to be converted back into real-world quantities. Thus two conversions are required, one from
the real-world to the digital domain and then back from the digital domain to the real-world.
Modern digitally controlled industrial units extensively use Analogue to Digital (A/D) and
Digital to Analogue (D/A) converters to covert quantities represented as an analogue voltage
into an equivalent digital representation and vice versa. Consider the example of an industrial
controller that controls a chemical reaction vessel which is being heated to expedite the
chemical reaction. Figure 1.9. Temperature of the vessel is monitored to control the chemical
reaction. As the temperature of the vessel rises the heat has to be reduced by a proportional
CS302 - Digital Logic & Design
level. An electronic temperature sensor (transducer) converts the temperature into an
equivalent voltage value. This voltage value is continuous and proportion to the temperature.
The voltage representing the temperature is converted into a digital representation which is fed
to a digital controller that generates a digital value corresponding to the desired amount of
heat. The digitized output representing the heat is converted back to a voltage value which is
continuous and is used to control a valve that regulates the heat. An A/D converter converts
the analogue voltage value representing the temperature into a corresponding digital value for
processing. A D/A converter converts back the digital heat value to its corresponding
continuous value for regulating the heater.
Figure 1.9
Digitally Controlled Industrial Heater Unit
A/D and D/A converters are an important aspect of digital systems. These devices serve
as a bridge between the real and digital world allow the two to communicate and interact
Number Systems and Codes
Decimal Number System
The decimal number system has ten unique digits 0, 1, 2, 3... 9. Using these single digits,
ten different values can be represented. Values greater than ten can be represented by using
the same digits in different combinations. Thus ten is represented by the number 10, two
hundred seventy five is represented by 275 etc. Thus same set of numbers 0,1 2... 9 are
repeated in a specific order to represent larger numbers.
The decimal number system is a positional number system as the position of a digit
represents its true magnitude. For example, 2 is less than 7, however 2 in 275 represents 200,
whereas 7 represents 70. The left most digit has the highest weight and the right most digit
has the lowest weight. 275 can be written in the form of an expression in terms of the base
value of the number system and weights.
2 x 102 + 7 x 101 + 5 x 100 = 200 + 70 + 5 = 275
where, 10 represents the base or radix
102, 101, 100 represent the weights 100, 10 and 1 of the numbers 2, 7 and 5
CS302 - Digital Logic & Design
Fractions in Decimal Number System
In a Decimal Number System the fraction part is separated from the Integer part by a
decimal point. The Integer part of a number is written on the left hand side of the decimal
point. The Fraction part is written on the right side of the decimal point. The digits of the
Integer part on the left hand side of the decimal point have weights 100, 101, 102 etc.
respectively starting from the digit to the immediate left of the decimal point and moving away
from the decimal point towards the most significant digit on the left hand side. Fractions in
decimal number system are also represented in terms of the base value of the number system
and weights. The weights of the fraction part are represented by 10-1, 10-2, 10-3 etc. The
weights decrease by a factor of 10 moving right of the decimal point. The number 382.91 in
terms of the base number and weights is represented as
3 x 102 + 8 x 101 + 2 x 100 + 9 x 10-1 + 1 x 10-2 = 300 + 80 + 2 + 0.9 + 0.01 = 382.91
Caveman number system
A number system discovered by archaeologists in a prehistoric cave indicates that the
caveman used a number system that has 5 distinct shapes ∑, Ć, >, Ω and ↑. Furthermore it
has been determined that the symbols ∑ to ↑ represents the decimal equivalents 0 to 5
Centuries ago a caveman returning after a successful hunting expedition records his
successful hunt on the cave wall by carving out the numbers Ć↑. What does the number Ć↑
represent? The table 1.1 indicates that the Caveman numbers Ć↑ represents decimal number
Decimal Number
Caveman Number
Decimal Number
Caveman Number
Table 1.1
Decimal equivalents of the Caveman Numbers
The Caveman is using a Base-5 number system. A Base-5 number system has five
unique symbols representing numbers 0 to 4. To represent numbers larger than 4, a
combination of 2, 3, 4 or more combinations of Caveman numbers are used. Therefore, to
represent the decimal number 5, a two number combination of the Caveman number system is
used. The most significant digit is Ć which is equivalent to decimal 1. The least significant digit
is ∑ which is equivalent to decimal 0. The five combinations of Caveman numbers having the
most significant digit Ć, represent decimal values 5 to 9 respectively. This is similar to the
Decimal Number system, where a 2-digit combination of numbers is used to represent values
CS302 - Digital Logic & Design
greater than 9. The most significant digit is set to 1 and the least significant digit varies from 0
to 9 to represent the next 10 values after the largest single decimal number digit 9.
The Caveman number Ć↑ can be written in expression form based on the Base value 5
and weights 50, 51, 52 etc.
= Ć x 51 + ↑ x 50 = Ć x 5 + ↑ x 1
Replacing the Caveman numbers Ć and ↑ with equivalent decimal values in the expression
= Ć x 51 + ↑ x 50 = 1 x 5 + 4 x 1 = 9
The number ĆΩ↑∑ in decimal is represented in expression form as
Ć x 53 + Ω x 52 + ↑ x 51 + ∑ x 50 = Ć x 125 + Ω x 25 + ↑ x 5 + ∑ x 1
Replacing the Caveman numbers with equivalent decimal values in the expression yields
= (1) x 125 + (3) x 25 + (4) x 5 + (0) x 1 = 125 + 75 + 20 + 0 = 220
Binary Number System
The Caveman Number system is a hypothetical number system introduced to explain
that number system other than the Decimal Number system can exist and can be used to
represent and count numbers. Digital systems use a Binary number system. Binary as the
name indicates is a Base-2 number system having only two numbers 0 and 1. The Binary digit
0 or 1 is known as a `Bit'. Table 1.2
Decimal Number
Binary Number
Decimal Number
Binary Number
Table 1.2
Decimal equivalents of Binary Number System
Counting in Binary Number system is similar to counting in Decimal or Caveman
Number systems. In a decimal Number system a value larger than 9 has to be represented by
2, 3, 4 or more digits. In the Caveman Number System a value larger than 4 has to be
represented by 2, 3, 4 or more digits of the Caveman Number System. Similarly, in the Binary
Number System a Binary number larger than 1 has to be represented by 2, 3, 4 or more binary
CS302 - Digital Logic & Design
Any binary number comprising of Binary 0 and 1 can be easily represented in terms of
its decimal equivalent by writing the Binary Number in the form of an expression using the
Base value 2 and weights 20, 21, 22 etc.
The number 100112 (the subscript 2 indicates that the number is a binary number and
not a decimal number ten thousand and eleven) can be rewritten in terms of the expression
100112 = (1 x 24) + (0 x 23) + (0 x 22) + (1 x 21) + (1 x 20)
= (1 x 16) + (0 x 8) + (0 x 4) + (1 x 2) + (1 x 1)
= 16 + 0 + 0 + 2 + 1
= 19
Fractions in Binary Number System
In a Decimal number system the Integer part and the Fraction part of a number are
separated by a decimal point. In a Binary Number System the Integer part and the Fraction
part of a Binary Number can be similarly represented separated by a decimal point. The Binary
number 1011.1012 has an Integer part represented by 1011 and a fraction part 101 separated
by a decimal point. The subscript 2 indicates that the number is a binary number and not a
decimal number. The Binary number 1011.1012 can be written in terms of an expression using
the Base value 2 and weights 23, 22, 21, 20, 2-1, 2-2 and 2-3.
1011.1012 = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) + (1 x 2-1) + (0 x 2-2) + (1 x 2-3)
= (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1) + (1 x 1/2) + (0 x 1/4) + (1 x 1/8)
= 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125
= 11.625
Computers do handle numbers such as 11.625 that have an integer part and a fraction
part. However, it does not use the binary representation 1011.101. Such numbers are
represented and used in Floating-Point Numbers notation which will be discussed latter. | {"url":"http://www.zeepedia.com/read.php?an_overview_number_systems_digital_logic_design&b=9&c=1","timestamp":"2014-04-19T19:34:52Z","content_type":null,"content_length":"250911","record_id":"<urn:uuid:bb15236c-1f19-438b-8985-8b39ac28060c>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00228-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to Calculate Pattern Multiples
Okay, it's time for some crochet math!
Wait! Come back! It won't be that hard, I promise!
Sometimes you might want to make a crochet pattern a different size than the pattern dictates. This is especially common for afghans. The pattern might be for a baby afghan, but you'd like to make an
adult-size afghan, for instance. Really helpful pattern writers will include a line stating what the multiple for the pattern is. This is usually given as something like "chain a multiple of 4 + 2,"
which means to make a number of chain stitches that is a multiple of 4, then chain 2 more. This is often abbreviated as "mult 4 + 2."
If the pattern doesn't tell you what the multiple is, though, can you figure it out for yourself?
Yes! To figure out the multiple, you need to count the chains used in the pattern repeat. Let's use the following pattern snippet as an example.
The pattern starts:
Chain 133.
Row 1: work shell (3dc, ch1, dc) in 6th chain from hook, *skip next 3 chs, work shell in next ch; repeat from * across to last 3 chs, skip next 2 chs, dc in last ch; 32 shells.
The pattern repeat in this pattern is the part between the *'s:
*skip next 3 chs, work shell in next ch; repeat from *
This tells you to skip 3 chains, then do a shell in the 4th chain, so the repeat uses four chains.
Next, you need to add the chains at the beginning and the end of the first row:
work shell (3dc, ch1, dc) in 6th chain from hook
across to last 3 chs
So you need 6 chains at the beginning and 3 chains at the end, or 9 total. Thus your starting chain is 9 chains plus a multiple of 4, usually written:
multiple of 4 plus 9
mult 4 + 9
In the example, the starting chain was 133, which is 124 + 9, or (31 * 4) + 9. The (31 * 4) is 31 shells. The 32nd shell is the first one in the 6th chain.
Sometimes you need to look at the second or third or later rows to be sure about your repeat. For example, you might need, say, an even number of shells to work the second row. If so, you'd still
have the same repeat (mult 4 + 9), but you'd have to figure out how many shells you want first. If you wanted 16 shells instead of the original 32, do this: Remember that the + 9 part includes one
shell, so you need to figure out only the number of stitches needed for 15 shells. This would be 15 * 4, or 60. So you'd need (15 * 4) + 9 or 60 + 9 or 69 stitches.
There now, that wasn't so bad, was it?
22 Comments:
Multiples used to give me fits, but I've been figuring it out lately (suddenly I'm getting used to math???). This is great, to have it all down and set out! Now if I forget about multiples, I can
check out your blog!! Thank you!
barbara fraser said:
This is wonderful, I always have problem making a larger size so anymore hint on doing so would be greatly appreciated here in Ny,
Happy Birthday, Lisa!! Hope they just get better and better. :)
--Lisa S. :)
It seems as if I've been crocheting for a million years, but I've never been able to calculate multiples. Thanks so much for adding to my crochet knowledge and making it easy.
Thank you SO much for posting this! I was trying to figure out how to do the multiple math to make a baby granny ripple afghan pattern a larger afghan.
i made your crinkle textured baby blanket and it was lovely. Now i want to make another one but i want to use a different stitch. can i use the same instructions but substitute lets say a double
crochet instead of the single crochet?
I made your crinkle textured baby blanket and it was lovely. Now i want to make another one but i want to use a different stitch. can i use the same instructions but substitute lets say a double
crochet instead of the single crochet?
Tuesday, October 13, 2009 8:38:00 PM
Ellen - Give it a try on a test swatch and see how you like it. That's how you make up a new pattern! :-)
I have been searching all over the internet to figure out multiples in patterns so I could get a gauge without doing the entire pattern using the amount of chains the pattern required. I was just
about to give it all up when I found this.Thank you, thank you, thank you
Janie - you're welcome! :-)
When a pattern states 1 pattern repeat to determine gauge, can I assume the repeat is everything between the **'s. My problem is if I am beginning a pattern with 100 chain stitches, I am trying
to obtain the number of stitches I need to use to determine if my gauge is correct. There has to be an easier way than making a swatch with all the 100 stitches (I hope)
I'd appreciate it if you could help me on this.
Janie - you don't need to make all 100 stitches for a gauge swatch. Just use the explanation above to determine the pattern multiple, and then make a few repeats for your swatch. I hope this
I am trying to figure out the multiple of a lattice stitch using a ch 3, sc, ch 3, sc instead of the ch5 normally called for since I want a smaller mesh--suggestions.
SunShinyDa - you might try to rewrite your pattern the way you want it, and then using your new pattern, go through the directions given above. I hope that helps!
Hey Lisa, can u please help! I have a pattern for a Bernat ripple afghan and I'm needing to make it wider. Here is the first row.
Note: Ch 3 at beg of row counts as dc throughout. With A, ch 157. 1st row: (RS). 1 dc in 4th ch from hook. Ch 1. Miss next ch. 1 dc in next ch. Ch 1. Miss next ch. 3 dc in next ch. Ch 3. 3 dc in
next ch. Ch 1. Miss next ch. 1 dc in next ch. Ch 1. Miss next ch. *(Yoh and draw up a loop in next ch. Yoh and draw through 2 loops on hook) 3 times. Yoh and draw through all 4 loops on hook –
cluster made. Ch 1. Miss next ch. 1 dc in next ch. Ch 1. Miss next ch. 3 dc in next ch. Ch 3. 3 dc in next ch. Ch 1. Miss next ch. 1 dc in next ch. Ch 1. Miss next ch. Rep from * to last 2 ch. 1
dc in each of next 2 ch. Turn.
Do you think you can help?
I was pulling my hair out trying to figure out the multiple of a pattern I wanted to reduce. Just couldn't come up with a viable multiple until I read your calculation instructions and BINGO - I
got it. Thank you soooooooooooo much. I'll never have this problem again. Kudos from Cathedral City, CA!
does this also work if the repeat is not until row 2? I need to make a ripple pattern smaller but the first row is a stright hdc to the end.
reba m. thanks so much for this information. This has been quite helpful. Thanks Lisa!
So I'm getting confused. I have a pattern
Row 1 Sc in 2nd ch from hook and each chain across.
Row 2: Ch 1, DC in next, sc in next sc
Row 3: CH 1, DC in next dc, sc in next sc
Row 4: Ch 1, dc in next dc, sc in next sc
I can't figure out what the multiple is?
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THIS IS AN AWESOME HELP! THANKS so much! Your response to Jane was a BIG help as well! | {"url":"http://ledzeplisa.blogspot.com/2005/10/how-to-calculate-pattern-multiples.html","timestamp":"2014-04-20T08:15:46Z","content_type":null,"content_length":"46963","record_id":"<urn:uuid:d8d738b3-d769-4a6e-90ef-d7b2ff3ddae2>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00227-ip-10-147-4-33.ec2.internal.warc.gz"} |
If the LHC does not find the Higgs Boson or other "new physics", then Einstein's "last twenty years" will eventually be fully vindicated, including the most natural cosmological extension of general relativity to a finite, deterministic universe.
Duration ? years (02009-???)
“If the LHC does not find the Higgs Boson or other "new physics", then Einstein's "last twenty years" will eventually be fully vindicated, including the most natural cosmological extension of general
relativity to a finite, deterministic universe.”
Voting has been temporarily disabled.
Ryals's Argument
Einstein didn't know about the real, massive, particle potential of the quantum vacuum, or he never would have abandoned this finite "quasi-static" cosmological model, because there is no instability
when matter generation from the negative energy states drives vacuum expansion, because this "anti-gravity" effect is obviously offset by the gravity of the newly created massive particle, and that
makes all the difference in the world to the Dirac Equation, which can be reformulated in this vacuum as it was originally intended, to unify General Relativity and Quantum Theory, as Paul Dirac did
with SR and QM.
This leads to a new cosmology that predicts that increasing tension between the expanding vacuum and ordinary matter is what eventually causes the forces to be compromised, aka., "big bangs", so
there is no need for a naked singularity, nor extra-ordinarily rapid inflation when a universe with certain volume "evolves" information inherently forward to a higher order of the same basic
Our Darwinian Universe
Physics for the prediction is discussed briefly, here:
And in greater detail, here: | {"url":"http://longbets.org/476/","timestamp":"2014-04-16T13:19:05Z","content_type":null,"content_length":"11859","record_id":"<urn:uuid:311b92a8-b625-490a-aafc-e40a0bcbb882>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00534-ip-10-147-4-33.ec2.internal.warc.gz"} |
How can I analyze multiple mediators in Stata?
Stata FAQ
How can I analyze multiple mediators in Stata?
Preacher and Hayes (2008) show how to analyze models with multiple mediators in SPSS and SAS, how can I analyze multiple mediators in Stata? Here is the full citation: Preacher, K.J. and Hayes, A.F.
2008. Asymptotic and resampling strategies for assessing and
comparing indirect effects in multiple mediator models.
Behavioral Research Methods
, 40, 879-891. NOTE: If running the code on this page, please copy it all into a do-file and run all of it. Mediator variables are variables that sit between independent variable and dependent
variable and mediate the effect of the IV on the DV. A model with two mediators is shown in the figure below.
In the figure above
represents the regression coefficient for the IV when the MV is regressed on the IV while
is the coefficient for the MV when the DV is regressed on MV and IV. The symbol
represents the direct effect of the IV on the DV. Generally, researchers want to determine the indirect effect of the IV on the DV through the MV. One common way to compute the indirect effect is by
using the product of the coefficients method. This method determines the indirect effect by multiplying the regression coefficients, for example,
a1*b1 = a1b1
. In addition to computing the indirect effect we also want to obtain the standard error of
. Further, we want to be able to do this for each of the mediator variables in the model. Thus, we need the
coefficients for each of the mediator variable in the model. We will obtain all of the necessary coefficients using the
(seemingly unrelated regression) command as suggested by Maarten Buis on the Statalist. The general form of the
command will look something like this:
sureg (mv1 iv)(mv2 iv)(dv mv1 mv2 mv3 iv)
Example 1
hsb2 dataset with science as the dv, math as the iv and read and write as the two mediator variables. We will need the coefficients for read on math and write on math as well as the coefficients for
science on read and write from the equation that also includes math.
use http://www.ats.ucla.edu/stat/data/hsb2, clear
sureg (read math)(write math)(science read write math)
Seemingly unrelated regression
Equation Obs Parms RMSE "R-sq" chi2 P
read 200 1 7.662848 0.4386 156.26 0.0000
write 200 1 7.437294 0.3812 123.23 0.0000
science 200 3 6.983853 0.4999 199.96 0.0000
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
read |
math | .724807 .0579824 12.50 0.000 .6111636 .8384504
_cons | 14.07254 3.100201 4.54 0.000 7.996255 20.14882
write |
math | .6247082 .0562757 11.10 0.000 .5144099 .7350065
_cons | 19.88724 3.008947 6.61 0.000 13.98981 25.78467
science |
read | .3015317 .0679912 4.43 0.000 .1682715 .434792
write | .2065257 .0700532 2.95 0.003 .0692239 .3438274
math | .3190094 .0759047 4.20 0.000 .170239 .4677798
_cons | 8.407353 3.160709 2.66 0.008 2.212476 14.60223
Now we have all the coefficients we need to compute the indirect effect coefficients and their standard errors. We can do this using the nlcom (nonlinear combination) command. We will run nlcom three
times: Once for each of the two specific indirect effects for read and write and once for the total indirect effect. To compute an indirect direct we specify a product of coefficients. For example,
the coefficient for read on math is [read]_b[math] and the coefficient for science on read is [science]_b[read]. Thus, the product is [read]_b[math]*[science]_b[read]. To get the total indirect
effect we just add the two product terms together in the nlcom command.
/* indirect via read */
nlcom [read]_b[math]*[science]_b[read]
1: [read]_b[math]*[science]_b[read]
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
1 | .2185523 .05229 4.18 0.000 .1160659 .3210388
-- indirect via write */
nlcom [write]_b[math]*[science]_b[write]
1: [write]_b[math]*[science]_b[write]
----- | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-----
1 | .1290183 .0452798 2.85 0.004 .0402715 .2177651
-------- indirect */
nlcom [read]_b[math]*[science]_b[read]+[write]_b[math]*[science]_b[write]
1: [read]_b[math]*[science]_b[read]+[write]_b[math]*[science]_b[write]
----------- | Coef. Std. Err. z P>|z| [95% Conf. Interval]
----------- .3475706 .0594916 5.84 0.000 .2309693 .4641719
--------------suggest that each of the separate indirect effects as well as the total indirect effect are significant. From the above results it is also possible to compute the ratio of indirect
to direct effect and the proportion due to the indirect effect. These computations require an estimate of the direct effect, which can be found in the sureg output. In this example the direct effect
is given by the coefficient for math in the last equation (.3190094). Here are the manual computations for the ratio of indirect to direct and the proportion of total effect that is mediated.
/* ratio of indirect to direct */
display .3475706/.3190094
1.0895309
of total effect that is mediated */
display .3475706/(.3475706+.3190094)
.52142369
nlco standard errors using the delta method which assumes that the estimates of the indirect effect are normally distributed. For many situations this is acceptable but it does not work
well for the indirect effects which are usually positively skewed and kurtotic. Thus the z-test and p-values for these indirect effects generally cannot be trusted. Therefore, it is recommended that
bootstrap standard errors and confidence intervals be used. Below is a short ado-program that is called by the bootstrap command. It computes the indirect effect coefficients as the product of sureg
coefficients (as before) but does not use the nlcom command since the standard errors will be computed using the bootstrap.bootmm is an rclass program that produces three return values which we have
called "indread", "indwrite" and "indtotal." These are the local names for each of the indirect effect coefficients and for the total indirect effect. We run bootmm with the bootstrap command. We
give the bootstrap command the names of the three return values and select options for the number of replications and to omit printing dots after each replication. Since we selected 5,000
replications you may need to be a bit patient depending upon the speed of your computer.
capture program drop bootmm
program bootmm, rclass
syntax [if] [in]
sureg (read math)(write math)(science read write math) `if' `in'
return scalar indread = [read]_b[math]*[science]_b[read]
return scalar indwrite = [write]_b[math]*[science]_b[write]
return scalar indtotal = [read]_b[math]*[science]_b[read]+ ///
dread) r(indwrite) r(indtotal), bca reps(5000) nodots: bootmm
Bootstrap results Number of obs = 200
Replications = 5000
command: bootmm
_bs_1: r(indread)
_bs_2: r(indwrite)
_bs_3: r(indtotal)
-------------- Observed Bootstrap Normal-based
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+ .2185523 .0544617 4.01 0.000 .1118094 .3252953
_bs_2 | .1290183 .0498037 2.59 0.010 .0314048 .2266318
_bs_3 | .3475706 .0653076 5.32 0.000 .2195701 .4755711
--------------ootstrap standard errors to see if the indirect effects are significant but it is usually recommended that bias-corrected or percentile confidence intervals be used instead. These
confidence intervals are nonsymmetric reflecting the skewness of the sampling distribution of the product coefficients. If the confidence interval does not contain zero than the indirect effect is
considered to be statistically significant.
estat boot, percentile bc bca
Bootstrap results Number of obs = 200
Replications = 5000
command: bootmm
_bs_1: r(indread)
_bs_2: r(indwrite)
_bs_3: r(indtotal)
-------------- Observed Bootstrap
| Coef. Bias Std. Err. [95% Conf. Interval]
-------------+ .21855231 -.0009252 .05446169 .1116576 .3263005 (P)
| .1140179 .3286456 (BC)
| .1140179 .3286456 (BCa)
_bs_2 | .12901828 .0009822 .04980373 .0375536 .2286579 (P)
| .0375377 .22842 (BC)
| .0333511 .2264691 (BCa)
_bs_3 | .34757059 .000057 .0653076 .2181866 .4773324 (P)
| .2209776 .4805473 (BC)
| .2158857 .4752103 (BCa)
--------------ile confidence interval
(BC) bias-corrected confidence interval
(BCa) bias-corrected and accelerated confidence interval
In this example, the total indirect effect of math through read and write is significant as are the individual indirect effects.
Example 2
What do you do if you also have control variables? You just add them to each of the equations in the sureg model. Let's say that socst is a covariate. Here is how the bootstrap process would work.
capture program drop bootmm
program bootmm, rclass
syntax [if] [in]
sureg (read math socst)(write math socst)(science read write math socst) `if' `in'
return scalar indread = [read]_b[math]*[science]_b[read]
return scalar indwrite = [write]_b[math]*[science]_b[write]
return scalar indtotal = [read]_b[math]*[science]_b[read] + ///
dread) r(indwrite) r(indtotal), bca reps(5000) nodots: bootmm
Bootstrap results Number of obs = 200
Replications = 5000
command: bootmm
_bs_1: r(indread)
_bs_2: r(indwrite)
_bs_3: r(indtotal)
-------------- Observed Bootstrap Normal-based
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+ .1561855 .040306 3.87 0.000 .0771872 .2351837
_bs_2 | .0890589 .0352121 2.53 0.011 .0200444 .1580733
_bs_3 | .2452443 .0477817 5.13 0.000 .1515939 .3388947
--------------rcentile bc bca
Bootstrap results Number of obs = 200
Replications = 5000
command: bootmm
_bs_1: r(indread)
_bs_2: r(indwrite)
_bs_3: r(indtotal)
-------------- Observed Bootstrap
| Coef. Bias Std. Err. [95% Conf. Interval]
-------------+ .15618546 -.0016606 .04030598 .0784972 .2359464 (P)
| .0836141 .2407494 (BC)
| .0838816 .2413034 (BCa)
_bs_2 | .08905886 .0000963 .0352121 .0241053 .163005 (P)
| .0274379 .1664222 (BC)
| .0260387 .164438 (BCa)
_bs_3 | .24524432 -.0015643 .0477817 .1536668 .341307 (P)
| .1581453 .3477974 (BC)
| .1581453 .3477974 (BCa)
--------------ile confidence interval
(BC) bias-corrected confidence interval
(BCa) bias-corrected and accelerated confidence interval
Although the total and individual indirect are much smaller in the model with the covariate, they are still statistically significant using the 95% confidence intervals.
The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California. | {"url":"http://www.ats.ucla.edu/stat/stata/faq/mulmediation.htm","timestamp":"2014-04-18T20:43:07Z","content_type":null,"content_length":"31547","record_id":"<urn:uuid:75873064-2b1e-448a-968a-bf64dcf7d548>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00071-ip-10-147-4-33.ec2.internal.warc.gz"} |
Should numpy.sqrt(-1) return 1j rather than nan?
pearu at cens.ioc.ee pearu at cens.ioc.ee
Thu Oct 12 02:45:37 CDT 2006
PS: I am still sending this message to numpy list only because the
proposal below affects numpy code, not scipy one.
I think Fernando points make sense, numpy.foo(x) != scipy.foo(x) can
cause confusion and frustration both among new numpy/scipy users and
developers (who need to find explanations for the choises made).
