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jfreechart plotting line chart on candlestick chart
10-07-2012, 01:18 AM
jfreechart plotting line chart on candlestick chart
Hi everyone, new to this forum.
Im learning about plotting graphs in Java, and have had success plotting a candlestick graph in jfreecharts. I have now plotted a line graph on top of that. The problem is my line graph is
currently plotting from the same dataSet as my candlestick chart and is plotting the closing points. I want to be able to decide which points of the OHLCDataItem[] to plot. I.e I may want to plot
the "open" variable as a line chart or whatever. Also, OHLCDataItem OHLCDataItem[] want four data points, whereas for my line graph, I just want the date, and another variable of my choice.
These are the lines that are relevant:
List<OHLCDataItem> data = getData(stockSymbol);
OHLCDataItem[] dataItems = data.toArray(new OHLCDataItem[data.size()]);
XYDataset generalDataSet = new DefaultOHLCDataset(stockSymbol, dataItems);
XYPlot mainPlot = new XYPlot(generalDataSet, domainAxis, rangeAxis, candleStickRenderer);
mainPlot.setDataset(1, generalDataSet);
mainPlot.setRenderer(1, LineRenderer);
so the data comes into data, each item containing a date followed by five doubles.
generalDataSet turns that into data that the candlestick graph plots. mainPlot plots that data on the graph, then the next two lines change the renderer to the line graph plotter, but still using
the same data set, and this is where Im no sure how to select which of the five doubles to plot against the date.
Any one done this?
Any examples would be greatly appreciated. | {"url":"http://www.java-forums.org/new-java/63619-jfreechart-plotting-line-chart-candlestick-chart-print.html","timestamp":"2014-04-19T04:22:21Z","content_type":null,"content_length":"5878","record_id":"<urn:uuid:f02856da-7537-4ef5-af66-92cd4f5d659c>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00028-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> EFA in CFA Framework using Dichotomous Variables
EmilyLeckman posted on Monday, March 27, 2000 - 7:22 am
I heard at your short course (day 1) of using the CFA framework to conduct an EFA. I was hoping that you could help me w/ the correct restrictions to place on the model to be able to work it, if it
is even possible w/ dichotomous variables.
bmuthen@ucla.edu posted on Monday, March 27, 2000 - 8:22 am
EFA within CFA works the same for categorical outcomes as for the continuous case we discussed in our Mplus short course.
Yuan H Li posted on Thursday, August 02, 2001 - 8:13 am
ear Dr. Muthen:
I am working on the multidimensional IRT (MIRT) data modeling. Conceptually, MIRT is the same as the Factor analysis with categories variables. Item discriminations in MIRT is similar to factor
loadings and item difficulty in MIRT is similar to the item mean in FA analysis. I am wondering how many categories variables MPLUS can handle under the confirmatory FA with mean estimates? Thanks.
Yuan H LI
Bengt O. Muthen posted on Thursday, August 02, 2001 - 9:04 am
Yes, MIRT can be seen as factor analysis with categorical observed variables in Mplus. Item difficulty is related to the threshold parameter. Mplus allows for a maximum of 10 categories for ordered
polytomous variables, that is 9 thresholds.
Anonymous posted on Sunday, May 01, 2005 - 10:33 pm
Can you explain " what do u mean by EFA within CFA frame work?" and how to do it? Actually I read this help to get the factor scores.
Linda K. Muthen posted on Monday, May 02, 2005 - 6:16 am
EFA within a CFA framework is a CFA model where an EFA model is specified. Restrictions equal to the number of factors squared must be specified to identify the model. For example, in a 3 factor
model, nine restrictions must be placed. Three of them can be achieved by fixing the factor variances to one. The other 6 are achieved by fixing factor loadings to zero. How to do this is described
in the Day 1 course handout.
Anonymous posted on Monday, June 13, 2005 - 10:35 am
Hi Linda,
Can you give an intuitive explnation for EFA within CFA framework? I have hard time understanding the why we are doing this and what we can gain from it intuitively.
Thank you.
Linda K. Muthen posted on Tuesday, June 14, 2005 - 9:09 am
It is setting up an EFA model in the CFA framework. This gives you the advantage of seeing standard errors for the factor loadings and also seeing modification indices including those for correlated
Anonymous posted on Wednesday, June 22, 2005 - 9:12 pm
Hello, is my understanding from Bengt's UCLA web lecture correct? Thank you.
For an efa in a cfa framework, the appropriate K^2 restrictions (for oblique and orthogonal solutions) are for identification purposes, achieve the least restrictve K-factor CFA models possible, and
the models are therefore considered unrestricted by Bengt, and Karl Joreskog (1969). These models will have the same chi-sq and df as ML EFA solutions. If the K^2 restrictions are distributed
appropriately, the solutions are also unique. When models have > K^2 restrictions they are considered restricted.
BMuthen posted on Thursday, June 23, 2005 - 3:44 am
All of what you say is correct.
Cintia posted on Friday, March 30, 2007 - 6:05 pm
I am analyzing the structure of an established questionnaire. It has 31 dichotomous items. I first did an EFA (ULS estimator) that yielded a five-factor solution. Then I wanted to study the
correlation between these factors and other variables such as gender, age, etc. So I did an EFA in CFA framework to find the factor scores but I got the message:
IN A TOTAL OF 0.75938E+06 INTEGRATION POINTS.
Would you have any suggestions as to how I should proceed?
Thank you.
Linda K. Muthen posted on Saturday, March 31, 2007 - 5:50 am
It sounds like you are using the maximum likelihood estimator. You can try INTEGRATION = MONTECARLO; or you can switch to the WLSMV estimator.
Cintia posted on Sunday, April 01, 2007 - 8:59 am
Thank you for the suggestions. I used WLSMV as the estimator and received the message:
With INTEGRATION = MONTECARLO, the output stated:
Could you please explain what might be going on? Would there be other alternatives to obtain the factor scores?
Linda K. Muthen posted on Sunday, April 01, 2007 - 9:19 am
You have a negative residual variance which makes the model inadmissible. You need to change your model. If this does not help, you need to send your input, data, output, and license number to
Xuan Huang posted on Friday, July 06, 2007 - 1:34 pm
Dear Professors, Could you help me with the syntax of an EFA within CFA model?I am running factor analyses on 9 acculturation items. The EFA reveals one factor. I followed the Mplus handout 1 that
was ordered from Mplus website to set up an EFA within CFA model. Because I am new to Mplus, I am not so sure whether my syntax is right.
TITLE: acculturation
data: file is 'acculturationw1.dat';
format is (F4.0, 09F10.2);
names are id facclt01 facclt03 facclt05 facclt09 facclt11 facclt13
facclt15 facclt17 facclt19;
MISSING = blank;
USEVARIABLES ARE facclt01-facclt19;
analysis: estimator = ml;
model: acc BY facclt01-facclt19*0
output: standardized modindices(3.84) sampstat fsdeterminacy;
Could you take a look to see whether the syntax is OK? Thank you so much for your time and help!
Linda K. Muthen posted on Friday, July 06, 2007 - 1:46 pm
EFA in a CFA framework is not needed for only one factor.
Ken Cor posted on Thursday, May 07, 2009 - 4:22 pm
I've successfully run an EFA in CFA using your workshop framework but am unsure if I can trust the results. The following is a portion of my input code:
NAMES ARE q1-q14;
CATEGORICAL ARE q1-q14;
ESTIMATOR = ML;
f1 by q1-q14*0 q2*1;
f2 by q1-q14*0 q12*1 q2@0 q4@0 q6@0;
I get the following message in my output:
Pameter 25 is the factor loading of factor 2 on item 14 (Lambda 2,14)
I also get some hard to believe model fit results:
Pearson Chi-Square
Value 9742.828
Degrees of Freedom 16322
P-Value 1.0000
One final note, I've tried running this with the WLSMV estimator and I get the message that standard errors could not be calculated because my model is not identified.
Any insights on how I've gone wrong and ways I can correct the problem would be much appreciated.
Thank you,
Ken Cor
Bengt O. Muthen posted on Thursday, May 07, 2009 - 5:39 pm
With the introduction of "ESEM", the new Exploratory Structural Equation Modeling approach in Mplus, you no longer need to do "EFA in a CFA framework". See the Mplus home page about ESEM and its
User's Guide example addendum.
One possible reason for your problem is that you might need more than one item per factor to have a good starting value of 1.
The Pearson chi-2 is not trustworthy with 14 categorical items because of too many empty cells. Around 8 is probably the limit.
Cengiz Erisen posted on Saturday, August 20, 2011 - 4:58 pm
I am having difficulty in making the better fit decision between one- and two-factor models (on binary data) thru chi-square tests.
Here is what I did: I conducted CFA models (on both models in line with Mplus directions) with DIFFTEST and received a significant chi-square difference indicating that constraining the parameters
(two factor model) worsens the fit of the model.
Here is where I have the problem: Brown (2006) on CFA suggests a different conclusion for the comparison of chi-square values for the nested models. He interprets a significant chi-square difference
as an indicator that the two-factor model fits the data better.
So, which one is correct?
Linda K. Muthen posted on Monday, August 22, 2011 - 7:50 am
Two factors should not worsen fit, please send the two relevant outputs and your license number to registration@statodel.com.
Carmelo Callueng posted on Monday, May 21, 2012 - 7:49 am
Hello Dr. Muthen,
I am examining the factor structure of a measure with 63 items in 23 countries and subsequently, do a test of invariance for countries that meet model fit. The items have 2 options and hence, I used
Mplus CFA with WLSMV as estimator. As an initial step, run a CFA with variance of the latent factors fixed @ 1. In 4 countries, I got this message in the output:
Kindly help me to fix this problem.
Also, In all the 23 countries the CFI and TLI were below .90 but the RMSEA was above .o6. Is there any problem that you can see on this? I would appreciate if you can check one of my CFA output.
Linda K. Muthen posted on Monday, May 21, 2012 - 11:10 am
Please send the outputs with this message and your license number to support@statmodel.com.
It sounds like the model does not fit based on those fit statistics.
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1011 Submissions
[76] viXra:1011.0077 [pdf] replaced on 15 Jan 2011
On "Discovering and Proving that π is Irrational"
Authors: Li Zhou
Comments: 7 pages.
We discuss the logical fallacies in an article appeared in The American Mathematical Monthly [6], and present the historical origin and motivation of the simple proofs of the irrationality of π.
Category: Number Theory
[75] viXra:1011.0076 [pdf] submitted on 30 Nov 2010
Implementing Proper Length on a Graph in R3 Via Polymerization Produces Negligible Entropy When Accounting for the Distinctness of Scattering Probability Distributions
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Comments: 6 pages.
A graph model of general relativity is discussed. The relationship between entropy and the incompressibility of information is highlighted.
Category: Quantum Gravity and String Theory
[74] viXra:1011.0075 [pdf] replaced on 2012-05-03 03:28:42
Is Mass Constant in a Gravitational Field?
Authors: Zhan Likui
Comments: 14 pages, 4 figures, 3586 words, In English
In a black hole, no force can prevent matter from collapsing into a point, a gravitational singularity. The problem of the singularity is always a difficult one. We attempt to discussion gravitation
from another angle in this work. Based on experimental analysis and theoretical verification, a hypothesis regarding mass in gravitational fields is presented. It can avoid the problem of
singularity, and the meaning of gravitation is simple to determine.
Category: Relativity and Cosmology
[73] viXra:1011.0074 [pdf] submitted on 30 Nov 2010
Evolutionary Sequence of Spacetime/intrinsic Spacetime and Associated Sequence of Geometry in a Metric Force Field. Part II.
Authors: Akindele J. Adekugbe
Comments: 20 pages. Submitted to Progress in Physics
Graphical analysis of the geometry of a curved 'three-dimensional' absolute intrinsic metric space, (an absolute intrinsic Riemannian metric space) φM^3, which is curved onto the absolute time/
absolute intrinsic time 'dimensions' (along the vertical), as a curved hyper-surface, and projects a flat three-dimensional proper intrinsic metric space φE'^3 underlying its outward manifestation
namely, the flat proper physical Euclidean 3-space φE'^3, both as flat hyper-surfaces along the horizontal, isolated in part one of this paper, is done. Two absolute intrinsic tensor equations, one
of which is of the divergenceless form of Einstein free-space field equations and the other which is a tensorial statement of local Euclidean invariance on φM^3, are derived. Simultaneous (algebraic)
solution of the equations yields the absolute intrinsic metric tensor and absolute intrinsic Ricci tensor of absolute intrinsic Riemann geometry on the curved absolute intrinsic metric space φM^3, in
terms of an isolated absolute intrinsic curvature parameter. Relations for absolute intrinsic coordinate projections into the underlying flat proper intrinsic space are derived. A superposition
procedure that yields resultant absolute intrinsic metric tensor and resultant absolute intrinsic Ricci tensor, as well as resultant absolute intrinsic coordinate projection relations when two or a
larger number of absolute intrinsic Riemannian metric spaces co-exist, are developed. Finally the fact that a curved 'three-dimensional' absolute intrinsic metric space φM^3 is perfectly isotropic
(that is, all directions are perfectly the same) and is consequently contracted to a 'onedimensional' absolute intrinsic metric space denoted by φρ, which is curved onto the absolute time/absolute
intrinsic time 'dimensions' along the vertical and that the underlying projective three-dimensional flat proper intrinsic metric space φE'^3 is perfectly isotropic and is consequently contracted to a
straight line one-dimensional isotropic proper intrinsic metric space φρ' along the horizontal, with respect to observers in the physical proper Euclidean 3-space φE'^3 that overlies φρ', are
Category: Relativity and Cosmology
[72] viXra:1011.0073 [pdf] submitted on 30 Nov 2010
Evolutionary Sequence of Spacetime/intrinsic Spacetime and Associated Sequence of Geometries in a Metric Force Field. Part I.
Authors: Akindele J. Adekugbe
Comments: 11 pages. Submitted to Progress in Physics
Having isolated a four-world picture in which four symmetrical universes in different spacetime domains coexist and in which an isolated two-dimensional intrinsic spacetime underlies the
four-dimensional spacetime in each universe, and having shown that the special theory of relativity rests on a four-world background elsewhere, we review the geometry of spacetime/intrinsic spacetime
in a long-range metric force field within the four-world picture in the four parts of this paper. We show within an elaborate programme that the four-dimensional metric spacetime and its underlying
two-dimensional intrinsic metric spacetime undergo two stages of evolution in the sequence of absolute spacetime/absolute intrinsic spacetime → proper spacetime/proper intrinsic spacetime →
relativistic spacetime/relativistic intrinsic spacetime in all finite neighborhood of a long-range metric force field and that these are supported by a sequence of spacetime /intrinic spacetime
geometries. The programme takes off in this first paper by isolating two classes of three-dimensional Riemannian metric space namely, the conventional three-dimensional Riemannian metric space and a
new 'three-dimensional' absolute intrinsic Riemannian metric space.
Category: Relativity and Cosmology
[71] viXra:1011.0072 [pdf] submitted on 29 Nov 2010
The Age and Size of the Universe.
Authors: J. Dunning-Davies
Comments: 3 pages.
The possible implications of some reported high-valued red-shifts for both the age and size of the Universe are examined on the basis of presently accepted theory.
Category: Astrophysics
[70] viXra:1011.0071 [pdf] submitted on 29 Nov 2010
Can the Universe be Represented by a Superposition of Spacetime Manifolds?
Authors: Raymond Jensen
Comments: 15 pages, To appear in 2011 SPESIF Proceedings.
In this article it is argued, that the universe cannot be modeled as a space-time manifold. A theorem of geometry provides that null geodesics on a space-time manifold which begin at the same point
with the same initial tangent vector are unique. But in reality, light originating from a single point with a given initial direction does not travel along a unique null geodesic path when a massive
object attracts it, in particular when the massive object is in an indefinite location. Therefore, the universe cannot be described as a space-time manifold. It is then argued that the universe is a
superposition of space-time manifolds, where the manifolds form a Hilbert space over the complex numbers.
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[69] viXra:1011.0070 [pdf] submitted on 29 Nov 2010
Determinants of Population Growth in Rajasthan: An Analysis
Authors: V.V. Singh, Alka Mittal, Neetish Sharma, Florentin Smarandache
Comments: 12 pages
Rajasthan is the biggest State of India and is currently in the second phase of demographic transition and is moving towards the third phase of demographic transition with very slow pace. However,
state's population will continue to grow for a time period. Rajasthan's performance in the social and economic sector has been poor in past. The poor performance is the outcome of poverty, illiteracy
and poor development, which co-exist and reinforce each other. There are many demographic and socio-economic factors responsible for population growth. This paper attempts to identify the demographic
and socio-economic variables, which are responsible for population growth in Rajasthan with the help of multivariate analysis.
Category: Statistics
[68] viXra:1011.0069 [pdf] submitted on 29 Nov 2010
PDEs and Symmetry : an Open Problem
Authors: Elemér E Rosinger
Comments: 7 pages
A simple and basic problem is formulated about symmetric partial differential operators. The symmetries considered here are other than Lie symmetries.
Category: General Mathematics
[67] viXra:1011.0068 [pdf] replaced on 14 Dec 2010
Nonassociative Octonionic Ternary Gauge Field Theories
Authors: Carlos Castro
Comments: 20 pages, submitted to J. Phys. A : Math and Theor.
A novel (to our knowledge) nonassociative and noncommutative octonionic ternary gauge field theory is explicitly constructed that it is based on a ternary-bracket structure involving the octonion
algebra. The ternary bracket was defined earlier by Yamazaki. The field strengths F[μ][ν] are given in terms of the 3-bracket [B[μ],B[ν], Φ] involving an auxiliary octonionic-valued scalar field Φ =
Φ^ae[a] which plays the role of a "coupling" function. In the concluding remarks a list of relevant future investigations are briefly outlined.
Category: Quantum Gravity and String Theory
[66] viXra:1011.0067 [pdf] replaced on 2013-08-27 08:35:09
The Origin of Space and Time
Authors: John A. Gowan
Comments: 12 Pages. adding new section
Fundamentally, the dimensionality of spacetime is a matter of energy conservation: three dimensions are sufficient to establish an entropic domain in which the basic thermodynamic requirements
necessary to conserve the energy of free forms of electromagnetic energy (light, EM radiation, etc.) are present; likewise, four dimensions are necessary to meet the conservation requirements of
bound forms of electromagnetic energy (mass/matter).
Category: Relativity and Cosmology
[65] viXra:1011.0066 [pdf] submitted on 26 Nov 2010
On the Accelerating Universal Expansion
Authors: R. Wayte
Comments: 25 pages
Repulsive gravity at large distances has been included in the universal solution of Einstein's equations by introducing a cosmological constant, which excludes the dark energy interpretation. For an
external-coordinate-observer cosmological model, the big-bang singularity has been replaced by a granular primeval particle, and expansion is controlled by the velocity of light. Then problems
inherent in the standard model do not arise, and no inflation phase is necessary. It is advantageous to truncate the graviton field at a maximum radius, which is related to proton dimensions through
the ratio (e^2/Gm^2). This governs the onset of universal repulsion at around 7Gyr, in rough agreement with observations of Type Ia supernovae.
Category: Relativity and Cosmology
[64] viXra:1011.0065 [pdf] replaced on 2013-06-16 08:31:43
Apertures for Excitation of Algebraically Self-Dual E/M Fields
Authors: T. E. Raptis
Comments: 15 Pages. Corrected Typos in main text
We propose a technique for setting up parallel (self-dual) stationary electromagnetic fields in the context of transformation optics. Several paradoxes that may appear in this regime are discussed. A
particular type of communication based on stationary patterns through Aharonov-Bohm interferometry is also introduced as an alternative to a previous proposal by Putthof.
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[63] viXra:1011.0064 [pdf] replaced on 29 Nov 2010
Why Natural Selection Cannot Explain Biological Evolution
Authors: Stephen P. Smith
Comments: 17 pages
The indifferent process of natural selection has been dubbed "the blind watchmaker" by Richard Dawkins. Arguments against natural selection are presented that relate to both ontology (reason-based)
and epistemology (evidence-based), and the belief that the blind watchmaker drives evolution is revealed to be only a stipulation, at best. The belief is found coming from a metaphysical preference
towards naturalism. A new account of evolution is presented that does not hold naturalism as a preference, and permits teleological (or guided) evolution and vitalism. This new account departs from
the hidden agenda of naturalism, and fully discloses its preference towards self-evidence in its pursuit of truth.
Category: Mind Science
[62] viXra:1011.0063 [pdf] replaced on 23 May 2011
The Intrinsic Motions of Matter
Authors: John A. Gowan
Comments: 10 pages
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[61] viXra:1011.0062 [pdf] submitted on 25 Nov 2010
The Black Hole Catastrophe: A Reply to J. J. Sharples
Authors: Stephen J. Crothers
Comments: 29 pages
A recent Letter to the Editor (Sharples J. J., Coordinate transformations and metric extension: a rebuttal to the relativistic claims of Stephen J. Crothers, Progress in Physics, v.1, 2010) has
analysed a number of my publications in Progress in Physics. There are serious problems with this treatment which should be brought to the attention of the Journal's readership. Dr. Sharples has
committed errors in both mathematics and physics. For instance, his notion that r = 0 in the so-called "Schwarzschild solution" marks the point at the centre of the related manifold is false, as is
his related claim that Schwarzschild's actual solution describes a manifold that is extendible. His post hoc introduction of Newtonian concepts and related mathematical expressions into
Schwarzschild's actual solution are invalid; for instance, Newtonian two-body relations into what is alleged to be a one-body problem. Each of the objections are treated in turn and their invalidity
fully demonstrated. Black hole theory is riddled with contradictions. This article provides definitive proof that black holes do not exist.
Category: Relativity and Cosmology
[60] viXra:1011.0061 [pdf] replaced on 24 Jan 2011
Reflections on the Future of Particle Theory
Authors: Ervin Goldfain
Comments: 50 pages
Quantum Field Theory (QFT) lies at the foundation of the Standard Model for particle physics (SM) and is built in compliance with a number of postulates called consistency conditions. The remarkable
success of SM can be traced back to a unitary, local, renormalizable, gauge invariant and anomaly-free formulation of QFT. Experimental observations of recent years suggest that developing the theory
beyond SM may require a careful revision of conceptual foundations of QFT. As it is known, QFT describes interaction of stable or quasi-stable fields whose evolution is deterministic and
time-reversible. By contrast, behavior of strongly coupled fields or dynamics in the Terascale sector is prone to become unstable and chaotic. Nonrenormalizable interactions are likely to proliferate
and prevent full cancellation of ultraviolet divergences. A specific signature of this transient regime is the onset of long-range dynamic correlations in space-time, the emergence of strange
attractors in phase space and transition from smooth to fractal topology. Our focus here is the impact of fractal topology on physics unfolding above the electroweak scale. Arguments are given for
perturbative renormalization of field theory on fractal space-time, breaking of discrete symmetries, hierarchical generation of particle masses and couplings as well as the potential for highly
unusual phases of matter which are ultra-weakly coupled to SM. A surprising implication of this approach is that classical gravity emerges as a dual description of field theory on fractal space-time.
Category: High Energy Particle Physics
[59] viXra:1011.0060 [pdf] submitted on 24 Nov 2010
New Lewis Structures Through the Application of the Hypertorus Electron Model
Authors: Omar Yépez
Comments: 12 pages
The hypertorus electron model is applied to the chemical bond. As a consequence, the bond topology can be determined. A linear correlation is found between the normalized bond area and the bond
energy. The normalization number is a whole number. This number is interpreted as the Lewis's electron pair. A new electron distribution in the molecule follows. This discovery prompts to review the
chemical bond, as it is understood in chemistry and physics.
Category: Chemistry
[58] viXra:1011.0059 [pdf] submitted on 23 Nov 2010
A Possible Connection Between "Inflation" and the "Big Crunch"
Authors: John A. Gowan
Comments: 4 pages
In Alan Guth's theory of inflation, "repulsive gravity" is produced by the "negative pressure" of a supercooled "Higgs inflaton field" in a "false vacuum". This "repulsive gravity" drives the
expansion of space (restoring the "true vacuum") during the brief period of inflation which initiates the Universe. There is a curious similarity between this conception of the mechanics of inflation
and the notion of a "rebound" at the end of a cosmic life cycle or "Big Crunch" in a closed, cyclic Universe. This "rebound" is the explosion of a cosmic-sized black hole which has exhausted all
external supplies of space, and so can no longer create the gravitational field and time dimension necessary to contain its energy content, which exists largely or wholly in the form of
gravitationally bound light (due to proton decay within the black hole's event horizon). (See: "Entropy, Gravitation, and Thermodynamics".)
Category: Relativity and Cosmology
[57] viXra:1011.0058 [pdf] submitted on 23 Nov 2010
New Perspectives on the Classical Theory of Motion, Interaction and Geometry of Space-Time
Authors: A. R. Hadjesfandiari
Comments: 73 pages
By examining the theory of relativity, as originally proposed by Lorentz, Poincare and Einstein, a fundamental theory of general motion is developed. From this, the relationship between space-time
and matter is discovered. As a result, the geometrical theory of interaction is introduced. The corresponding geometrical theory of electrodynamics resolves the origin of electromagnetic interaction,
as a vortex-like field, and clarifies some of the existing ambiguities.
Category: Relativity and Cosmology
[56] viXra:1011.0057 [pdf] replaced on 5 Dec 2010
Is an Algebraic Cubic Equation the Primitive Instinct Beyond Electromagnetic and Nuclear World?
Authors: Jin He
Comments: 16 pages
Everyone lives his or her life instinctively. Does the instinct originate from the natural world? If the instinct is a rational process, is the natural world rational? Unfortunately, people have not
found any rational principle behind the natural world. Because human activities are realized directly through electromagnetic and nuclear forces of entropy-increase, people are difficult to recognize
the principle. Compared to the large-scale structure of galaxies, human bodies and their immediate environment are the "microscopic" world. The electromagnetic and nuclear forces which rule the
world, however, disappear in the formation of large-scale galaxy structures. Similarly they disappear in the formation of the solar system. My previous papers found many evidences that galaxies are
rational. This paper shows that large-scale galaxy structure must originate from an algebraic cubic equation.
Category: Astrophysics
[55] viXra:1011.0056 [pdf] submitted on 23 Nov 2010
The Energetic Aspect of the First Axiom of Mechanics
Authors: Edmundas Jauniškis
Comments: 4 pages
A new approach to the axiom of mechanics. The energetic approach of the first axiom of mechanics. The principles and the concept of mechanical movement keeps changing. Mechanics is closely related to
Category: Classical Physics
[54] viXra:1011.0055 [pdf] submitted on 23 Nov 2010
The Hearth Wave Equation (Mass Formation and Evolution)
Authors: Vincenzo Sicari
Comments: 1 page
There are various physics phenomenon which can find a simple explanation in linking gravity and electromagnetism. Einstein's Relativity can simply explain only the mass because he considered time as
a scalar rather than a vector. The intension of this paper is to propose a new point of view, treating time as a three-dimensional vector, finding three vector value formula by three-dimensional
space-time formula (curvature formula), finding a new symmetry on the plane for a wave equation substitute for Maxwell's symmetrical wave, only E and only B, not E and B linked. This linking is in
error; in fact they are a two sum effect. The electromagnetic wave in space has, as well known, three energy component: B field E field and wave length (frequency). This energy is acknowledged, but
we must see the wave like an elastic chain of single wavelets (a string of individual wavelets).
Category: Quantum Gravity and String Theory
[53] viXra:1011.0054 [pdf] replaced on 24 Nov 2010
Four Comments on "The Road to Reality" by R Penrose
Authors: Elemér E Rosinger
Comments: 8 pages
Four comments are presented on the book of Roger Penrose entitled "The Road to Reality, A Complete Guide to the Laws of the Universe". The first comment answers a concern raised in the book. The last
three point to important omissions in the book.
Category: General Mathematics
[52] viXra:1011.0053 [pdf] submitted on 22 Nov 2010
Champs, Vide, et Univers Miroir
Authors: L. Borissova, D. Rabounski
Comments: 263 pages, première édition en langue française, traduit de l'anglais et édité par Patrick Marquet, American Research Press, Rehoboth (NM), USA
Cet ouvrage est la traduction française du livre "Fields, Vacuum and the Mirror Universe" publié originalement en anglais en 2009, par les physiciens Larissa Borissova et Dmitri Rabounski, enrichi de
nouveaux exposés. Le livre propose une analyse physico-mathématique nouvelle en élaborant une théorie des observables dans le cadre de la relativité générale. Dans leur célèbre livre de référence
"Théorie des Champs", Lev Landau et Evgeny Lifshitz ont décrit de manière très complète le mouvement des particules dans les champs électromagnétique et gravitationnel. Les méthodes d'analyse
covariante alors en vigueur depuis le milieu des années 30 ne prenaient pas encore en compte les concepts de quantités physiquement observables (grandeurs chronologiquement invariantes ou plus
précisément grandeurs dites "chronométriques") de la relativité générale. Les auteurs ont donc voulu insister sur la nécessité d'étendre cette perspective mathématique à la théorie physique existante
en l'appliquant au mouvement des particules se déplaçant dans les champs électromagnétiques et gravitationnels. De plus, l'étude des mouvements d'une particule douée de moment de rotation
intrinsèque, n'a pas été entreprise dans ce contexte par Landau et Lifshitz. C'est pourquoi un exposé séparé du livre a été entièrement consacré à ce type de mouvement particulier. Les auteurs ont
également ajouté un chapitre redéfinissant les éléments d'algèbre tensorielle et d'analyse dans le cadre des invariants chronométriques. L'ensemble de cet ouvrage se présente alors comme une
contribution supplémentaire à la "Théorie des Champs".
Category: Relativity and Cosmology
[51] viXra:1011.0052 [pdf] replaced on 22 Dec 2010
Space, Time and Units-Fundamental Vision
Authors: G. V. Sharlanov
Comments: 13 pages
The change of the units between two "time-spatial domains" with different gravitational potentials is analyzed in the article, as a consequence of the nature of space and time which are mutually
connected with each other in the warped spacetime of the Universe. The main topic of the article is: New definition of the postulate "invariance of the speed of light" - "uncertainty principle of the
macro-world". Other topics concerned: The change of the SI base units (second and metre) and the speed of light in the Gravitational Field are analyzed while discussing of the article of Albert
Einstein "On the Influence of Gravitation on the Propagation of Light". The change of the SI base units (second and metre) in Field without Gravity is analyzed, and then conclusion about Special
Theory of Relativity is made. new universal hierarchical structure of the SI System is proposed, where the "second" and "metre" stay at the highest level. The main principle of the hierarchical place
of any unit is also proposed. The consequences of this article directly reflect on cosmology and give explanations of a lot of problems (such as: "the accelerated expansion of the Universe", "the
problem with the dark matter", etc.).
Category: Relativity and Cosmology
[50] viXra:1011.0051 [pdf] submitted on 21 Nov 2010
Adiabatic Expansion of the Universe
Authors: Shehrin Sayed
Comments: 4 pages
The expansion of the universe is proved to be an adiabatic process. The proposed idea shows good agreement with the observable universe and can explain some basic characteristics of the universe. It
is also showed that the first kind of Friedmann model contradicts with the observable universe. The proposed idea also indicates that the size of the universe is finite.
Category: Relativity and Cosmology
[49] viXra:1011.0050 [pdf] submitted on 20 Nov 2010
Recording and Reproduction of Pattern Memory Trace in EEG by Direct Electrical Stimulation of Brain Cortex
Authors: Andrey G. Shapkin, Michael V. Taborov, Yuriy G. Shapkin
Comments: 9 pages
This study demonstrates the capability of external signal recording into memory and the reproduction of memory trace of this pattern in EEG by direct AC electrical stimulation of rat cerebral cortex.
Additionally, we examine shifts of the DC potential level related to these phenomena. We show that in the course of memory trace reproduction, consecutive phases of engram activation and relaxation
are registered and accompanied by corresponding negative and positive DC shifts. The observed electrophysiological changes may reflect consecutive activation and inhibition phases of neural ensembles
participating in engram formation.