So, let me propose the following solution so that all parties will get
the same results without sacrifying numpy.sqrt speed on non-negative input
and scipy.sqrt backward compatibility:
Define numpy.sqrt as follows:
def sqrt(x):
r = nx.sqrt(x)
if nx.nan in r:
i = nx.where(nx.isnan(r))
r = _tocomplex(r)
r[i] = nx.sqrt(_tocomplex(x[i]))
return r
and define
that takes only non-negative input, this is for those users who expect
sqrt to fail on negative input (as Numeric.sqrt and math.sqrt do).
Using Tomcat but need to do more? Need to support web services, security?
Get stuff done quickly with pre-integrated technology to make your job easier
Download IBM WebSphere Application Server v.1.0.1 based on Apache Geronimo
More information about the Numpy-discussion mailing list | {"url":"http://mail.scipy.org/pipermail/numpy-discussion/2006-October/023740.html","timestamp":"2014-04-18T03:24:19Z","content_type":null,"content_length":"3935","record_id":"<urn:uuid:08846ab9-4ea3-4cf3-a86a-a21c015af5e7>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00015-ip-10-147-4-33.ec2.internal.warc.gz"} |
Davenport, CA Prealgebra Tutor
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Anybody know how to work this very very hard question out? (EQO) :) (would be nice)!
March 19th 2013, 05:38 AM
Anybody know how to work this very very hard question out? (EQO) :) (would be nice)!
1) A local supermarket sells beauty bar soaps which are ordered from a manufacturer. Annual Demand for beauty bar soaps is 5000. The supermarket incurs an annual holding cost of 20% of purchase
price and incurs a fixed order placement, transportation and receiving costs of £49 each time an order for the soaps is placed, regardless of the order quantity. The price charged by the
manufacturer varies according to the following discount pricing; ( my task is to determine the quantity of beauty bar soaps that the supermarket should order each time it makes an order.)
order quantity unit price
0-999 £5.00
1000-2499 £4.85
2500 and over £4.75
March 19th 2013, 05:58 AM
Re: Anybody know how to work this very very hard question out? (EQO) :) (would be nic
I assume you know that
$Q= \sqrt{\frac{2DS}{H}}$
S= 49
H= 0.2Qu
Where u is the unit price
Find Q for each unit price and then find the total annual cost for each EOQ at those unit prices.
If the EOQ for unit price £5 is 1100 it is outside the order range so just choose the quantity to be the nearest limit of the order range (so the EOQ for unit price £5 would be 999)
March 19th 2013, 06:08 AM
Re: Anybody know how to work this very very hard question out? (EQO) :) (would be nic
aww thankyou for replyin hun :)
well it really confuses me im doing a Business and IT degree and I really hate my maths module but theres no way out of them I have to do them!
but I though the formulae was q*= square root of 2KD/ h (dunno how to do the little symbols)
q is order quantity
D is number of units demanded per year
K is setup or ordering costs
h is cost of holding one unit of inventory for one unit of time
p is unit price
March 19th 2013, 06:40 AM
Re: Anybody know how to work this very very hard question out? (EQO) :) (would be nic
aww thankyou for replyin hun :)
well it really confuses me im doing a Business and IT degree and I really hate my maths module but theres no way out of them I have to do them!
but I though the formulae was q*= square root of 2KD/ h (dunno how to do the little symbols)
q is order quantity
D is number of units demanded per year
K is setup or ordering costs
h is cost of holding one unit of inventory for one unit of time
p is unit price
The is the correct equation, I just used S instead of K because wikipedia used that. And i used u instead of p.
But I made a mistake, I took H as being the total holding cost, not holding cost per unit. As you say "The supermarket incurs an annual holding cost of 20% of purchase price" The purchase price
is Qu, 20% of this is 0.2Qu and this holding cost per unit would be (0.2Qu)/Q= 0.2u
So H= 0.2u
D= 5000
K= 49
Once again, from
$Q= \sqrt{\frac{2DK}{H}}$
$Q= \sqrt{\frac{2\times 5000 \times 49}{0.2u}}$
Find the EOQ for each value of u and then find out which has the lowest total cost.
Total annual cost is given by
$Cost_{total}= 5000u+49 \frac{5000}{Q}+ 0.2Qu$
March 19th 2013, 06:56 AM
Re: Anybody know how to work this very very hard question out? (EQO) :) (would be nic
Okii ill give it a go thankyou xxx | {"url":"http://mathhelpforum.com/advanced-statistics/215068-anybody-know-how-work-very-very-hard-question-out-eqo-would-nice-print.html","timestamp":"2014-04-18T09:09:36Z","content_type":null,"content_length":"9522","record_id":"<urn:uuid:da486617-8621-4946-8d7b-0ccb7c2a1d09>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00015-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Acceleration of the Universe - A.R. Liddle
4.4.2. Gaussianity and adiabaticity
While gaussianity and adiabaticity are predictions of the simplest models, it is well advertised that there exist inflationary models which violate these hypotheses. Whether observations violating
gaussianity or adiabaticity would exclude inflation depends on the type observed. For example, chi-squared distributed perturbations are relatively easy to generate from inflation, by ensuring that
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Another Optimization problem
Hello again this problem is giving me trouble
Find the point on the line y=mx+b that is closest to the orign. A complete answer should use calculus to show that the point is closest. We assume that m not= 0 and m,b are Reals.
i know your supposed to use the distance formula but i'm not sure how to go about it. Any help would be greatly appreciated. Thanks | {"url":"http://mathhelpforum.com/calculus/55658-another-optimization-problem.html","timestamp":"2014-04-18T01:07:43Z","content_type":null,"content_length":"43212","record_id":"<urn:uuid:6b28d592-1d72-40a3-a4b5-496d0c06d9e4>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00068-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mathwords: Third Quartile
Third Quartile
High Quartile
Higher Quartile
For a set of data, a number for which 75% of the data is less than that number. The third quartile is the same as the median of the part of the data which is greater than the median. Same as 75th
See also
First quartile, five-number summary | {"url":"http://www.mathwords.com/t/third_quartile.htm","timestamp":"2014-04-20T20:55:59Z","content_type":null,"content_length":"13102","record_id":"<urn:uuid:0341cb68-3a78-4a7a-acb7-fb7cf68daab6>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00492-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Patent application title: CIRCUIT AND METHOD FOR GAIN ERROR CORRECTION IN ADC
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Gain errors are corrected in an ADC chip including an integrator (17), a comparator (30), and a digital filter (37) by storing a gain-adjusted LSB size based on measured gain error in a memory (44).
The gain-adjusted LSB size is applied to the digital filter to cause gain-adjusted LSB size values to be added to or subtracted from accumulated content of the digital filter in accordance with a
first or second state, respectively, of the comparator (30) during each cycle of the ADC. The final accumulated content after all required cycles of the ADC is a gain-corrected digital output signal
An analog to digital converter (ADC) comprising:(a) an integrator for sampling an input signal, having an input coupled to receive an input signal and also having an output;(b) a comparator having a
signal input coupled to the output of the integrator, and an output representing a present state of the comparator depending on a relationship of an output voltage on the output of the integrator to
a threshold voltage;(c) a digital filter having an input coupled to the output of the comparator and also having an output for conducting a gain-corrected digital output signal representative of the
input signal;(d) a memory for storing an externally determined gain-adjusted LSB size number; and(e) circuitry coupled to the memory for applying an integral number of the gain-adjusted LSB size
numbers to the input of the digital filter to cause the integral number of gain-adjusted LSB size numbers to be applied to content of the digital filter in accordance with the present state of the
comparator to accumulate a value in the digital filter during successive cycles of the ADC so as to generate the gain-corrected digital output signal.
The ADC of claim 1 wherein the comparator is a window comparator having a threshold input which receives a plurality of threshold voltages, wherein the window comparator produces a first state which
is a +1 state, a second state which is a -1 state, and wherein the window comparator produces a third state which is a 0 state and which results in a zero input to the digital filter, wherein the
integral number of gain-adjusted LSB size numbers is applied to an increment input of the digital filter in accordance with the +1 state of the comparator so as to cause the integral number of
gain-adjusted LSB size numbers to be added to the accumulated content of the digital filter, and wherein the integral number of gain-adjusted LSB size numbers is applied to a decrement input of the
digital filter in accordance with the -1 state of the comparator so as to cause the integral number of gain-adjusted LSB size numbers to be subtracted from the accumulated content of the digital
The ADC of claim 1 wherein the integrator and the comparator form a delta-sigma modulator.
The ADC of claim 3 wherein the delta sigma modulator is a first order delta-sigma modulator.
The ADC of claim 1, wherein the ADC has a mixed delta-sigma/cyclic architecture.
The ADC of claim 4 wherein the digital filter is configured as an accumulator.
The ADC of claim 6 wherein the accumulator is configured as an up-down counter the content of which is incremented or decremented in the amount of the integral number of gain-adjusted LSB size
The ADC of claim 1 wherein the digital filter includes an SAR (successive approximation register) result register, an output of which provides the gain-corrected digital output.
The ADC of claim 1 wherein the integrator includes an operational amplifier having a non-inverting input coupled by a first sampling capacitor to receive a first input signal and also coupled by a
first integrating capacitor to an inverting output of the operational amplifier, the operational amplifier also having an inverting input coupled by a second sampling capacitor to receive a second
input signal and also coupled by a second integrating capacitor to a non-inverting output of the operational amplifier.
The ADC of claim 9 wherein the non-inverting input of the operational amplifier is coupled by a third sampling capacitor to a first reference voltage, a second reference voltage, or is unconnected to
any reference voltage in accordance with an output state of the comparator, and wherein the inverting input of the operational amplifier is coupled by a fourth sampling capacitor to the first
reference voltage, the second reference voltage, or is unconnected to any reference voltage in accordance with that output state of the comparator, to prevent overflow of a delta-sigma modulator.
The ADC of claim 10 wherein the input signal is equal to the first input signal minus the second input signal.
The ADC of claim 1 wherein the integral number is
1. 13.
The ADC of claim 1 wherein the memory is one of the group including a fuse link memory, a one time programmable memory, and an electrically erasable programmable memory.
The ADC of claim 1 wherein the memory includes means for receiving the gain-adjusted LSB size number from an external test and trim system.
The ADC of claim 1 wherein the memory and the output of the comparator are coupled to an increment input of the digital filter by a first ANDing circuit and are coupled to a decrement input of the
digital filter by a second ANDing circuit, wherein the first ANDing circuit effectuates adding of the integral number of gain-adjusted LSB size numbers from the memory to the content of the digital
filter in accordance with a first state of the comparator, and wherein the second ANDing circuit effectuates subtracting of the integral number of gain-adjusted LSB size numbers from the memory to
the decrement input in accordance with a second state of the comparator.
A method for operating an analog to digital converter which includes
an integrator having an input coupled to receive an input signal and also having an output,
a comparator having a signal input coupled to the output of the integrator, and an output, the output of the comparator representing a first state or a second state, depending on a relationship of an
output voltage on the output of the integrator to a threshold voltage, and
a digital filter having an input coupled to the output of the comparator, the method comprising:(a) storing an externally determined gain-adjusted LSB size number in a memory;(b) applying the
gain-adjusted LSB size number to the input of the digital filter to cause an integral number of gain-adjusted LSB size numbers to be applied to content of the digital filter in accordance with a
present state of the comparator so as to accumulate a value in the digital filter; and(c) after all required cycles of the analog to digital converter, providing the accumulated value at an output of
the digital filter as a gain-corrected digital output signal that represents the input signal.
The method of claim 16 including adding the integral number of gain-adjusted LSB size numbers from the memory to the content of the digital filter in accordance with a first state of the comparator,
and subtracting the integral number of gain-adjusted LSB size numbers from the memory in accordance with a second state of the comparator.
The method of claim 16 including operating the integrator and the comparator as a delta-sigma modulator and operating the digital filter as an up-down counter the content of which is incremented or
decremented by the integral number of gain-adjusted LSB size numbers.
The method of claim 16 wherein the ADC has a mixed delta-sigma/cyclic architecture and the method includes providing an output of a SAR result register as the gain-corrected digital output.
The method of claim 16 including providing additional bits of resolution in an analog to digital conversion process such that no missing code appears during generation of a smaller number of bits
which are utilized as the gain-corrected digital output signal.
An analog to digital converter comprising:(a) an integrator having an input coupled to receive an input signal and also having an output;(b) a comparator having a signal input coupled to the output
of the integrator, and an output, the output representing a first state or a second state, depending on a relationship of an output voltage on the output of the integrator to a threshold voltage;(c)
a digital filter having an input coupled to the output of the comparator;(d) means for storing an externally determined gain-adjusted LSB size number in a memory; and(e) means for applying the
gain-adjusted LSB size number to content of the digital filter so as to cause an integral number of gain-adjusted LSB size numbers to be applied to content of the digital filter in response to a
present state of the comparator so as to accumulate a gain-corrected digital value in the digital filter that, the after all required cycles of the analog-to-digital converter, is a gain-corrected
digital value that represents the input signal.
BACKGROUND OF THE INVENTION [0001]
The present invention relates generally to integrated circuit ADCs (analog to digital converters), and particularly to ADC circuits and methods for correcting the gain of an ADC with minimum circuit
complexity, a minimum amount of integrated circuit chip area, and a minimum amount of time required in addition to the basic ADC conversion time in order to accomplish the gain error correction.
Every ADC has gain error. There are two basic ways to correct the ADC gain error, namely analog correction and digital correction. One analog gain error correction technique is trimming of the
reference voltage. That controls the transfer function of the converter so as to achieve whatever correction is needed for its gain. Another analog ADC gain correction technique is scaling the ratios
of the sampling capacitors of the integration stage of the ADC. However, the known analog techniques for correcting ADC gain error require increased amounts of integrated circuit die area, and they
typically require use of costly laser trimming techniques. Furthermore, the gain correction accuracy which can be achieved using analog ADC gain error correction techniques is less than can be
achieved using digital ADC gain error correction techniques. Also, digital gain error correction techniques are easier to implement during final integrated circuit testing procedures than the analog
gain error correction techniques.
Several digital methods for trimming ADC gain error have been employed. FIG. 1 illustrates a prior art delta-sigma ADC 20A which includes an integrator 17, the (-) output 27A of which is coupled to
the (-) input of a window comparator 30. The (+) output 27B of integrator 17 is coupled to the (+) input of window comparator 30. Window comparator 30 may be composed of two or more conventional
comparators which receive various threshold voltages V
on the various conductors of window comparator threshold bus 29, respectively. An input signal voltage Vin=Vin
-Vin.sup.- is coupled between plates of sampling capacitors 23A and 23B, the other plates of which are connected to the (+) and (-) inputs, respectively, of an operational amplifier 18 of integrator
17. The (+) input conductor 26A of operational amplifier 18 is coupled by an integrating capacitor 24A to the (-) output conductor 27A of operational amplifier 18, and the (-) input conductor 26B of
operational amplifier 18 is coupled by an integrating capacitor 24B to the (+) output conductor 27B of operational amplifier 18. Input conductor 26A is coupled by a sampling capacitor 28A to the pole
of a single pole, triple throw switch circuit S1, and input conductor 26B is coupled by a sampling capacitor 28B to the pole of a single pole, triple throw switch circuit S2. Switch circuit S1
couples the (+) input of operational amplifier 18 through sampling capacitor 28A to Vref
, Vref.sup.- or an open terminal, depending upon whether the output CompOut of window comparator 30 is +1, -1, or 0, respectively, so the reference Vref can be integrated in the correct direction if
necessary, depending on the decision of window comparator 30. Similarly, switch circuit S2 couples the (-) input of operational amplifier 18 through sampling capacitor 28B to Vref.sup.-, Vref
, or an open terminal, depending upon whether CompOut is +1, -1, or 0, respectively, so Vref can be integrated in the correct direction if necessary.
The output conductors 31 of window comparator 30 are coupled to the input of a digital filter 37, the output conductors 40 of which produce the basic un-corrected digital representation Dout of the
input voltage Vin. The uncorrected signal Dout is multiplied by a digital gain correction coefficient by means of a digital multiplier 38 to produce a gain-corrected digital output signal Dout
[0005]FIG. 2
is a flowchart which indicates the operation of ADC 20A of FIG. 1. The part of
FIG. 2
within dashed line A generally indicates how the delta-sigma modulator 15 consisting of integrator 17, window comparator 30, and switch circuits S1 and S2 operate in conjunction with window
comparator 30 so as to cause the accumulation of new values in digital filter 37. As indicated in block 1 of
FIG. 2
, integrator 17 integrates only the input voltage Vin during the first integration cycle of delta-sigma modulator 15, since initially window comparator output CompOut is 0. Then, as indicated in
block 2, window comparator 30 determines whether the output of integrator 17 is greater than a threshold voltage V
, less than V
.sup.-, or between them. Depending on the results of this comparison, the output of window comparator 30 is "1", "-1", or "0", respectively. The threshold voltages V
and V
.sup.- can be derived from Vref and the scaling of various capacitors in integrator 17 so as to provide the various desired window comparator threshold voltages V
(not shown) of the standard comparators of which window comparator 30 is comprised.
In any case, window comparator output CompOut is coupled into and added to the accumulated result in digital filter 37. If modulator 15 is a first order delta-sigma modulator as shown in FIG. 1,
digital filter 37 can be implemented as an up-down counter, i.e., as an accumulator. For first-order delta-sigma modulator 15 as shown in FIG. 1, the up-down counter or accumulator of digital filter
37 is incremented by 1 if the output of window comparator 30 is +1 as indicated in decision block 3 and block 4, is decremented by 1 if the output of window comparator 30 is -1 as indicated in
decision block 5 and block 6, or remains unchanged if the output of window comparator 30 is "0" as indicated in block 7 of
FIG. 2
Then, as indicated in decision block 9, the process being performed in ADC 20A determines if all of the integration cycles required by ADC 20A have been performed. If this determination is negative,
the integrate and compare loop including blocks 1-9 is repeated as needed to complete the basic analog-to-digital conversion. On the second and each following required integration cycle after the
second, integrator 17 samples not only the input voltage Vin, but also samples the reference voltage value Vref
or Vref.sup.- in accordance with window comparator output CompOut. In the case wherein CompOut is too high, i.e., the input to window comparator 30 is greater than the upper threshold voltage
Vref.sup.-, window comparator 30 makes a decision to generate a +1 output level, and during the next integration cycle integrator 17 samples the lower reference voltage value V
.sup.- in addition to sampling the input voltage Vin, and this causes the output of integrator 17 to decrease. Similarly, in the case wherein CompOut is too low, i.e., the input to window comparator
30 is less than the lower threshold voltage V
.sup.-, window comparator 30 makes a decision to generate a -1 output level, and during the next integration cycle integrator 17 samples the upper reference voltage value V
in addition to sampling the input voltage Vin, and this causes the output of integrator 17 to increase.
After the process of sampling Vin and Vref the required numbers of times, the determination of decision block 9 eventually is affirmative. The window comparator output CompOut then is coupled for the
last time into digital filter 37, which then produces the uncorrected digital output Dout on bus 40, and the basic analog-to-digital conversion is complete.
After the basic conversion of Vin to the uncorrected digital output signal Dout on bus 40 has been completed, it is multiplied by a gain error trim coefficient to produce the corrected final digital
output Dout(gain-corrected) of ADC 20A, as indicated in blocks 11, 12, and 13 in the flow chart of
FIG. 2
As an example, assume that an input voltage Vin of 1.0 volt and a reference voltage Vref are applied to ADC 20A of FIG. 1. Assume also that the analog-to-digital conversion of the applied 1.0 volt
input signal is performed, and the result is not the desired 1.0 volt digital output voltage value, but instead is a 0.9 volt digital output voltage value because of gain error of ADC 20A. One way to
correct the gain error in the digital domain is to determine the value of a digital gain correction coefficient, which in this simplified example is approximately 1.1, and store it in the ADC
integrated circuit die. Then the digital output conversion value of 0.9 volt produced by digital filter 37 on bus 40 is automatically multiplied by the stored digital gain error trim coefficient of
approximately 1.1. In this example of Vin being equal to 1.0 volt, the multiplication result is a corrected output value of Dout very close to 1 volt (actually, 0.99 volts). From then on, every time
an analog to digital conversion of Vin is performed, the resulting digital output Dout on bus 40 is multiplied by the digital gain error trim coefficient 1.1 (in this simplified example) to thereby
obtain a corrected digital output voltage Dout(gain-corrected).
A problem with the above described prior art technique is that implementing a digital multiplier is always expensive because it requires complex circuitry and a large amount of integrated circuit die
area, and also because it requires a large amount of quiescent current and hence a large amount of power dissipation. The above described prior art technique also is very time-consuming, because the
analog-to-digital conversion must be performed first, and then the slow digital multiplication of the ADC conversion result must be multiplied, bit by bit, by the digital error correction
coefficient, which adds a substantial amount to the time required for the basic ADC conversion process in order to obtain a gain-error-corrected digital output value Dout(gain-corrected) which
accurately represents Vin. An additional drawback of this digital multiplication process is the possibility that the ADC transfer function may have missing codes, due to round-off errors.
Another known digital technique for trimming ADC gain error is by changing the number of integration cycles in a first order delta sigma ADC. This method does not provide adequate trim resolution if
the number of integration cycles is too low. (A 0.1% resolution change is often considered sufficient to be an acceptable increase in resolution order to achieve an acceptable ADC gain error
correction.) In some ADC architectures, the number of integration cycles can be very low, so the resolution of the digital gain error correction is poor if the method of changing the number of
integration cycles is used to accomplish the gain error correction. These architectures all have the problem that it is quite difficult to correct the ADC gain error by changing the number of
Primary shortcomings of the prior art technique of multiplying the conversion result by a gain error correction coefficient are that it requires too much power dissipation and too much circuit
complexity, and also requires too much chip area, and too much total analog-to-digital conversion time in order to obtain the gain-error-corrected digital output value.
Thus, there is an unmet need for an ADC which corrects its ADC transfer characteristic for gain error without increasing the amount of time required for obtaining a gain-corrected digital output
substantially beyond the amount of time required for a basic analog to digital conversion.
There also is an unmet need for an ADC which corrects its ADC transfer characteristic for gain error without increasing the amount of integrated circuit chip area for obtaining a gain-corrected
digital output substantially beyond the amount of chip area required for a basic analog to digital conversion.
There also is an unmet need for an ADC which corrects its ADC transfer characteristic for gain error without increasing the amount of power consumption required in obtaining a gain-corrected digital
output substantially beyond the amount of power consumption required for a basic analog to digital conversion.
There also is an unmet need for an ADC which corrects its ADC transfer characteristic for gain error without the high level of circuit complexity generally associated with using analog circuit
techniques to correct a transfer characteristic of an ADC for gain error.
There also is an unmet need for an ADC gain error correction technique which is particularly suitable for use in a delta-sigma/cyclic ADC architecture, and is also useful in a SAR ADC architecture
and in a higher-order delta-sigma ADC architecture.
SUMMARY OF THE INVENTION [0019]
It is an object of the invention to provide an ADC which corrects its ADC transfer characteristic for gain error without increasing the amount of time required for obtaining a gain-corrected digital
output substantially beyond the amount of time required for a basic analog to digital conversion.
It is another object of the invention to provide an ADC which corrects its ADC transfer characteristic for gain error without increasing the amount of integrated circuit chip area for obtaining a
gain-corrected digital output substantially beyond the amount of chip area required for a basic analog to digital conversion.
It is another object of the invention to provide an ADC which corrects its ADC transfer characteristic for gain error without increasing the amount of power consumption required in obtaining a
gain-corrected digital output substantially beyond the amount of power consumption required for a basic analog to digital conversion.
It is another object of the invention to provide an ADC which corrects its ADC transfer characteristic for gain error while avoiding the high level of circuit complexity generally associated with
using analog circuit techniques to correct a transfer characteristic of an ADC for gain error.
It is another object of the invention to provide an ADC gain error correction technique which is particularly suitable for use in a delta-sigma/cyclic ADC architecture, and is also useful in a SAR
ADC architecture and in a higher-order delta-sigma ADC architecture.
Briefly described, and in accordance with one embodiment, the present invention provides correction of errors in an ADC chip including an integrator (17), a comparator (30), and a digital filter (37)
by storing a gain-adjusted LSB size based on measured gain error in a memory (44). The gain-adjusted LSB size is applied to the digital filter to cause gain-adjusted LSB size values to be added to or
subtracted from accumulated content of the digital filter in accordance with a first or second state, respectively, of the comparator (30) during each cycle of the ADC. The final accumulated content
after all required cycles of the ADC is equal to a gain-corrected digital output signal (Dout(gain-corrected)).
In one embodiment, the invention provides an analog to digital converter (ADC) (20B) including an integrator (17) for sampling an input signal (Vin), the integrator having an input (26A,B) coupled to
receive an input signal (Vin) and also having an output (27A,B). A comparator (30) has a signal input coupled to the output (27A,B) of the integrator (17), and an output (33A,B). The output of the
comparator represents a present state of the comparator depending on a relationship of an output voltage on the output (27A,B) of the integrator (17) to a threshold voltage (V
). A digital filter (37) has an input (48A,B) coupled to the output (33A,B) of the comparator (30) and also has an output (49) for conducting a gain-corrected digital output signal (Dout
(gain-corrected)) representative of the input signal (Vin). A memory (44) stores an externally determined gain-adjusted LSB size number. Circuitry (47A,B) coupled to the memory (44) applies an
integral number of the gain-adjusted LSB size numbers to the input (48A,B) of the digital filter to cause the integral number of gain-adjusted LSB size numbers to be applied to content of the digital
filter (37) in accordance with the present state of the comparator (30) to accumulate a value in the digital filter (37) during successive cycles of the ADC (20B) so as to generate the gain-corrected
digital output signal (Dout(gain-corrected)).
In one embodiment, the comparator is a window comparator (30) having a threshold input (29) which receives a plurality of threshold voltages (V
's), wherein a first state of the window comparator is a +1 state, a second state of the window comparator is a -1 state, and wherein the window comparator (30) produces a third state which is a 0
state and which results in a zero input to the digital filter (37). The integral number of gain-adjusted LSB size numbers is applied to an increment input (48A) of the digital filter (37) in
accordance with the +1 state of the comparator (30) so as to cause the integral number of gain-adjusted LSB size numbers to be added to an accumulated content of the digital filter (37). The integral
number of gain-adjusted LSB size numbers is applied to a decrement input (48B) of the digital filter (37) in accordance with the -1 state of the comparator (30) so as to cause the integral number of
gain-adjusted LSB size numbers to be subtracted from the accumulated content of the digital filter (37).
In one embodiment, integrator (17) and the comparator (30) form a delta-sigma modulator (15). In one embodiment, the delta sigma modulator (15) is a first order delta-sigma modulator. In one
embodiment, the ADC has a mixed delta-sigma/cyclic architecture.
In one embodiment, the digital filter includes an accumulator which is configured as an up-down counter the content of which is incremented or decremented in the amount of the integral number of
gain-adjusted LSB size numbers.
In one embodiment, the digital filter includes an SAR (successive approximation register) results register (450), an output of which provides the gain-corrected digital output.
In a described embodiment, the integrator includes an operational amplifier (18) having a non-inverting input coupled by a first sampling capacitor (23A) to receive a first input signal (Vin
) and also coupled by a first integrating capacitor (24A) to an inverting output (27A) of the operational amplifier (18), the operational amplifier (18) also having an inverting input coupled by a
second sampling capacitor (23B) to receive a second input signal (Vin.sup.-) and also coupled by a second integrating capacitor (24B) to a non-inverting output (27B) of the operational amplifier
(18). The non-inverting input of the operational amplifier (18) is coupled by a third sampling capacitor (28A) to a first reference voltage (Vref
), a second reference voltage (Vref.sup.-), or is unconnected to any reference voltage in accordance with an output state of the comparator (30), and wherein the inverting input of the operational
amplifier (18) is coupled by a fourth sampling capacitor (28B) to the first reference voltage (Vref
), the second reference voltage (Vref.sup.-), or is unconnected to any reference voltage in accordance with that output state of the comparator (30), to prevent overflow of the delta-sigma modulator.
The memory (44) can be a fuse link memory, a one time programmable memory, or an electrically erasable programmable memory, and can be a user-accessible register which allows a user to control the
transfer characteristic of the ADC. The memory can receive the gain-adjusted LSB size number from an external test and trim system.
In a described embodiment, the memory (44) and the output (33A,B) of the window comparator (30) are coupled to an increment input (INC) of the digital filter (37) by a first ANDing circuit (47A) and
are coupled to a decrement input (DEC) of the digital filter (37) by a second ANDing circuit (47B), wherein the first ANDing circuit (47A) effectuates adding of the integral number of gain-adjusted
LSB size numbers from the memory (44) to the content of the digital filter (37) in accordance with a first state of the window comparator (30), and wherein the second ANDing circuit (47B) effectuates
subtracting of the integral number of gain-adjusted LSB size numbers from the memory (44) to the decrement input (DEC) in accordance with a second state of the window comparator (30).
In one embodiment, the invention provides a method for operating an analog to digital converter which includes an integrator (17) having an input (26A,B) coupled to receive an input signal (Vin) and
also having an output (27A,B), a comparator (30) having a signal input coupled to the output (27A,B) of the integrator (18), and an output (33A,B), the output of the comparator representing a first
state or a second state, depending on a relationship of an output voltage on the output (27A,B) of the integrator (17) to a threshold voltage (V
), and a digital filter (37) having an input (48A,B) coupled to the output (33A,B) of the comparator (30), wherein the method includes storing an externally determined gain-adjusted LSB size number
in a memory (44), applying the gain-adjusted LSB size number to the input (48A,B) of the digital filter (37) to cause an integral number of gain-adjusted LSB size numbers to be applied to content of
the digital filter (37) in accordance with a present state of the comparator (30) so as to accumulate a value in the digital filter (37), and after all required cycles of the analog to digital
converter, providing the accumulated value at an output of the digital filter (37) as a gain-corrected digital output signal (Dout(gain-corrected)) that represents the input signal (Vin).
In one embodiment, the method includes adding the integral number of gain-adjusted LSB size numbers from the memory (44) to the content of the digital filter (37) in accordance with a first state of
the comparator (30), and subtracting the integral number of gain-adjusted LSB size numbers from the memory (44) in accordance with a second state of the comparator (30).
In one embodiment, the method includes operating the integrator (17) and the comparator (30) as a delta-sigma modulator (15) and operating the digital filter (37) as an up-down counter the content of
which is incremented or decremented by the integral number of gain-adjusted LSB size numbers. In one embodiment the ADC has a mixed delta-sigma/cyclic architecture and the method includes providing
an output of a SAR result register as the gain-corrected digital output.
In one embodiment, the method includes providing additional bits of resolution in an analog to digital conversion process such that no missing code appears during generation of a smaller number of
bits which are utilized as the gain-corrected digital output signal.
In one embodiment, the invention provides an analog to digital converter (20B) including an integrator (17) having an input (26A,B) coupled to receive an input signal (Vin) and also having an output
(27A,B), a comparator (30) having a signal input coupled to the output (27A,B) of the integrator (18), and an output (33A,B), the output representing a first state or a second state, depending on a
relationship of an output voltage on the output (27A,B) of the integrator (17) to a threshold voltage (V
), a digital filter (37) having an input (48A,B) coupled to the output (33A,B) of the comparator (30), means (44) for storing an externally determined gain-adjusted LSB size number in a memory (44),
and means (47A, 47B) for applying the gain-adjusted LSB size number to content of the digital filter (37) so as to cause an integral number of gain-adjusted LSB size numbers to be applied to content
of the digital filter (37) in response to a present state of the comparator (30) so as to accumulate a gain-corrected digital value (Dout(gain-corrected)) in the digital filter (37) that, the after
all required cycles of the analog-to-digital converter, is a gain-corrected digital value (Dout(gain-corrected)) that represents the input signal (Vin).
BRIEF DESCRIPTION OF THE DRAWINGS [0038]
FIG. 1 is a block diagram of a conventional delta-sigma ADC including a digital multiplier for providing DC gain error correction.
[0039]FIG. 2
is a flow chart illustrating a conventional digital ADC gain error correction technique used in the delta-sigma ADC of FIG. 1.
FIG. 3 is a block diagram illustrating an integrated circuit first order delta-sigma ADC utilizing a digital gain error correction technique according to the invention by, in effect, digitally
modifying the value of the LSB (least significant bit) of the basic ADC conversion result.
[0041]FIG. 4
is a flow chart illustrating a digital ADC gain error correction technique used in the delta-sigma ADC of FIG. 3.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS [0042]
Referring to FIG. 3, integrated circuit delta-sigma ADC chip 20B includes integrator 17, the (-) output 27A of which is coupled to the (-) input of a window comparator 30. The (+) output 27B of
integrator 17 is coupled to the (+) input of window comparator 30. Window comparator 30 may be composed of two or more conventional comparators which may receive various thresholds voltages V
on the conductors of window comparator threshold bus 29, respectively. (By way of definition, it should be understood that if a conventional comparator receives a differential input signal, it may be
considered to have a zero threshold voltage. The term "threshold voltage" as used herein is intended to encompass a threshold voltage applied to one input of a 2-input comparator the other input of
which receives an input signal voltage that is to be compared with the threshold voltage, and is intended to also encompass a zero threshold voltage of a comparator which receives a differential
input signal.)
An input signal voltage Vin=Vin
-Vin.sup.- is coupled between plates of input sampling capacitors 23A and 23B, the other plates of which are connected to the (+) and (-) inputs, respectively, of operational amplifier 18 of
integrator 17. The (+) input conductor 26A of operational amplifier 18 is coupled by an integrating capacitor 24A to the (-) output conductor 27A of operational amplifier 18, and the (-) input
conductor 26B of operational amplifier 18 is coupled by an integrating capacitor 24B to the (+) output conductor 27B of operational amplifier 18.
Just as in Prior Art FIG. 1, input conductor 26A is coupled by a reference sampling capacitor 28A to the pole of a single pole, triple throw switch circuit S1, and input conductor 26B is coupled by a
reference sampling capacitor 28B to the pole of a single pole, triple throw switch circuit S2. Switch circuit S1 couples the (+) input of operational amplifier 18 through sampling capacitor 28A to
, Vref.sup.- or an open terminal, depending upon whether the output CompOut of window comparator 30 is +1, -1, or 0, respectively, so the reference Vref can be integrated in the correct direction if
necessary, depending on the decision of window comparator 30. Similarly, switch circuit S2 couples the (-) input of operational amplifier 18 through sampling capacitor 28B to Vref.sup.-, Vref
or an open terminal, depending upon whether CompOut is +1, -1, or 0, respectively, so Vref can be integrated in the correct direction if necessary.
Note that more details of the conventional delta-sigma modulator 15 appear in FIG. 3a of the assignee's subsequently mentioned pending patent application Ser. No. 11/738,566, which is incorporated
herein by reference. However, those skilled in the art can readily understand the basic operation of integrator 15 as described and as shown in FIGS. 1 and 3 hereof.
By way of definition, the term "integrator" as used herein is intended is intended to encompass not only a conventional integrator such as integrator 17 shown in FIG. 3, but also an integrator in a
cyclic SAR ADC, wherein the integrator only samples once per conversion. The present invention is as applicable to that kind of ADC as to a delta-sigma ADC.
Output conductor 33A of window comparator 30 is connected to an enable input of an ANDing circuit 47A which also has multiple gain error data inputs connected to a "gain-adjusted LSB size bus" 46
which is connected to the output of a "gain-adjusted LSB size memory" 44. Gain-adjusted LSB size memory 44 can be implemented by means of blowable fuses, an EEPROM (electrically erasable programmable
memory), or an OTP (one time programmable) memory or the like which is loaded from a conventional external test and trim system (not shown) with a "gain-adjusted LSB size number", expressed in binary
format. The output of ANDing circuit 47A is connected by conductor 48A, on which a signal INCREMENT BY GAIN-ADJUSTED LSB SIZE is applied to an increment input (INC) of digital filter 37, conventional
internal circuitry (not shown) of which causes a digital number equal to the gain-adjusted LSB size to be added or "incremented" to the contents of the up-down counter or accumulator in digital
filter 37. Similarly, output conductor 33B of window comparator 30 is connected to an enable input of another ANDing circuit 47B, which also has multiple inputs connected to "LSB size bus" 46. The
output of ANDing circuit 47B is connected by a bus 48B, on which a signal DECREMENT BY GAIN-ADJUSTED LSB SIZE is applied to a decrement input (DEC) of digital filter 37, conventional internal
circuitry (not shown) of which causes a digital number equal to the gain-adjusted LSB size to be subtracted or "decremented" from the contents of the up-down counter or accumulator in digital filter
37. After all of the required integration cycles have been performed, the output of digital filter 37 is the gain-corrected digital output signal Dout(gain-corrected).
By way of definition, the term "up-down counter" as used herein is intended to encompass accumulating circuitry in a digital filter the contents of which can be incremented by adding a non-integral
LSB size number thereto or decremented by subtracting the non-integral LSB size number therefrom.
It should be understood that ANDing circuits 47A and 47B or equivalent circuitry could be included within digital filter 37 so as to provide adding or incrementing of integral numbers of LSB size
numbers to or subtraction of integral numbers of LSB size numbers from the content of digital filter 37. Gain-adjusted LSB size memory 44 could be incorporated as a register which, if desired, could
be considered to be within digital filter 37. Gain-adjusted LSB size memory 44 also could be implemented as a user-accessible register, which would allow the user to adjust the transfer
characteristic of the ADC.
It should be appreciated that the output bus of window comparator 30 can have as many conductors as is needed to conduct the number of output states that can be generated by window comparator 30. For
example, if window comparator 30 generates 3 states, a two-wire bus may be required as the output of window comparator 30. If window comparator 30 generates 5 states, a three-wire bus is required as
the output of window comparator 30. (However, if an ordinary non-window type of comparator is used, then a single conductor can be used as the output of that comparator to represent the two states,
"1" and "0".)
If the window comparator 30 has more than 3 states, appropriate integral multiples of the gain-adjusted LSB size are added to or subtracted from the contents of digital filter 37. For example, for a
5-level window comparator 30, its possible outputs are +2, +1, 0, 1, and -2. In this case, the gain-adjusted quantities available to be added to or subtracted from the contents of digital filter 37
will include 2×(gain-adjusted LSB size), which is implemented by shifting the quantity "gain-adjusted LSB size" by one binary bit position, 1×(gain-adjusted LSB size), 0, -1×(gain-adjusted LSB size),
and -2×(gain-adjusted LSB size).
Those skilled in the art know that a window comparator can be thought of as a kind of flash ADC. For example, if a window comparator can generate 64 different states, it would be comprised of 64
ordinary comparators coupled in parallel, the 64 comparators having 64 different input threshold voltages. The output bus of a window comparator, for example a window comparator that can generate 64
different states, ordinarily is coupled to circuitry that performs an encoding function which converts the 64 comparator outputs into a 6-bit code. For example, window comparator 30 of FIG. 3, with
its three output states +1, 0, and 1, is composed of 2 basic comparators having 2 separate input threshold voltages (e.g., V
and V
.sup.-), respectively, so that one basic comparator changes state when the window comparator input voltage exceeds the upper threshold voltage V
and the other basic comparator changes state when the window comparator input voltage is less than the lower threshold voltage V
.sup.-, and when the window comparator input voltage is between the upper and lower threshold voltages, neither basic comparator changes state. Two comparator output conductors 33A and 33B are
required to represent the three states.
[0053]FIG. 4
is a flowchart which indicates the process of obtaining the gain-adjusted LSB size and also indicates the operation of ADC 20B of FIG. 3. As indicated in block 50 of
FIG. 4
, the first step in the overall procedure is to determine the gain error trim coefficient.
The gain error coefficient value that needs to be multiplied by the measured digital output of ADC 20B is determined on the basis of actual measurements of Dout and the desired full scale value
thereof, to determine the percentage of gain error. During final testing of ADC 20B, a known value of input voltage Vin is applied to the integrated circuit chip including ADC 20B, and the resulting
analog-to-digital Dout is measured. Then the resulting error is calculated and the error is used to calculate how much the size of the LSB needs to be adjusted so that the error of the measured
conversion result Dout after gain correction will be zero. The error is expressed as a percentage of the known correct value of Dout, in binary format.
As indicated in block 51 of
FIG. 4
, the gain-adjusted LSB size is determined after the gain error trim coefficient has been determined. The relationship between the gain error trim coefficient and the gain-adjusted LSB size can be
expressed as (gain correction coefficient)×(uncorrected value of Dout)=(gain-corrected value of Dout). That value of the gain correction coefficient, which is equal to the gain-adjusted LSB size, is
entered into and permanently stored in gain-adjusted LSB size memory 44, for example by blowing certain fuses or links or writing it into an OTP memory inside the integrated circuit ADC. The initial
contents of gain-adjusted LSB size memory 44 can be a default value of 1.000 . . . . Then, when the amount of ADC gain error is determined, that default value is modified to a value different than
1.000 . . . , so as to generate the number GAIN-ADJUSTED LSB SIZE which is added to the accumulator or up-down counter contents if the output of window comparator 30 is +1 or is subtracted from the
accumulator or up-down counter contents if the output of window comparator 30 is -1.
The part of
FIG. 4
within dashed line A generally indicates how the first-order delta-sigma modulator 15 consisting of integrator 17, window comparator 30, and switch circuits S1 and S2 in FIG. 3 operate in conjunction
with window comparator 30 so that during every integration cycle the input voltage Vin is integrated from the input capacitors to the holding or feedback capacitors, and this causes the value of the
output of integrator 17 to go in one direction. For example, if input voltage Vin is positive, the output voltage of integrator 17 increases in response to Vin. At some point, integrator would
"overflow" and stop integrating except for the fact that during every integration cycle, window comparator 30 makes a decision as to whether the output of integrator 17 is beyond a particular upper
threshold voltage V
. When V
is reached, then during the next integration cycle, integrator 17 also samples (in addition to the input voltage Vin) the reference voltage Vref in the appropriate direction, so that if the output of
integrator 17 is too high, then reference voltage Vref is sampled in the negative direction (i.e., V
.sup.- is sampled) and as result the output voltage of integrator 17 decreases.
As indicated in block 1 of
FIG. 4
, integrator 17 integrates only the input voltage Vin during the first integration cycle of delta-sigma modulator 15, since initially window comparator output CompOut is 0. Then, as indicated in
block 2, window comparator 30 determines whether the output of integrator 17 is greater than V
, less than V
.sup.-, or between V
and V
.sup.-. Depending on the results of this comparison, the output of window comparator 30 is "1", "-1", or "0", respectively. (The threshold voltages V
and V
.sup.- can be derived from Vref and the scaling of various capacitors in integrator 17 so as to provide the various desired window comparator threshold voltages V
of the standard comparators of which window comparator 30 is comprised.)
In any case, window comparator output voltage CompOut is coupled to and causes the gain-adjusted LSB size to be added to or subtracted from the accumulated result in digital filter 37. If modulator
15 is a first order delta-sigma modulator as shown in FIG. 3, digital filter 37 can be implemented as an up-down counter, i.e., as an accumulator. For first-order delta-sigma modulator 15 as shown in
FIG. 3, the up-down counter or accumulator of digital filter 37 is incremented by the value of the digital number GAIN-ADJUSTED LSB SIZE stored in gain-adjusted LSB size memory 44 if the output of
window comparator 30 is +1, as indicated in decision block 3 and in block 4A. Similarly, the up-down counter in digital filter 37 is decremented by the value of the digital number GAIN-ADJUSTED LSB
SIZE stored in gain-adjusted LSB size memory 44 if the output of window comparator 30 is -1, as indicated in decision block 5 and in block 6A, or remains unchanged if the output of window comparator
30 is "0", as indicated in block 7 of
FIG. 4
Then, as indicated in block 8, the resulting value of the window comparator output voltage CompOut is used to update the accumulated result in the up-down counter in digital filter 37 either by the
amount +GAIN-ADJUSTED LSB SIZE or by the amount -GAIN-ADJUSTED LSB SIZE.
Then, as indicated in decision block 9, the process being performed in ADC 20B determines if all of the integration cycles required by ADC 20B have been performed. If this determination is negative,
the integrate and compare loop including blocks 1-9 is repeated to complete the analog-to-digital conversion. On the second and each following required integration cycle after the second, integrator
17 samples not only the input voltage Vin, but also samples the reference voltage value Vref
or Vref.sup.- in accordance with window comparator output CompOut. In the case wherein CompOut is too high, i.e., the input to window comparator 30 is greater than upper threshold voltage V
, window comparator 30 makes the decision to generate a +1 output level, and during the next integration cycle integrator 17 samples the lower reference voltage value Vref in addition to sampling
input voltage Vin, and this causes the output of integrator 17 to decrease. Similarly, in the case wherein CompOut is too low, i.e., the input to window comparator 30 is less than lower threshold
voltage V
.sup.-, window comparator 30 makes a decision to generate a -1 output level, and during the next integration cycle integrator 17 samples the upper reference voltage value Vref
in addition to sampling input voltage Vin, and this causes the output of integrator 17 to increase. (It should be appreciated that the input voltage Vin, the reference voltage Vref, and the
integrator output voltage each can be either differential or single-ended-voltages, but in most practical implementations they are differential signals or are sampled as differential signals. For
example, the input voltage Vin and the integrator output voltage typically are differential signals, and the reference voltage Vref is referenced to ground but is sampled as a differential signal.)
After the process of sampling Vin and Vref the required numbers of times for ADC 20B, the determination of decision block 9 is affirmative. The window comparator output CompOut then is coupled for
the last time into digital filter 37, which produces the gain-error-corrected digital output Dout, and the gain-error-adjusted analog-to-digital conversion of Vin to Dout(gain-corrected) is complete.
This avoids the prior art process of digital multiplying of the basic uncorrected value of Dout by a gain error trim coefficient to produce the corrected final digital output Dout(gain-corrected).
It should be noted that only blocks 50, 51, 4A and 6B as shown in the flowchart of
FIG. 4
are substantially different than in the flowchart of Prior Art
FIG. 2
As an example, assume that an input voltage Vin of 1.0 volt and a reference voltage Vref are applied to ADC 20B of FIG. 3. Assume also that the analog-to-digital conversion of the applied 1.0 volt
input signal is performed, and the result is not the desired 1.0 volt digital output voltage value, but instead is a 0.9 volt digital output voltage value because of gain error of ADC 20B.
The value of the needed digital gain correction coefficient, which in this simplified example is approximately 1.1, is determined and then stored in gain-adjusted LSB size memory 44. Then the digital
output conversion value of 0.9 volt produced by the integrator causes the gain-corrected digital output Dout(gain-corrected) which is accumulated in digital filter 37 and produced on bus 49 to be
very close to 1 volt (actually, 0.99 volts in this simplified example).
Thus, instead of performing a digital multiplication of the basic uncorrected ADC conversion result Dout at the output of the digital filter in order to correct the ADC gain error as in Prior Art
FIGS. 1 and 2, the present invention, in effect, digitally adds an integral multiple of a gain-adjusted LSB size number to the accumulated contents in the up-down counter of digital filter 37 in
response to each +1 level produced by window comparator 30, and also, in effect, digitally subtracts an integral multiple of the gain-adjusted LSB size number from the accumulated contents in the
up-down counter of digital filter 37 in response to each -1 level produced by window comparator 30. The gain correction function is achieved without use of a complex, slow digital multiplier as
required by the prior art, and is achieved simultaneously, on a bit-to-bit basis as the analog-to-digital conversion progresses. Therefore, the substantial additional amount of time for digital
multiplication by the gain correction coefficient after the basic ADC conversion operation has completely occurred as required in the prior art is avoided by the present invention.
The above mentioned technique of adjusting the size of the LSB of the digital output Dout of ADC 20B can cause "missing codes". If the LSB size is changed from 1.0 to a gain-adjusted value and that
value then is repeatedly added to and/or subtracted from the accumulator or up-down counter of ADC 20B after each integration cycle, the result in the accumulator will always be a value that is equal
to an integral number of gain-adjusted LSB sizes. At the end of all of the integration cycles required by ADC 20B for a single ADC conversion, there will be a need to perform a round-off of the value
in the accumulator or op-down counter. Assume that the LSB size is not 1.0, but is 1.1 because it is necessary to trim the ADC gain error. Also assume that the value of Vin, and hence the value of
Dout, is very close to zero wherein window comparator 30 goes to +1 only once during the entire integration process. Then the final result in the accumulator of digital filter 37 will be 1.1 because
its output was incremented only once by GAIN-ADJUSTED LSB SIZE. Also assume that the part of the up-down counter content located after the decimal point is discarded, so the final value of Dout
(gain-corrected) is 1.0.
Next, assume that Vin is twice as large as in the foregoing example. Then, window comparator 30 switches to a +1 twice during the entire integration process so at the end of the entire integration
process the final result in the accumulator or up-down counter is 2×1.1, or 2.2. The part of this located after the decimal point is rounded off, i.e. discarded, and the final result in digital
filter 37 is 2.
Next assume that Vin is 9 times greater than Vin in the first example above. Then the result in accumulator after the entire integration process is 9×1.1, i.e., 9.9. Discarding the part located after
the decimal point results in a final value of value 9.
Next, assume Vin is 10 times greater than in the first foregoing example. Then the result in the accumulator of digital filter 37 is exactly 11 after all of the integration cycles required by ADC 20B
have been completed. In this case there is nothing to round off, and the final result in the accumulator is taken to be 11. Thus, a missing code, namely 10, has been encountered. (Note that a missing
code will occur regardless of what rounding off procedure is used.) A solution to the above described missing code problem is to add more bits of resolution to the analog to digital conversion
process such that no missing code appears during the generation of the smaller number of bits which are utilized as the output Dout. That is, adding more bits of resolution to the conversion process
means that the lower amount of resolution actually desired for Dout is achieved before a missing code has a chance to occur.
The technique of the present invention is equally applicable for other methods of integration, such as higher order delta-sigma conversion or a pure SAR (successive approximation register)
conversion. In some ADC architectures, additional bits of resolution can be obtained at very little expense.
In one ADC architecture referred to as a "mixed delta-sigma/cyclic architecture", conventional integration is utilized to generate some of the output bits, after which a SAR mode of operation is
utilized and a residue on the output of the integrator is multiplied by a factor of 2 during each cycle. The binary content of the SAR filter is also shifted by one bit. (By way of definition, the
term "cycles" can refer to sub-cycles of the SAR process.)
Details of such a "mixed delta-sigma/cyclic ADC architecture", including additional details of conventional first order delta-sigma modulator 15 shown in FIGS. 1 and 3 herein, are shown in FIG. 3a of
the assignee's pending patent application Ser. No. 11/738,566 entitled "Hybrid Delta-Sigma/SAR Analog to Digital Converter and Methods for Using Such", filed Apr. 23, 2007 by Jerry Doorenbos, Marco
Gardner and Dimitar Trifonov, and incorporated herein by reference.
A cyclic SAR ADC works as follows. First, the integrator samples the ADC input voltage. Then the ADC performs a number of SAR successive approximation analog to digital conversion cycles. In the case
of a cyclic SAR such as one included as part of the mixed delta-sigma/cyclic ADC architecture in the above mentioned patent application Ser. No. 11/738,566, the residue on the output of the
integrator 310 shown in FIG. 3a after a predetermined number of integration cycles therein is multiplied by 2 each cycle. If the comparator output is equal to +1, then together with the
multiplication by 2, a negative reference voltage Vref is sampled. If the comparator output is equal to -1, then together with the multiplication by 2, a positive reference voltage Vref is sampled.
If the comparator output is equal to 0, then only the multiplication by 2 is performed.
The procedure performed during each SAR cycle, can be summarized by the following algorithm:
-US-00001 If (Vresidue > +Vthreshold) Then Vout = 2*(Vresidue - Vref) Else If (Vresidue < -Vthreshold) Then Vout = 2*(Vresidue + Vref) Else Vout = 2*(Vresidue) End If
The foregoing algorithm can be modified for various SAR ADC
Next, an example is provided to show how the result is accumulated in the SAR result register 450 in FIG. 3a of Ser. No. 11/738,566, (which can be a digital filter that in this particular case is
more like a shift register). Assume that the ADC is a 5 bit SAR ADC and that in accordance with 5 corresponding SAR cycles, comparator 414 generates a sequence of 5 decisions or states which are +1,
0, +1, -1, +1. The conversion results can be produced in two different ways that yield the same result.
The first way, which can be referred to as a "shift register approach", is as follows. Since the comparator decision during the first SAR cycle is +1, so the initial value (00000) plus the comparator
output +1 is loaded into the shift register. The result is 00000+1=00001.
For the second cycle, the comparator decision is 0, so the present contents of the shift register are shifted one position to the left and the 00001 becomes 00010 and is added to the comparator
output 0, producing the result 00010+0=00010.
For the third cycle, the comparator decision is +1, so the present contents of the shift register are shifted one position to the left and the 00010 becomes 00100 and is added to the comparator
output +1, producing the result 00100+1=00101.
For the fourth cycle, the comparator decision is -1, so the present contents of the shift register are shifted one position to the left and the 00101 becomes 01010 and is added to the comparator
output -1, producing the result 01010-1=01001.
For the fifth cycle, the comparator decision is +1, so the present contents of the shift register is shifted one position left and of the 01001 becomes 10010 and is added to the comparator output +1,
producing the result 10010+1=10011. Thus, the final conversion result is 10011.
The second way, which is referred to as the "digital filter approach", is as follows. Since the comparator decision during the first SAR cycle is +1, the comparator output is scaled by the integer
value 16, and the scaled comparator output becomes +10000. The initial value 00000 plus the scaled comparator output +10000 is loaded into the result register 450, producing the result 00000+10000=
For the second SAR cycle, the comparator decision is 0, so the comparator output is scaled by the integer value 8. Then the previous value 10000 in the result register plus the scaled comparator
output 0 is loaded into the result register, producing the result 10000+0=10000.
For the third cycle, the comparator decision is +1, so the comparator output is scaled by the integer value 4, and the scaled comparator output becomes +100. Then the previous value 10000 in the
result register plus the scaled comparator output+100 is loaded into the result register, producing the result 10000+100=10100.