Category: Mind Science
[48] viXra:1011.0049 [pdf] submitted on 21 Mar 2010
Ultra Polemics with Upper and Lower Cases
Authors: Florentin Smarandache
Comments: 129 pages, in Romanian
A short history on Smarandache's avant-garde movement called "paradoxism" in Romanian, English, Portuguese, French, Spanish. Also, polemics and manifestos on paradoxism, new literary species
introduced by the author (paradoxist distichs, dualistic distichs, tautological distichs, etc.), reviews, interviews.
Category: General Science and Philosophy
[47] viXra:1011.0048 [pdf] submitted on 21 Mar 2010
Besides and Behind Paradoxism
Authors: Florentin Smarandache
Comments: 127 pages, in Romanian
"Besides and Behind Paradoxism" comprises essays, in Romanian language, on Florentin Smarandache's literary non-paradoxist work (especially his diaries and metaphoric verses), by Silviu Popescu,
Marian Barbu, Titu Popescu, Daniel Deleanu, Alexandru Lungu, Evelina Oprea, Lucian Chisu, Ion Radu Zagreanu, Ion Rotaru. Interviews by Florentin Smarandache with Octavian Blaga, Ada Cirstoiu, Mihail
I. Vlad, A. D. Rachieru, Emil Burlacu, Veronica Balaj, Ion Stanica (Radio France Internationale). Florentin Smarandache in correspondence with 32 writers, among them: Andre Peragallo, Nancy Wilson,
Gloria Badarau, Bernardo Schiavetta, Beverly J. Kleikamp, Jessie Hraska, Olof G. Tandberg, Harriet G. Hunt, Al. Cistelecan, Paul Goma, Dan Danila, Constantin Corduneanu, etc.
Category: General Science and Philosophy
[46] viXra:1011.0047 [pdf] submitted on 21 Mar 2010
On New Functions in Number Theory
Authors: Florentin Smarandache
Comments: 98 pages, in Romanian
New functions introduced in number theory by the author are presented, studied, generalized some of them, and contributions of other mathematicians to these functions are also showed up.
Category: Number Theory
[45] viXra:1011.0046 [pdf] submitted on 21 Mar 2010
Hermeneutics of Paradoxism
Authors: Florentin Smarandache
Comments: 122 pages, in Romanian
"Paradoxism's Hermeneutics" includes selected by the editor articles on paradoxism, in Romanian language, articles written by Marian Barbu, George Bajenaru, Radu Enescu, Ovidiu Ghidirmic, Dumitru
Ichim, Alexandru Lungu, Mircea Marinescu, Ion Rotaru, Geo Vasile, etc. Paradoxism in science, arts and letters was set up by Florentin Smarandache in 1980 and then used in many creations.
Category: General Science and Philosophy
[44] viXra:1011.0045 [pdf] submitted on 21 Mar 2010
New Functions in Number Theory
Authors: Florentin Smarandache
Comments: 120 pages, in Romanian
Definitions, constructions, properties, and solved and unsolved problems on Smarandache type functions are presented in this book.
Category: Number Theory
[43] viXra:1011.0044 [pdf] submitted on 19 Nov 2010
[42] viXra:1011.0043 [pdf] replaced on 13 Dec 2010
Cosmological and Intrinsic Redshifts
Authors: José Francisco García Juliá
Comments: 5 pages
In a recent article, a single tired light mechanism, based in the interaction between electromagnetic waves, has been proposed for explaining both redshifts: cosmological (without expansion of the
universe) and intrinsic. A second paper specifies that said interaction would be the scattering. This article is to reinforce and clarify the whole idea.
Category: Astrophysics
[41] viXra:1011.0042 [pdf] replaced on 20 Nov 2010
[40] viXra:1011.0041 [pdf] submitted on 18 Nov 2010
Photon Diffraction
Authors: Daniele Sasso
Comments: 6 pages, 4 figures
In this article two types of diffraction are considered: the macroscopic diffraction occurs whether with electromagnetic waves or with photon beams like light, the photon diffraction or nanoscopic
diffration occurs only with photons. The difference between the two diffractions depends on both the width of the slot and the wavelength of radiation. This consideration proves further that
electromagnetism and optics, including in optics infrared, visible, ultraviolet and X radiation, have a different physical nature: electromagnetism is characterized by waves, optics by photon beams.
The frequency band employed by infrared radiation represents the point of connection between the two diffractions and the two physical behaviours.
Category: Classical Physics
[39] viXra:1011.0040 [pdf] submitted on 17 Nov 2010
Rayleigh Benard Convection: a Hydrodynamics Analysis
Authors: Zhe Wu
Comments: 7 pages, the article will be submitted to 'Journal of Fluids and Structures'
see paper
Category: Classical Physics
[38] viXra:1011.0039 [pdf] submitted on 17 Nov 2010
Some Orbital and Other Properties of the 'Special Gravitating Annulus'
Authors: Guy Moore, Richard Moore
Comments: 40 pages
Our obtaining the analytical equations for the gravitation of a particular type of mathematical annulus, which we called a 'Special Gravitating Annulus' (SGA), greatly facilitates studying its
orbital properties by computer programming. This includes isomorphism, periodic and chaotic polar orbits, and orbits in three dimensions. We provide further insights into the gravitational properties
of this annulus and describe our computer algorithms and programs. We study a number of periodic orbits, giving them names to aid identification. 'Ellipses extraordinaires' which are bisected by the
annulus, have no gravitating matter at either focus and represent a fundamental departure from the normal association of elliptical orbits with Keplerian motion. We describe how we came across this
type of orbit and the analysis we performed. We present the simultaneous differential equations of motion of 'ellipses extraordinaires' and other orbits as a mathematical challenge. The 'St.Louis
Gateway Arch' orbit contains two 'instantaneous static points' (ISP). Polar elliptical orbits can wander considerably without tending to form other kinds of orbit. If this type of orbit is favoured
then this gives a similarity to spiral galaxies containing polar orbiting material. Annular oscillatory orbits and rotating polar elliptical orbits are computed in isometric projection. A 'daisy'
orbit is computed in stereo-isometric projection. The singularity at the centre of the SGA is discussed in relation to mechanics and computing, and it appears mathematically different from a black
hole. In the Appendix, we prove by a mathematical method that a thin plane self-gravitating Newtonian annulus, free from external influence, exhibiting radial gravitation that varies inversely with
the radius in the annular plane, must have an area mass density which also varies inversely with the radius and this exact solution is the only exact solution.
Category: Mathematical Physics
[37] viXra:1011.0038 [pdf] submitted on 17 Nov 2010
Interval Linear Algebra
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 249 pages
This Interval arithmetic or interval mathematics developed in 1950's and 1960's by mathematicians as an approach to putting bounds on rounding errors and measurement error in mathematical
computations. However no proper interval algebraic structures have been defined or studies. In this book we for the first time introduce several types of interval linear algebras and study them.
Category: Algebra
[36] viXra:1011.0037 [pdf] submitted on 14 Nov 2010
Lattice Rationals
Authors: Nathaniel S. K. Hellerstein
Comments: 17 pages
This paper redefines the addition of rational numbers, in a way that allows division by zero. This requires defining a "compensator" on the integers, plus extending least-common-multiple (LCM) to
zero and negative numbers. "Compensated addition" defines ordinary addition on all ratios, including the 'infinities' n/0, and also 'zeroids' 0/n. The infinities and the zeroids form two 'double
ringlets'. The lattice rationals modulo the zeroids yields the infinities plus the 'wheel numbers'. Due to the presence of the 'alternator' @ = 0/-1, double-distribution does not apply, but
triple-distribution still does.
Category: Algebra
[35] viXra:1011.0036 [pdf] replaced on 2012-12-05 14:03:26
The Higgs Boson and the Weak Force IVBs: Part V
Authors: John A. Gowan
Comments: 10 Pages.
The IVBs (Intermediate Vector Bosons) are the field vectors (force carriers) of the weak force. The IVBs reconstitute (or revisit) the very energy dense, early metric of spacetime (during the "Big
Bang"), and their mass is the probable consequence of the binding energy necessary to condense, compact, and/or convolute the spacetime metric to a particular symmetric energy state, defined by a
specific force-unification era (such as the Electroweak Era, for instance), with a specific energy density and temperature. Originally, the "W" IVBs were indistinguishable from the early dense metric
of which they were a part - the energy level of electroweak unification. The "Electroweak Era" (EW) existed from 10(-12) to 10(-35) seconds after the Big Bang, when collision energy exceeded 100 GEV
and the temperature exceeded 10(15) Kelvins. During this time (a tiny fraction of a second in human terms) the whole of spacetime - the whole Cosmos - was in effect a single huge "W" IVB within which
all the transitions of "identity" within the lepton family of particles (including the heavy leptons), and all the transitions of "flavor" within the quark family of particles (including quarks of
the heavy baryons or "hyperons"), could take place freely without restriction or energy barriers (during the EW Era, quark and lepton families were unified among themselves, but quarks remained
separate from leptons.) (See: Brian Greene: "The Fabric of the Cosmos", page 270, Knopf, 2004.)
Category: High Energy Particle Physics
[34] viXra:1011.0034 [pdf] submitted on 20 Mar 2010
Le Mouvement Littéraire Paradoxiste
Authors: Constantin M. Popa
Comments: 39 pages
The author studies the paradoxism, a movement originated by the dissident mathematician Florentin Smarandache in 1980's and based on usage of paradoxes and contradictions in arts, literature,
philosophy, mathematics, science. The author compare paradoxism with other avant-gardes of the first part of the twenty's century.
Category: General Science and Philosophy
[33] viXra:1011.0033 [pdf] submitted on 20 Mar 2010
Chinese Neutrosophy and Taoist Natural Philosophy
Authors: Florentin Smarandache, Jiang Zhengjie
Comments: 152 pages
While Taoism is based on the union of opposites, Neutrosophy considers the union of opposites and the neutralities in between them. We thought that Neutrosophy and traditional Chinese Dialectics may
be combined, and establish the Chinese Neutrosophy concept, with its premise about the existence of the universal absolute main body, whose existence may be theoretically proven through the
establishment of Taoist Natural Philosophy.
Category: General Science and Philosophy
[32] viXra:1011.0032 [pdf] submitted on 20 Mar 2010
Subjective Questions and Answers for a Mathematics Instructor of Higher Education
Authors: Florentin Smarandache
Comments: 15 pages
This article of mathematical education reflects author's experience with job applications and teaching methods and procedures to employ in the American Higher Education. It is organized as a standard
Category: General Mathematics
[31] viXra:1011.0031 [pdf] submitted on 20 Mar 2010
[30] viXra:1011.0030 [pdf] submitted on 20 Mar 2010
[29] viXra:1011.0029 [pdf] submitted on 20 Mar 2010
A Generalization of the Inequality of H&oulm;lder
Authors: Florentin Smarandache
Comments: 2 pages
One generalizes the inequality of Hödler thanks to a reasoning by recurrence. As particular cases, one obtains a generalization of the inequality of Cauchy-Buniakovski-Scwartz, and some interesting
Category: Number Theory
[28] viXra:1011.0028 [pdf] submitted on 20 Mar 2010
[27] viXra:1011.0027 [pdf] submitted on 20 Mar 2010
A Generalization of the Inequality Cauchybouniakovski-Schwarz
Authors: Florentin Smarandache
Comments: 2 pages
Let us consider the real numbers...
Category: Number Theory
[26] viXra:1011.0026 [pdf] submitted on 12 Nov 2010
The Effects of Gravity on the Mind's Perception
Authors: Jeffrey S. Keen
Comments: 6 pages
This paper demonstrates that by using Noetics, the mind can quantitatively track the earth's annual elliptical orbit around the sun, due to the change in the earth-sun gravitational attraction. These
measurements have a remarkable correlation coefficient of 0.9999 to the inverse of the Newtonian gravitational force raised to the power of 6.
Category: Mind Science
[25] viXra:1011.0025 [pdf] submitted on 12 Nov 2010
The Parameters of S. Marinov's Curve (Evidence for My Three-dimensional Time and my new Wave Formula)
Authors: Vincenzo Sicari
Comments: 2 pages
There are various physics phenomenon which can find a simple explanation in linking gravity and electromagnetism. Einstein's Relativity can simply explain only the mass because he considered time as
a scalar rather than a vector. The intension of this paper is to propose a new point of view, treating time as a three-dimensional vector, finding three vector value formula by three-dimensional
space-time formula (curvature formula), finding a new symmetry on the plane for a wave equation substitute for Maxwell's symmetrical wave, only E and only B, not E and B linked. This linking is in
error; in fact they are a two sum effect. The electromagnetic wave in space has, as well known, three energy components: B field E field and wave length (frequency). This energy is acknowledged, but
we must see the wave like an elastic chain of single wavelets (a string of individual wavelets).
Category: Quantum Gravity and String Theory
[24] viXra:1011.0024 [pdf] replaced on 2012-07-22 23:10:55
Introduction to the Higgs Boson Papers
Authors: John A. Gowan
Comments: 15 Pages.
Although I had heard about, read about, and wondered about the "Higgs boson" for years, I simply couldn't get a "feel" for this particle, mostly because I was unable to place it within any overall,
coherent scheme of physical phenomena. I didn't want to believe in its reality, but I hadn't wanted to believe in the reality of the "W" and "Z" IVBs, either. Having eaten a large serving of humble
pie with the discovery of these particles in the early 1980s at CERN, I was not eager for second helpings from the Higgs, so I kept searching for its conservation role. What finally broke the impasse
for me was the article by Gordon Kane in Scientific American (and there is much else in this article I don't agree with), which mentioned there could be more than one Higgs boson. (See: "The
Mysteries of Mass" by Gordon Kane, Scientific American , July 2005, pp. 41-48.) That idea allowed me almost immediately to "do my thing", which is the construction of General Systems hierarchies,
using the "phase transition" energy levels, or force- unification symmetric energy states, as benchmarks for the four sequential steps of a weak force decay "cascade" from the "Multiverse" to "ground
state" atomic matter in our universe, with one step allotted to each of the four forces as they joined (or separated from) the unification hierarchy, and one Higgs boson identifying each
unified-force energy plateau. (See: "Table of the Higgs Cascade".) (On July 4, 2012, CERN announced the tentative discovery of a massive, Higgs-like boson, at 126 GEV on the LHC at Geneva,
Category: High Energy Particle Physics
[23] viXra:1011.0023 [pdf] replaced on 11 Nov 2010
The Dirac Equation in Accelerating Frames
Authors: Jack Sarfatti
Comments: 4 pages
I predict a new translational-rotational coupling in rotating frames that may have been missed hitherto. This eq. 1.12 below should give rise to new physics of clamped charges in rotating capacitors
for example. Since accelerating frames are also locally equivalent to Newton's gravity force there may be some new quantum mechanical effects here as well.
Category: Mind Science
[22] viXra:1011.0022 [pdf] submitted on 11 Nov 2010
Consciousness-mediated Spin Theory: The Transcendental Ground of Quantum Reality
Authors: Huping Hu, Maoxin Wu
Comments: 34 pages
It is our comprehension that Conciousness is both transcendent and immanent as simialarly understood in Hinuism. The transcendetal aspect of Conciousness produced and influences reality through
self-referential spin as the interactive output of Conciousness. In turn, relaiuty produces and influences immanent aspect of Conciousness as the interactive inout to Conciousness through
self-referential spin. The spin-mediated copnsciousness theory as originally proposed has mainly dealt with the immenant aspect of Conciousness which is driven by the self-referential spin processes.
This paper focuses and "e;regurgitates" on the transcendental aspect of Conciousness which drives the self0referential spin processes.
Category: Mind Science
[21] viXra:1011.0021 [pdf] submitted on 11 Nov 2010
Experimental Support of Spin-mediated Consciousness Theory from Various Sources
Authors: Huping Hu, Maoxin Wu
Comments: 29 pages
This paper summarizes experimental support to spin-mdeiated conciousness theory from variuos sources including the results of our own. Ind oing so we also provide explanations bases on this theory to
experimental phenomena such as out-of-body experience and sensed presence. quantum-like cognitive functions and optical illusions. Whether one agrees or not with the spin-mediated consciousness
theory is for one alone to judge. In any event, the importance of the experimental results mentioned in this paper is obvious: quantum effects play important roles in brain/cognitive functions
despite of the denials and suspicions of the naysayer and skeptics.