For the fourth cycle, the comparator decision is -1, so the comparator output is scaled by the integer value 2, and the scaled comparator output becomes -10. Then the previous value 10100 in the
result register plus the scaled comparator output (-10) is loaded into the result register, producing the result 10100-10=10010.
For the fifth cycle, the comparator decision is +1, so the comparator output is scaled by the integer value 1, and scaled comparator output remains +1. Then the previous value 10010 plus the scaled
comparator output +1 is loaded into the result register, producing the result 10010+1=10011. Thus, the final conversion result is 10011.
Next, an example be provided to show how the basic technique of the present invention can be applied to the mixed delta-sigma/cyclic architecture shown in above mentioned pending patent application
Ser. No. 11/738,566. In the present invention, instead of sending the comparator values +1, 0, -1 to the result register 450 as described above, the properly scaled "LSB size number" is sent to the
result register. Assume that the ideal conversion result from the ADC for the same input voltage as in the previous example is 10100, but because of ADC gain error the actual measured result is
10011. Then the needed LSB size number is calculated to be 1.0001, which is a binary number representing a 6.25% gain error correction. If this gain-corrected "LSB size number" is applied to the
above mentioned algorithm above with the same sequence of decisions from the comparator 414, the result for each of the previously described examples will be as follows.
To apply the present invention to the previously described the "shift register approach", for the first cycle, of the comparator decision is +1, the initial value 00000.0000 in the SAR result
register 450 plus the "LSB size number" scaled by the comparator output +1 is loaded into the shift register, producing the result 00000.0000+1.0001=00001.0001.
For the second cycle, the comparator decision is 0 so for present contents of the shift register is shifted one position to the left and the 00001.0001 becomes 00010.0010 and is added to the "LSB
size number" scaled by the comparator output 0, producing the result 00010.0010+0=00010.0010.
For the third cycle, the comparator decision is +1, so the present contents of the shift register are shifted one position to the left and the 00010.0010 becomes 00100.0100 and is added to the "LSB
size number" scaled by the comparator output +1, producing a result 00100.0100+1.0001=00101.0101.
For the fourth cycle, the comparator decision is -1, so the present contents of the shift register are shifted one position to the left and the 00101.0101 becomes 01010.1010 and is added to the "LSB
size number" scaled by the comparator output -1, producing the result 01010.1010-1.0001=01001.1001.
For the fifth cycle, the comparator decision is +1, so the present contents of the contents of the shift register are shifted one position to the left and the 01001.1001 becomes 10011.0010 and is
added to the "LSB size number" scaled by the comparator output (+1), producing the result 10011.0010+1.0001=10100.0011.
After rounding off (discarding) the numbers after the decimal point, the final gain-adjusted ADC conversion result is 10100.
To apply the present invention to the previously described "digital filter approach", for the first SAR cycle, the comparator decision is +1, so the "LSB size number" is scaled by the comparator
output and by the integer value 16 so the scaled value becomes +10001.0000. The initial value 00000.0000 plus the scaled comparator output +10001.0000 is loaded into the digital filter, producing the
result 00000.0000+10001.0000=10001.0000.
For the second SAR cycle, the comparator decision is 0, so the "LSB size number" is scaled by the comparator output and by the integer value 8 so the scaled value becomes 0.0000 since the comparator
output states is 0. Then the previous value 10001.0000 plus the scaled "LSB size number" 0.0000 is loaded into the digital filter, producing the result 10001.0000+0=10001.0000.
For the third cycle, the comparator decision is +1, so the "LSB size number" is scaled by the comparator output and by the integer value 4 so the scaled value becomes +100.0100. Then the previous
value 10001.0000 plus the scaled "LSB size number"+100.0100 is loaded into the digital filter, producing the result 10001.0000+100.0100=10101.0100.
For the fourth cycle, the comparator decision is -1, so the "LSB size number" is scaled by the comparator output and by the integer value 2 so the scaled value becomes -10.0010. Then the previous
value 10101.0100 plus the scaled "LSB size number"-10.0010 is loaded into the digital filter, producing the result 10101.0100-10.0010=10011.0010.
For the fifth cycle, the comparator decision is +1, so the "LSB size number" is scaled by the comparator output and by the integer value 1 so the scaled value remains +1.0001. Then the previous value
(10011.0010) plus the scaled "LSB size number"+1.0001 is loaded into the digital filter, producing the result 10011.0010+1.0001=10100.0011.
After rounding off or discarding the numbers after the decimal point, the final gain-corrected ADC conversion result is 10100.
There are various implementations of SAR ADC but all of them accumulate the desired digital result in the ways exemplified above, and all of them can be corrected for gain error by means of the above
described algorithm.
For application to the architecture described in above mentioned Ser. No. 11/738,566, the present invention works in the same way. First, in the integration mode the "LSB size number" is applied
during every integration cycle to the digital filter contents as scaled by the comparator output (+1, 0 or -1). No shifting to the left is performed. When the mixed integrating/SAR circuit switches
from integrating mode to SAR mode, the accumulated number in the register (i.e., the DS Result Counter 460 in FIG. 3a of Ser. No. 11/738,566) is taken as "initial value" and the gain-corrected
conversion proceeds in the above described SAR mode.
As an example, assume that there are 8 integration cycles followed by 5 SAR cycles. Also assume that during the integration cycles the comparator switched states six times to +1, then once to 0, and
then to -1. Then the integration result will be 6-1=5 (i.e., binary 101). In SAR mode, the initial value of the result register will be 00000101 using the "shift register" algorithm or 10100000 using
the "digital filter" approach.
According to the present invention, and with the same "LSB size number"+1.0001 after the integration mode operation and before the SAR mode operation, the initial value of the result register will be
00000101.0101 using the above described "shift register" algorithm or 10101010.0000 using the "digital filter" approach. As in the "shift register" approach, the initial value is shifted one position
left every SAR cycle so the final effect of the integration cycles (which determine the MSBs of the result) on the result is the same.
Fundamentally, the invention provides an LSB size number and then scales it by an integer and then adds or subtracts the scaled amount to the contents of the digital filter, according to the state of
the comparator. For example, in the case of using the invention in a successive approximation ADC, the LSB size number might be successively scaled by numbers such as 16,384, 8192, 4096, 2048, 1024,
512, 256 etc., before the LSB size number is added to or subtracted from the content of the SAR register according to the status of the comparator.
The invention thus provides a method of correction of the gain error of an ADC by changing the value of its LSB which is added to or subtracted from the digital filter contents in response to the
output states generated by the window comparator. For example, if the nominal value of the LSB of ADC 20B in FIG. 3 is "1", its value may be adjusted by an amount in the range from "0.111110000000"
to "1.000001111111". With 8 bits available to trim the value of the LSB, this provides a +/-3% trim range and 0.024% trim resolution.
An advantage of the above described embodiment of the invention is that it does not require an additional digital multiplier as required by the prior art, and therefore does not require additional
time and additional integrated circuit die area for the multiplication required using the prior art techniques, and therefore also reduces the amount of power consumption required for a DC gain
correction. The technique of the present invention can be easily implemented as a so-called final test trimming operation which takes into account all shifts of the ADC performance during the usual
wafer back grinding process and integrated circuit packaging operations. The only penalty is the increased size of the accumulation register of the digital filter. The described embodiments of the
invention also provide a convenient way of dealing with missing output codes which occur if the LSB size is trimmed to a value greater than 1, by simply increasing the resolution. This is especially
easily accomplished in a mixed delta sigma/cyclic architecture and in a SAR ADC architecture, because each additional bit of resolution requires only one additional integration cycle.
Other advantages of the above described embodiment of the present invention and adjusting LSB size of the ADC to trim gain error include the fact that it is in an entirely digital correction and does
not require any additional analog circuitry. It is capable of providing any desirable trim range and trim resolution, does not substantially increase the overall gain-adjusted ADC conversion time,
does not require additional mathematical operations, and requires less integrated circuit chip area and less power consumption than the prior art techniques for correcting gain error of an ADC. The
technique of the present invention also improves DNL (differential non-linearity) and INL (integral non-linearity) of the ADC by adding more bits of resolution to the conversion result, and can be
applied to various ADC architectures, such as SAR ADCs and high order delta sigma ADCs.
While the invention has been described with reference to several particular embodiments thereof, those skilled in the art will be able to make various modifications to the described embodiments of
the invention without departing from its true spirit and scope. It is intended that all elements or steps which are insubstantially different from those recited in the claims but perform
substantially the same functions, respectively, in substantially the same way to achieve the same result as what is claimed are within the scope of the invention. For example, the invention applies
not only to first order delta-sigma ADC architecture as described above, but also to a mixed delta-sigma/cyclic architecture, or to a complete SAR ADC, or a higher order delta-sigma ADC. It applies
to all ADC architectures that require a low number of integration cycles and to accomplish an analog to digital conversion. It should be appreciated that if the modulator 15 is a SAR ADC, then the
digital filter 37 is more like a shift register than an up-down counter. If modulator 15 is a higher order delta-sigma modulator, then the digital filter typically is a much more complex digital
filter that depends on the architecture of the ADC.
Patent applications by Dimitar T. Trifonov, Vail, AZ US
Patent applications by Jerry L. Doorenbos, Tucson, AR US
Patent applications in class CONVERTER COMPENSATION
Patent applications in all subclasses CONVERTER COMPENSATION
User Contributions:
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Banach Tarski Paradox in the Plane
March 29th 2013, 06:21 AM #1
Junior Member
Apr 2010
Banach Tarski Paradox in the Plane
Hello everyone,
Not quite sure that this is the right place for this question, but I couldn't find a topology or functional analysis section anywhere so here goes..
Basically, I have been given an assignment to prove that there is no Banach-Tarski paradox in R^2. I have decided to take the approach of first showing that there exists a nonnegative, finitely
additive set function for all subsets of the unit circle that is invariant under rotations, up to here everything seems clear and I have been able to follow the proofs given in Lax's book
"Functional analysis" which using the Hahn-Banach Theorem. Apparently there is a way of showing that the existence of such a function immediately proves that the Banach-Tarksi doesn't exist in R^
2 but I don't see it.. Is the proof more complicated than I am thinking? Could someone please recommend a place where I can find a proof of this or at least offer me an explanation of why it
works? Cheers
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Having trouble with quotient rule - Calc help please. (x^2 + 1)^4 / x^4
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I have to derive that
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The answer is supposed to be [4(x^2-1)(x^2+1)]/x^5 which is not what I get
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sorry the answer is [4(x^2-1)(x^2+1)^3]/x^5
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First write it in the equation editor to get a better look at it:\[\frac{ (x^2+1)^4 }{ x^4 }=\left( \frac{ x^2+1 }{ x } \right)^4\] Now you can also use the Chain Rule, because we have the 4th
power as a second step. The derivative is:\[4\left( \frac{ x^2+1 }{ x } \right)^3 \cdot \frac{ x \cdot 2x - 1 \cdot(x^2+1) }{ x^2 }=4\left( \frac{ x^2+1 }{ x } \right)^3 \cdot \frac{ x^2- 1}{ x^2
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Can I use chain rule where there is a fraction?
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Although we're done differentiating, we could still simplify:\[\frac{ 4(x^2+1)^3(x^2-1) }{ x^5 }\]
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@pottersheep: as long as there is a chain, you can (have to) use the Chain Rule! The chain here is: 1. u=(x^2+1)/x 2. y=u^4
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Ah ok
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Thank you for your explanation
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You could have done it without Chain Rule, if you leave the original function as it was:\[\left( \frac{ (x^2 + 1)^4 }{ x^4 }\right)'=\frac{ x^4 \cdot 4(x^2+1)^3 \cdot 2x-(x^2+1)^4 \cdot 4x^3 }{ x
^8 }\] (still used Chain Rule - see factor 2x!) This now also has to be simplified (begin with dividing everything by x³):\[\frac{ 8x^2(x^2+1)^3-4(x^2+1)^4 }{ x^5 }=\]\[\frac{ 4(x^2+1)^3(2x^2-(x^
2+1)) }{ x^5 }=\frac{ 4(x^2+1)^3(x^2-1) }{ x^5 }\]So we come to the same answer, eventually. You decide which method is easier!
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OHhhhh that's what I did before! I see my mistake now! :)
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Thanks again!
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is replying to Can someone tell me what button the professor is hitting...
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Reconstructing Position From Depth, Continued
Picking up where I left off here…
As I mentioned, you can also reconstruct a world-space position using the frustum ray technique. The first step is that you need your frustum corners to be rotated so that they match the current
orientation of your camera. You can do this by transforming the frustum corners by a “camera world matrix”, which is a matrix representing the camera’s position and orientation in world-space. If
you don’t have this available you can just invert your view matrix. I’ll demonstrate doing it right in the vertex shader for the sake of simplicity, but you’d probably want to do it ahead of time in
your application code.
// Vertex shader for rendering a full-screen quad
void QuadVS ( in float3 in_vPositionOS : POSITION,
in float3 in_vTexCoordAndCornerIndex : TEXCOORD0,
out float4 out_vPositionCS : POSITION,
out float2 out_vTexCoord : TEXCOORD0,
out float3 out_vFrustumCornerWS : TEXCOORD1 )
// Offset the position by half a pixel to correctly
// align texels to pixels. Only necessary for D3D9 or XNA
out_vPositionCS.x = in_vPositionOS.x - (1.0f/g_vOcclusionTextureSize.x);
out_vPositionCS.y = in_vPositionOS.y + (1.0f/g_vOcclusionTextureSize.y);
out_vPositionCS.z = in_vPositionOS.z;
out_vPositionCS.w = 1.0f;
// Pass along the texture coordinate and the position
// of the frustum corner in world-space. This frustum corner
// position is interpolated so that the pixel shader always
// has a ray from camera->far-clip plane
out_vTexCoord = in_vTexCoordAndCornerIndex.xy;
float3 vFrustumCornerVS = g_vFrustumCornersVS[in_vTexCoordAndCornerIndex.z];
out_vFrustumCornerWS = mul(vFrustumCornerVS, g_matCameraWorld);
So what we’ve done here is we’ve rotated (not translated, since vFrusumCornerVS is only a float3) the view-space frustum corner so that it’s now matches the camera’s orientation. However it’s still
centered around <0,0,0> and not the camera’s world-space position, so when we reconstruct position we’ll also add the camera’s world-space position:
// Pixel shader function for reconstructing world-space position
float3 WSPositionFromDepth(float2 vTexCoord, float3 vFrustumRayWS)
float fPixelDepth = tex2D(DepthSampler, vTexCoord).r;
return g_vCameraPosWS + fPixelDepth * vFrustumRayWS;
And there it is. Easy peasy, lemon squeezy.
The other bit I hinted at was using this same technique with arbitray geometry, for example the bounding volumes for a local light source. For this we once again need a ray that points from the
camera position through the pixel position to the far-clip plane. We can do this in the pixel shader by using the view-space position of the pixel.
void VSBoundingVolume( in float3 in_vPositionOS : POSITION,
out float4 out_vPositionCS : POSITION,
out float3 out_vPositionVS : TEXCOORD0 )
out_vPositionCS = mul(in_vPositionOS, g_matWorldViewProj);
// Pass along the view-space vertex position to the pixel shader
out_vPositionVS = mul(in_vPositionOS, g_matWorldView);
Then in our pixel shader, we calculate the ray and reconstruct position like this:
float3 VSPositionFromDepth(float2 vTexCoord, float3 vPositionVS)
// Calculate the frustum ray using the view-space position.
// g_fFarCip is the distance to the camera's far clipping plane.
// Negating the Z component only necessary for right-handed coordinates
float3 vFrustumRayVS = vPositionVS.xyz * (g_fFarClip/-vPositionVS.z);
return tex2D(DepthSampler, vTexCoord).x * vFrustumRayVS;
So there you go, I did your homework for you. Now stop beating me up in the schoolyard!
EDIT: Fixed the code and explanation so that it actually works now! Big thanks to Bill and Josh for pointing out the mistake.
UPDATE: More position from depth goodness here
15 comments
2. I don’t think your solution for arbitrary geometry works. You alluded to the problems yourself in your previous post.
Interpolationg (xyz / z) per-vertex doesn’t work as it is not a linear operation. You have to do the division in the pixel shader for this to work.
3. Phil was correct. I tried doing the calculation in the vertex shader for my code, and it introduced very nasty visual artifacts. When I moved the calculation to the fragment shader, it produced
the correct results.
4. Yup, you guys are right. For a while I was trying to figure out why I wasn’t getting artifacts…and then I realized that in code I was calculating the ray in the pixel shader too. Whoops. :-D
Thanks everyone for pointing it out, much appreciated.
5. A heads up to those using the first technique, it expects coordinates in the 0 to 1 range. Which should have been apparent with the texCoord parameter.
My own world space arbitrary implementation:
float3 GetFrustumRay(in float2 screenPosition)
float2 sp = sign(screenPosition);
return float3(Camera.FrustumRay.x * sp.x, Camera.FrustumRay.y * sp.y, Camera.MaxDepth);
The Camera.FrustumRay is calculated in the application using the following:
Vector2 frustumRay = new Vector2();
frustumRay.Y = (float)Math.Tan(Math.PI / 3.0 / 2.0) * camera.Viewport.MaxDepth;
frustumRay.X = -(frustumRay.Y * camera.Viewport.AspectRatio);
6. Forgot to mention in that last one, you alike the first one multiply view space depth (negated if your not using floating point buffers) then add camera position.
7. Hi!
I have a problem getting the last technique to work. VSPositionFromDepth gets the position in view space right? So in order to obtain the reconstructed world space position, I multiply the
resulting value with an inverse view matrix like this:
//View space position
float3 wsPos = VSPositionFromDepth(tex, input.vsPos);
//Transform to world space
//wsPos = mul(wsPos, InvertView);
Where the texcoords are:
input.ssPos.xy /= input.ssPos.w;
//Transforming from [-1,1]->[1,-1] to [0,1]->[1,0]
float2 tex = (0.5f * (float2(input.ssPos.x, -input.ssPos.y) + 1)) – halfPixel;
Where SsPos equals csPos in the example. Well…the problem is that it doesn’t work, the ws positionons are incorrect. Any ideas of what I’m doing wrong?
8. You need to do this:
wsPos = mul(float4(wsPos, 1.0f), InvertView);
If you don’t convert to a float4 and set w = 1.0, then your view-space position won’t get transformed by the translation part of your inverse view matrix (in other words, it will only get
9. Pingback: Game Rendering » Position Reconstruction
11. Hi,
I have a clever solution for you. It allows reconstruction of the view position in only two mul and no computation at all in the application.
First you need to output the depth in view space of your image. You can choose a R32F a G16R16F or anything you want.
Second, when you need to retrieve the pixel position in view space. Like in a full screen post process, draw a quad with those vertices (-1,-1) (-1,1) (1,1) (1,-1). If you need it in a real
geometry, just send the xyw of the projected position and divide xy by w in the pixel shader
Let’s go with some math now:
1. we have a well know projection matrix with lot of zero and some value
A 0 0 0
0 B 0 0
0 0 C -1
0 0 D 0
let’s write the process to transform a view space position to projection space ( Vw == 1 )
Px = Vx * A + Vy * 0 + Vz * 0 + 1 * 0
Py = Vx * 0 + Vy * B + Vz * 0 + 1 * 0
Pw = Vx * 0 + Vy * 0 – 1 * Vz + 1 * 0
Now let’s reconstruct Vx. What we know in your fragment program is the interpolated pixel position in projected space ( Px/Pw ). let’s call it Ix for InterpolatedX
Px/Pw = Ix
Vx * A / -Vz = Ix
Vx = Ix * Vz * ( -1 / A)
if we do the same for Vy we got
Vy = Iy * Vz * ( -1 / B).
Let’s write this in a HLSL form.
// Vertex part
out.position = in.position;
out.projpos.xy = in.position.xy;
out.magiccoef = -1.f / float2( gProj(0][0], gProj[1][1]
// Pixel part
float3 viewposition;
viewposition.z = tex2D( /***/ ).x;
viewposition.xy = viewposition.zz * in.magiccoef.xy * in.projpos.xy;
You know have your view space position, you can transform it to World space with a transform with the inverse view or keep it this way and do your math in view space.
Voilà :)
12. Interesting trick, GALOP1N.
On the whole, this actually seems like it would be cheaper than the prevalent method from Crytek. The “magiccoef” can be calculated outside the shaders, but is still significantly cheaper to do
than the frustum corner. Then the actual stored depth needn’t be normalized & negated, so that is another savings. Of course, anything relying on the depth being in the range [0,1] might be
affected, but that will be situation-specific.
I’m gonna try using this from now on, and see how well it holds up. Thanks. :)
13. Some more questions lol:
-g_matCameraWorld is the inverse of view matrix or inverse of world-view matrix?
-g_vCameraPosWS is the camera position multiplied by the world matrix or simply the camera position vector?
-How do I calculate vFrustumRayWS?
float3 vFrustumRayWS = vPositionWS.xyz * (g_fFarClip/-vPositionWS.z);
14. This is a really good tip especially to those fresh to
the blogosphere. Short but very accurate information… Appreciate your sharing this one.
A must read article!
15. I rarely leave comments, but after looking at a bunch of responses
on Reconstructing Position From Depth, Continued | The Danger Zone.
I do have a couple of questions for you if you don’t mind. Is it simply me or does it appear like some of these remarks look as if they are left by brain dead folks? :-P And, if you are posting
at other online sites, I’d like to follow you.
Would you post a list of every one of your shared sites like your
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center of mass
February 15th 2009, 11:17 PM #1
Feb 2009
center of mass
Consider the thin plate, of constant density p , which lies above the x-axis and between the two ellipses $x^2+y^2/4=1 , x^2/4+y^2/16=1$ Find its center of mass(HINT: In order to reduce the
amount of integration required, use symmetry and the formula which states that the area A of the region enclosed by the ellipse $A = piab$ )
Last edited by heneri; February 15th 2009 at 11:18 PM. Reason: Wrong formula
To locate the center of mass, you'll need the area of the region, the moment about the x-axis and the moment about the y-axis. Use the hint your problem gave to solve for the area. The formula
for the moments should be in your book, if not look here
February 15th 2009, 11:40 PM #2
MHF Contributor
Oct 2005 | {"url":"http://mathhelpforum.com/calculus/73834-center-mass.html","timestamp":"2014-04-21T08:02:17Z","content_type":null,"content_length":"32602","record_id":"<urn:uuid:b62757cc-337d-4705-9f47-2f6b07c5f18f>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00541-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: August 2004 [00521]
[Date Index] [Thread Index] [Author Index]
Re: Re: Publicon problems converting sample document to LaTeX
• To: mathgroup at smc.vnet.net
• Subject: [mg50334] Re: [mg50313] Re: Publicon problems converting sample document to LaTeX
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Fri, 27 Aug 2004 02:57:57 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <cg20f3$od7$1@smc.vnet.net> <cgcicp$eo7$1@smc.vnet.net> <200408241022.GAA06691@smc.vnet.net> <cghfrv$jeo$1@smc.vnet.net> <200408261051.GAA16395@smc.vnet.net>
• Reply-to: murray at math.umass.edu
• Sender: owner-wri-mathgroup at wolfram.com
Yes, the procedure below handles the graphics fine.
Note, still, the missing \maketitle in the LaTeX file generated. Is it
unreasonable to expect that to be included for a Publicon cell that's of
Style title?
Steve Luttrell wrote:
> ...I had a bit more
> success with the graphics because I did a DVI->PS translation before viewing
> the results on screen.
> Here's what I did:
> 1. Create a DefaultSample.nb document using the "S" button at the top-left
> of the Default palette.
> 2. Save As LaTeX.
> 3. Use MiKTeX to compile the LaTeX source.
> 4. Use the YAP to look at the DVI. Graphics do NOT display.
> 5. Process DVI->PS, and use GSVIEW to look at the PS. Graphics DO display.
> Steve Luttrell
> "Murray Eisenberg" <murray at math.umass.edu> wrote in message
> news:cghfrv$jeo$1 at smc.vnet.net...
>>My first attempt at translating a Publicon document into LaTeX and
>>printing the result .... I tried MiKTeX....
>>(1) There was no \maketitle in the generated LaTeX, so no title appeared!
>>(2) None of the .ps files (originating from the \includegraphics
>>commands) rendered in the Yap viewer and none printed. (See below for
>>the Yap report.)
Murray Eisenberg murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305
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Matrix Computations
This tutorial reviews the functions that Mathematica provides for carrying out matrix computations. Further information on these functions can be found in standard mathematical texts by such authors
as Golub and van Loan or Meyer. The operations described in this tutorial are unique to matrices; an exception is the computation of norms, which also extends to scalars and vectors.
Basic Operations
This section gives a review of some basic concepts and operations that will be used throughout the tutorial to discuss matrix operations.
The norm of a mathematical object is a measurement of the length, size, or extent of the object. In Mathematica norms are available for scalars, vectors, and matrices.
Computing norms in Mathematica.
Vector Norms
For vector spaces, norms allow a measure of distance. This allows the definition of many familiar concepts such as neighborhood, closeness of approximation, and goodness of fit. A vector norm is a
function that satisfies the following relations.
Typically this function uses the notation . The subscript is used to distinguish different norms, of which the p-norms are particularly important. For , the p-norm is defined as follows.
Some common p-norms are the 1-, 2-, and -norms.
Matrix Norms
Matrix norms are used to give a measure of distance in matrix spaces. This is necessary if you want to quantify what it means for one matrix to be near to another, for example, to say that a matrix
is nearly singular.
Matrix norms also use the double bar notation of vector norms. One of the most common matrix norms is the Frobenius norm (also called the Euclidean norm).
Other common norms are the p-norms. These are defined in terms of vector p-norms as follows.
Thus the matrix p-norms show the maximum expansion that a matrix can apply to any vector.
One of the fundamental subspaces associated with each matrix is the null space. Vectors in the null space of a matrix are mapped to zero by the action of the matrix.
The rank of a matrix corresponds to the number of linearly independent rows or columns in the matrix.
For an m×n matrix A the following relations hold: Length[NullSpace[A]]+MatrixRank[A]n, and Length[NullSpace[A^T]]+MatrixRank[A^T]m. From this it follows that the null space is empty if and only if
the rank is equal to n and that the null space of the transpose of A is empty if and only if the rank of A is equal to m.
If the rank is equal to the number of columns, it is said to have full column rank. If the rank is equal to the number of rows, it is said to have full row rank. One way to understand the rank of a
matrix is to consider the row echelon form.
Reduced Row Echelon Form
A matrix can be reduced to a row echelon form by a combination of row operations that start with a pivot position at the top-left element and subtract multiples of the pivot row from following rows
so that all entries in the column below the pivot are zero. The next pivot is chosen by going to the next row and column. If this pivot is zero and any nonzero entries are in the column beneath, the
rows are exchanged and the process is repeated. The process finishes when the last row or column is reached.
A matrix is in row echelon form if any row that consists entirely of zeros is followed only by other zero rows and if the first nonzero entry in row is in the column , then elements in columns from 1
to in all rows below are zero. The row echelon form is not unique but its form (in the sense of the positions of the pivots) is unique. It also gives a way to determine the rank of a matrix as the
number of nonzero rows.
A matrix is in reduced row echelon form if it is in row echelon form, each pivot is one, and all the entries above (and below) each pivot are zero. It can be formed by a procedure similar to the
procedure for the row echelon form, also taking steps to reduce the pivot to one (by division) and reduce entries in the column above each pivot to zero (by subtracting multiples of the current pivot
row). The reduced row echelon form of a matrix is unique.
The reduced row echelon form (and row echelon form) give a way to determine the rank of a matrix as the number of nonzero rows. In Mathematica the reduced row echelon form of a matrix can be computed
by the function RowReduce.
The inverse of a square matrix A is defined by
where is the identity matrix. The inverse can be computed in Mathematica with the function Inverse.
The matrix inverse can in principle be used to solve the matrix equation
by multiplying it by the inverse of .
However, it is always better to solve the matrix equation directly. Such techniques are discussed in the section "Solving Linear Systems".
When the matrix is singular or is not square it is still possible to find an approximate inverse that minimizes . When the 2-norm is used this will find a least squares solution known as the
The solution found by the pseudoinverse is a least squares solution. These are discussed in more detail under "Least Squares Solutions".
The determinant of an n×n matrix is defined as follows.
It can be computed in Mathematica with the function Det.
Solving Linear Systems
One of the most important uses of matrices is to represent and solve linear systems. This section discusses how to solve linear systems with Mathematica. It makes strong use of LinearSolve, the main
function provided for this purpose.
Solving a linear system involves solving a matrix equation . Because is an × matrix this is a set of linear equations in unknowns.
When the system is said to be square. If there are more equations than unknowns and the system is overdetermined. If there are fewer equations than unknowns and the system is underdetermined.
Classes of linear systems represented by rectangular matrices.
Note that even though you could solve the matrix equation by computing the inverse via , this is not a recommended method. You should use a function that is designed to solve linear systems directly
and in Mathematica this is provided by LinearSolve.
Solving linear systems with LinearSolve.
If no solution can be found, the system is inconsistent. In this case it might still be useful to find a best-fit solution. This is often done with a least squares technique, which is explored under
"Least Squares Solutions".
Options for LinearSolve.
LinearSolve has three options that allow users more control. The Modulus and ZeroTest options are used by symbolic techniques and are discussed in the section "Symbolic and Exact Matrices". The
Method option allows you to choose methods that are known to be suitable for a particular problem; the default setting of Automatic means that LinearSolve chooses a method based on the input.
Singular Matrices
In those cases where a solution cannot be found, it is still possible to find a solution that makes a best fit to the problem. One important class of best-fit solutions involves least squares
solutions and is discussed in "Least Squares Solutions".
Homogeneous Equations
The homogeneous matrix equation involves a zero right-hand side.
This equation has a nonzero solution if the matrix is singular. To test if a matrix is singular, you can compute the determinant.
Estimating and Calculating Accuracy
An important way to quantify the accuracy of the solution of a linear system is to compute the condition number . This is defined as follows for a suitable choice of norm.
It can be shown that for the matrix equation the relative error in is times the relative error in and . Thus, the condition number quantifies the accuracy of the solution. If the condition number of
a matrix is large, the matrix is said to be ill-conditioned. You cannot expect a good solution from an ill-conditioned system. For some extremely ill-conditioned systems it is not possible to obtain
any solution.
In Mathematica an approximation of the condition number can be computed with the function .
Symbolic and Exact Matrices
LinearSolve works for all the different types of matrices that can be represented in Mathematica. These are described in more detail in "Matrix Types".
In order to simplify the intermediate expressions, the ZeroTest option might be useful.
There are a number of methods that are specific to symbolic and exact computation: , , and . These are discussed in the section "Symbolic Methods".
Row Reduction
Saving the Factorization
The one-argument form of LinearSolve works in a completely equivalent way to the two-argument form. It works with the same range of input matrices, for example, returning the expected results for
symbolic, exact, or sparse matrix input. It also accepts the same options.
LinearSolve provides a number of different techniques to solve matrix equations that are specialized to particular problems. You can select between these using the option Method. In this way a
uniform interface is provided to all the functionality that Mathematica provides for solving matrix equations.
The default setting of the option Method is Automatic. As is typical for Mathematica, this means that the system will make an automatic choice of method to use. For LinearSolve if the input matrix is
numerical and dense, then a method using LAPACK routines is used; if it is numerical and sparse, a multifrontal direct solver is used. If the matrix is symbolic then specialized symbolic routines are
The details of the different methods are now described.
LAPACK is the default method for solving dense numerical matrices. When the matrix is square and nonsingular the routines dgesv, dlange, and dgecon are used for real matrices and zgesv, zlange, and
zgecon for complex matrices. When the matrix is nonsquare or singular dgelss is used for real matrices and zgelss for complex matrices. More information on LAPACK is available in the references.
If the input matrix uses arbitrary-precision numbers, then LAPACK algorithms extended for arbitrary-precision computation are used.
The multifrontal method is a direct solver used by default if the input matrix is sparse; it can also be selected by specifying a method option.
If the input matrix to the multifrontal method is dense, it is converted to a sparse matrix.
The implementation of the multifrontal method uses the "UMFPACK" library.
The Krylov method is an iterative solver that is suitable for large sparse linear systems, such as those arising from numerical solving of PDEs. In Mathematica two Krylov methods are implemented:
conjugate gradient (for symmetric positive definite matrices) and BiCGSTAB (for nonsymmetric systems). It is possible to set the method that is used and a number of other parameters by using
appropriate suboptions.
Suboptions of the Krylov method.
The default method for Krylov, BiCGSTAB, is more expensive but more generally applicable. The conjugate gradient method is suitable for symmetric positive definite systems, always converging to a
solution (though the convergence may be slow). If the matrix is not symmetric positive definite, the conjugate gradient may not converge to a solution.
At present, only the ILU preconditioner is built-in. You can still define your own preconditioner by defining a function that is applied to the input matrix. An example that involves solving a
diagonal matrix is shown next.
Generally, a problem will not be structured so that it can benefit so much from such a simple preconditioner. However, this example is useful since it shows how to create and use your own
If the input matrix to the Krylov method is dense, the result is still found because the method is based on matrix/vector multiplication.
The Krylov method can be used to solve systems that are too large for a direct solver. However, it is not a general solver, being particularly suitable for those problems that have some form of
diagonal dominance.
For dense matrices the Cholesky method uses LAPACK functions such as dpotrf and dpotrs for real matrices and zpotrf and zpotrs for complex matrices. For sparse matrices the Cholesky method uses the "
TAUCS" library.
Symbolic Methods
Least Squares Solutions
In these cases it is possible to find a best-fit solution that minimizes . A particularly common choice of p is 2; this generates a least squares solution. These are often used because the function
is differentiable in and because the 2-norm is preserved under orthogonal transformations.
If the rank of is (i.e., it has full column rank), it can be shown (Golub and van Loan) that there is a unique solution to the least squares problem and it solves the linear system.
These are called the normal equations. Although they can be solved directly to get a least squares solution, this is not recommended because if the matrix is ill-conditioned then the product will be
even more ill-conditioned. Ways of finding least squares solutions for full rank matrices are explored in "Examples: Full Rank Least Squares Solutions".
It should be noted that best-fit solutions that minimize other norms, such as 1 and , are also possible. A number of ways to find solutions to overdetermined systems that are suitable for special
types of matrices or to minimize other norms are given under "Minimization of 1 and Infinity Norms".
A general way to find a least squares solution to an overdetermined system is to use a singular value decomposition to form a matrix that is known as the pseudoinverse of a matrix. In Mathematica
this is computed with PseudoInverse. This technique works even if the input matrix is rank deficient. The basis of the technique follows.
Data Fitting
Scientific measurements often result in collections of ordered pairs of data, . In order to make predictions at times other than those that were measured, it is common to try to fit a curve through
the data points. If the curve is a straight line, then the aim is to find unknown coefficients and for which
for all the data pairs. This can be written in a matrix form as shown here and is immediately recognized as being equivalent to solving an overdetermined system of linear equations.
Fitting a more general curve, for example,
is equivalent to the matrix equation
In this case the left-hand matrix is a Vandermonde matrix. In fact any function that is linear in the unknown coefficients can be used.
The function FindFit is quite general; it can also fit functions to data that is not linear in the parameters.
Eigensystem Computations
The solution of the eigenvalue problem is one of the major areas for matrix computations. It has many applications in physics, chemistry, and engineering. For an × matrix the eigenvalues are the
roots of its characteristic polynomial, . The set of roots, , are called the spectrum of the matrix. For each eigenvalue, , the vectors, , that satisfy
are described as eigenvectors. The matrix of the eigenvectors, , if it exists and is nonsingular, may be used as a similarity transformation to form a diagonal matrix with the eigenvalues on the
diagonal elements. Many important applications of eigenvalues involve the diagonalization of matrices in this way.
Mathematica has various functions for computing eigenvalues and eigenvectors.
Certain applications of eigenvalues do not require all of the eigenvalues to be computed. Mathematica provides a mechanism for obtaining only some eigenvalues.
Options for Eigensystem.
Eigensystem has four options that allow users more control. The Cubics, Quartics, and ZeroTest options are used by symbolic techniques and are discussed under "Symbolic and Exact Matrices". The
Method option allows knowledgeable users to choose methods that are particularly appropriate for their problems; the default setting of Automatic means that Eigensystem makes a choice of a method
that is generally suitable.
Eigensystem Properties
This section describes certain properties of eigensystem computations. It should be remembered that because the eigenvalues of an × matrix can be associated with the roots of an eigenvalues.
Diagonalizing a Matrix
Similarity transformations preserve a number of useful properties of a matrix such as determinant, rank, and trace.
Symbolic and Exact Matrices
As is typical for Mathematica, if a computation is done for a symbolic or exact matrix it will use symbolic computer algebra techniques and return a symbolic or exact result.
Generalized Eigenvalues
For × matrices the generalized eigenvalues are the roots of its characteristic polynomial, . For each generalized eigenvalue, , the vectors, , that satisfy
are described as generalized eigenvectors.
Generalized eigenvalues and eigenvectors.
Eigenvalue computations are solved with a number of different techniques that are specialized to particular problems. You can select between these using the option Method. In this way a uniform
interface is provided to all the functionality that Mathematica provides.
The default setting of the option Method is Automatic. As is typical for Mathematica, this means that the system will make an automatic choice of method to use. For eigenvalue computation when the
input is an × matrix of machine numbers and the number of eigenvalues requested is less than 20% of an Arnoldi method is used. Otherwise, if the input matrix is numerical then a method using LAPACK
routines is used. If the matrix is symbolic then specialized symbolic routines are used.
The details of the different methods are now described.
LAPACK is the default method for computing the entire set of eigenvalues and eigenvectors. When the matrix is unsymmetric, dgeev is used for real matrices and zgeev is used for complex matrices. For
symmetric matrices, dsyevr is used for real matrices and zheevr is used for complex matrices. For generalized eigenvalues the routine dggev is used for real matrices and zggev for complex matrices.
More information on LAPACK is available in the references section.
If the input matrix uses arbitrary-precision numbers, LAPACK algorithms extended for arbitrary-precision computation are used.
The Arnoldi method is an iterative method used to compute a finite number of eigenvalues. The implementation of the Arnoldi method uses the "ARPACK" library. The most general problem that can be
solved with this technique is to compute a few selected eigenvalues for
Because it is an iterative technique and only finds a few eigenvalues, it is often suitable for sparse matrices. One of the features of the technique is the ability to apply a shift and solve for a
given eigenvector and eigenvalue (where ).
This is effective for finding eigenvalues near to .
The following suboptions can be given to the Arnoldi method.
Suboptions of the Arnoldi method.
For many large sparse systems that occur in practical computations the Arnoldi algorithm is able to converge quite quickly.
Symbolic Methods
Symbolic eigenvalue computations work by interpolating the characteristic polynomial.
Matrix Decompositions
This section will discuss a number of standard techniques for working with matrices. These are often used as building blocks for solving matrix problems. The decompositions fall into the categories
of factorizations, orthogonal transformations, and similarity transformations.
LU Decomposition
The LU decomposition of a matrix is frequently used as part of a Gaussian elimination process for solving a matrix equation. In Mathematica there is no need to use an LU decomposition to solve a
matrix equation, because the function LinearSolve does this for you, as discussed in the section "Solving Linear Systems". Note that if you want to save the factorization of a matrix, so that it can
be used to solve the same left-hand side, you can use the one-argument form of LinearSolve, as demonstrated in the section "Saving the Factorization".
The LU decomposition with partial pivoting of a square nonsingular matrix involves the computation of the unique matrices , , and such that
where is lower triangular with ones on the diagonal, is upper triangular, and is a permutation matrix. Once it is found, it can be used to solve the matrix equation by finding the solution to the
following equivalent system.
This can be done by first solving for in the following.
Then solve for to get the desired solution.
Because both and are triangular matrices it is particularly easy to solve these systems.
Despite the fact that in Mathematica you do not need to use the LU decomposition for solving matrix equations, Mathematica provides the function LUDecomposition for computing the LU decomposition.
LUDecomposition returns a list of three elements. The first element is a combination of upper and lower triangular matrices, the second element is a vector specifying rows used for pivoting (a
permutation vector which is equivalent to the permutation matrix), and the third element is an estimate of the condition number.
One of the features of the LU decomposition is that the lower and upper matrices are stored in the same matrix. The individual and factors can be obtained as follows. Note how they use a vectorized
operation for element-by-element multiplication with a sparse matrix. This makes the process more efficient; these techniques are discussed further in "Programming Efficiency".
Cholesky Decomposition
The Cholesky decomposition of a symmetric positive definite matrix is a factorization into a unique upper triangular such that
This factorization has a number of uses, one of which is that, because it is a triangular factorization, it can be used to solve systems of equations involving symmetric positive definite matrices.
(The way that a triangular factorization can be used to solve a matrix equation is shown in the section "LU Decomposition".) If a matrix is known to be of this form it is preferred over the LU
factorization because the Cholesky factorization is faster to compute. If you want to solve a matrix equation using the Cholesky factorization you can do this directly from LinearSolve using the
Cholesky method; this is described in a previous section.
The Cholesky factorization can be computed in Mathematica with the function CholeskyDecomposition.
One way to test if a matrix is positive definite is to see if the Cholesky decomposition exists.
Cholesky and LU Factorizations
The Cholesky factorization is related to the LU factorization as
where is the diagonal matrix of pivots.
Finally, this computes ; its transpose is equal to the Cholesky decomposition.
Orthogonalization generates an orthonormal basis from an arbitrary basis. An orthonormal basis is a basis, , for which
In Mathematica a set of vectors can be orthogonalized with the function Orthogonalize.
By default a Gram-Schmidt orthogonalization is computed, but a number of other orthogonalizations can be computed.
QR Decomposition
The QR decomposition of a rectangular matrix with linearly independent columns involves the computation of matrices and such that
where the columns of form an orthonormal basis for the columns of and is upper triangular. It can be computed in Mathematica with the function QRDecomposition.
Solving Systems of Equations
For a square nonsingular matrix the QR decomposition can be used to solve the matrix equation , as is also the case for the LU decomposition. However, when the matrix is rectangular, the QR
decomposition is also useful for solving the matrix equation.
One particular application of the QR decomposition is to find least squares solutions to overdetermined systems, by solving the system of normal equations
Because , this can be simplified as
Thus the normal equations simplify to
and because is nonsingular this simplifies to
Because is triangular this is a particularly easy system to solve; sample code to implement this technique is given in "Examples: Least Squares QR".
Singular Value Decomposition
The singular value decomposition of a rectangular matrix involves the computation of orthogonal matrices and and a diagonal matrix such that
The diagonal elements of the matrix are called the singular values of . In Mathematica, the function SingularValueList computes the singular values and the function SingularValueDecomposition
computes the full singular value decomposition.
Note that if the matrix is complex, the definition of the singular value decomposition uses the conjugate transpose.
There are many applications of the singular value decomposition. The singular values of a matrix give information on the linear transformation represented by . For example, the action of on a unit
sphere generates an ellipsoid with semi-axes given by the singular values. The singular values can be used to compute the rank of a matrix; the number of nonzero singular values is equal to the rank.
The singular values can be used to compute the 2-norm of a matrix, and the columns of the matrix that correspond to zero singular values are the null space of the matrix.
Certain applications of singular values do not require all of the singular values to be computed. Mathematica provides a mechanism for obtaining only some singular values.
Generalized Singular Values
For an × matrix and a × matrix the generalized singular values are given by the pair of factorizations
where is ×, is ×, and is ×; and are orthogonal, and is invertible.
The functions SingularValueList and SingularValueDecomposition both take a Tolerance option.
The option controls the size at which singular values are treated as zero. By default, values that are tol times smaller than the largest singular value are dropped, where tol is 100×$MachineEpsilon
for machine-number matrices. For arbitrary-precision matrices, it is , where is the precision of the matrix.
Schur Decomposition
The Schur decomposition of a square matrix involves finding a unitary matrix that can be used for a similarity transformation of to form a block upper triangular matrix with 1×1 and 2×2 blocks on the
diagonal (the 2×2 blocks correspond to complex conjugate pairs of eigenvalues for a real matrix ). A block upper triangular matrix of this form can be called upper quasi-triangular.
The Schur decomposition always exists and so the similarity transformation of to upper triangular always exists. This contrasts with the eigensystem similarity transformation, used to diagonalize a
matrix, which does not always exist.
Generalized Schur Decomposition
For × matrices and , the generalized Schur decomposition is defined as
where and are unitary, is upper triangular, and is upper quasi-triangular.
For real input, a result involving complex numbers and an upper triangular result can be obtained with the option .
SchurDecomposition takes two options.
Options for SchurDecomposition.
The option can be used to allow pivoting and scaling to improve the quality of the result. When it is set to True, pivoting and scaling may be used and a matrix that represents the changes to is
returned. With this new matrix the definition of the Schur decomposition can be seen as follows.
Jordan Decomposition
The Jordan decomposition of a square matrix involves finding the nonsingular matrix that can be used for a similarity transformation of to generate a matrix (known as the Jordan form) that has a
particularly simple triangular structure.
The Jordan decomposition always exists, but it is hard to compute with floating-point arithmetic. However, computation with exact arithmetic avoids these problems.
Functions of Matrices
The computation of functions of matrices is a general problem with applications in many areas such as control theory. The function of a square matrix is not the same as the application of the
function to each element in the matrix. Clearly element-wise application would not maintain properties consistent with the application of the function to a scale.
There are a number of ways to define functions of matrices; one useful way is to consider a series expansion. For the exponential function this works as follows.
One way to compute this series involves diagonalizing , so that and . Therefore, the exponential of can be computed as follows.
This technique can be generalized to functions of the eigenvalues of . Note that while this is one way to define functions of matrices, it does not provide a good way to compute them.
Mathematica does not have a function for computing general functions of matrices, but it has some specific functions.
A technique for computing parametrized functions of matrices by solving differential equations is given in "Examples: Matrix Functions with NDSolve". | {"url":"http://reference.wolfram.com/mathematica/tutorial/LinearAlgebraMatrixComputations.html","timestamp":"2014-04-17T07:29:18Z","content_type":null,"content_length":"369278","record_id":"<urn:uuid:a15d7f90-583e-4662-84b2-d58fa2c5ebea>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00459-ip-10-147-4-33.ec2.internal.warc.gz"} |
Modus ponens
From Uncyclopedia, the content-free encyclopedia
Modus ponens is a valid logical structure for argumentation and also a way to make yourself look smart.^[1]^[2]^[3] In logic, modus ponens takes the logically valid form:
1. P → Q
2. P
3. ∴ Q
Modus ponens declares that for any "P" and "Q," the following is a valid way of reasoning:
1. If P, then Q.
2. P.
3. Therefore, Q.
4. Modus pwned!
The first is an if-then statement, that conditions the truth of "Q" on the truth of "P," represented by the arrow leading "P" to "Q." The second asserts that "P" is true. Following logically from the
first two, the third finds "Q" to be true.
Because of the intellectual high-level^[4] logical nature of modus ponens, merely repeating its structure causes an argumentative opponent to shut the fuck up and subsequently lose the argument.^[5]
Declaring QED crushes a failed opponent further.^[6]
There is much debate among philosophers about what exactly P and Q are. Some philosophers, such as those in the ancient Alphabetical School, hold that P and Q refer to the corresponding letters in
the alphabet, and modus ponens is meant to explain the inescapable progression from the sixteenth letter to the seventeenth letter. The ancient Greek philosopher Socrates famously said, "But,
Critias, it only makes sense that P and Q refer to the respective letters in our alphabet."^[7]
In this school of thought, therefore, it is always true that P → Q, while it is never true that Q → P, except at the request of a police officer at a sobriety checkpoint.
However, modern philosophers have shifted from the classical Alphabetical School of the ancients to a more figurative explanation of the mysterious P and Q. Bertrand Russell, for instance, formed
what is now called the Bathroom School, hold that P actually refers to the act of urination, Q to the act of forming a queue, such that when one needs to urinate, he or she finds the nearest bathroom
and attaches himself to the end of an extant queue. Abandoning the Platonic theory, this is the most widely-accepted view among modern philosophers.
Modus tollens
Alongside modus ponens (Latin for "method of putting in," like one's two cents), there is modus tollens ("method of taking out"). This natural symmetry is based on the philosophical postulate that
"Whatever goes up must come down."
It is formulated by logicians^[8] as:
1. If P, then Q
2. Not Q
3. Not P
4. Therefore... nah, nevermind.
The reader can keep this in mind whenever he puts in his two cents but decides he'd rather have them back in his pocket.
Featured Article (read another featured article) Featured version: 6 April 2013
This article has been featured on the front page. — You can vote for or nominate your favourite articles at | {"url":"http://uncyclopedia.wikia.com/wiki/Modus_ponens?redirect=no","timestamp":"2014-04-16T14:38:17Z","content_type":null,"content_length":"41824","record_id":"<urn:uuid:f37500dd-aac8-462a-a187-d5abedb47bfc>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00596-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: "sound fields" (jan schnupp )
Subject: Re: "sound fields"
From: jan schnupp <jan.schnupp(at)PHYSIOL.OX.AC.UK>
Date: Mon, 4 Dec 2000 12:00:05 +0000
In applied mathematics people talk about a "field" whenever a ("dependent")
scalar variable which varies as a function of one or several other
"independent" variables. If you pick your dependent variable to be sound
pressure level and time, frequency, spatial coordinates of source
position, and so on as your independent variables, then you get a
definition of sound field which is very "inclusive", but nevertheless
"mathematically exact". That definition may be a bit broad, but I find any
further restricitions would be very arbitrary, and wouldn't gain us
anything. Does that help?
At 00:32 03/12/00 -0000, you wrote:
>I wonder if the list(s) could help;
>I'm looking for opinions on reasonably inclusive definitions of the term :
>"sound field", in order to draw distinctions between this concept and the
>more general "sound environment".
>I'm sorry if this is somewhat 'off topic', but a variety of perspectives
>seems appropriate for this particular question.
>thanks (in advance)
>Peter Lennox
>Hardwick House
>tel: (0114) 2661509
>e-mail: peter(at)lennox01.freeserve.co.uk
>or:- ppl100(at)york.ac.uk
Dr. Jan Schnupp
Oxford University, Laboratory of Physiology, Parks Road, Oxford OX1 3PT, U.K.
Tel (+44-1865) 272 513 Fax (+44-1865) 272 469
This message came from the mail archive
maintained by: DAn Ellis <dpwe@ee.columbia.edu>
Electrical Engineering Dept., Columbia University | {"url":"http://www.auditory.org/postings/2000/496.html","timestamp":"2014-04-21T09:39:00Z","content_type":null,"content_length":"2319","record_id":"<urn:uuid:c6087cc4-eb06-49ed-862e-89500e7b2bdd>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00561-ip-10-147-4-33.ec2.internal.warc.gz"} |
The vertical distribution of the mean shear stress inside the surf zone is compared to the terms in the time-averaged horizontal momentum equation using one set of laboratory measurements of the free
surface elevations and fluid velocities u and w induced by regular waves spilling on a plane slope. The vertical distribution of the eddy viscosity is estimated directly from the measured mean shear
stress and velocity. The shear stress distribution in the surf zone is shown to vary linearly with depth until the bottom boundary layer where it reached a nearly constant, negative value. The shear
stress variation in the transition region differs distinctly from the inner surf zone. The vertical variation of uw is shown to be small outside the surf zone except near the bottom. Inside the surf
zone, it is shown that the uw term of the horizontal momentum equation is likely to be important in the transition region and that its importance diminishes in the inner surf zone. The vertical
distribution of the eddy viscosity has a form which is small near trough level, increases to a maximum value about one-third of the depth below trough level, and then decreases toward the bottom. The
eddy viscosity in the middle of the bottom boundary layer is two orders of magnitude less than the eddy viscosity in the interior.
breaking waves; undertow; undertow profiles; bottom boundary layer
Full Text:
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Creative Commons Attribution 3.0 License | {"url":"http://journals.tdl.org/icce/index.php/icce/article/view/5461","timestamp":"2014-04-18T08:17:40Z","content_type":null,"content_length":"16369","record_id":"<urn:uuid:cfe08d57-3d41-4d4c-b334-eec5040cfd30>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00280-ip-10-147-4-33.ec2.internal.warc.gz"} |
Immaculata Algebra 2 Tutor
Find a Immaculata Algebra 2 Tutor
...I favor the Socratic Method of teaching, asking questions of the student to help him/her find her/his own way through the problem rather than telling what the next step is. This way the student
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85th Percentile Speed Question - Page 2 - Transportation
Please cite your reference in your statement above. I believe you're fabricating the definition of the 85th percentile in this example where the sample set of 100 values. Regardless of the number
of values in a sample set, percentiles and actual values are not the same.
Here's one source of MANY definition references. Pick any one you'd like:
You can also read about percentiles at
And then there's FHWA-SA-10-001: "85th percentile speed – the speed at or below which 85 percent of vehicles travel" | {"url":"http://engineerboards.com/index.php?showtopic=18562&p=6946878&page=2","timestamp":"2014-04-19T11:57:27Z","content_type":null,"content_length":"193962","record_id":"<urn:uuid:0256205d-b35c-46c7-90e2-8ce40352f1a1>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00132-ip-10-147-4-33.ec2.internal.warc.gz"} |
● ArizMATYC Sessions Summary - Joint Conference: SUnMaRC, MAA and ArizMATYC
Schedule >
Friday Saturday Sunday
Listed in alphabetical order by presenter name.
Attanucci, Frank J. A note on the proportional partitioning of line segments, triangles and tetrahedra [View Presentation]
In the first part of this paper I solve the following problem: Where can one place a point G inside or on a triangle so that line segments from G to each of the vertices divide the triangle into
three sub triangles whose areas A1, A2 and A3, respectively, satisfy the proportion: A1:A2:A3 = w1:w2:w3, where the wi's are non-negative constants with positive sum? I then state and prove an
analogous result for tetrahedra. I finish with a theorem concerning the centroid of n! points. Along the way, everything is made more intriguing by allowing the wi's to be parameterized functions.