Category: Mind Science
[20] viXra:1011.0020 [pdf] submitted on 11 Nov 2010
Current Landscape and Future Direction of Theoretical & Experimental Quantum Brain/Mind/Consciousness Research
Authors: Huping Hu, Maoxin Wu
Comments: 10 pages
The issues surrounding quantum brain/mind/conciousness research are both confusing and complex. If one can manage to grasp these issues, one may find that the past of this field has been fruitful and
its future is indeed very promising. The current landscape and past achievements in this filed have already been discussed by our colleagues as pointed herein. This editorial mainly attempts to
classify/clarify some of the major issues and discuss what are lying ahead. Whatever difficulties may still remain, recent experimental results by several groups including those of the aauthors' own
make it very clear that quantum effects play important roles in brain functions despite of the denials and suspicions of the naysayer and skeptics
Category: Mind Science
[19] viXra:1011.0019 [pdf] submitted on 11 Nov 2010
On Reduction
Authors: Nathaniel S. K. Hellerstein
Comments: 33 pages
In this paper I discuss "reduction", a.k.a. "reciprocal addition"; addition conjugated by reciprocal. I discuss reduction's definition, its laws, its graphs, its geometry, its algebra, its calculus,
and its practical applications. This paper contains a problem set with answer key.
Category: Algebra
[18] viXra:1011.0018 [pdf] replaced on 21 Dec 2010
The Higgs Boson and the Weak Force IVBs (Intermediate Vector Bosons): A General Systems Perspective (parts II, III, IV)
Authors: John A. Gowan
Comments: 10 pages
The IVBs (Intermediate Vector Bosons) are the field vectors (force carriers) of the weak force. The IVBs reconstitute the very dense, early metric of spacetime (during the "Big Bang"), and their mass
is the probable consequence of the binding energy necessary to condense, compact, and/or convolute the spacetime metric. Originally, the "W" IVBs were indistinguishable from the early dense metric of
which they were a part - the energy level of electroweak unification. The "Electroweak Era" existed from 10(-12) to 10(-35) seconds after the Big Bang, when collision energy exceeded 100 GEV and the
temperature exceeded 10(15) Kelvins. During this time (a tiny fraction of a second in human terms) the whole of spacetime - the whole Cosmos - was in effect a single huge "W" IVB within which all the
transitions of "identity" within the lepton family of particles (including the heavy leptons), and all the transitions of "flavor" within the quark family of particles (including quarks of the heavy
baryons or "hyperons"), could take place freely without restriction or energy barriers (quark and lepton families were unified among themselves, but quarks remained separate from leptons.) (See:
Brian Greene: "The Fabric of the Cosmos", page 270, Knopf, 2004.)
Category: Relativity and Cosmology
[17] viXra:1011.0017 [pdf] submitted on 10 Nov 2010
Another Look at the Cosmological Model of Omnès
Authors: Henry D. May
Comments: 7 pages
The cosmological model of Roland Omnès was abandoned more than 30 years ago because it failed to show that coalescence was able to continue long enough to produce aggregations as large as the masses
of galaxies. It was also determined that a universe containing antimatter galaxies is inconsistent with observations of cosmic annihilation radiation. This paper explores the implications of a simple
assumption that suggests a reevaluation of those objections is needed.
Category: Relativity and Cosmology
[16] viXra:1011.0016 [pdf] replaced on 6 Apr 2011
In Support of Comte-Sponville
Authors: Elemér E Rosinger
Comments: 165 pages.
When I was a small boy, about two years old, on occasion, I had some nightmares. One morning I mentioned that to my Mother, and she replied, in a most natural matter of course manner, as if it had
been about a simple and trivial issue, that next time, when I would again have such a nightmare, I should simply remember that I was dreaming, that all it would only be in a dream, and then I should
just wake up ... Since my Mother's reply came so instantly, smoothly, and without the least emotion, let alone dramatization, I simply took it as such ... And next time, when a nightmare came upon me
during my dream, I simply did, and yes, I managed rather naturally to do, what my Mother had told me to do ... And never ever I would again have nightmares for more than a mere moment, before I would
manage to wake up from them ... Later, nightmares would, so to say, even avoid me completely...
Category: Religion and Spiritualism
[15] viXra:1011.0015 [pdf] submitted on 8 Nov 2010
Gravitational Energy Cannot Be Negative: a Comment on The Grand Design
Authors: Ron Bourgoin
Comments: 3 pages
The authors of The Grand Design treat the gravitational energy of a large astronomical object as a negative quantity. The formula used to calculate gravitational energy is GM 2 r , in which all
quantities are positive. Even if mass were considered to be negative, the square of the mass is a positive quantity. All other quantities in the formula are positive because G is positive, and since
there is no such physical thing as a negative radius, r is positive. The authors have simply affixed a negative sign in front of the formula, which is neither correct mathematically nor physically.
We show in this paper that the correct treatment of gravitational energy leads to the conclusion that Earth's gravitational energy produces Earth's space. Earth has its own space, as does every
astronomical object in the Universe. The general overall space of the Universe, therefore, is a series of interacting spaces.
Category: Quantum Gravity and String Theory
[14] viXra:1011.0014 [pdf] submitted on 8 Nov 2010
Identity Charge and the Origin of Life
Authors: John A. Gowan
Comments: 4 pages
"Identity" charge (also known as "number" charge) is the fundamental charge of the weak force and the most important of the particle charges. Identity charge is the symmetry debt of light's
anonymity, or complete lack of identity. One photon cannot be distinguished from another, but the elementary leptonic particles are distinct from photons and from each other, and hence carry identity
charges. Neutrinos are the explicit or "bare" form of identity charge, which is also carried in a "hidden" or implicit form by the massive leptonic elementary particles - electrons and their heavier
kin. Single elementary particles cannot enter or leave the 4-dimensional realm of manifest reality without a conserving identity charge - the functional equivalent of a human "soul" or a citizen's
passport. The utility of identity charge (in terms of symmetry conservation) is to facilitate particle-antiparticle annihilations by helping particles identify their appropriate "anti-mates" in a
timely fashion - ensuring a conserved pathway for elementary particles returning to their original state of symmetry (light). For more on the function of identity charge see: "Identity Charge and the
Weak Force", and "The Origin of Matter and Information".
Category: Biochemistry
[13] viXra:1011.0013 [pdf] submitted on 8 Nov 2010
Nonlinear Theory of Elementary Particles: 5.the Electron and Positron Equations (Linear Approach)
Authors: Alexander G. Kyriakos
Comments: 18 pages.
The purpose of this chapter is to describe the mechanism of generation of massive fermions - electron and positron. The presented below theory describes the electron and positron mass production by
means of breakdown of massive intermediate boson without the presence of Higgs's boson. It is shown that nonlinearity is critical for the appearance of fermions' currents and masses. Here is
considered only the linear form of equations. The analysis of nonlinear forms will be making in the following chapter.
Category: High Energy Particle Physics
[12] viXra:1011.0012 [pdf] submitted on 7 Nov 2010
Mach Principle Applied to Rotary Motion: Centrifugal Force in a Spinning Reference Frame as a Consequence of Gravitational Interaction with the Universe
Authors: Leonid I. Filippov
Comments: 11 pages, 8 figures, In Russian
The mass point is moving along the circle of a radius at angular velocity of ω and interacts with the spherical domain of 3.10^26m radius, with a uniform mass and 10^-26 kg/m^3density. As a result of
the treatment we arrive at the formula for a centrifugal force F = m,ω^2.a.
Category: Relativity and Cosmology
[11] viXra:1011.0011 [pdf] submitted on 6 Nov 2010
Re-Identification of the Many-World Background of Special Relativity as Four-World Background. Part II.
Authors: Akindele O. J. Adekugbe
Comments: 15 pages. Paper to appear in Progress in Physics vol.1, 2011, pp. 25-39.
The re-identification of the many-world background of the special theory of relativity (SR) as four-world background in the first part of this paper (instead of two-wold background isolated in the
initial papers), is concluded in this second part. The flat two-dimensional proper intrinsic spacetime, which underlies the flat four-dimensional proper spacetime in each universe, introduced as
ansatz in the initial paper, is derived formally within the four-world picture. The identical magnitudes of masses, identical sizes and identical shapes of the four members of every quartet of
symmetry-partner particles or objects in the four universes are shown. The immutability of Lorentz invariance on flat spacetime of SR in each of the four universes is shown to arise as a consequence
of the perfect symmetry of relative motion at all times among the four members of every quartet of symmetry-partner particles and objects in the four universes. The The perfect symmetry of relative
motions at all times, coupled with the identical magnitudes of masses, identical sizes and identical shapes, of the members of every quartet of symmetry-partner particles and objects in the four
universes, guarantee perfect symmetry of state among the universes.
Category: Relativity and Cosmology
[10] viXra:1011.0010 [pdf] submitted on 6 Nov 2010
Re-Identification of the Many-World Background of Special Relativity as Four-World Background. Part I.
Authors: Akindele O. J. Adekugbe
Comments: 22 pages. Paper to appear in Progress in Physics, vol.1, 2011, pp. 3-24.
The pair of co-existing symmetrical universes, referred to as our (or positive) universe and negative universe, isolated and shown to constitute a two-world background for the special theory of
relativity (SR) in previous papers, encompasses another pair of symmetrical universes, referred to as positive time-universe and negative time-universe. The Euclidean 3-spaces (in the context of SR)
of the positive time-universe and the negative time-universe constitute the time dimensions of our (or positive) universe and the negative universe respectively, relative to observers in the
Euclidean 3-spaces of our universe and the negative universe and the Euclidean 3-spaces of or our universe and the negative universe constitute the time dimensions of the positive time-universe and
the negative time-universe respectively, relative to observers in the Euclidean 3-spaces of the positive time-universe and the negative time-universe. Thus time is a secondary concept derived from
the concept of space according to this paper. The one-dimensional particle or object in time dimension to every three-dimensional particle or object in 3-space in our universe is a three-dimensional
particle or object in 3-space in the positive time-universe. Perfect symmetry of natural laws is established among the resulting four universes and two outstanding issues about the new spacetime/
intrinsic spacetime geometrical representation of Lorentz transformation/intrinsic Lorentz transformation in the two-world picture, developed in the previous papers, are resolved within the larger
four-world picture in this first part of this paper.
Category: Relativity and Cosmology
[9] viXra:1011.0009 [pdf] replaced on 13 Dec 2010
Generalized Uncertainty Principle
Authors: Saurav Dwivedi
Comments: 10 pages.
Quantum theory brought an irreducible lawlessness in physics. This is accompanied by lack of specification of state of a system. We can not measure states even though they ever existed. We can
measure only transition from one state into another. We deduce this lack of determination of state mathematically, and thus provide formalism for maximum precision of determination of mixed states.
However, the results thus obtained show consistency with Heisenberg's uncertainty relations.
Category: Quantum Physics
[8] viXra:1011.0008 [pdf] replaced on 3 May 2010
Global and Local Gauge Symmetries: Part V
Authors: John A. Gowan
Comments: 9 pages.
We observe that the "local gauge symmetry currents" (the field vectors of the forces, or components thereof), are in the case of both the spacetime metric (time) and the electromagnetic force
(magnetism), derived from an implicit expression embedded in the original global symmetry state. Magnetism, for example, occurs in its primordial state as the magnetic half of an electromagnetic
wave, or light. Likewise, time occurs in an implicit and suppressed state ("frequency") in the global metric of space, again as gauged by the global energy and metric constant c (frequency x
wavelength = c). This naturally leads us to suspect the same relationship should hold for the two particle forces (strong and weak forces) - we should find the precursors of their local, material
gauge currents in the primordial and symmetric global metric of light and space.
Category: Quantum Gravity and String Theory
[7] viXra:1011.0007 [pdf] submitted on 4 Nov 2010
A New Face of the Multiverse Hypothesis: Bosonic-Phononic Inflaton Quantum Universes
Authors: Lukasz Glinka
Comments: 12 pages, published in Prespacetime Journal 1(9), pp. 1395-1402 (November 2010)
The boson-phonon duality due to inflaton energy is presented in the context of quantum universes discussed recently by the author. The duality leads to bonons, i.e. the bosonic-phononic quantum
universes. This state of things manifestly corresponds to the Lewis-Kripke modal realism, and physical presence of Multiverse in Nature.
Category: Relativity and Cosmology
[6] viXra:1011.0006 [pdf] submitted on 2 Nov 2010
Equivalent Photon Model of Matter
Authors: Juan Carlos Alcerro Mena
Comments: 11 pages
An unpublished observation made in the theory of relativity, reveals a reality of higher spatial dimensions which implies a new model of matter, called equivalent photon model of matter, which
displays every particle with rest mass greater than zero, as a photon in a fourdimensional space (without referring to the space-time). This gives an alternative derivation of quantum physics, also
can observe the reality behind special relativity, arise physical predictions that not are covered by the existing physical theory and spontaneously arises the concept of fivedimensional space-time,
with its fourth spatial axis of compact type.
Category: Relativity and Cosmology
[5] viXra:1011.0005 [pdf] submitted on 3 Nov 2010
La Prova! a Last (T.o.e.)
Authors: Vincenzo Sicari
Comments: 38 pages
My theory states that our space-time (energy) is 6-dimensional; three spatial and three time, but only if the energy and thus the mass is considered as the effect of curvature.
Category: Quantum Gravity and String Theory
[4] viXra:1011.0004 [pdf] submitted on 3 Nov 2010
The Economical Expression of the Muon-, Neutron-, and Proton-Electron Mass Ratios
Authors: J. S. Markovitch
Comments: 6 pages
It is demonstrated that the proton-, neutron-, and muon-electron mass ratios may be expressed precisely and economically with the aid of two constants that derive from twin approximations of the fine
structure constant.
Category: High Energy Particle Physics
[3] viXra:1011.0003 [pdf] submitted on 3 Nov 2010
Zero Kelvin Big Bang, an Alternative Paradigm: III. The Big Bang
Authors: Royce Haynes
Comments: 11 pages, Paper 3 in series of 3, "Zero Kelvin Big Bang, an Alternative Paradigm", submitted to Apeiron, November 2, 2010.
In the first paper in this series, we described a "cosmic fabric" which served as the birthplace of our universe: spin-oriented hydrogen atoms at zero Kelvin in a matrix perhaps infinite and (almost)
eternal. In the second paper we described how a portion of the cosmic fabric ultimately condensed into a Bose-Einstein condensate (BEC), the "primeval atom". In this third paper we describe the Big
Bang itself, an implosion-explosion event involving nuclear fusion of hydrogen into the primordial mix of elements. Using the ZKBB model, one can calculate the approximate temperature of the Big Bang
as 5.7 billion K. The explosion fragmented the remaining BEC, propelling billions of fragments of "cosmic shrapnel" out from the locus of the Big Bang, which ultimately evolved into the structures we
see in our present universe.
Category: Relativity and Cosmology
[2] viXra:1011.0002 [pdf] submitted on 3 Nov 2010
Zero Kelvin Big Bang, an Alternative Paradigm: II. Bose-Einstein Condensation and the Primeval Atom
Authors: Royce Haynes
Comments: 8 pages, Paper 2 in series of 3, "Zero Kelvin Big Bang, an Alternative Paradigm", submitted to Apeiron, November 2, 2010.
In the first paper in this series, we described the logic suggesting a "cosmic fabric", which served as the birthplace of our universe: a sparse distribution of spin-oriented hydrogen atoms at zero
Kelvin, perhaps infinite and (almost) eternal. This second paper describes how a portion of this cosmic fabric could have condensed into a Bose-Einstein condensate (BEC). This cold ball of highly
concentrated matter may be the "primeval atom", proposed by Georges Lemaître in 1931, as the starting point for our universe.