Beaudrie, Brian Improving the mathematical readiness of middle-achieving, college-bound students [View Presentation]
Based on more than six years of research, this presentation will discuss a practical two-tiered strategy designed to help college-bound students be better prepared for credit-bearing college
mathematics courses before beginning their post-secondary education.
Dudley, Anne & Watkins, Laura Active math
The presenters will share a variety of activities to engage students in active math. These activities will require students to get up and move to help them retain the math. The presenters will share
activities from many levels of mathematics and invite attendees to share their favorite activities as well.
Dumitrascu, Gabriela Generalization in mathematics from elementary school to college
The program will be divided into two parts. First, I will use a power point presentation format to describe a way to define the practice of generalization in mathematics. The presentation will
include a theoretical definition and examples from textbooks how this definition may be used to organize mathematics instruction from elementary school level to high school and college levels. In the
second part of the program, the participants will discuss how the practice of generalization is reflected in the Common Core State Standards for Mathematics.
Hughes-Hallett, Deborah & Lozano, Guadalupe Mathematics and sustainability
The emphasis in the Common Core Curriculum Standards on mathematical structure and modeling, and the worldwide concern with sustainability, can be combined into engaging materials for use in
pre-calculus and calculus, as well as in quantitative reasoning and college algebra.
Knapp, Jessica Teaching foundations: new courses to prepare future teachers
A statewide Project NEXT grant has developed a set of course materials to improve the mathematical content knowledge for pre-service elementary school teachers in Arizona. We will discuss the rubric
of higher order thinking developed by the Teaching Foundations faculty to serve as a guide in the course preparation as well as present some of the courses and course materials. We will explain the
courses which have been developed and discuss some of the early results from the piloters. We will also give examples of the course materials being used and compare them to recommendations made by
other mathematics education research projects.
Kozak, Kathryn AMATYC proctored testing position statement
This session will be an open forum to collect input on the draft of the
AMATYC Position Statement
on proctored testing.
Mayo, Tim Lucky Larry and lines of verse
A sampling of "Lucky Larry" problems will be worked out live, with humorous anecdotes and poetry lines used by the instructor in his classes. The pedagogical merits of these problems will be
discussed. Attendees will have the opportunity to present "Lucky Larry" problems they have encountered.
McCallum, William The Illustrative Mathematics Project
The Illustrative Mathematics Project is building a community of mathematicians, educators, and teachers dedicated to designing high quality tasks to illustrate the Common Core State Standards in
Mathematics. The core of the project is an interactive website where tasks are submitted, reviewed, edited and published by the community.
Mendel, Marilou WeBWorK demonstration [View Presentation]
This session will provide attendees with an overview of WeBWorK, an open-source online homework system. WeBWorK is hosted by the MAA and is accompanied by a National Problem Library that consists of
more than 20,000 homework problems. Students are provided with individualized problems, immediate feedback on the correctness of their responses, and an opportunity to make repeated attempts to
obtain the correct answers. Instructors are provided with automatic grading of assignments and detailed statistical information about the performance of the students. We will examine the student and
instructor interfaces, explore the National Problem Library, and discuss how to obtain WeBWorK.
Schettler, Jordan
& Charles Collingwood
Occupy Calculus! The math behind the 99% [View Presentation]
We will explore Lorenz curves for income and explain how tools from integral and differential calculus can be used to analyze trends in the unequal distribution of wealth. We will discuss a powerful
student project which helps build analytical skills and social consciousness.
Welch, Eric Unlocking the secrets of counting problems
Participants will solve problems suitable for 4th through 8th grade students that clarify the overarching concepts and elicit the habits of mind used to solve systematic listing and counting
problems. The concepts range from the addition principle of counting to the meaning behind the formulas for permutations and combinations. Bring your willingness to change hats, approaching problems
like a middle grades student while thinking about instructional moves like a middle school teacher. | {"url":"https://sites.google.com/site/2012march30/schedule/summary-of-talks---arizmatyc","timestamp":"2014-04-19T06:57:17Z","content_type":null,"content_length":"31114","record_id":"<urn:uuid:d986f8b5-f614-4993-8c4d-5a6b9d565942>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00398-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is the physical form of binary data in a computer processor?
what about the one from D-Wave?
Is that more than an announcement?
Their flux qubits would use the direction of current flow (or, equivalently, the direction of a magnetic field) to decode 0, 1 and superpositions of those.
the sort of digital logic circuits found in most PCs?
In all.
what sort of circuits use current?
Well, current can correspond to data in analog circuits. | {"url":"http://www.physicsforums.com/showthread.php?p=4233534","timestamp":"2014-04-19T19:37:32Z","content_type":null,"content_length":"46384","record_id":"<urn:uuid:a5f8701c-b381-4ded-932e-68c876144a0a>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00568-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hermosa Beach Math Tutor
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...Mastering the fundamentals is not about memorizing rules, but understanding the underlying logic involved. My own memory is terrible and I did very poorly in subjects requiring a good memory -
basically all subjects that did not involve math. I did well in math because I was able to remember a core set of postulates, which (thankfully) never changed, and work from there.
41 Subjects: including geometry, ACT Math, LSAT, writing
...I have helped students prepare for many standardized tests including the ISEE, HSPT, ACT, PSAT, SAT (math, reading, & writing), SAT Subject Tests (Bio E/M & Chem), as well as AP Tests (Biology,
Chemistry, and Environmental Science-APES), and the GED. Excellent references and resume available. T...
24 Subjects: including prealgebra, ACT Math, algebra 1, algebra 2
...I was the coordinator of Biola University's Writing Center, where I worked one on one with undergraduate and graduate students to improve their essays so their voices came through. I am well
acquainted with test preparation skills. I am qualified and able to teach SAT Critical Reading, Math, and Writing.
22 Subjects: including SAT math, English, trigonometry, ACT Math
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6 Subjects: including prealgebra, biology, Japanese, physical science
...The high scores are greater than three times the national average. What makes this even more impressive is that I only saw my students an average of 2.5 times per week (2 days one week, 3 days
the next, alternating). I was trained as a Classicist, a Chemist, and a Mathematician. I have enjoyed...
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mean,median,mode etc.
June 4th 2009, 05:51 PM
mean,median,mode etc.
I'm sure this problem is easy, but the months are throwing me off. can someone help?
Month Jan Feb Mar Apr May June
students hired 3 2 4 2 6 0
(i) Determine the mean, the median, the mode, and the range for the above data.
(ii) Compute the first and the third quartiles
(iii) Compute the z-scores for the months of may.
i have attempted this but, i'm not sure if i'm doing it right
June 4th 2009, 06:17 PM
I'm sure this problem is easy, but the months are throwing me off. can someone help?
Month Jan Feb Mar Apr May June
students hired 3 2 4 2 6 0
(i) Determine the mean, the median, the mode, and the range for the above data.
(ii) Compute the first and the third quartiles
(iii) Compute the z-scores for the months of may.
i have attempted this but, i'm not sure if i'm doing it right
The months are merely the periods. You could view the months as the x values, and the numerical data as the y values, and generate a plot.
June 4th 2009, 10:30 PM
sorry I cannot comment on 'months are merely the periods.'
June 6th 2009, 04:39 PM
are the mean, median and mode suppose to be months or numbers?
I'm sure this problem is easy, but the months are throwing me off. can someone help?
Month Jan Feb Mar Apr May June
students hired 3 2 4 2 6 0
(i) Determine the mean, the median, the mode, and the range for the above data.
(ii) Compute the first and the third quartiles
(iii) Compute the z-scores for the months of may.
i have attempted this but, i'm not sure if i'm doing it right
June 20th 2009, 07:47 PM | {"url":"http://mathhelpforum.com/advanced-statistics/91824-mean-median-mode-etc-print.html","timestamp":"2014-04-18T05:35:42Z","content_type":null,"content_length":"7125","record_id":"<urn:uuid:c8255c12-8400-4e20-9626-9a51f253bf9d>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00475-ip-10-147-4-33.ec2.internal.warc.gz"} |
Matlab stats utilities
• Matlab stats utilities (see documentation and examples within the functions)
AffinityPropagationClusteringInterface.m - Cluster vectors using Frey's affinity-propagation algorithm.
AssignStatsConstants.m - A script assigining several statistical constants used by other functions
DisplayPCA.m - Display first principal components of data
FindDistWithGivenKL.m - Compute a distribution with a given KL distance from P
FindIndependentDistWithGivenKLFromUniform.m - Compute an indep. dist. with a given KL distance from uniform
KL_distances_rand_points.m - Compute distances between many pairs of 'random' points (distributions).
NextBestTree.m - Computes the 'next best' spanning tree
NormalizeData.m - Normalization - Subtracts the mean of each row and divide by standard deviation (from Tal Shay)
Phi.m - Standard Gaussian Phi
PhiInv.m - Inverse standard Gaussian Phi
PolishMean.m - Calculate mean after without lower/upper t
add_noise_to_vec.m - Adds white gaussian noise to a vector
bernoulli_sum_prob.m - Compute the probability distribution of sum of N independent bernoulli random
cdf_hist.m - Compute cdf(t) for a histogram
chinese_restaurant_process.m - Gerenate an instance of chinese restaurant process
cond_mean.m - Compute conditional mean of a distribution given that value is > alpha
convolution_prob.m - Return the convolution of two probabilities: p+q-2pq
cumsum_hist.m - Perform cumulative summation for a histogram
decide_success_or_failure_online.m - Internal function for deciding when to stop trying (if number of failures
derich_correct.m - Add a relative derichlet correction
displaysamplespca.m - Display a PCA of the samples.
distributions_diff.m - Compute the difference between two distributions.
distrnd.m - Simulate data from any distribution (like normrnd but for general dists.)
diststat.m - Compute mean and var of a general distribution
enrichment_fdr_plot.m - Plot the 'enrichment' FDR when comparing two sets,
entropy.m - Compute entropy (in log base 2)
ewen_sampling_formula.m - Generate a sample from Ewen's distribution
gauss_smooth.m - Simple 1d Gaussian smoothing of a histogram.
generate_points.m - Generate a set of points on the plane
h.m - Binary entropy function
hypergeometric_for_many_sets.m - Give hypergeometric-like p-value for pairwise intersections of many sets
hypergeometric_for_three_sets.m - Give hypergeometric p-value for 3-way intersection, given pairwise intersections
indian_buffet_process.m - Gerenate an instance of chinese restaurant process
integral_hist.m - Perform integral for a histogram (we use simple rectangle integration)
integral_hist2d.m - Perform two-dimensional integral for a histogram (we use simple rectangle integration)
isbimodal_hist.m - Test if a distribution is bimodal
kurtosis_hist.m - Compute kurtosis for a histogram
log_binom.m - Compute log of binomial coefficient N over M.
log_factorial_vec.m - Compute log-factorial for a vector.
maxnormcdf.m - Cumulative distribution function of maximum of N gaussians
maxnormpdf.m - Density of maximum of N standard Gaussians
maxnormstat.m - Compute mean and st.d. of maximum of N standard Gaussians
mean_hist.m - Compute mean for a histogram
mean_not_nan.m - unction vec = mean_not_nan(mat)
mean_single.m - A function for computing the mean of a large single array that
med_hist.m - Compute median for a histogram
median_hist.m - Compute median for a histogram
median_mad.m - unction [median_vec, mad_vec] = median_mad(mat)
moment_hist.m - Compute central moment of any order for a histogram
my_cov.m - Compute covariance for two matrices
my_least_squares.m - Find the best a and b such that the sum of squares of y - (ax+b) is minimized
my_quantile.m - Like Matlab's quantile but enables to get quantiles outside [0,1].
my_smooth.m - Performs moving sum (default)/average
ncx2power_to_ncp.m - Compute the non-centrality parameter needed to achieve a desired power
normalize_hist.m - Normalize a histogram to have sum one
normalize_hist2d.m - Normalize a two-dimensional histogram to have sum one
plane_check.m - Check if certain points are on the same plain
poisson_point_process_rnd.m - Generate data from a Poisson point process in the cube [0,1]^n
posdefrnd - Copy.m - Sample a semi positive-definite matrix
posdefrnd.m - Sample a semi positive-definite matrix
powernormstat.m - compmute moments of |z|^p when p is standard Gaussian
pvalPearson.m - VALPEARSON Tail probability for Pearson's linear correlation.
quantile_hist.m - Compute quantile for a histogram
quantile_normalize.m - Perform quantile normalization on a matrix.
rand_nchoosek.m - Pick at random k indices out of n
rand_normalized.m - Randomize an mXn matrix such that each column sums to 1
rand_seed.m - A script for seeding Matlab's rand
randcircle.m - Draw random points in a circle
random_Tree_to_point.m - Get the closest point on a random tree (in KL) to a point (distribution).
randsphere.m - Generate random points inside a 3-d sphere
ranksum1side.m - Wilcoxon rank sum test but with one-sided p-value
rationormpdf.m - Ratio of two Gaussians density distriburion
rationormstat.m - First two moments of ratio of gaussians
relative_entropy.m - Compute the relative entropy between distributions P and Q
sample_gaussian_integral.m - A Gaussin integral
set_windows_pval.m - Assume n variables are uniformly distributed on [1, N]
set_windows_pval_old.m - Assume n variables are uniformly distributed on [1, N]
simple_binom_windows_pval.m - Assume n variables are uniformly distributed on [1, N]
simulate_by_normalization.m - Simulate uniform r.v.s. normalized by their sum
simulate_correlated_bernoulli.m - Simulates a set of correlated bernoulli random variables
simulate_correlated_gaussians.m - Simulates a set of correlated gaussian random
simulate_quasi_regression.m - Simulate linear regression model and quasi regression
simulate_rand_points.m - A script for simulating throwing random points in the unit interval
skewness_hist.m - Compute skewness for a histogram
std_hist.m - Compute std for a histogram
std_to_q_binary.m - Compute Bernoulli probability for a given standard deviation
sum_hist.m - Add values in two histograms
table_to_marginal_probs.m - Compute product of marginals for a probability table
test_KL_distances_rand_points.m - Script for runnning the test_KL_distances_rand_points function
test_MI_computation.m - Test inference and mutual information computation for trees.
test_chisquare.m - Test for conditional indepndence via chi-square test (from Kevin Murphy's BNT)
test_correlated_gaussians.m - Generate correlated gaussians. Then perform de-convulution
test_distributions_diff.m - A script for testing the distributions_diff function
test_goodness_of_fit.m - function test_goodness_of_fit()
test_hypergeometric_for_many_sets.m - Test the function hypergeometric_for_many_sets
test_hypergeometric_for_three_sets.m - Give p-value for the hypergeometric score of intersection of three sets,
test_hypergeometric_scores.m - Test all hypergeometric functions
test_probit.m - Run a test of matlab probit regression. Determine coefficients, simulate
test_regression.m - unction test_regression
test_windows_pval.m - Test the p-values for significance, using both windows methods,
uniform_log_sum_reciprocal_mean.m - Compute the mean of -1/(sum_{i} log(p_i))
uniform_spanning_tree.m - Generate a uniform spanning tree on n vertices
uniform_sum_reciprocal_mean.m - Compute the mean of 1/(sum_{i} p_i)
var_hist.m - Compute variance for a histogram
weighted_hist.m - Builds a histogram by binning the elements of val into containers, one
weighted_mean.m - Compute weighted average
weighted_rand.m - Produce random numbers drawn according to some weights.
stats.tgz - Download the whole package. (107 .m files)
fdr - Go to the fdr package.
hmp - Go to the hmp package.
mog - Go to the mog package. | {"url":"http://www.broadinstitute.org/~orzuk/matlab/libs/stats/matlab_stats_utils.html","timestamp":"2014-04-20T10:06:46Z","content_type":null,"content_length":"30832","record_id":"<urn:uuid:adab6cb5-61ee-4a38-8d0f-b91cb37d247a>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00225-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: July 1997 [00243]
[Date Index] [Thread Index] [Author Index]
Exponent function
• To: mathgroup at smc.vnet.net
• Subject: [mg7807] Exponent function
• From: Raya Firsov-Khanin <raya at mech.ed.ac.uk>
• Date: Sat, 12 Jul 1997 02:45:49 -0400
• Sender: owner-wri-mathgroup at wolfram.com
I've noticed the following strange feature of
Exponent[expr, arg, List]
function of Mathematica. Suppose I want to
find all powers of x in the expression of the type
test = x + x^(1+a) + x^(-1+a) + x^2
then Exponent[test, x, List] gives the correct result
{1, 2, 1-a, -1+a}.
Suppose now, I add x^(-1) to test:
test1 = test + x^(-1)
Exponent[test1, x, List] gives a result {-1, -1, 1, 1, 2}.
For expressions like
test2 = test + x^(-n) the result of Exponent[] is correct
{1, 2, 1-a, -1+a, -n}, while for expressions where n is specified,
e.g. n=1, 2, 3, ... the result is always wrong.
I need to use Exponent function in a program that
should solve nonlinear problems using Multiple Scale Method. I need to
identify exponential frequencies of different terms in the equations, and
to study under what conditions they become resonant.
Any suggestions on how I can make Exponent function to give correct
results? Or may be there are some other ways to find powers of x in the
expressions of the type written above.
Any help would be appreciated. I already spent a morning trying to solve
this puzzle.
Thanks a lot. | {"url":"http://forums.wolfram.com/mathgroup/archive/1997/Jul/msg00243.html","timestamp":"2014-04-20T01:04:41Z","content_type":null,"content_length":"35102","record_id":"<urn:uuid:48088d7b-53ef-44d4-aa1e-b4cc94c9dd98>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00266-ip-10-147-4-33.ec2.internal.warc.gz"} |
Using mixture models with known class membership to address incomplete covariance structures in multiple-group growth models
Multi-group latent growth modelling in the structural equation modelling framework has been widely utilized for examining differences in growth trajectories across multiple manifest groups. Despite
its usefulness, the traditional maximum likelihood estimation for multi-group latent growth modelling is not feasible when one of the groups has no response at any given data collection point, or
when all participants within a group have the same response at one of the time points. In other words, multi-group latent growth modelling requires a complete covariance structure for each observed
group. The primary purpose of the present study is to show how to circumvent these data problems by developing a simple but creative approach using an existing estimation procedure for growth mixture
modelling. A Monte Carlo simulation study was carried out to see whether the modified estimation approach provided tangible results and to see how these results were comparable to the standard
multi-group results. The proposed approach produced results that were valid and reliable under the mentioned problematic data conditions. We also present a real data example and demonstrate that the
proposed estimation approach can be used for the chi-square difference test to check various types of measurement invariance as conducted in a standard multi-group analysis. | {"url":"http://onlinelibrary.wiley.com/doi/10.1111/bmsp.12008/full","timestamp":"2014-04-16T22:17:17Z","content_type":null,"content_length":"216514","record_id":"<urn:uuid:02ccf929-49a6-481c-a492-3d15ffba289b>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00095-ip-10-147-4-33.ec2.internal.warc.gz"} |
Metric Measurement
The metric system of measurement is based on 10 and powers of 10. The prefixes used for length, capacity, and mass tell what part of the basic unit is being considered.
kilo- 1,000 kilo- k as in km for kilometers
hecto- 100 hecto- h as in hm for hectometers
deka- 10 deka- da as in dam for dekameters
deci- 0.1 deci- d as in dm for decimeters
centi- 0.01 centi- c as in cm for centimeters
milli- 0.001 milli- m as in mm for millimeters
At this grade level, the only prefixes that will be used are kilo-, deci-, centi-, and milli-.
The common units of length include millimeters, centimeters, decimeters, meters, and kilometers. Students should know the following equivalencies and abbreviations. Remind students that abbreviations
do not take a plural s.
100 centimeters (cm) = 1 meter (m)
10 decimeters (dm) = 1 meter (m)
1,000 meters (m) = 1 kilometer (km)
In the metric system of measurement, the common units of measure and abbreviations for capacity and mass* are shown below. Students should know the following equivalencies and abbreviations.
1,000 milliliters (mL) = 1 liter (L)
1,000 grams (g) = 1 kilogram (kg)
*Note that the metric system most commonly uses a measure of mass rather than weight. These terms are often used interchangeably, but there is a difference. Mass measures the amount of matter in an
object. Weight measures the gravitational pull on the object. In space, an astronaut who is “weightless” still has the same mass as on Earth.
Teaching Model 14.2: Meter and Kilometer | {"url":"http://www.eduplace.com/math/mw/models/overview/3_14_2.html","timestamp":"2014-04-17T18:42:41Z","content_type":null,"content_length":"5419","record_id":"<urn:uuid:890b5ca6-15f3-461a-ba42-7aa71effc211>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00141-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Boy On The Tower Of Height H = 18 M In The ... | Chegg.com
The boy on the tower of height h = 18 m in the figure below throws a ball a distance of x = 56 m, as shown. At what speed, in m/s, is the ball thrown?
1. m/s | {"url":"http://www.chegg.com/homework-help/questions-and-answers/boy-tower-height-h-18-m-figure-throws-ball-distance-x-56-m-shown-speed-m-s-ball-thrown-1-m-q1616076","timestamp":"2014-04-25T06:17:26Z","content_type":null,"content_length":"20207","record_id":"<urn:uuid:cfa91aa2-7129-40b6-8376-845de9525398>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00596-ip-10-147-4-33.ec2.internal.warc.gz"} |
Search for 2^nd-Generation Leptoquarks in the
µj+µj Channel in ppbar Collisions at sqrt{s} = 1.96 TeV
The observed symmetry in the spectrum of elementary particles between leptons and quarks motivates the existence of leptoquarks [1]. Leptoquarks (LQ) are bosons carrying both quark and lepton quantum
numbers and fractional electric charge. Although leptoquarks could in principle decay into into any combination of a lepton and a quark, indirect experiemental limits lead to the assumtion that there
would be three different generations of leptoquarks, each coupling to only one quark and lepton family and therefore individually conserving the family lepton numbers [2].
At the TeVatron, leptoquarks would mainly be produced in pairs. In this analysis, the selection of di-muon + di-jet events is presented, and about 104 pb^-1 of proton-antiproton collisions, collected
during Run IIa with the DØ detector between September 2002 and June 2003, have been compared to Monte-Carlo Simulations of the expected Standard Model background as well as simulated scalar
leptoquark events. The data have been shown to be compatible with the expected background, arising predominantly from Drell-Yan Z/gamma production.
One event is remaining in the data after the last cut while 1.59 ± 0.47 are expected from the background. Therefore, no excess of the data over the expected background was found, and the 95 %
confidence level (C.L.) upper limits on the cross-section of scalar leptoquarks have been calculated, assuming 100 % branching fraction into charged leptons, beta = BF(LQ[2]->µ j) = 1. These limits
have been compared to theoretically calculated cross-sections, and a lower bound of the mass of scalar 2^nd-generation leptoquarks of M(LQ[2]) > 186 GeV was extracted.
A detailed description is available as DØ note 4190 (restricted access; non-DØ-speakers, please contact the DØ NP Convenors).
Scalar sum of the transverse energies of the
system for all
di-muon + di-jet events with
> 110 GeV.
Reconstructed mu+j mass for all di-muon + di-jet events with M(µµ) > 110 GeV. There are two
possibilities to combine the two highest-p[T] muons with the two highest-E[T] jets. Only the
combination with the smaller mass difference of the two leptoquark candidates of the event is chosen,
and the reconstructed "leptoquark mass" of this event is the mean of the masses of the two µ+j systems.
Calculated cross-sections for scalar 2^nd-generation leptoquarks as compared to the 95 % C.L.
upper limit on the cross-section, assuming 100 % branching fraction into charged leptons. The
LO and NLO cross-sections are calculated using reference [3] (taking into accound the correct
center-of-mass energy). The error band has been estimated by varying the renormalization and
factorization scale between M(LQ)/2 and 2 M(LQ).
Event diplays of the highest-S[T] candidate, remaining in the data after
the last cut: M(µµ) = 112 GeV, S[T] = 372 GeV.
[1] J.C Pati and A. Salam, Phys. Rev. D 10, 275 (1974);
E. Eichten et al., Phys. Rev. D 34, 1547 (1986);
W. Buchmüller and D. Wyler, Phys. Lett. B 177, 377 (1986);
E. Eichten et al., Phys. Rev. Lett. 50, 811 (1983);
H. Georgi and S. Glashow, Phys. Rev. Lett. 32, 438 (1974).
[2] M. Leurer, Phy. Rev. D 49, 333 (1994).
[3] M. Krämer, T. Plehn M. Spira and P.M. Zerwas, Phys. Rev. Lett. 79, 341-344 (1997). | {"url":"http://www-d0.fnal.gov/Run2Physics/np/results/LP2003/lq_mumu/","timestamp":"2014-04-21T12:37:28Z","content_type":null,"content_length":"7589","record_id":"<urn:uuid:7fb3745d-d0d3-401a-82a3-3bb13609f9e0>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00185-ip-10-147-4-33.ec2.internal.warc.gz"} |
Banked curve involving friction
Hi could some of you guys please help me with question, its question 77 in Understand Physics By Cummings.
For those that don't have access to the book its a question dealing with a banked curve with radius R and a angle alpha. Now there is also a friction force stoping the car from sliding of the banked
curve, now my question is how do you derive the equation to explain the situation. I know for a normal banked curve without friction the answer is just, Nsin(angle) = m(v^2)/r and Ncos(angle) = mg
So therefore tan(angle) = (v^2)/rg
Now how do you derive the equation is friction is also a force helping the car sliding down, i know the answer is suppose to be vmax = (rgtan(angle + tan^-1(co-efficient of f)))
The friction makes it a little more difficult.
The horizontal forces are:
(1)[tex]F_{xfriction} = \mu_sF_Ncos\theta[/tex]
(2)[tex]F_{Nx} = F_Nsin\theta[/tex]
(3)[tex]F_Nsin\theta + \mu_sF_Ncos\theta = mv^2/r[/tex]
One has to look at the vertical components of the forces to find the normal force. These have to sum to zero (since there is no vertical acceleration). The friction force has a downward vertical
component and this, together with gravity, equals the vertical component of the normal force:
[tex]mg + \mu_sF_Nsin\theta = F_Ncos\theta[/tex]
(4)[tex]F_N = mg/(cos\theta - \mu_ssin\theta)[/tex]
So substituting into (3):
[tex]\frac{mg}{(cos\theta - \mu_ssin\theta)}(sin\theta + \mu_scos\theta) = mv^2/r[/tex]
(5)[tex]v = \sqrt{\frac{rg(sin\theta + \mu_scos\theta)}{(cos\theta - \mu_ssin\theta)}[/tex] | {"url":"http://www.physicsforums.com/showthread.php?t=69626","timestamp":"2014-04-16T22:08:04Z","content_type":null,"content_length":"46953","record_id":"<urn:uuid:d15bcd27-dc7a-4396-b229-febb14a2fcc3>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00579-ip-10-147-4-33.ec2.internal.warc.gz"} |
Academic Bulletin
School of Sciences
Bachelor of Science in Mathematics
The Bachelor of Science degree in Mathematics is designed to prepare individuals to understand the nature of truth and the concept of proof in the discipline of mathematics, to understand the
application of mathematical techniques to other fields, and to formulate and solve problems mathematically. The Bachelor of Science places a greater emphasis on mathematical knowledge and its
relation to the sciences through additional coursework and potential research opportunities. Students have greater opportunities to complete coursework for either graduate school in mathematics or
entry into business or industry.