Category: Relativity and Cosmology
[1] viXra:1011.0001 [pdf] submitted on 3 Nov 2010
Zero Kelvin Big Bang, an Alternative Paradigm: I. Logic and the Cosmic Fabric
Authors: Royce Haynes
Comments: 13 pages, Paper 1 in series of 3, "Zero Kelvin Big Bang, an Alternative Paradigm", submitted to Apeiron, November 2, 2010.
This is the first in a series of papers describing an alternative paradigm for the history of the universe. The Zero Kelvin Big Bang (ZKBB) theory is compared to the prevailing paradigm of the
Standard Big Bang (SBB), and challenges the notion that the universe is "all there is". Logic suggests that the Big Bang was not a creation event, but that the universe did have a beginning: a
"cosmic fabric" of pre-existing matter, in pre-existing space. Instead, the ZKBB was a transitional event between that "beginning" and our present universe. Extrapolating entropy back in time (as SBB
does for matter and energy) and applying simple logic suggests a "cosmic fabric" consisting of the simplest, stable particles of matter, at the lowest energy state possible: singlet state,
spin-oriented atomic hydrogen at zero Kelvin, at a density of, at most, only a few atoms per cubic meter of space, infinite and (almost) eternal. Papers II and III in this series describe formation
of an atomic hydrogen Bose-Einstein condensate as Lemaître's primeval atom, followed by an implosion-explosion Big Bang.
Category: Relativity and Cosmology | {"url":"http://vixra.org/all/1011","timestamp":"2014-04-20T05:44:21Z","content_type":null,"content_length":"92426","record_id":"<urn:uuid:90d72eec-acec-49fa-8ea3-4d34cf628c07>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00403-ip-10-147-4-33.ec2.internal.warc.gz"} |
normed division algebra
Algebraic theories
Algebras and modules
Higher algebras
Model category presentations
Geometry on formal duals of algebras
Normed division algebras
A normed division algebra is a generalisation of the real numbers, complex numbers, quaternions, and octonions. Amazingly, there are no other examples!
A normed division algebra is a Banach algebra that is also a division algebra.
While the norm in a Banach algebra is normally only submultiplicative (${\|x y\|} \leq {\|x\|} {\|y\|}$), the norm in a normed division algebra must be multiplicative (${\|x y\|} = {\|x\|} {\|y\|}$).
Accordingly, this norm is considered to be an absolute value and often written ${|{-}|}$ instead of ${\|{-}\|}$. There is also a converse: if the norm on a Banach algebra is multiplicative (including
${\|1\|} = 1$), then it must be a division algebra. While the term ‘normed division algebra’ does not seem to include the completeness condition of a Banach algebra, in fact the only examples have
finite dimension and are therefore complete.
The only normed division algebra over the complex numbers is the algebra of complex numbers themselves. Up to isomorphism, there are exactly four finite-dimensional normed division algebras over the
real numbers:
Each of these is produced from the previous one by the Cayley–Dickson construction; when applied to $\mathbb{O}$, this construction produces the algebra of sedenions, which do not form a division
The Cayley–Dickson construction applies to an algebra with involution; by the abstract nonsense of that construction, we can see that the four normed division algebras above have these properties:
• $\mathbb{R}$ is associative, commutative, and with trivial involution,
• $\mathbb{C}$ is associative and commutative but has nontrivial involution,
• $\mathbb{H}$ is associative but noncommutative and with nontrivial involution,
• $\mathbb{O}$ is neither associative, commutative, nor with trivial involution.
However, these algebras do all have some useful algebraic properties; in particular, they are all alternative (a weak version of associativity). They are also all composition algebras.
A normed field is a commutative normed division algebra; it follows from the preceding that the only normed fields are $\mathbb{R}$ and $\mathbb{C}$.
see division algebra and supersymmetry | {"url":"http://ncatlab.org/nlab/show/normed+division+algebra","timestamp":"2014-04-18T03:37:12Z","content_type":null,"content_length":"32847","record_id":"<urn:uuid:2e69873a-6639-42f8-8055-0c2c6d028ada>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00603-ip-10-147-4-33.ec2.internal.warc.gz"} |
Michael Grady
Professor and Chair
Department of Physics
SUNY College at Fredonia
Director of Cooperative
• Rockefeller University, Ph.D. 1982
• University of Washington, MS, 1978
• University of Michigan, BGS, 1973
Research Interests:
• Lattice Field Theory
• Phase Transitions and Critical Phenomena
• Strong Interactions
• Non-Perturbative Methods
• The Early Universe
Teaching Interests and Specialties:
• Lab and Lecture Demonstration Based Instruction
• Thermal Physics
• Acoustics
• Optics
• Computer Spreadsheet Applications in Education
Selected Publications:
(most available electronically from Physics Preprint Archive, http://arxiv.org)
• "Reconsidering gauge-Higgs continuity", hep-lat/0507037, Physics Letters B, 626, 161 (2005).
• "Critical or tricritical point in mixed-action SU(2) lattice gauge theory?", hep-lat/0404015, Nuclear physics B, 713, 204 (2005).
• "Evidence for layered symmetry breaking in SU(2) lattice gauge theory", hep-lat/0409163.
• "Quantum Mechanics, Quantum Gravity, and Approximate Lorentz Invariance from a Classical Phase-Boundary Universe", hep-th/0104237 (to appear as Ch. 3 in "Trends in Quantum Gravity Research," NOVA
Science Publishers, 2006).
• "Gauge Invariant SO(3)-Z2 Monopole as Possible Source of Confinement in SU(2) Lattice Gauge Theory" hep-lat/9806024.
• "Universe as a Phase Boundary in a Four-Dimensional Euclidean Space", gr-qc/9805076.
• "Monopole Loop Distribution and Confinement in SU(2) Lattice Gauge Theory", hep-lat/9802035, Physics Letters B, 455, 239 (1999).
• "Monopole Loop Suppression and Loss of Confinement in Restricted-Action SU(2) Lattice Gauge Theory", hep-lat/9801016.
• "Deconfinement from Action Restriction", Nuclear Physics B (Proc. Suppl.) 53, 599 (1997) (LATTICE 96).
• "Finite Temperature Phase Transition in SU(2) Lattice Gauge Theory with Extended Action" (with Rajiv Gavai and Manu Mathur), Nuclear Physics B 423, 123 (1994).
• "Can a Logarithmically Running Coupling Mimic a String Tension?", Physical Review D 50, 6009 (1994).
• "Cabibbo Mixing from Strong-interaction Condensates", Nuovo Cimento 105A, 1065 (1992).
• "Spurious First-Order Phase Transitions from Topological Effects on the Lattice", Nuclear Physics B 365, 699 (1991).
Other Interests:
Contact me at:
NEW phone:(716)-673-4624, fax (716)-673-3347, or e-mail at grady@fredonia.edu | {"url":"http://www.fredonia.edu/department/physics/grady.asp","timestamp":"2014-04-18T03:13:07Z","content_type":null,"content_length":"18297","record_id":"<urn:uuid:b6ff186f-838a-40b0-8e60-95556fe81c53>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00299-ip-10-147-4-33.ec2.internal.warc.gz"} |
Asset Pricing: (Revised)
More About This Textbook
"An excellent survey of asset pricing theory and applications from the modern viewpoint of stochastic discount factors and their associated geometry. This book was already a classic among finance
scholars and on Ph.D. syllabi when it circulated in the form of class notes. It will also prove highly useful to practitioners who seek an in-depth introduction to these tools."—Yacine At-Sahalia,
Princeton University
"This is a beautiful book that uses the elegant simplicity of the stochastic discount factor to present a general theory of the pricing of stocks, bonds, and derivatives and a practical approach to
estimating particular models derived from the general theory. It will help experts in the field to consolidate their knowledge and beginners to appreciate the unity of asset pricing theory. Cochrane
uses his mastery of the subject to present it in a clear and compelling manner that is easily accessible."—Michael Brennan, Anderson School, University of California, Los Angeles
"This is an impressive treatise of very high quality. It is a serious scholarly monograph, of interest to those who are working to advance financial theory, and it can also serve as a textbook in an
advanced finance course. It is thoughtful, inductive, and comprehensive."—Robert J. Shiller, author of Irrational Exuberance
"This is a sparkling, intuitive, makes-it-look-easier-than-it really-is, gem of a book . . . Cochrane's focus is the classical asset pricing models of frictionless markets and rational expectations.
But the lessons learned are relevant in many empirical contexts. Cochrane's clever intuition and easy, informal writing style make the book a joy to read."—Wayne Ferson, Boston College
"This book represents an exciting step forward in the exposition of financial economics. The last twenty years of finance research have advanced and enriched the field, and textbook treatments have
lagged behind these developments. This text will replace the previous generation of books and should have a broad market. It is written in an informal, almost breezy style that will appeal to
students and is divided into small, easily digested chapters. . . . The book moves easily between discrete-time and continuous-time models. This is an excellent thing as it encourages students to see
beyond the formalism to the underlying economics. I strongly recommend it as an advanced finance text."—John Y. Campbell, coauthor of The Econometrics of Financial Markets
Read More Show Less
Editorial Reviews
From the Publisher
Co-Winner of the 2001 Paul A. Samuelson award
"This is a brilliant and useful book, well-deserving of the TIAA-CREF Samuelson Award. . . . The clever intuition and informal writing style make it a joy to read. Like a star athlete does with the
sport, Cochrane makes it look easier than it really is."—
Journal of Economic Literature
Journal of Economic Literature
This is a brilliant and useful book, well-deserving of the TIAA-CREF Samuelson Award. . . . The clever intuition and informal writing style make it a joy to read. Like a star athlete does with the
sport, Cochrane makes it look easier than it really is. Read More Show Less
Product Details
• ISBN-13: 9780691121376
• Publisher: Princeton University Press
• Publication date: 1/3/2005
• Edition description: REV
• Edition number: 1
• Pages: 568
• Sales rank: 948,771
• Product dimensions: 6.30 (w) x 9.30 (h) x 1.50 (d)
Read an Excerpt
Asset Pricing
By John H. Cochrane
Princeton University Press
Copyright © 2000 Princeton University Press
All right reserved.
ISBN: 0-691-07498-4
Chapter One
Consumption-Based Model and Overview
An investor must decide how much to save and how much to consume, and what portfolio of assets to hold. The most basic pricing equation comes from the first-order condition for that decision. The
marginal utility loss of consuming a little less today and buying a little more of the asset should equal the marginal utility gain of consuming a little more of the asset's payoff in the future. If
the price and payoff do not satisfy this relation, the investor should buy more or less of the asset. It follows that the asset's price should equal the expected discounted value of the asset's
payoff, using the investor's marginal utility to discount the payoff. With this simple idea, I present many classic issues in finance.
Interest rates are related to expected marginal utility growth, and hence to the expected path of consumption. In a time of high real interest rates, it makes sense to save, buy bonds, and then
consume more tomorrow. Therefore, high real interest rates should be associated with an expectation of growing consumption.
Most importantly, risk corrections to asset prices should be driven by the covariance of asset payoffs with marginal utility and hence by the covariance of asset payoffs with consumption. Other
things equal, an asset that does badly in states of nature like a recession, in which the investor feels poor and is consuming little, is less desirable than an asset that does badly in states of
nature like a boom in which the investor feels wealthy and is consuming a great deal. The former asset will sell for a lower price; its price will reflect a discount for its "riskiness," and this
riskiness depends on a co-variance, not a variance.
Marginal utility, not consumption, is the fundamental measure of how you feel. Most of the theory of asset pricing is about how to go from marginal utility to observable indicators. Consumption is
low when marginal utility is high, of course, so consumption may be a useful indicator. Consumption is also low and marginal utility is high when the investor's other assets have done poorly; thus we
may expect that prices are low for assets that covary positively with a large index such as the market portfolio. This is a Capital Asset Pricing Model. We will see a wide variety of additional
indicators for marginal utility, things against which to compute a convariance in order to predict the risk-adjustment for prices.
* * *
1.1 Basic Pricing Equation
An investor's first-order conditions give the basic consumption-based model,
[p.sub.t] = [E.sub.t] [[beta] u'([c.sub.t+1])/u'([c.sub.t]) [x.sub.t+1]].
Our basic objective is to figure out the value of any stream of uncertain cash flows. I start with an apparently simple case, which turns out to capture very general situations.
Let us find the value at time t of a payoff [x.sub.t+1]. If you buy a stock today, the payoff next period is the stock price plus dividend, [x.sub.t+1] = [p.sub.t+1] + [d.sub.t+1]. [x.sub.t+1] is a
random variable: an investor does not know exactly how much he will get from his investment, but he can assess the probability of various possible outcomes. Do not confuse the payoff [x.sub.t+1] with
the profit or return; [x.sub.t+1] is the value of the investment at time t + 1, without subtracting or dividing by the cost of the investment.
We find the value of this payoff by asking what it is worth to a typical investor. To do this, we need a convenient mathematical formalism to capture what an investor wants. We model investors by a
utility function defined over current and future values of consumption,
U([c.sub.t], [c.sub.t+1]) = u([c.sub.t]) + [beta][E.sub.t][u([c.sub.t+1])],
where [c.sub.t] denotes consumption at date t. We often use a convenient power utility form,
u([c.sub.t]) = [1 / 1 - y] [[c.sup.1-y.sub.t].
The limit as y [right arrow] 1 is
u(c) = ln(c).
The utility function captures the fundamental desire for more consumption, rather than posit a desire for intermediate objectives such as mean and variance of portfolio returns. Consumption
[c.sub.t+1] is also random; the investor does not know his wealth tomorrow, and hence how much he will decide to consume tomorrow. The period utility function u(·) is increasing, reflecting a desire
for more consumption, and concave, reflecting the declining marginal value of additional consumption. The last bite is never as satisfying as the first.
This formalism captures investors' impatience and their aversion to risk, so we can quantitatively correct for the risk and delay of cash flows. Discounting the future by [beta] captures impatience,
and [beta] is called the subjective discount factor. The curvature of the utility function generates aversion to risk and to intertemporal substitution: The investor prefers a consumption stream that
is steady over time and across states of nature.
Now, assume that the investor can freely buy or sell as much of the payoff [x.sub.t+1] as he wishes, at a price [p.sub.t]. How much will he buy or sell? To find the answer, denote by e the original
consumption level (if the investor bought none of the asset), and denote by [xi] the amount of the asset he chooses to buy. Then, his problem is
max u([c.sub.t]) + [E.sub.t][[beta]u([c.sub.t+1]) s.t. [xi]
[c.sub.t] + [e.sub.t] - [p.sub.t][xi],
[c.sub.t+1] = [e.sub.t+1] + [x.sub.t+1][xi].
Substituting the constraints into the objective, and setting the derivative with respect to [xi] equal to zero, we obtain the first-order condition for an optimal consumption and portfolio choice,
[p.sub.t]u'([c.sub.t]) = [E.sub.t][[beta]u'([c.sub.t+1])[x.sub.t+1]], (1.1)
[p.sub.t] = [E.sub.t][[beta] u' ([c.sub.t+1])/u'([c.sub.t]) [x.sub.t+1]]. (1.2)
The investor buys more or less of the asset until this first-order condition holds.
Equation (1.1) expresses the standard marginal condition for an optimum: [p.sub.t]u'([c.sub.t]) is the loss in utility if the investor buys another unit of the asset; [E.sub.t][[beta]u'{[c.sub.t+1])
[x.sub.t+1]] is the increase in (discounted, expected) utility he obtains from the extra payoff at t+1. The investor continues to buy or sell the asset until the marginal loss equals the marginal
Equation (1.2) is the central asset pricing formula. Given the payoff [x.sub.t+1] and given the investor's consumption choice [c.sub.t], [c.sub.t+1], it tells you what market price [p.sub.t] to
expect. Its economic content is simply the first-order conditions for optimal consumption and portfolio formation. Most of the theory of asset pricing just consists of specializations and
manipulations of this formula.
We have stopped short of a complete solution to the model, i.e., an expression with exogenous items on the right-hand side. We relate one endogenous variable, price, to two other endogenous
variables, consumption and payoffs. One can continue to solve this model and derive the optimal consumption choice [c.sub.t], [c.sub.t+1] in terms of more fundamental givens of the model. In the
model I have sketched so far, those givens are the income sequence [e.sub.t], [e.sub.t+1] and a specification of the full set of assets that the investor may buy and sell. We will in fact study such
fuller solutions below. However, for many purposes one can stop short of specifying (possibly wrongly) all this extra structure, and obtain very useful predictions about asset prices from (1.2), even
though consumption is an endogenous variable.
* * *
1.2 Marginal Rate of Substitution/Stochastic Discount Factor
We break up the basic consumption-based pricing equation into
p = E(mx).
m = [beta] u'([c.sub.t+1]) / u'([c.sub.t]),
where [m.sub.t+1] is the stochastic discount factor.
A convenient way to break up the basic pricing equation (1.2) is to define the stochastic discount factor [m.sub.t+1]
[m.sub.t+1] = [beta] u'([c.sub.t+1]) / u'([c.sub.t]). (1.3)
Then, the basic pricing formula (1.2) can simply be expressed as
[p.sub.t] = [E.sub.t]([m.sub.t+1][x.sub.t+1]). (1.4)
When it is not necessary to be explicit about time subscripts or the difference between conditional and unconditional expectation, I will suppress the subscripts and just write p = E(mx). The price
always comes at t, the payoff at t + 1, and the expectation is conditional on time-t information.
The term stochastic discount factor refers to the way m generalizes standard discount factor ideas. If there is no uncertainty, we can express prices via the standard present value formula
[p.sub.t] = 1 / [R.sup.f] [x.sub.t+1], (1.5)
where [R.sup.f] is the gross risk-free rate. 1/[R.sup.f] is the discount factor. Since gross interest rates are typically greater than one, the payoff [x.sub.t+1] sells "at a discount." Riskier
assets have lower prices than equivalent risk-free assets, so they are often valued by using risk-adjusted discount factors,
[[p.sup.i.sub.t] = 1 / [R.sup.i][E.sub.t]([x.sup.i.sub.t+1]).
Here, I have added the i superscript to emphasize that each risky asset i must be discounted by an asset-specific risk-adjusted discount factor 1/[R.sup.i].
In this context, equation (1.4) is obviously a generalization, and it says something deep: one can incorporate all risk corrections by defining a single stochastic discount factor-the same one for
each asset-and putting it inside the expectation. [m.sub.t+1] is stochastic or random because it is not known with certainty at time t. The correlation between the random components of the common
discount factor m and the asset-specific payoff [x.sup.i] generate asset-specific risk corrections.
[m.sub.t+1] is also often called the marginal rate of substitution after (1.3). In that equation, [m.sub.t+1] is the rate at which the investor is willing to substitute consumption at time t + 1 for
consumption at time t. [m.sub.t+1] is sometimes also called the pricing kernel. If you know what a kernel is and you express the expectation as an integral, you can see where the name comes from. It
is sometimes called a change of measure or a state-price density.
For the moment, introducing the discount factor m and breaking the basic pricing equation (1.2) into (1.3) and (1.4) is just a notational convenience. However, it represents a much deeper and more
useful separation. For example, notice that p = E(mx) would still be valid if we changed the utility function, but we would have a different function connecting m to data. All asset pricing models
amount to alternative ways of connecting the stochastic discount factor to data. At the same time, we will study lots of alternative expressions of p = E(mx), and we can summarize many empirical
approaches by applying them to p = E(mx). By separating our models into these two components, we do not have to redo all that elaboration for each asset pricing model.
* * *
1.3 Prices, Payoffs, and Notation
The price [p.sub.t] and payoff [x.sub.t+1] seem like a very restrictive kind of security. In fact, this notation is quite general and allows us easily to accommodate many different asset pricing
questions. In particular, we can cover stocks, bonds, and options and make clear that there is one theory for all asset pricing.
For stocks, the one-period payoff is of course the next price plus dividend, [x.sub.t+1] = [p.sub.t+1] + [d.sub.t+1]. We frequently divide the payoff [x.sub.t+1] by the price [p.sub.t] to obtain a
gross return
[R.sub.t+1] = [x.sub.t+1] / [p.sub.t].
We can think of a return as a payoff with price one. If you pay one dollar today, the return is how many dollars or units of consumption you get tomorrow. Thus, returns obey
1 = E(mR),
which is by far the most important special case of the basic formula p = E(mx). I use capital letters to denote gross returns R, which have a numerical value like 1.05. I use lowercase letters to
denote net returns r = R - 1 or log (continuously compounded) returns r = ln(R), both of which have numerical values like 0.05. One may also quote percent returns 100 x r.
Returns are often used in empirical work because they are typically stationary over time. (Stationary in the statistical sense; they do not have trends and you can meaningfully take an average.
"Stationary" does not mean constant.) However, thinking in terms of returns takes us away from the central task of finding asset prices. Dividing by dividends and creating a payoff of the form
[x.sub.t+1] = (1 + [p.sub.t+1]/[d.sub.t+1]) [d.sub.t+1]/[d.sub.t]
corresponding to a price [p.sub.t]/[d.sub.t] is a way to look at prices but still to examine stationary variables.
Not everything can be reduced to a return. If you borrow a dollar at the interest rate [R.sup.f] and invest it in an asset with return R, you pay no money out-of-pocket today, and get the payoff R -
[R.sup.f]. This is a payoff with a zero price, so you obviously cannot divide payoff by price to get a return. Zero price does not imply zero payoff. It is a bet in which the value of the chance of
losing exactly balances the value of the chance of winning, so that no money changes hands when the bet is made. It is common to study equity strategies in which one short-sells one stock or
portfolio and invests the proceeds in another stock or portfolio, generating an excess return. I denote any such difference between returns as an excess return, [R.sup.e]. It is also called a
zero-cost portfolio.
In fact, much asset pricing focuses on excess returns. Our economic understanding of interest rate variation turns out to have little to do with our understanding of risk premia, so it is convenient
to separate the two phenomena by looking at interest rates and excess returns separately.
We also want to think about the managed portfolios, in which one invests more or less in an asset according to some signal. The "price" of such a strategy is the amount invested at time t, say
[z.sub.t], and the payoff is [z.sub.t][R.sub.t+1]. For example, a market timing strategy might make an investment in stocks proportional to the price-dividend ratio, investing less when prices are
higher. We could represent such a strategy as a payoff using [z.sub.t] = a - b([p.sub.t]/[d.sub.t]).
When we think about conditioning information below, we will think of objects like [z.sub.t] as instruments.
Excerpted from Asset Pricing by John H. Cochrane Copyright © 2000 by Princeton University Press. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Read More Show Less
Table of Contents
1 Consumption-based model and overview 3
2 Applying the basic model 35
3 Contingent claims markets 49
4 The discount factor 61
5 Mean-variance frontier and beta representations 77
6 Relation between discount factors, betas, and mean-variance frontiers 99
7 Implications of existence and equivalence theorems 121
8 Conditioning information 131
9 Factor pricing models 149
10 GMM in explicit discount factor models 189
11 GMM : general formulas and applications 201
12 Regression-based tests of linear factor models 229
13 GMM for linear factor models in discount factor form 253
14 Maximum likelihood 267
15 Time-series, cross-section, and GMM/DF tests of linear factor models 279
16 Which method? 293
17 Option pricing 313
18 Option pricing without perfect replication 327
19 Term structure of interest rates 349
20 Expected returns in the time series and cross section 389
21 Equity premium puzzle and consumption-based models 455
App. A.1 Brownian motion 489
App. A.2 Diffusion model 491
App. A.3 Ito's lemma 494
Read More Show Less | {"url":"http://www.barnesandnoble.com/w/asset-pricing-john-h-cochrane/1101640465?ean=9780691121376&itm=1&usri=9780691121376","timestamp":"2014-04-20T20:20:22Z","content_type":null,"content_length":"144704","record_id":"<urn:uuid:5086e2ed-ec85-443b-927e-09658c47f145>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00249-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pythagoras and His Contributions to the Math World, Mathematics - CollegeTermPapers.com
Pythagoras and His Contributions to the Math World
Pythagoras and His Contributions to the Math World Although Pythagoras was not the best known Greek mathematician, he made many contributions to the way we use math today. Pythagoras is credited
with inventing the Pythagorean theorem. He also founded the Pythagorean
brotherhood. Pythagoras also invented a lot of number patterns. Plato and
Aristotle were influenced by Pythagoras's way of thinking. Also, he was a
Greek religious leader who made huge developments in math that may have
changed the math world.
Pythagoras of Samos is often described as the first pure mathematician.
He is an extremely important figure in the development of modern mathematics
yet we know relatively few facts of his life. We are not exactly sure of his birth
and death date.
Pythagoras was born in about 569 BC in Samos, Ionia. He died in about
475 BC but his death place is not known. Little is known of Pythagoras's
childhood. All accounts of his physical appearance seem to be false except the
description of a birthmark which he had on his thigh. Pythagoras's father was
Mnesarchus who was a merchant from Tyre. There is a story told that
Mnesarchus brought corn to Samos at a time of famine and was granted
citizenship of Samos as a mark of gratitude. Pythagoras's mother was Pythais
and she was a native of Samos.
Pythagoras took many trips in his life. His first came when he was only a
child. He visited Italy with his father. In about 535 BC Pythagoras went to Egypt.
This happened a few years after the Tyrant Polycrates seized control of Samos.
There is evidence to suggest that Pythagoras and Polycrates were friends at first
but when Polycrates abandoned his alliance with Egypt and attacked it, their
friendship abruptly ended. Soon after Polycrates death, Pythagoras returned to
Samos. Pythagoras invented many theorems. Probably his most popular
theorem is the Pythagorean Theorem. This is used for a right angled triangle.
This theorem enables you to find the length of the third side of a right triangle
when only knowing the length of two sides. This is considered his most important
contribution to math.
Pythagoras also invented the five regular solids. It is thought that
Pythagoras himself knew how to construct the first three but it is unlikely that he
knew how to construct the other two.
Pythagoras also founded a philosophical and religious school in Croton
(now Crotone, on the east of the heal of southern Italy) that had many followers.
Pythagoras was the head of the society with an inner circle of followers known as
mathematikoi. The mathematikoi lived permanently with the Society, had no
personal possessions and were vegetarians. They were taught by Pythagoras
himself and obeyed strict rules. The beliefs that Pythagoras held were:
(1) At its deepest level, reality is mathematical in nature,
(2) Philosophy can be used for spiritual purification,
(3) The soul can rise to union with the divine,
(4) Certain symbols have a mystical significance, and
(5) All brothers of the order should observe strict loyalty and secrecy.
This society defined figurate numbers to be the number of dots in certain
geometrical configurations (Mathematical Structures for Computer Science, Pg. 145).
Of Pythagoras's actual work nothing is known. His school practiced
secrecy and communalism making it hard to distinguish between the work of
Pythagoras and that of his followers. Certainly his school made outstanding
contributions to mathematics, and it is possible to be fairly certain about some of
Pythagoras's mathematical contributions. Pythagoras's accomplishments have
changed the math world tremendously and his contributions to the math world
are truly incredible.
Some topics in this essay
Croton Crotone | Science Pg | Little Pythagoras's | Pythagorean Theorem | Pythagoras Samos | Samos Pythagoras | Pythagoras Greek | Plato Aristotle | Pythagoras Polycrates | Tyrant Polycrates |
math world | contributions math | pythagoras invented | changed math world | samos pythagoras | world pythagoras | pythagorean theorem | math world pythagoras | changed math | contributions math
world | math pythagoras |
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Conceptualizing Concepts using the Lattice Formalism
In Proceedings of the 15th International FLAIRS Conference (Special Track 'Categorization and Concept Representation: Models and Implications'), Pensacola Florida, May 14-17, 2002, American
Association for Artificial Intelligence. (Note: due to page limit somne things are not fully explained. For details see forthcoming invited paper to Journal of Experimental and Theoretical Artificial
Contextualizing Concepts
Liane Gabora and Diederik Aerts
Center Leo Apostel for Interdisciplinary Studies (CLEA)
Free University of Brussels (VUB)
Krijgskundestraat 33, Brussels
B1160, Belgium, EUROPE
lgabora@vub.ac.be , diraerts@vub.ac.be
http://www.vub.ac.be/CLEA/liane/ , http://www.vub.ac.be/CLEA/aerts/
ABSTRACT: To cope with problems arising in the description of (1) contextual interactions, and (2) the generation of new states with new properties when quantum entities become entangled, the
mathematics of quantum mechanics was developed. Similar problems arise with concepts. We use a generalization of standard quantum mechanics, the mathematical lattice theoretic formalism, to develop a
formal description of the contextual manner in which concepts are evoked, used, and combined to generate meaning.
1. The Problem of Conjunctions
According to the classical theory of concepts, there exists for each concept a set of defining features that are singly necessary and jointly sufficient (e.g. Sutcliffe 1993). Extensive evidence has
been provided against this theory (see Komatsu 1992 and Smith & Medin 1981 for overviews). Two major alternatives have been put forth. According to the prototype theory (Rosch, 1975a, 1978, 1983;
Rosch and Mervis 1975), concepts are represented by a set of, not defining, but characteristic features, which are weighted in the definition of the prototype. A new item is categorized as an
instance of the concept if it is sufficiently similar to this prototype. According to the exemplar theory, (e.g., Heit and Barsalou 1996; Medin et al. 1984; Nosofsky 1988, 1992) a concept is
represented by, not defining or characteristic features, but a set of instances of it stored in memory. A new item is categorized as an instance of a concept if it is sufficiently similar to one or
more of these previous instances. We use the term representational theories to refer to both prototype and exemplar theories since concepts take the form of fixed representations (as opposed to
changing according to context).
Representational theories are adequate for predicting experimental results for many dependent variables including typicality ratings, latency of category decision, exemplar generation frequencies,
and category naming frequencies. However, they run into problems trying to account for the creative generation of, and membership assessment for, conjunctions of concepts. They cannot account for
phenomena such as the so-called guppy effect, where guppy is not rated as a good example of pet, nor of fish, but it is rated as a good example of pet fish (Osherson & Smith 1981). This is
problematic because if (1) activation of pet does not cause activation of guppy, and (2) activation of fish does not cause activation of guppy, how is it that (3) pet fish, which activates both pet
AND fish, causes activation of guppy? (In fact, it has been demonstrated experimentally that other conjunctions are better examples of the ‘guppy effect’ than pet fish (Storms et al. 1998), but since
the guppy example is well-known we will continue to use it here as an example.)
Zadeh (1965, 1982) tried, unsuccessfully, to solve the conjunction problem using a minimum rule model, where the typicality of an item as a conjunction of two concepts (conjunction typicality) equals
the minimum of the typicalities of the two constituents. Storms et al. (2000) showed that a weighted and calibrated version of this model can account for a substantial proportion of the variance in
typicality ratings for conjunctions exhibiting the guppy effect, suggesting the effect could be due to the existence of contrast categories. However, another study provided negative evidence for
contrast categories (Verbeemen et al., in press).
Conjunction cannot be described with the mathematics of classical physical theories because it only allows one to describe a composite or joint entity by means of the product state space of the state
spaces of the two subentities. Thus if X[1] is the state space of the first subentity, and X[2] the state space of the second subentity, the state space of the joint entity is the Cartesian product
space X[1 ]* X[2]. So if the first subentity is ‘door’ and the second is ‘bell’, one can give a description of the two at once, but they are still two. The classical approach cannot even describe the
situation wherein two entities generate a new entity that has all the properties of its subentities, let alone a new entity with certain properties of one subentity and certain of the properties of
the other. The problem can be solved ad hoc by starting all over again with a new state space each time there appears a state that was not possible given the previous state space. However, in so
doing we fail to include exactly those changes of state that involve the generation of novelty. Another possibility would be to make the state space infinitely large to begin with. However, since we
hold only a small number of items in mind at any one time, this is not a viable solution to the problem of describing what happens in cognition.