Degree Requirements:
1. Students must complete a minimum of 120 credit hours with a cumulative grade point average of 2.0 or higher.
2. Entering freshmen must take SSCI-E 105 Science Freshmen Learning Community (1 cr.).
3. General Education. Students must complete all of the requirements of the Indiana University Kokomo campus-wide general education curriculum. The General Education requirements in quantitative
literacy, critical thinking, and physical and life sciences are satisfied by the major.
4. Required Science and Informatics Courses
□ PHYS-P 221 Physics 1 (5 cr.)
□ One of the following: CHEM-C105/125 Principles of Chemistry I and Laboratory (5 cr.), BIOL-L 105 Intro to Biology (5 cr.), or GEOL-G 100 General Geology (5 cr.)
□ INFO-I 101 Introduction to Informatics (4 cr.)
□ Five additional credits in the School of Sciences outside of Mathematics/statistics.
5. Mathematics Courses-Students must complete a minimum of 41 credit hours in mathematics with a grade point average of at least 2.0. The following courses are required: MATH-M 215-216 Calculus I-II
(10cr.), MATH-M 311 Calculus III (4 cr.), MATH-M303 Linear Algebra for Undergraduates (3 cr.). In addition, students must complete two sequences from Group A and an additional 12 credit hours
from Groups A or B.
□ Group A: MATH-M 403/404 Introduction to Modern Algebra I-II (6 cr.), MATH-M 413/414 Introduction to Analysis I-II (6 cr.), MATH-M 413/415 Introduction to Analysis I/Elementary Complex
Variables with Applications (6 cr.), MATH-M 447/448 Mathematical Models and Applications I-II (6 cr.), MATH-M 471/472 Numerical Analysis I-II (6 cr.)
□ Group B: MATH-M 313 Elementary Differential Equations with Applications (3 cr.), MATH-T 336 Topics in Euclidean Geometry (3 cr.), MATH-M 347 Discrete Mathematics (3 cr.), MATH-M 360 Elements
of Probability (3 cr.), MATH-M 366 Elements of Statistical Inference (3 cr.), MATH-M 415 Elementary Complex Variables with Applications (3 cr.)
6. General Examination—Students must pass a written examination covering the entire undergraduate mathematics program. The examination will be given near the end of the semester in which the student
is expected to graduate. The mathematics faculty may permit a student who does not perform satisfactorily on the written examination to take an oral examination that same semester. Students who
still do not perform satisfactorily may take the general examination the next time it is offered. Those who do not pass the general examination on the second attempt must petition the mathematics
faculty to take the general examination a third time, and are expected to document additional preparation in mathematics.
7. Students must complete 30 of the last 60 credit hours, including at least 9 credit hours of mathematics from Groups A or B, and the general examination at Indiana University Kokomo. | {"url":"http://www.iuk.edu/~bulletin/iuk/2013-2014/majors-minors/school-of-sciences/bachelor-of-science-in-mathematics.shtml","timestamp":"2014-04-20T04:23:55Z","content_type":null,"content_length":"12281","record_id":"<urn:uuid:374b5602-4f28-409c-ae15-d894436e44d1>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00056-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Need help with these convergence Tests?
November 28th 2010, 06:35 PM
Need help with these convergence Tests?
Do these converge absolutely, conditionally, or diverge?
sigma from n = 1 to infinity of (n + 1)^2 / n!
sigma from n = 1 to infinity of (1/ (arctan(n))^(2n)
explain please!
November 28th 2010, 06:38 PM
I first break it up. (n+1)^2=n^2+2n+1
Now you can evaluate three separate Summations.
I think the 2nd one can be evaluate like a geometric since ArcTan(Infinity)=pi/2
November 28th 2010, 06:39 PM
Prove It
For the first, try the ratio test.
Evaluate $\displaystyle \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|$. If this is $\displaystyle <1$ the series converges absolutely, if this is $\displaystyle >1$ the series diverges. If
this $\displaystyle =1$ the test is inconclusive.
November 28th 2010, 07:08 PM
so for the first one I applied the ratio test and got
lim (as n goes to infinity) of (n^2 + 2n + 1) / ( (n+1)(n^2) ) + lim (as n goes to infinity) (2n + 1)/( (n+1)(n^2) + lim (as n goes to infinity) of 1/(n+1)
and since all of these go to zero then the sum converges absolutely?
and for the second one you have
sum (from 1 to infinity) 1 / (pi/2)^2n
and because pi/2 is greater than 1 it just diverges by geometric series test?
November 28th 2010, 07:14 PM
You are mixing up two separate suggestions. I thought you wanted to solve the series but you just want to know if they converge. Just follow Post #3
$\displaystyle a_{n+1}=\frac{(n+2)^2}{(n+1)!}$
$\displaystyle \frac{\frac{(n+2)^2}{(n+1)!}}{\frac{(n+1)^2}{n!}}\ rightarrow \frac{(n+2)^2}{(n+1)^3}\rightarrow \frac{n^2}{n^3}\rightarrow\lim_{n\to\infty}\frac{1 }{n}=0$
November 28th 2010, 07:19 PM
ahh i see
so lim (as n goes to infinity) (n+2)^2 / ( (n+1)(n+1)2 which is just 0 so it converges abs
what about the second one?
November 28th 2010, 08:36 PM
Try the nth root test | {"url":"http://mathhelpforum.com/calculus/164682-need-help-these-convergence-tests-print.html","timestamp":"2014-04-17T22:24:19Z","content_type":null,"content_length":"7504","record_id":"<urn:uuid:970fc475-39f5-43ac-abb6-9f0a8fb44d91>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00527-ip-10-147-4-33.ec2.internal.warc.gz"} |
Patent US7047047 - Non-linear observation model for removing noise from corrupted signals
The present invention relates to noise reduction. In particular, the present invention relates to reducing noise in signals used in pattern recognition.
A pattern recognition system, such as a speech recognition system, takes an input signal and attempts to decode the signal to find a pattern represented by the signal. For example, in a speech
recognition system, a speech signal is received by the recognition system and is decoded to identify a string of words represented by the speech signal.
However, input signals are typically corrupted by some form of additive noise. Therefore, to improve the performance of the pattern recognition system, it is often desirable to estimate the additive
noise and use the estimate to provide a cleaner signal.
Spectral subtraction has been used in the past for noise removal, particularly in automatic speech recognition systems. Conventional wisdom holds that when perfect noise estimates are available,
basic spectral subtraction should do a good job of removing the noise; however, this has been found not to be the case.
Standard spectral subtraction is motivated by the observation that noise and speech spectra mix linearly, and therefore, their spectra should mix according to
|Y[k]| ^2 =|X[k]| ^2 +|N[k]| ^2
Typically, this equation is solved for a |X[k]|^2, and a maximum attenuation floor F is introduced to avoid producing negative power special densities.
$ X ^ [ k ] 2 = Y [ k ] 2 max ( Y [ k ] 2 - N [ k ] 2 Y [ k ] 2 , F ) EQ . 1$
Several experiments were run to examine the performance of Equation 1 using the true spectra of n, and floors F from e^−20 to e^−2. The true noise spectra were computed from the true additive noise
time series for each utterance. All experiments were conducted using the data, code and training scripts provided within the Aurora 2 evaluation framework described by H. G. Hirsch and D. Pearce in
“The Aurora Experimental Framework for the Performance Evaluations of Speech Recognition Systems Under Noisy Conditions,” ISCA ITRW ASR 2000 “Automatic Speech Recognition: Challenges for the Next
Millennium”, Paris, France, September 2000. The following digit error rates were found for various floors:
e^−20 e^−10 e^−5 e^−3 e^−2
87.50 56.00 34.54 11.31 15.56
From the foregoing, it is clear that even when the noise spectra is known exactly, spectral subtraction does not perform perfectly and improvements can be made. In light of this, a noise removal
technique is needed that is more effective at estimating the clean speech spectral features.
A new statistical model describes the corruption of spectral features caused by additive noise. In particular, the model explicitly represents the effect of unknown phase together with the unobserved
clean signal and noise. Development of the model has realized three techniques for reducing noise in a noisy signal as a function of the model.
Generally, as an aspect of the present invention and utilized in two techniques, a frame of a noisy input signal is converted into an input feature vector. An estimate of a noise-reduced feature
vector uses a model of the acoustic environment. The model is based on a non-linear function that describes a relationship between the input feature vector, a clean feature vector, a noise feature
vector and a phase relationship indicative of mixing of the clean feature vector and the noise feature vector. Inclusion of a mathematical representation of the phase relationship renders an accurate
model. One unique characteristic of the phase relationship is that it is in the same domain as the clean feature vector and the noise feature vector. Another separate distinguishing characteristic is
that the phase relationship includes a phase factor with a statistical distribution.
In another aspect, a method for reducing noise in a noisy input signal includes converting a frame of the noisy input signal into an input feature vector; and obtaining a noise-reduced feature vector
by using an equation of the form
{circumflex over (x)}=y+ln|1−e ^n−y|
where y is the input feature vector and n comprises a noise estimate.
FIG. 1 is a block diagram of one computing environment in which the present invention may be practiced.
FIG. 2 is a block diagram of an alternative computing environment in which the present invention may be practiced.
FIG. 3A is a plot of conditional observation probability p(y|x, n) with normal approximation for p[α](α).
FIG. 3B is a plot of a sample distribution for a filter bank.
FIG. 3C is a plot of normal approximation of p(y|x, n) as a function of v.
FIG. 3D is a plot of distributions of α for several frequency bins.
FIG. 4A is a plot of output SNR to input SNR for known spectral subtraction.
FIG. 4B is a plot of output SNR to input SNR for a new spectral subtraction method of the present invention.
FIG. 5 is a method illustrating steps for obtaining a weighted Gaussian approximation in the model of the present invention.
FIG. 6 is a flow diagram of another method of estimating clean speech.
FIG. 7 is a block diagram of a pattern recognition system in which the present invention may be used.
Before describing aspects of the present invention, a brief description of exemplary computing environments will be discussed.
FIG. 1 illustrates an example of a suitable computing system environment 100 on which the invention may be implemented. The computing system environment 100 is only one example of a suitable
computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention. Neither should the computing environment 100 be interpreted as having any
dependency or requirement relating to any one or combination of components illustrated in the exemplary operating environment 100.
The invention is operational with numerous other general purpose or special purpose computing system environments or configurations. Examples of well-known computing systems, environments, and/or
configurations that may be suitable for use with the invention include, but are not limited to, personal computers, server computers, hand-held or laptop devices, multiprocessor systems,
microprocessor-based systems, set top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, telephony systems, distributed computing environments that include any
of the above systems or devices, and the like.
The invention may be described in the general context of computer-executable instructions, such as program modules, being executed by a computer. Generally, program modules include routines,
programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Tasks performed by the programs and modules are described below and
with the aid of figures. Those skilled in the art can implement the description and figures as computer-executable instructions, which can be embodied on any form of computer readable media discussed
The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed
computing environment, program modules may be located in both local and remote computer storage media including memory storage devices.
With reference to FIG. 1, an exemplary system for implementing the invention includes a general-purpose computing device in the form of a computer 110. Components of computer 110 may include, but are
not limited to, a processing unit 120, a system memory 130, and a system bus 121 that couples various system components including the system memory to the processing unit 120. The system bus 121 may
be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. By way of example, and not
limitation, such architectures include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video Electronics Standards Association (VESA) local
bus, and Peripheral Component Interconnect (PCI) bus also known as Mezzanine bus.
Computer 110 typically includes a variety of computer readable media. Computer readable media can be any available media that can be accessed by computer 110 and includes both volatile and
nonvolatile media, removable and non-removable media. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage
media includes both volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data
structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or
other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and
which can be accessed by computer 110. Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a
carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in
such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless
media such as acoustic, RF, infrared and other wireless media. Combinations of any of the above should also be included within the scope of computer readable media.
The system memory 130 includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM) 131 and random access memory (RAM) 132. A basic input/output
system 133 (BIOS), containing the basic routines that help to transfer information between elements within computer 110, such as during start-up, is typically stored in ROM 131. RAM 132 typically
contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit 120. By way of example, and not limitation, FIG. 1 illustrates operating
system 134, application programs 135, other program modules 136, and program data 137.
The computer 110 may also include other removable/non-removable volatile/nonvolatile computer storage media. By way of example only, FIG. 1 illustrates a hard disk drive 141 that reads from or writes
to non-removable, nonvolatile magnetic media, a magnetic disk drive 151 that reads from or writes to a removable, nonvolatile magnetic disk 152, and an optical disk drive 155 that reads from or
writes to a removable, nonvolatile optical disk 156 such as a CD ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media that can be used in the
exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the
like. The hard disk drive 141 is typically connected to the system bus 121 through a non-removable memory interface such as interface 140, and magnetic disk drive 151 and optical disk drive 155 are
typically connected to the system bus 121 by a removable memory interface, such as interface 150.
The drives and their associated computer storage media discussed above and illustrated in FIG. 1, provide storage of computer readable instructions, data structures, program modules and other data
for the computer 110. In FIG. 1, for example, hard disk drive 141 is illustrated as storing operating system 144, application programs 145, other program modules 146, and program data 147. Note that
these components can either be the same as or different from operating system 134, application programs 135, other program modules 136, and program data 137. Operating system 144, application
programs 145, other program modules 146, and program data 147 are given different numbers here to illustrate that, at a minimum, they are different copies.
A user may enter commands and information into the computer 110 through input devices such as a keyboard 162, a microphone 163, and a pointing device 161, such as a mouse, trackball or touch pad.
Other input devices (not shown) may include a joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit 120 through a user
input interface 160 that is coupled to the system bus, but may be connected by other interface and bus structures, such as a parallel port, game port or a universal serial bus (USB). A monitor 191 or
other type of display device is also connected to the system bus 121 via an interface, such as a video interface 190. In addition to the monitor, computers may also include other peripheral output
devices such as speakers 197 and printer 196, which may be connected through an output peripheral interface 190.
The computer 110 may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 180. The remote computer 180 may be a personal computer, a
hand-held device, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computer 110. The
logical connections depicted in FIG. 1 include a local area network (LAN) 171 and a wide area network (WAN) 173, but may also include other networks. Such networking environments are commonplace in
offices, enterprise-wide computer networks, intranets and the Internet.
When used in a LAN networking environment, the computer 110 is connected to the LAN 171 through a network interface or adapter 170. When used in a WAN networking environment, the computer 110
typically includes a modem 172 or other means for establishing communications over the WAN 173, such as the Internet. The modem 172, which may be internal or external, may be connected to the system
bus 121 via the user input interface 160, or other appropriate mechanism. In a networked environment, program modules depicted relative to the computer 110, or portions thereof, may be stored in the
remote memory storage device. By way of example, and not limitation, FIG. 1 illustrates remote application programs 185 as residing on remote computer 180. It will be appreciated that the network
connections shown are exemplary and other means of establishing a communications link between the computers may be used.
FIG. 2 is a block diagram of a mobile device 200, which is an exemplary computing environment. Mobile device 200 includes a microprocessor 202, memory 204, input/output (I/O) components 206, and a
communication interface 208 for communicating with remote computers or other mobile devices. In one embodiment, the afore-mentioned components are coupled for communication with one another over a
suitable bus 210.
Memory 204 is implemented as non-volatile electronic memory such as random access memory (RAM) with a battery back-up module (not shown) such that information stored in memory 204 is not lost when
the general power to mobile device 200 is shut down. A portion of memory 204 is preferably allocated as addressable memory for program execution, while another portion of memory 204 is preferably
used for storage, such as to simulate storage on a disk drive.
Memory 204 includes an operating system 212, application programs 214 as well as an object store 216. During operation, operating system 212 is preferably executed by processor 202 from memory 204.
Operating system 212, in one preferred embodiment, is a WINDOWS® CE brand operating system commercially available from Microsoft Corporation. Operating system 212 is preferably designed for mobile
devices, and implements database features that can be utilized by applications 214 through a set of exposed application programming interfaces and methods. The objects in object store 216 are
maintained by applications 214 and operating system 212, at least partially in response to calls to the exposed application programming interfaces and methods.
Communication interface 208 represents numerous devices and technologies that allow mobile device 200 to send and receive information. The devices include wired and wireless modems, satellite
receivers and broadcast tuners to name a few. Mobile device 200 can also be directly connected to a computer to exchange data therewith. In such cases, communication interface 208 can be an infrared
transceiver or a serial or parallel communication connection, all of which are capable of transmitting streaming information.
Input/output components 206 include a variety of input devices such as a touch-sensitive screen, buttons, rollers, and a microphone as well as a variety of output devices including an audio
generator, a vibrating device, and a display. The devices listed above are by way of example and need not all be present on mobile device 200. In addition, other input/output devices may be attached
to or found with mobile device 200 within the scope of the present invention.
Under one aspect of the present invention, a system and method are provided that remove noise from pattern recognition signals. To do this, this aspect of the present invention uses a new statistical
model, which describes the corruption to the pattern recognition signals, and in particular to speech recognition spectral features, caused by additive noise. The model explicitly represents the
effect of unknown phase together with the unobserved clean speech and noise as three hidden variables. The model is used to produce robust features for automatic speech recognition. As will be
described below, the model is constructed in the log Mel-frequency feature domain. Advantages of this domain include low dimensionality, allowing for efficient training in inference. Logarithmic
Mel-frequency spectral coefficients are also linearly related to Mel-Frequency Cepstrum Coefficients (MFCC), which correspond to the features used in a recognition system. Furthermore, corruption
from linear channels and additive noise are localized within individual Mel-frequency bins, which allows processing of each dimension of the feature independently.
As indicated above, the model of the present invention is constructed in the logarithmic Mel-frequency spectral domain. Each spectral frame is processed by passing it through a magnitude-squared
operation, a Mel-frequency filterbank, and a logarithm.
Generally, the noisy or observation (Y) signal is a linear combination of speech (X) and noise (N) as represented by Y[k]=X[k]+N[k]. Accordingly, the noisy log Mel-spectral features y[i ]can be
directly related to the unobserved spectra X[k] and N[k], which can be represented as:
$exp y i = ∑ k w k i X [ k ] 2 + ∑ k w k i N [ k ] 2 + ∑ k w k i X [ k ] N [ k ] cos θ k EQ . 2$
where, w[k] ^i is the kth coefficient in the ith Mel-frequency filterbank. The variable θ[k ]is the phase difference between X[k] and N[k]. When the clean signal and noise are uncorrelated, the θ[k ]
are uncorrelated and have a uniform distribution over the range [−π, π].
Eq. 2 can be re-written to show how the noisy log spectral energies y[i ]are a function of the unobserved log spectral energies x[i ]and n[i].
$exp y i = exp x i + exp n i + 2 α i exp x i + n i 2 EQ . 3 α i = ∑ k w k i X [ k ] N [ k ] cos θ k ∑ k w k i X [ k ] 2 ∑ k w k i
N [ k ] 2 exp x i = ∑ k w k i X [ k ] 2 exp n i = ∑ k w k i N [ k ] 2 EQ . 4$
As a consequence of this model, when y[i ]is observed there are actually three unobserved random variables. The first two include the clean log spectral energy and the noise log spectral energy that
would have been produced in the absence of mixing. The third variable, α[i], accounts for the unknown phase between the two sources.
In the general case, α[i ]will be a function of X[k] and N[k]. However, if the magnitude spectra are assumed constant over the bandwidth of a particular filterbank, the definition of α[i ]collapses
to a weighted sum of several independent random variables:
$α i ≈ ∑ k w k i ∑ j w j i cos θ k . EQ . 5$
FIG. 3D shows the true distributions of α for a range of frequency bins. They were estimated from a set of joint noise, clean speech, and noisy speech data by solving for the unknown α. The higher
frequency, higher bandwidth filters produce α distributions that are more nearly Gaussian. By design, the low frequency bins in a Mel-frequency filterbank have a narrow bandwidth, and the bandwidth
increases with frequency. This means that the effective number of terms in Eq. 5 also increases with frequency. As a result, a Gaussian assumption is quite bad for the lowest frequency bins, and
becomes much better as the bandwidth of the filters increase. As the bandwidth increases, so does the number of effective terms in Eq. 5, and the central limit theorem begins to apply. In practice, a
frequency-dependent Gaussian approximation P[αi](α[i])=N(α[i]; 0, σ[αi] ^2) works well.
At this point it should be noted, that the parameter σ[α] ^2 can be estimated from a small set of training data. The estimate is the sample variance computed from all the sample values of α^(l)'s
(for l=1, 2, . . . L, L=the total number Mel-filter banks). A linear regression line is fit to the computed σ[α] ^2 as a function of l.
Conditional Observation Probability
Eq. 3 places a hard constraint on the four random variables, in effect yielding three degrees of freedom. This can be expressed by solving for y and writing the conditional probability distribution,
$p ( y | x , n , α ) = δ ( y - ln ( e x + e n + 2 α e x + n 2 ) ) . EQ . 6$
The conditional probability p(y|x, n) is found by forming the distribution p(y, α|x, n) and marginalizing over α. Note p(α|x,n)=p(α) is assumed, which is reasonable.
$p ( y | x , n ) = ∫ - ∞ ∞ p ( y | x , n , α ) p α ( α ) ⅆ α = ∫ - ∞ ∞ δ ( y - ln ( e x + e n + 2 α e x + n 2 ) ) p α ( α ) ⅆ α$
The identity
$∫ - ∞ ∞ δ ( f ( α ) ) p α ( α ) ⅆ α = ∑ { α : f ( α ) = 0 } p α ( α ) ⅆ ⅆ α f ( α ) $
can then be used to evaluate the integral in closed form:
$p ( y | x , n ) = e x + e n + 2 α e x + n 2 2 e x + n 2 p α ( α ) | α = e y - e x - e n 2 e x + n 2 = 1 2 exp ( y - x + n 2 ) p α ( e y - e x - e n 2 e x + n 2
) EQ . 7$
When the Gaussian approximation for p[α](α) is introduced, the likelihood function becomes EQ. 7A
$ln p ( y | x , n ) = y - x + n 2 - 1 2 ln 8 π σ α 2 - ( e y - e x - e n ) 2 8 σ α 2 e ( x + n ) .$
Shift Invariance
The conditional probability (Eq. 7) appears at first glance to have three independent variables. Instead, it is only a function of two: the relative value of speech and observation (x−y), and the
relative value of noise and observation (n−y).
Instead of working with absolute values of y, x, and n, Eq. 7 can be rearranged as
$p ( y | x , n ) = 1 2 exp ( - x _ + n _ 2 ) p α ( 1 - e x _ - e n _ 2 e x _ + n _ 2 )$
{overscore (x)}=x−y, {overscore (n)}=n−y
By working with the normalized values, three independent variables have been reduced to two.
Model Behavior
FIG. 3A contains a plot of this conditional probability distribution. FIG. 3B shows an equivalent plot, directly estimated from data, thereby confirming Eq. 7A.
Faith in the model can be built through either examining its ability to explain existing data, or by examining its asymptotic behavior.
FIG. 3A contains a plot of the conditional probability distribution of the model represented by Eq. 7A. Note that due to the shift invariance of this model, there are only two independent terms in
the plot.
Compare this to FIG. 3B, which is a histogram of x−y versus n−y for a single frequency bin across all utterances in set A, subway noise, 10 dB SNR. It is clear that the model is an accurate
description of the data. It should be noted some previous models, have an error model that is independent of SNR. The new model automatically adjusts its variance for different SNR hypotheses.
As we move left along n=y in the graph of FIG. 3A, the variance perpendicular to this line decreases. This corresponds to more and more certainty about the noise value as the ratio of speech to
observation decreases. In the limit of this low SNR hypothesis, the model is reducing it's un-certainty that n=y to zero. If the prior probability distributions for speech and noise are concentrated
in this area, the model reduces to
p(y|x)=p [n](y).
Symmetrically, as we move down along x=y in the same graph, the variance perpendicular to this line decreases. As the ratio of noise to observation decreases, the model has increasing certainty that
x=y. We refer to this region as the high SNR hypothesis. If the priors for speech and noise are concentrated in this area, the model reduces to
The graph also has a third region of interest, starting from the origin and moving in a positive x and n direction. In this region, both speech and noise are greater than the observation. This occurs
most frequently when x and n have similar magnitudes, and are destructively interfering with each other. In this case the relevant θ exist in the region
$ θ > π 2 .$
Relationship to Spectral Subtraction
Eq. 7A can be used to derive a new formula for spectral subtraction. The first step is to hold n and y fixed, and find a maximum likelihood estimate for x. Taking the derivative with respect to x in
Eq. 7A and equating it to zero results in EQ. 8
e ^x−n=√{square root over ((e ^y−n−1)^2+(2σ[α] ^2)^2)}{square root over ((e ^y−n−1)^2+(2σ[α] ^2)^2)}−2σ[α] ^2.
This formula is already more well-behaved than standard spectral subtraction. The first term is always real because the square root is taken of the sum of two positive numbers. Furthermore, the
magnitude of the second term is never larger than the magnitude of the first term, so both sides of Eq. 8 are non-negative. The entire formula has exactly one zero, at n=y. This automatically
prevents taking the logarithm of any negative numbers during spectral subtraction, allowing the maximum attenuation floor F to be relaxed.
When σ[α] ^2=0 and Eq. 8 is solved for x, the result is a new spectral subtraction equation with an unexpected absolute value operation. EQ. 9
{circumflex over (x)}=y+ln|1−e ^n−y|
The difference between Eq. 1 and Eq. 9 is confined to the region y<n, as illustrated in FIGS. 4A and 4B.
Spectral subtraction assumes any observation below y=n is equivalent to a signal to noise ratio of the floor F, and produces maximum attenuation.
The data illustrated in FIG. 3B contradicts this assumption, showing a non-zero probability mass in the region n>y. The end result is that, even with perfect knowledge of the true noise, spectral
subtraction treats these points inappropriately.
Eq. 9 has more reasonable behavior in this region. As the observation becomes much lower than the noise estimate, the function approaches x=n. The new model indicates the most likely state is that x
and n have similar magnitudes and axe experiencing destructive phase interference.
Table II compares the relative accuracy of using Equations 1 and 9 for speech recognition, when the true noise spectra are available. Although the new method does not require a floor to prevent
taking the logarithm of a negative number, it is included because it does yield a small improvement in error rate.
TABLE II
Method e^−20 e^−10 e^−5 e^−3 e^−2
Standard 87.50 56.00 34.54 11.31 15.56
(Eq. 1)
Proposed 6.43 5.74 4.10 7.82 10.00
(Eq. 9)
Regardless of the value chosen for the floor, the new method outperforms the old spectral subtraction rule. Although the old method is quite sensitive to the value chosen, the new method is not,
producing less than 10% digit error rate for all tests.