2. The Problem of Contextuality
The problems that arise with conjunctions reflects a more general problem with representational theories (see Riegler, Peschl and von Stein 1999 for overview). As Rosch (1999) puts it, they do not
account for the fact that concepts "have a participatory, not an identifying function in situations"; that is, they cannot explain the contextual way in which concepts are evoked and used (see also
Gerrig and Murphy 1992; Hampton, 1987; Komatsu, 1992; Medin and Shoben, 1988; Murphy and Medin, 1985). Not only does a concept give meaning to a stimulus or situation, but the situation evokes
meaning in the concept, and when more than one is active they evoke meaning in each other. It is this contextuality that makes them difficult to model when two or more arise together, or follow one
another, as in a creative construction such as a conjunction, invention, or sentence.
3. The Formalism
This story has a precedent. The same two problems–that of conjunctions of entities, and that of contextuality–arose in physics in the last century. Classical physics could not describe what happens
when quantum entities interact. According to the dynamical evolution described by the Schrödinger equation, quantum entities spontaneously enter an entangled state that contains new properties the
original entities did not have. To describe the birth of new states and new properties it was necessary to develop the formalism of quantum mechanics.
The shortcomings of classical mechanics were also revealed when it came to describing the measurement process. It could describe situations where the effect of the measurement was negligible, but not
situations where the measurement intrinsically influenced the evolution of the entity; it could not incorporate the context generated by a measurement directly into the formal description of the
quantum entity. This too required the quantum formalism.
First we describe the pure quantum formalism, and then we briefly describe the generalization of it that we apply to the description of concepts.
3.1 Pure Quantum Formalism
As in any mathematical model, we begin by cutting out a piece of reality and say this is the entity of interest, and these are its properties. The set of actual properties constitute the state of the
entity. We also define a state space, which delineates, given how the properties can change, the possible states of the entity. A quantum entity is described using not just a state space but also a
set of measurement contexts. The algebraic structure of the state space is given by the vector space structure of the complex Hilbert space: states are represented by unit vectors, and measurement
contexts by self-adjoint operators.
One says a quantum entity is entangled if it is a composite of subentities that can only be individuated by a separating measurement. When such a measurement is performed on an entangled quantum
entity, its state changes probabilistically, and this change of state is called quantum collapse.
In pure quantum mechanics, if H[1] is the Hilbert space representing the state space of the first subentity, and H[2] the Hilbert space representing the state space of the second subentity, the state
space of the composite is not the Cartesian product, as in classical physics, but the tensor product, i.e., H[1 ]Ä H[2]. The tensor product always generates new states with new properties,
specifically the entangled states. Thus it is possible to describe the spontaneous generation of new states with new properties. However, in the pure quantum formalism, a state can only collapse to
itself with a probability equal to one; thus it cannot describe situations of intermediate contextuality.
3.2 Generalized Quantum Formalism
The standard quantum formalism has been generalized, making it possible to describe entities with any degree of contextuality (Aerts 1993; Aerts and Durt 1994a, 1994b; Foulis and Randall 1981; Foulis
et al. 1983; Jauch 1968; Mackey 1963; Piron 1976, 1989, 1990; Pitowsky 1989; Randall and Foulis 1976, 1978). The generalized formalisms use lattice theory to describe the states and properties of
physical entities, and the result is referred to as a state property system. The approach is sufficiently general to be used to describe the different context-dependent states in which a concept can
exist, and the features of the concept manifested in these various states.
4. Incorporating Contextuality into a Theory of Concepts
One of the first applications of these generalized formalisms to cognition was modeling the decision making process. Aerts and Aerts (1996) proved that in situations where one moves from a state of
indecision to a decided state (or vice versa), the probability distribution necessary to describe this change of state is non-Kolmogorovian, and therefore a classical probability model cannot be
used. Moreover, they proved that such situations can be accurately described using these generalized quantum mathematical formalisms. Their mathematical treatment also applies to the situation where
the state of the mind changes from thinking about a concept to an instantiation of that concept, or vice versa. Once again, context induces a nondeterministic change of the state of the mind which
introduces a non-Kolmogorivian probability on the state space. Thus, a nonclassical (quantum) formalism is necessary.
In our approach, concepts are described using what to a first approximation can be viewed as an entangled states of exemplars, though this is not precisely accurate. For technical reasons (see Gabora
2001), the term potentiality state is used instead of entangled state. For a given stimulus, the probability that a potentiality state representing a certain concept will, in a given context,
collapse to another state representing another concept is related to the algebraic structure of the total state space, and to how the context is represented in this space. The state space where
concepts ‘live’ is not limited a priori to only those dimensions which appear to be most relevant; thus concepts retain in their representation the contexts in which they have, or even could
potentially be, evoked or collapsed to. It is this that allows their contextual character to be expressed. The stimulus situation plays the role of the measurement context by determining which state
is collapsed upon. Stimuli are categorized as instances of a concept not according to how well they match a static prototype or set of typical exemplars, but according to the extent to which they
correspond to, and thereby actualize or collapse upon, one the potential interpretations of the concept. (As a metaphorical explanatory aid, if concepts were apples, and the stimulus a knife, then
the qualities of the knife determine not just which apple to slice, but which direction to slice through it: changing the context in which a stimulus situation is embedded can cause a different
version of the concept to be elicited.) This approach has something in common with both prototype and exemplar theories. Like exemplar theory, concepts consist of exemplars, but the exemplars are in
a sense ‘woven together’ like a prototype.
5. Preliminary Theoretical Evidence of the Utility of the Approach
We present three sources of theoretical evidence of the utility of the approach.
5.1 A Proof that Bell Inequalities can be Violated by Concepts
The presence of entanglement can be tested for by determining whether correlation experiments on the joint entity violate Bell inequalities (Bell 1964). Using an example involving the concept cat and
specific instances of cats we proved that Bell inequalities are violated in the relationship between a concept, and specific instances of this concept (Aerts et al. 2000a; Gabora 2001). Thus we have
evidence that this approach indeed reflects the underlying structure of concepts.
5.2 Application to the Pet Fish Problem
We have applied the contextualized approach to the Pet Fish Problem (Aerts et al. 2000b; Gabora 2001). Conjunctions such as this are dealt with by incorporating context-dependency, as follows: (1)
activation of pet still rarely causes activation of guppy, and likewise (2) activation of fish still rarely causes activation of guppy. But now (3) pet fish causes activation of the potentiality
states petin the context of pet fish AND fish in the context of pet fish. Since for both, the probability of collapsing onto the state guppy is high, it is very likely to be activated. Thus we have a
formalism for describing concepts that is not stumped by a situation wherein an entity that is neither a good instance of A nor B is nevertheless a good instance of A AND B.
Note that whereas in representational approaches relations between concepts arise through overlapping context-independent distributions, in the present approach, the closeness of one concept to
another (expressed as the probability that its potentiality state will collapse to an actualized state of the other) is context-dependent. Thus it is possible for two states to be far apart from each
other with respect to a one context (for example ‘fish’ and ‘guppy’ in the context of just being asked to name a fish), and close to one another with respect to another context (for example ‘fish’
and ‘guppy’ in the context of both ‘pet’ and being asked to name a ‘fish’). Examples such as this are evidence that the mind handles nondisjunction (as well as negation) in a nonclassical manner
(Aerts et al. 2000b).
5.3 Describing Impossibilist Creativity
Boden (1990) uses the term impossibilist creativity to refer to creative acts that not only explore the existing state space but transform that state space. In other words, it involves the
spontaneous generation of new states with new properties. In (Gabora 2001) the contextual lattice approach is used to generate a mathematical description of impossibilist creativity using as an
example the invention of the torch. This example involves the spontaneous appearance of a new state (the state of mind that conceives of the torch) with a new property (the property of being able to
move fire).
6. Research in Progress
We are comparing the performance of the contextualized theory of concepts with prototype and examplar theories using previous data sets for typicality ratings, latency of category decision, exemplar
generation frequencies, category naming frequencies on everyday natural language concepts, such as ‘trees’, ‘furniture’, or ‘games’. The purpose of these initial investigations is to make sure that
the proposed formalism is at least as successful as representational approaches for the simple case of single concepts. Assuming this to be the case, we will concentrate our efforts on conjunctions
of concepts, since this is where the current approach is expected to supercede representational theories. We will re-analyze previously collected data for noun-noun conjunctions such as ‘pet fish’,
and relative clause conjunctions such as ‘pets that are also fish’ (Storms et al. 1996). A new study is being prepared which will compare the proposed approach with representational approaches at
predicting the results of studies using situations that are highly contextual. Typicality ratings for conjunctions will be compared with, not just their components, but with other conjunctions that
share these components. (Thus, for example, does ‘brainchild’ share features with ‘childbirth’ or ‘brainstorm’? Does ‘brainstorm’ share features with ‘birdbrain’ or ‘sandstorm’?)
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Inductive Data Analysis, 35-65. London: Academic Press.
Wisniewski, E. 1997. When Concepts Combine. Psychonomic Bulletin & Review 4:167-183.
Wisniewski, E. & Gentner, D. 1991. On the combinatorial semantics of noun pairs: Minor and major adjustments. In G. B. Simpson (Ed.), Understanding Word and Sentence, 241-284. Amsterdam: Elsevier. | {"url":"http://cogprints.org/2139/1/flairs.html","timestamp":"2014-04-18T23:17:56Z","content_type":null,"content_length":"27311","record_id":"<urn:uuid:3cb7a54a-b224-4e36-a4d6-be69546fee16>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00363-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Advantages of Hacking...
I dont know if I am posting in the right thread but here it goes...
I am having a debate at school with the title : hacking is bad .
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I have been racking my brains to look for points but not much have i gotten..
so any people out here to help me with it?~
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3D Math Primer for Game Programmers (Coordinate Systems)
In this article, I would like to provide a brief math primer for people who would like to get involved in game programming. This is not an exhaustive explanation of all the math theory that one will
have to know in order to be a successful game programmer, but it’s the very minimum amount of information that is necessary to know before you can begin as a game programmer.
This article assumes you have a minimum understanding vectors, and matrices. I will simply show applications of vectors and matrices and how they apply to game programming.
Coordinate Systems
Before we can talk about transformations, we must make a formal definition of what our coordinate system is. The default coordinate system used by DirectX is a left-handed coordinate system. The
default coordinate system used by OpenGL is a right-handed coordinate system.
The easiest way to remember what coordinate system you are using is to use your hands. If you point your thumb, your index finger, and your middle finger orthogonal to each other, then each finger
will point in the direction of a positive cardinal axis in your coordinate space. Using your left hand, your thumb points to the right (the $+X$ axis), your index finger points up (the $+Y$ axis),
and your middle finger points away from you (the $+Z$ axis).
Using your right-hand, your thumb (the $+X$ axis) and your index finger (the $+Y$ axis) still point in the same direction, but the middle finger (the $+Z$ axis) points in the opposite direction. If
we rotated our hand around the index finger (the $+Y$ axis) in order to get your thumb to point to the right, then your middle finger (the $+Z$ axis) would be pointing towards you.
Another important note to remember is the direction of rotation. If you point your thumb in the positive direction of the axis of rotation and curl your fingers around that imaginary axis, your
fingers will curl in the positive direction of rotation. If you do this on your left hand your fingers will curl in a clockwise direction when looking down at your thumb. However if you do this on
your right hand, your fingers will curl in a counter-clockwise direction when looking down at your thumb.
This can be seen in the images below:
The following table shows the direction of rotation for positive and negative rotations depending on the handedness of the coordinate system.
Left-hand coordinate system Right-hand coordinate system
Viewed From Positive Rotation Negative Rotation Positive Rotation Negative Rotation
The negative end looking toward the positive end Counter-Clockwise Clockwise Clockwise Counter-Clockwise
The positive end looking toward the negative end Clockwise Counter-Clockwise Counter-Clockwise Clockwise
Coordinate Spaces
In a 3D game engine, we usually deal with several different coordinates spaces. Among the most commonly used spaces are Object space, World space, Inertial space, and Camera space.
Object Space
In object space, also known as local space, or modeling space, an object’s vertices are expressed relative to the object that they describe. That is, the way the artist intended for them to be
displayed if you didn’t move the object from the origin.
The image below shows an example of an object in object space. As you can see from the image, the object is placed at it’s relative origin.
World Space
The world coordinate space is the global coordinate space for which all other object spaces are described. The image below shows an object described in world space. Notice how the object’s world
transformation places it away from the world-origin by some translation, and rotation.
Inertial Space
The inertial space is the “halfway” space between object space and world space. The origin of inertial space has the same origin as object space, and the axes of inertial space are parallel to that
of the world space axes.
To transform a point from object space to inertial space requires only a rotation and to transform a point from inertial space to world space requires only a translation.
The image blow shows an example of inertial space.
Camera Space
Camera space is the coordinate space that is associated with the observer. Camera space is considered to be the origin and orientation of what we are looking at. The camera’s coordinate axis
usually assume the positive $\mathbf{X}$ axis points right, the positive $\mathbf{Y}$ axis points up, and in a left-handed coordinate system, the positive $\mathbf{Z}$ axis points forward, or at the
scene. In a right-handed coordinate system, the $\mathbf{Z}$ axis is reversed, so the negative $\mathbf{-Z}$ axis points forward, or at the object in our scene. A more common name for these camera
axes are the “Right”, “Up”, and “At” axes.
The camera space transformation together with the projection transformation is useful for answering such questions as:
1. Is an object completely in view of the camera, partially in view, or not in view at all,
2. Is one object closer to the camera than the other,
3. Is an object directly in front of, above, below, to the left, or to the right of the camera.
Combining Coordinate Spaces
In 3D computer graphics, coordinate spaces are described using a homogeneous coordinate system. A homogeneous coordinate system allows us to represent all of our affine transformations (translation,
rotation, scale, and perspective projection) in a similar way so they can easily be combined into a single representation.
Any number coordinate spaces can be combined using matrix multiplication which results in a single matrix that can be applied to all the vertices of an object.
Even multiple world coordinate spaces can be combined in order to derive a final coordinate space that describes the location of all of our vertices in an object. This is useful for nested
coordinate spaces where the position of an object is expressed relative to a “parent” object. When the parent object’s world transform is changed, the transform of the child object is also changed
implicitly. Using this method, complex scenes can be constructed from several smaller scenes and placed into the larger scene to create a complete world.
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What Is the Last Largest Number You Can Count To?
If you started to count today and continued counting day and night, without stopping, for the rest of your life, you would never get to a last number, because there isn’t one.
Mathematicians tells us that no matter how large a number you would get to, there would always still be one larger.
This idea is important in science and has a special name, infinity, and its own special symbol, ∞.
1. Anonymous says
what about a million
or A MILLION AND ONE
2. Anonymous says
The largest number one can count to depends how how fast you can count and how long you have until you die. It has nothing to do with infinity.
3. Anonymous says
ur really stupid there are numbers bigger than a million and one what about a billion smarty
4. Anonymous says
Y two are sooooooo stupid! ITS INFINITY!!!!!!!!!!!!!!!!!!!! YOU STUPID BRAIN BOOGERS!!!!!!!!!!!!!
5. Anonymous says
He thinks infinity is a number. Haha
6. Kelli Winstead says
What about a milliomn and two? And, for those of you who do not know how to count very well, well I just got to say that numbers always end. No matter what. What about a zillion, by the way.
7. annonymous says
what about 2 zillion????
8. Your mum says
the biggest number ever counted to is no idea
9. jam5 says
10. martyn says
eh? the largest number YOU can count to is the number you reach when you die after counting all life, how is that stupid? maybe you are too stupid to understand the simple logic!
the largest number in the world is NaN because numbers are infinite as the universe
11. dave ham says
jerk don’t be mean it is two I think
12. someone says
your not so smart are you JAM5……..REALLY JAM5?
13. hfgydfhguh says
amen sister/brother
14. hfgydfhguh says
a googolplex is the largent # it has a 1 and a thousand zeros you nincom
15. Tisk tisk.... says
All of you’s are idiots….. the highest number known to man is the Googleplex….
16. My word..... says
Really people? The question is what is the largest number that YOU can count to, not what is the largest number!!!!!!!! The largest number you can count to is based on you. GOSH.
17. julimans grawnedds says
well hows abouts a googolplex and one missterr smatry pantss? one+a googloplex=googolplex and 1
18. To the dumb ones, says
NO ONE CARES WHAT THE LARGEST NUMBER IS. YOU CAN ALWAYS, YES ALWAYS ADD 1 TO WHATEVER NUMBER YOU’VE GOT! SO STOP YELLING AT EACH OTHER!