In one embodiment, noisy-speech frames are processed independently of each other. A sequential tracker for estimating the log spectrum of non-stationary noise can be used to provide a noise estimate
on a frame-by-frame basis. A suitable noise estimator is described in METHOD OF ITERATIVE NOISE ESTIMATION IN A RECURSIVE FRAMEWORK Ser. No. 10/237,162, filed on even date herewith.
IV. A Bayesian Approach
Another advantage of deriving the conditional observation probability, Eq. 7A, is that it can be embedded into a unified Bayesian model. In this model, the observed variable y is related to the
hidden variables, including x and n through a unified probabilistic model.
p(y, x, n)=p(y|x, n)p [x](x)p [n](n)
From this model, one can infer posterior distributions on the hidden variables x and n, including MMSE (minimum mean square error) and maximum likelihood estimates. In this way, noisy observations
are turned into probability distributions over the hidden clean speech signal.
To produce noise-removed features for conventional decoding, conditional expectations of this model are taken.
$E [ x | y ] = ∫ - ∞ ∞ x p ( x | y ) ⅆ x , where EQ . 10 p ( x | y ) = ∫ - ∞ ∞ p ( y , x , n ) ⅆ n ∫ - ∞ ∞ ∫ - ∞ ∞ p ( y , x , n ) ⅆ x ⅆ n EQ . 11$
The Bayesian approach can additionally produce a variance of its estimate of E[x|y]. This variance can be easily leveraged within the decoder to improve word accuracy. A suitable decoding technique
is described in METHOD OF PATTERN RECOGNITION USING NOISE REDUCTION UNCERTAINTY, filed May 20, 2002 and assigned Ser. No. 10/152,127. In this form of uncertainty decoding, the static feature stream
is replaced with an estimate of p(y|x). The noise removal process outputs high variance for low SNR features, and low variance when the SNR is high. To support this framework, the following are also
$E [ x 2 | y ] = ∫ - ∞ ∞ x 2 p ( x | y ) ⅆ x , and EQ . 12 p ( y ) = ∫ - ∞ ∞ ∫ - ∞ ∞ p ( y , x , n ) ⅆ x ⅆ n . EQ . 13$
Better results are achieved with a stronger prior distribution for clean speech, such as a mixture model.
$p x ( x ) = ∑ m p x ( x | m ) p m ( m ) .$
When a mixture model is used, Equations 10, 11, 12, and 13 are conditioned on the mixture m, evaluated, and then combined in the standard way:
$p ( y ) = ∑ m p ( y | m ) p ( m ) E [ x | y ] = ∑ m E [ x | y , m ] p ( m | y ) p ( m | y ) = p ( y | m ) p ( m ) p ( y ) E [ x 2 | y ] = ∑ m E [ x 2 | y , m ]
p ( m | y )$
Approximating the Observation Likelihood
As mentioned previously, the form derived for p(y|x, n) does not lend itself to direct algebraic manipulation. Furthermore, it is capable of producing joint distributions that are not well modeled by
a Gaussian approximation. As a result, some steps are performed to compute the necessary integrations for noise removal.
When computation is less of an issue, a much finer non-iterative approximation of p(y|x, n) can be used. The approximation preserves the global shape of the conditional observation probability so
that the usefulness of the model is not masked by the approximation.
One perfectly reasonable, although computationally intensive, option is to make no approximation of Eq. 7A. The joint probability p(y, x, n) can be evaluated along a grid of points in x and n for
each observation y. Weighted sums of these values could produce accurate approximations to all of the necessary moments. Selecting an appropriate region is a non-trivial task, because the region is
dependent on the current observation and the noise and speech priors. More specifically, to avoid unnecessary computation, the evaluation should be limited to the region where the joint probability
has the most mass. Essentially, a circular paradox is realized where it is necessary to solve the problem before choosing appropriate parameters for a solution.
Another reasonable approach is to approximate the joint probability with a single Gaussian. This is the central idea in vector Taylor series (VTS) approximation. Because the prior distributions on x
and n limit the scope of p(y|x, n), this local approximation may be more appropriate than a global approximation. However, there are two potential pitfalls associated with this method. First, even
though the prior distributions are unimodal, applying p(y|x, n) can introduce more modes to the joint probability. Second, the quadratic expansions along x and n do not capture the shape of p(y|x, n)
well when n<<y or x<<y.
Instead, one aspect of the present inventions is a compromise between these two methods. In particular, a Gaussian approximation is used to avoid summation over a two-dimensional grid, while at the
same time preserving the true shape of p(y|x, n). This is accomplished by collapsing one dimension with a Gaussian approximation, and implementing a brute force summation along the remaining
Normal Approximation to Likelihood
In this aspect of the present invention, a Gaussian approximation is used along one dimension only, which allows preservation of the true shape of p(y|x, n), and allows a numerical integration along
the remaining dimension.
The weighted Gaussian approximation is found in four steps illustrated in FIG. 5 at 250. The coordinate space is first rotated in step 252. An expansion point is chosen in step 254. A second order
Taylor series approximation is then made in step 256. The approximation is then expressed as the parameters of a weighted Gaussian distribution in step 258.
The coordinate rotation is necessary because expanding along x or n directly can be problematic. A 45 degree rotation is used, which makes p(y|x,n) approximately Gaussian along u for each value of v.
$u ( y , x , n ) = 1 2 ( x + n - 2 y ) , v ( y , x , n ) = 1 2 ( x - n )$
Although the new coordinates u and v are linear functions of y, x and n, the cumbersome functional notation at this point can be dropped.
After this change of variables, the conditional observation likelihood becomes,
$ln p ( y | x , n ) = - 1 2 u - 1 2 ln 8 π σ α 2 - ( 1 - ⅇ u 2 ( ⅇ v 2 - ⅇ - v 2 ) ) 2 8 σ α 2 exp ( 2 u ) .$
Next, v is held constant and a weighted Taylor series approximation along u is determined. For each v, the Taylor series expansion point is found by performing the change of variables on Eq. 3,
holding v constant, α=O, and solving for u. The
u [v] =v−√{square root over (2)} ln(1+exp√{square root over (2)}v).
result is,
The coefficients of the expansion are the derivatives of p(y|x,n) evaluated at u[v].
$p ( y | x , n ) | u = u v = ln ( 1 + cosh 2 v ) - ln 4 π σ α 2 2$
$ⅆ ⅆ u ln p ( y | x , n ) | u = u v = - 1 2 2 ⅆ 2 ⅆ u 2 ln p ( y | x , n ) | u = u v = - 1 + cosh 2 v 4 σ α 2$
The quadratic approximation of p(y|x,n) at each value of v can then be expressed as a Gaussian distribution along u. Our final approximation is given by:
$p ( y | x , n ) = ⅇ K v N ( u ; μ v , σ v 2 ) , where σ v 2 = 4 σ α 2 1 + cosh 2 v , μ v = u v - 1 2 σ v 2 , and K v = 1 2 ln 2 + σ α 2 1 + cosh 2 v . EQ . 15$
As FIG. 3C illustrates, this final approximation is quite good at capturing the shape of p(y|x,n). And, as discussed below, the Gaussian approximation along u can be leveraged to eliminate a
significant amount of computation. Building the joint probability
The approximation for p(y|x,n) is complete, and is now combined with the priors p[x](x) and p[n](n) to produce the joint probability distribution. To conform to the approximation of the conditional
observation probability, these prior distributions to the (u, v) coordinate space are transformed, and written as a Gaussian in u whose mean and variance are functions of v. EQ. 16
p [x,n](x, n)=p[x](x)p [n](n)=N(u; η [v], γ[v] ^2).
From the joint probability, Equations 10, 11, 12, and 13 are computed. Each equation requires at least one double integral over x and n, which is equivalent to a double integral over u and v. For
$∫ - ∞ ∞ ∫ - ∞ ∞ xp ( y , x , n ) ⅆ x ⅆ n ∫ - ∞ ∞ ∫ - ∞ ∞ ( u + υ 2 + y ) N ( u ; μ υ , σ υ 2 ) N ( u ; η υ , γ υ 2 ) ⅆ u ⅆ ∫ - ∞ ∞ ⅇ K . υ ( μ ^ υ + υ 2 + y ) N
( μ υ ; η υ , σ υ 2 + γ υ 2 ) ⅆ υ , where μ ^ υ = σ υ 2 η υ + γ υ 2 μ υ σ υ 2 + γ υ 2 . EQ . 17$
Here, the method makes use of Eq. 14 for x, as well as Eq. 15 and Eq. 16 for p(y,x,n). The Gaussian approximation enables a symbolic evaluation of the integral over u, but the integral over v
The integration in v is currently implemented as a numerical integration, a weighed sum along discrete values of v. In one embodiment, 500 equally spaced points in the range [−20, 20] are used. Most
of the necessary values can be pre-computed and tabulated to speed up computation.
MMSE Estimator Based on Taylor Series Expansion
The foregoing has described a new spectral subtraction formula (Eq. 9) and a numerical integration, after rotation of axis, to compute the MMSE estimate for speech removal. The following provides a
further technique, in particular, an iterative Taylor series expansion to compute the MMSE (minimum mean square error) estimate in an analytical form in order to remove noise using the
phase-sensitive model of the acoustic environment described above.
Given the log-domain noisy speech observation y, the MMSE estimator {circumflex over (χ)} for clean speech χ is the conditional expectation:
$x ^ = E [ x | y ] = ∫ xp ( x | y ) ⅆ x = ∫ xp n _ ( y | x ) p ( x ) ⅆ x p ( y ) , EQ . 18$
where p[{overscore (n)}](y|x)=p(y|x, {overscore (n)}) is determined by the probabilistic environment model just presented. The prior model for clean speech, p(x) in Eq. 18 is assumed to have the
Gaussian mixture PDF:
$p ( x ) = ∑ m = 1 M c m N ( x ; μ m , σ m 2 ) ︸ p ( x | m ) , EQ . 19$
whose parameters are pre-trained from the log-domain clean speech data. This allows Eq. 18 to be written as
$x ^ = ∑ m = 1 M c m ∫ x p ( x | m ) p ( y | x , n _ ) ︷ J m ( x ) ⅆ x p ( y ) , EQ . 20$
The main difficulty in computing {circumflex over (χ)} above is the non-Gaussian nature of p(y|x, {overscore (n)}). To overcome this difficulty, a truncated second-order Taylor series expansion is
used to approximate the exponent of
$J m ( x ) = N ( x ; μ m , σ m 2 ) × N ( α ( x , n _ , y ) ; 0 , σ α 2 ) 2 ⅇ n _ + x 2 - y = C σ m ⅇ - 0.5 ( x - μ m ) 2 / σ m 2 - 0.5 x - 0.5 α 2 ( x ) / σ α 2 . EQ . 21$
That is, the following function is approximated
b [m](x)=−0.5(x−μm)^2/σ[m] ^2−0.5x−0.5α^2(x)/σ[α] ^2
$b m ( x ) ≈ b m ( 0 ) ( x 0 ) + b m ( 1 ) ( x 0 ) ( x - x 0 ) + b m ( 2 ) ( x 0 ) 2 ( x - x 0 ) 2 . EQ . 22$
In Eq. 22, a single-point expansion point χ[0 ]is used (i.e., x[0 ]does not depend on the mixture component m) to provide significantly improved computational efficiency, and χ[0 ]is iteratively
updated to increase its accuracy to the true value of clean speech x. The Taylor series expansion coefficients have the following closed forms:
$b m ( 0 ) ( x 0 ) = b m ( x ) | x = x 0 = - ( x 0 - μ m ) 2 2 σ m 2 - x 0 2 - ( ⅇ y - ⅇ n _ - ⅇ x 0 ) 2 8 σ α 2 ⅇ n _ + x 0 , b m ( 1 ) ( x 0 ) = ∂ b m ( x ) ∂ x | x = x 0 =
- x 0 - μ m σ m 2 - 1 2 + ⅇ 2 y - n _ - x 0 - 2 ⅇ y - x 0 + ⅇ n _ - x 0 - ⅇ x 0 - n _ 8 σ α 2 , b m ( 2 ) ( x 0 ) = ∂ 2 b m ( x ) ∂ 2 x | x = x 0 = - 1 σ m 2 + - ⅇ 2 y - n _ -
x 0 + 2 ⅇ y - x 0 - ⅇ n _ - x 0 - ⅇ x 0 - n _ 8 σ α 2 .$
It should be noted Σ[α]=σ^α ^2. In other words, a zero-mean Gaussian distribution is used for the phase factor α in the new model described above, and which is used in this embodiment.
Fitting Eq. 22 into a standard quadratic form, the following is obtained
$b m ( x ) ≈ b m ( 2 ) ( x 0 ) 2 [ x - ( x 0 - b m ( 1 ) ( x 0 ) b m ( 2 ) ( x 0 ) ) ] 2 + w m ( x 0 ) , where w m ( x 0 ) = b m ( 0 ) ( x 0 ) + b m ( 2 ) ( x 0 ) 2 [ x 0 2 - 2
b m ( 1 ) b m ( 2 ) x 0 - ( x 0 - b m ( 1 ) b m ( 2 ) ) 2 ] .$
This then allows computing the integral of Eq. 20 in a closed form:
$I m ( x 0 ) = ∫ x J m ⅆ x = C σ m ∫ x ⅇ b m ( x ) ⅆ x ≈ C ′ σ m b m ( 2 ) ⅇ w m ( x 0 ) × ( x 0 - b m ( 1 ) ( x 0 ) b m ( 2 ) ( x 0 ) ) . EQ . 23$
The denominator of Eq. 20 is computed according to
$p ( y ) = ∑ m = 1 M c m ∫ J m ( x ) ⅆ x = ∑ m = 1 M c m C σ m ∫ ⅇ b m ( x ) ⅆ x ≈ ∑ m = 1 M c m C ′ σ m b m ( 2 ) ⅇ w m ( x 0 ) . EQ . 24$
Substituting Eqs. 23 and 24 into Eq. 20, the final MMSE estimator is obtained:
$x ^ ≈ ∑ m = 1 M γ m ( x 0 , n _ ) ( x 0 - b m ( 1 ) ( x 0 ) b m ( 2 ) ( x 0 ) ) , EQ . 25$
Where the weighting factors are
$γ m ( x 0 , n _ ) = c m σ m b m ( 2 ) ⅇ w m ( x 0 ) ∑ m = 1 M c m σ m b m ( 2 ) ⅇ w m ( x 0 ) .$
Note that γ[m], b[m] ^(1) (χ[0]), and b[m] ^(2) (χ[0]) in Eq. 24 are all dependent on the noise estimator {overscore (n)}, which can be obtained from any suitable noise tracking estimator such as
described in the co-pending application referenced above.
Under this aspect of the present invention, the clean speech estimate of the current frame, x[t+1], is calculated several times using an iterative method shown in the flow diagram of FIG. 6.
The method of FIG. 6 begins at step 300 where the distribution parameters for the prior clean speech mixture model are pretrained from a set of clean training data. In particular, the mean, μ[m],
covariance, σ[m], and mixture weight, c[m], for each mixture component m in a set of M mixture components is determined.
At step 302, the expansion point, x[0] ^j, used in the Taylor series approximation for the current iteration, j, can be set equal to the mean vector of the Gaussian mixture model of the clean speech
that best accounts for (in the maximum likelihood sense) the noisy speech observation vector y given the estimated noise vector n.
At step 304, the MMSE estimator for clean speech is calculated according to Eq. 25.
At step 306, the Taylor series expansion point for the next iteration, x[0] ^j+1, is set equal to the noise estimate found for the current iteration, x[t+1] ^j. In terms of an equation:
x [0] ^j+1 =x [t+1] ^jEQ. 26
The updating step shown in Eq. 26 improves the estimate provided by the Taylor series expansion and thus improves the calculation during the next iteration.
At step 308, the iteration counter j is incremented before being compared to a set number of iterations J at step 310. If the iteration counter is less than the set number of iterations, more
iterations are to be performed and the process returns to step 304 to repeat steps 304, 306, 308 and 310 using the newly updated expansion point.
After J iterations have been performed at step 310, the final value for the clean speech estimate of the current frame has been determined and at step 312, the variables for the next frame are set.
In one embodiment, J is set equal to three. Specifically, the iteration counter j is set to zero, the frame value t is incremented by one, and the expansion point χ[0 ]for the first iteration of the
next frame is set.
A method and system for using the present invention in speech recognition is shown the block diagram of FIG. 7. The method begins where a noisy speech signal is converted into a sequence of feature
vectors. To do this, a microphone 404 of FIG. 7, converts audio waves from a speaker 400 and one or more additive noise sources 402 into electrical signals. The electrical signals are then sampled by
an analog-to-digital converter 406 to generate a sequence of digital values, which are grouped into frames of values by a frame constructor module 408. In one embodiment, A-to-D converter 406 samples
the analog signal at 16 kHz and 16 bits per sample, thereby creating 32 kilobytes of speech data per second and frame constructor module 408 creates a new frame every 10 milliseconds that includes 25
milliseconds worth of data.
Each frame of data provided by frame constructor module 408 is converted into a feature vector by a feature extractor 410. Methods for identifying such feature vectors are well known in the art and
include 13-dimensional Mel-Frequency Cepstrum Coefficients (MFCC) extraction.
The feature vectors for the noisy speech signal are provided to a noise estimation module 411 in FIG. 7. Noise estimation module 411 estimates the noise in the current frame and provides a feature
vector representing the noise estimate, or a distribution thereof, together with the noisy speech signal to a noise reduction module 412.
The noise reduction module 412 uses any one of the techniques described above, (new spectral subtraction of Eq. 9, the Bayesian approach with weighted Gaussian Approximation, or an MMSE estimator)
with model parameters of the corresponding implementing equations, which are stored in noise reduction parameter storage 411, to produce a sequence of noise-reduced feature vectors from the sequence
of noisy feature vectors, or distributions thereof.
The output of noise reduction module 412 is a series of noise-reduced feature vectors. If the input signal is a training signal, this series of noise-reduced feature vectors is provided to a trainer
424, which uses the noise-reduced feature vectors and a training text 426 to train an acoustic model 418. Techniques for training such models are known in the art and a description of them is not
required for an understanding of the present invention.
If the input signal is a test signal, the noise-reduced feature vectors are provided to a decoder 414, which identifies a most likely sequence of words based on the stream of feature vectors, a
lexicon 415, a language model 416, and the acoustic model 418. The particular method used for decoding is not important to the present invention and any of several known methods for decoding may be
The most probable sequence of hypothesis words is provided to a confidence measure module 420. Confidence measure module 420 identifies which words are most likely to have been improperly identified
by the speech recognizer, based in part on a secondary acoustic model(not shown). Confidence measure module 420 then provides the sequence of hypothesis words to an output module 422 along with
identifiers indicating which words may have been improperly identified. Those skilled in the art will recognize that confidence measure module 420 is not necessary for the practice of the present
Although FIG. 7 depicts a speech recognition system, the present invention may be used in other noise removal applications such as removing noise of recordings, or prior to transmission of data in
order to transmit cleaner data. In this manner, the pattern recognition system is also not limited to speech.
Although the present invention has been described with reference to particular embodiments, workers skilled in the art will recognize that changes may be made in form and detail without departing
from the spirit and scope of the invention. | {"url":"http://www.google.com.au/patents/US7047047","timestamp":"2014-04-17T22:00:34Z","content_type":null,"content_length":"207774","record_id":"<urn:uuid:9bbac8b1-69df-4780-a797-140545b18009>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00364-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: Analysis books?
Replies: 7 Last Post: Nov 6, 2005 1:51 AM
Messages: [ Previous | Next ]
Re: Analysis books?
Posted: Nov 5, 2005 11:17 PM
Colleyville Alan wrote:
> I will be taking Calculus I, II, and III with the
> first class beginning this coming January. My goal
> is to study enough analysis (concurrent with the
> Calculus courses) that when I complete Calculus III,
> I can enroll in an analysis class that while still
> undergraduate, would be an upper-division, rather
> than an introductory, class.
> I've seen in the sci.math archives Rudin recommended
> for graduate-level work and some of the authors of the
> Dover books (Kolmogorov, Shilov, Rosenlicht) recommended
> for undergraduate, but this is self-study only for
> the purpose of avoiding a lower-division class and
> having an instructor for the upper-division stuff.
> Given that, what would you recommend?
I'll assume you're talking about a typical U.S.
Calculus I, II, and III sequence.
It sounds like you simply want to omit the now common
sophomore/junior level "transition to advanced mathematics"
course. This should take far less work than it sounds
like you're planning. For one thing, this type of course
only developed within the past 30 years or so. (I think
partly in response to pushing higher levels of abstract
algebra and other pure mathematics courses into the
undergraduate curriculum in the 1960's, and partly
because of the increase in the percentage of the U.S.
population that began attending college in the late
1960's and the 1970's.) In some colleges, this type
of course serves a dual role as a discrete math course
(including combinatorics and mathematical induction)
for computer science majors. Thus, this course is
probably not even necessary for a reasonably good
mathematics student
If your college offers honors versions of the elementary
calculus sequence and you're able to get into the honors
sections, then you can probably skip the elementary
analysis course without missing much. If this isn't
possible for some reason, here are three texts that
you should try to get a hold of and read through while
you're taking the calculus sequence:
Michael Spivak, "Calculus", 3'rd edition, 1994.
Tom Apostol, "Calculus", Volume 1, 2'nd edition, 1967.
Richard Courant and Fritz John, "Introduction to Calculus
and Analysis", Volume 1, 1965. [This has been recently
reprinted, I believe.]
Spivak and Apostol are widely used in honors classes
and at places like CalTech, MIT, Univ. of Toronto, etc.
Courant/John is a classic that is in some ways pitched
at an even higher level (the problems are harder, for one
thing), but it gives an outstanding coverage of *everything*,
from theoretical to applied, that someone planning to
continue studying mathematics ideally should be exposed
to in an elementary calculus course.
Incidentally, Apostol and Courant/John each have a 2'nd
volume covering multivariable calculus, but I think for
what you're looking for the 1'st volumes will suffice.
Finally, for what it's worth, the best (very) elementary
real analysis text I know of for someone looking for
something to read along with a traditional calculus text
(not the three texts above, but the sort of text U.S.
colleges usually use for their calculus sequence) is
Victor Bryant, "Yet Another Introduction to Analysis",
Dave L. Renfro | {"url":"http://mathforum.org/kb/message.jspa?messageID=4071934","timestamp":"2014-04-19T15:28:54Z","content_type":null,"content_length":"27423","record_id":"<urn:uuid:4809eb49-c7bc-4176-8405-c051baece4dc>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00538-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] Not importing polynomial implementation functions by default
Charles R Harris charlesr.harris@gmail....
Sun Mar 13 05:49:02 CDT 2011
On Sat, Mar 12, 2011 at 10:57 PM, David Warde-Farley <
wardefar@iro.umontreal.ca> wrote:
> On 2011-03-12, at 9:32 PM, Charles R Harris wrote:
> > I'd like to change the polynomial package to only import the Classes,
> leaving the large number of implementation functions to be imported directly
> from the different modules if needed. I always regarded those functions as
> implementation helpers and kept them separate from the class so that others
> could use them to build their own classes if they desired. For most purposes
> I think the classes are more useful. So I think it was a mistake to import
> the functions by; default and I'm looking for a graceful and acceptable way
> out. Any suggestions.
> I hope this wouldn't include polyfit, polyval, roots and vander at least
> (I'd also be -1 on removing poly{add,sub,mul,div,der,int}, but more weakly
> so). Those 4 seem useful and "basic" enough to leave in the default
> namespace.
I use *vander a lot myself, so I agree with you there. The rest can all be
reached within the Polynomial class. Maybe I should add the *vander stuff to
the classes also. But I guess I'm looking for feedback from people who might
be using these functions.
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CADDIS Volume 4: Data Analysis
Basic Principles & Issues
Interpreting Statistics
A complex ecological dataset is likely to exhibit features that can be attributed to variability of data and do not represent stable properties of a system under study. Basic statistical inference
procedures (tests and confidence intervals) can help the analyst identify conclusions that are relatively well-supported and associated with acceptable risks of error, taking into account the
quantity and variability of the data.
In causal analysis, significance tests and confidence intervals are primarily used to establish whether biological or environmental conditions at a test site differ from expectations, and to help
interpret estimates of stressor-response relationships from larger, regional datasets. Other examples of where such inference procedures might be used include:
• Evaluating whether the form of a stressor-response relationship is consistent among regions, study years, or species.
• Determining whether one can account for the biological effects of a land-use variable with measurements of proximate stressors.
A confidence interval provides a range for a mean or other parameter that can be viewed as in reasonable agreement with data, taking into account the quantity and variability of data (Figure 1).
Statistical tests can be used to avoid any temptation to over-interpret noisy data, by focusing the analysis on effects that are are unlikely to be attributed to variation in the data.
Confidence intervals and statistical tests may be useful in analyzing data to support a causal analysis. However, scoring of evidence in a causal assessment is based on multiple lines of evidence,
taking into account statistical and biological considerations. Scoring of evidence is not based mechanically on results of statistical tests or any other statistical procedures.
More information about statistical testing and confidence intervals is available here. | {"url":"http://www.epa.gov/caddis/da_principles_1.html","timestamp":"2014-04-20T14:01:37Z","content_type":null,"content_length":"13029","record_id":"<urn:uuid:d1fc09d7-674b-4c35-9cda-2bca31448f26>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00132-ip-10-147-4-33.ec2.internal.warc.gz"} |
The 'magic formula' for linearly edge-reinforced random walks
Seminar Room 1, Newton Institute
Linearly edge-reinforced random walk on a finite graph is a mixture of reversible Markov chains with an explicitly known mixing measure. We give a new proof of this fact. (Joint work with Silke
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible. | {"url":"http://www.newton.ac.uk/programmes/MPA/seminars/2008121610001.html","timestamp":"2014-04-19T17:09:56Z","content_type":null,"content_length":"5988","record_id":"<urn:uuid:e42ff494-cf79-43ba-901a-cb0da3b68612>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00519-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> How to test correlation
yuming ning posted on Thursday, November 13, 2008 - 10:25 am
i have a simple question for coding
my hypothesized model is
hence my codes will be
E on A B C D;
D on A B;
but how do i test A<->B (ie, cov(A,B))?
i tried adding (A with B;) into above codes, but the result gave me estimates for all of cov(A,B), cov(A,C), and cov(B,C). i only want cov(A,B) and assume
Linda K. Muthen posted on Thursday, November 13, 2008 - 11:00 am
In regression, the means, variances, and covariances of the covariates are not parameters in the model. The model is estimated conditioned on the covariates. You should not mention a WITH b. If you
want to know the correlation between a and b, see the sample statistics.
Anna posted on Tuesday, June 15, 2010 - 7:08 am
I have a question concerning the test of correlations. How do I test if correlations between two latent factors (e.g. between level and slope) are significantly different between groups or time
points? Is this possible in Mplus?
Thank you very much for your help!
Linda K. Muthen posted on Tuesday, June 15, 2010 - 9:24 am
You can test if the covariances between two growth factors are the same for two groups using either MODEL TEST of chi-square or loglikelihood difference testing.
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