19. To the dumb ones, says
Oh yeah? What about 2 googleplexes? Or a googleplex of googleplexes? THERE IS NO LARGEST NUMBER!
20. dragon says
just google – Milli-Millillion ;)
It is 10 followed by 3000003 zeroes…probably the largest number with a name :D
21. gooolplex says
10^2 = 100
10^100 = Googol
10^Googol = Googolplex
10^Googolplex = ???
3155760000 seconds in 100 years if you live that long so if u count every second then you get that far at best.
In words: three billion one hundred and fifty five million seven hundred and sixty thousand.
22. anonymous says
actually numbers has no end example if you’ve thought of something and think it is the last number what happens if add 1 to it and add another 1 then another and so on
23. no end says
some things we witness in life are endless, like space and indeed numbers, its just hard for some people to comprehend that not everything has an END.
It is built in to us through day to day life that everything has an end.
24. corydee says
there’s no last number
25. Take a guess says
I dont know about counting. But I would say easiest way to explain the last number is the number 9, good old simple 9. Reason 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, then you get 10 and that is only the
numbers 1 and 0, so after 9 numbers just start dobling up.
Also lets say 100 was the last number, the “1″ and the two “0″s can be 9′s (999)
Or the number 1000000 = 9999999. So with that 9 is and always will be the LAST NUMBER.
Thank you and good day.
26. josh bradley says
i’ve heard of a such number that was called like septuaterillion… 1 followed by 7 billion 0′s… googol and googolplex aren’t the real names of the numbers… i know for sure that googolplex is 1 &
600,000 0′s…
27. josh bradley says
actually a googolplex is a 1 with a googol 0′s… the next number name after sextillion (1 with 600,0000 0′s)
28. Sean Martinez says
Tell me what’s the last number
29. Juan says
There is no last number … they would always keep going … its just that the names get so complex, that people havent even made a name for that number. So all you jerks saying what the last
number?, or OMG the last number is infinity. … Infinity isnt a number …. its more like a word that describes infinite stuff .. heres an example according to numbers. Numbers are INFINITE.
30. nonywise says
sepuatertrillion, right? What bout octatrillion i.e 800,000 0′s. Use your brain.
31. IH8NIGS says
u are all dumb quit worrying about numbers and go ride a bike haha
32. Gags says
Its like asking the question, Where does space end?
33. zayn says
i can tell you what the largest number it is number 9 because you have to add a lot of numbers because 9 is big and after 2.9 you have to go to 3.0 so nine is the biggest number thats the proof
34. adam says
no, there is no last number.
35. mythbuster says
All stupid fools! There is no end or beginning to a number… think of it…go as far back your imagination can take you (-)???? (if you have imagination) and go forward as far as your imagine takes
you +???… there lies your number… and behold! there is no beginning or end… so stop bothering about it. The very reason you discuss this simple, common sense topic proves your “insanity”!!!
Infinity is just an imagined word to fill up this vacuum, but that’s also your imagination… got it, fools!!!
36. Jean says
I was researching this for my 7 year old….he always tells me that he loves me the last number. And that he loves God the second to last number. This morning he was wanting to know what the last
number was….so sometimes we look for things for many reasons. I’m glad I could do this with my son.
37. B.A says
space and time never ends and no one can catch up with it so just dont worry
38. Ysureican Fogarty says
The largest number one can count to is “One Google”. That is one million followed by one millioon zeros. At that point it becomes impractical to use our number system and the speed of light is
used to measure distances or measurements. Please note the speed of light is a vast number thus is shortened and squared etc. will be utilized.
by ysureican
39. nathan ciezki says
the biggest number is infinity to the infinity power
40. Yasir says
There’s no last number duh I can’t believe you didn’t know that yasir out
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finite dimensional real division algebras
up vote 12 down vote favorite
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using methods from
algebraic topology, such as K-Theory.
Now for any given natural number $n$ the existence of such an algebra of dimension $n$ is expressible as an assertion $\phi_n$ in the first-order language of field theory. Since the theory $RCF$ of
real closed fields is complete, it follows from the theorem above that $RCF \vdash \neg \phi_n$ for all $n\not\in$ {1,2,4,8}. Here the universal quantifier on $n$ is in the meta-theory: we might say
that for each $n$ there is an elementary proof of $\phi_n$.
Given such a theorem scheme, one might wonder whether there might be a uniform elementary proof. Informally this could mean a proof by induction on the relevant complexity parameter: for example,
$$RCF \vdash \mbox{ any degree } d \mbox{ polynomial has at most } d \mbox{ roots}.$$ I would like to imagine that there is some first order-theory which suitably contains both RCF and Peano
Arithmetic (in particular, so as to enable discussion of finite sequences of field elements) in which the assertion $$\forall n \;\phi_n\leftrightarrow(n=1 \vee n=2 \vee n=4 \vee n=8)$$ can be
legfitimately formalized. Are there standard constructions for supporting finite sequences? If so, it should follow from completeness of RCF that this assertion is equivalent (within such a larger
theory) to a sentence $\Phi$ in the language of arithmetic. As noted above, via difficult results from topology, $\Phi$ is true in the standard model of Peano Arithmetic. Consequently, it makes sense
to ask whether $\Phi$ is provable within Peano Arithmetic.
Some questions:
(1) Can such a recipe be formalized, and does it reasonably capture the notion of "uniform elementary proof" or "purely algebraic" proof for such theorem schemes? Here I am not necessarily claiming
that these conjectural notions are the same.
(2) In the given example of the 1,2,4,8 theorem, do we expect $\Phi$ to be provable in Peano Arithmetic?
Perhaps I have been looking in the wrong places, but all I have managed to find are a few comments by Kreisel about "unwinding", on pages 67-68 of this note: http://elib.mi.sanu.ac.rs/files/journals/
The situation could be compared with what is known in the special cases of commutative division algebras (dimensions 1,2) and associative division algebras (dimensions 1,2,4). Hopf's proof of the
(1,2) theorem also uses some topology, namely that the $n$-dimensional sphere and $n$-dimensional projective space are not homeomorphic when $n>1$; in fact it suffices to show that a specific map
between these spaces is not a homeomorphism. Perhaps there is an elementary way to formulate this consideration? On the other hand, there is a different and "purely algebraic" proof, via Bezout's
Theorem. I don't have the reference at hand, but it there is a citation (froom the 1950s, as I recall) in the Springer-Verlag book Numbers (Ebbinghaus et. al.). I've seen this proof dismissed as
unreadable or unenlightening, but when I examined it years ago it seemed like it might qualify. The Frobenius proof of the (1,2,4) theorem is quite evidently purely algebraic, as is the later
extension (1,2,4,8) to alternative division algebras.
1 A commutative division algebra is also known as a field. Thus the $(1,2)$ theorem is equivalent to the algebraic closure of $\mathbb C$. I might hope the standard Galois theory proof of this fact
can be done entirely in RCF theory. – Will Sawin Aug 25 '12 at 21:04
We certainly know that the quadratic extension of a real closed field is algebraically closed. See for instance ncatlab.org/nlab/show/fundamental+theorem+of+algebra – Todd Trimble♦ Aug 26 '12 at
1 Is it obvious why there is a first-order sentence in the language of fields which asserts "There is a division algebra of dimension n?" I'm not sure I believe this. – Noah S Aug 26 '12 at 4:51
1 Also, re: David's comment, it seems to me that second-order arithmetic (which is still a first-order theory) might be a better setting for this, since then one can easily say "there is a division
algebra of dimension n." – Noah S Aug 26 '12 at 4:55
1 >A commutative division algebra is also known as a field. In case it is also associative. But there are nonassociative examples: see "When is R^2 a Division Algebra? Steven C. Althoen and Lawrence
D. Kugler The American Mathematical Monthly Vol. 90, No. 9 (Nov., 1983) (pp. 625-635) – Adam Epstein Jan 8 '13 at 20:53
show 6 more comments
1 Answer
active oldest votes
If you ask for a purely algebraic proof, algebraic K-theory is (almost) OK. Indeed for a compact Hausdorff space $X$, it's topological K-theory $K(X)$ is the same as algebraic K-theory $K_0
(C(X;\mathbb{C}))$. So rewrite the proof of Bott-Milnor-Kervaire we will get a (almost) purely algebraic proof.
EDIT(More detail):
The method topologists solve this problem (according to my knowledge) is constructing an complex line bundle $\xi$ over real projective space $\mathbb{R}\mathbb{P}^{n-1}$, and claim that $n
up vote 1 (\xi-\varepsilon)=0$ in $K(\mathbb{R}\mathbb{P}^{n-1})$, where $\varepsilon$ is the (complexification of) trivial line bundle over $\mathbb{R}\mathbb{P}^{n-1}$. Since we can compute $K(\
down vote mathbb{R}\mathbb{P}^{n-1})=\mathbb{Z}/2^{[\frac{n-1}{2}]}\mathbb{Z}$ with a generator $\xi-\varepsilon$, we must have $2^{[\frac{n-1}{2}]}|n$, and thus $n=1,2,4,8$.
Now let $A=C(\mathbb{R}\mathbb{P}^{n-1};\mathbb{C})$, the ring of continuous functions from $\mathbb{R}\mathbb{P}^{n-1}$ to $\mathbb{C}$. As above we can construct a finite generated
projective module $\Gamma(\xi)$ over $A$ and so on. The idea of topological proof now becomes purely algebraic except that the concept of continuous.
I've wondered whether algebraic K-Theory might provide a useful workaround, but I don't understand enough to have any idea what that would really mean. Could you elaborate? – Adam Epstein
Jul 30 '13 at 6:59
@Adam Epstein I have no idea whether it is exactly what you want, but I add something. – asatzhh Aug 1 '13 at 6:16
add comment
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Re: The natural spectrogram
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Re: The natural spectrogram
At 11:09 29.01.2004 -0800, Julius Smith wrote:
>At 01:48 AM 1/28/2004, Eckard Blumschein wrote:
>>...So far I can neither imagine the
>>STFT itself to be natural nor a spectrogram based on it. Wouldn't this
>>require to naturally choose size of the window?
>Yes -- and as a function of frequency. We normally call it a
>"multiresolution" STFT.
In this sense and due to the absence of arbitrary windows, both the actual
cochlear functions and the suggested natural spectrogram are distinguished
by a steplessly sliding rather than just a stepping 'multi' resolution. Use
of STFT at least requires an arbitrary decision how many windows to choose
for every moment. Perhaps there is no natural preference for a particular
variant of choices.
>>Wouldn't one have to decide further arbitrary parameters like the degree
>>of overlap?
>This is just a sampling-rate issue. If computational cost is no object,
>one can simply choose maximum overlap (i.e., a "sliding FFT" instead of a
>"hopping FFT"). On the other hand, FFT filter banks can usually be
>downsampled quite a lot and still give equivalent end results. In this
>context, your window is your anti-aliasing filter for
>downsampling. Reference: Jont B. Allen, "Short Term Spectral Analysis,
>Synthesis, and Modification by Discrete Fourier Transform", IEEE ASSP-25(3).
Cochlea is not subject to a sample rate, and the natural spectrogram adapts
to the given sampled input in a similar manner as does the 'sliding' FFT.
Neither cochlea nor the natural spectrogram require an anti-aliasing filter.
You are quite right: The wider the frequency range, the more it makes sense
to downsample the input to the low-frequency window. Isn't it a calamity:
Perhaps, there is no way of sliding down-sampling. Even with most powerful
computers, a multidimensional field of FFTs which might be further
complicated by a sophisticated structure of downsampling looks anything but
practical and parsimonious. Its first dimension are the frequency steps of
multiresolution, its second one are the temporal steps of overlap, its
third one are frequency dependent ratios of downsampling. Jont's paper
dates back to 1977 when one could not yet imagine PCs to do that job. So I
consider it of historical value.
>> Doesn't any usual spectrogram incompletely represent the information?
>STFTs are normally invertible, in my experience, even in the presence of
>aliasing due to downsampling (it gets canceled in the reconstruction). The
>classic spectrogram discards phase, so it is not exactly invertible. Of
>course, it is well known that phase can be reconstructed from STFT
>magnitude to a large extent for typical signals and analysis conditions.
Cochlea as well as the natural spectrogram are not subject to such
consequences of inappropriate theory.
>>Isn't the usual spectrogram subject to the notorious trade-off beween
>>spectral and temporal resolution?
>Well sure, but we can let the human ear tell us where to be on that
This might be not quite correct for several reasons. The smallest product
delta t times delta f of hearing is much better than according to the
uncertainty principle.
Aren't about 10 microseconds and 1 Hz realistic? The product is 10^-5 << 1.
Frequency resolution of the natural spectrogram is not at all restricted,
in principle.
>>Was there any physiological justification for STFT which could include the
>>Is there close similarity to measurement of BM motion and neural pattern?
>I don't understand the first question. My understanding of rectification
>that this is the nature of how the hair cells respond to basilar membrane
>vibration. Firing increases when the membrane pushes one way, but not the
That is my understanding of rectification, too.
>The STFT implements a filter bank, and the output of that filter
>bank can be rectified accordingly (applied to real time-domain signals at
>the STFT filter-bank output, of course).
The latter is the point. Complex FT including STFT does not at all deliver
a real time-domain signal but magnitude and phase. The usual spectrogram
shows magnitude vs. time. A magnitude cannot be rectified. Therefore, the
usual spectrogram fails to convey the information responsible for audible
effects of polarity, eg between positive and negative clicks.
The natural spectrogram is more natural in any comparison with the usual
one because it is based on Fourier cosine transform (FCT) which directly
provides the input to rectification.
Proponents of complex FT might claim the FCT to be just the real part of
FT. This is formally largely correct. However, it ignores several important
flaws arising from arbitrary preconditions of complex calculus. Let's skip
the two most basic arbitrary choices (origin and sign of imaginary part).
Complex FT always presumes tacit introduction of redundancy. The most
'correct' input to the FT in case of what I suggest to call an 'effectual
signal' fills a window that is located symmetrical with respect to zero but
padded with zeros for the time to come in which the signal is unknown. The
word 'effectual' indicates correspondence to the so called causal signal.
The effectual signal differs from a simply time-mirrored (anti-causal) one.
The zeros introduce two mutually cancelling fictive components each of
which alone would violate causality. That's why the traditional spectrogram
exhibits non-causality. Being obvious nonsense, non-causal output before
any input is the more strikingly to be seen the wider the window has been
chosen. Any Fourier transform of a causal signal or an effectual signal
shows Hermitian symmetry. In other words, its real part is symmetrical over
frequency. Negative frequency does not have any particular physical
meaning. It is just an artefact of complex calculus.
>I suppose you're posting to the right list!
In 1844, Ohm dismissed Seebeck's observation. He supposed that a missing
fundamental cannot be heared. Let's be open to the insight that cochlea
performs a real-valued rather than complex-valued frequency analysis. Then
we are in position to deal with the further steps of physiological signal
processing on a more sound basis. As a first result, I offered a 'joint
autocorrelation' hypothesis. Hopefully, it will reconcile Peter and
Christian. Also it could, for the first time, plausibly tell to Ohm why
Seebeck was right. To this list, Chen-Gia Tsai posted his observation of a
pitch at 9f0/4 (if I recall correctly). I imagine hundreds of experts here
on the lurk for something new. Of course, corrections are always and
anywhere unwelcome. I do not intend to hurt anybody. If someone feels
offended I apologize for that. I will sincerely try going on responding
privately to all objections and request. | {"url":"http://www.auditory.org/mhonarc/2004/msg00096.html","timestamp":"2014-04-16T20:12:48Z","content_type":null,"content_length":"11650","record_id":"<urn:uuid:91a9b3ba-c9ff-400b-8f88-0089f2036cfe>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00157-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cryptology ePrint Archive: Report 2009/305
Improved generic algorithms for 3-collisionsAntoine Joux and Stefan LucksAbstract: An $r$-collision for a function is a set of $r$ distinct inputs with identical outputs. Actually finding
$r$-collisions for a random map over a finite set of cardinality $N$ requires at least about $N^{(r-1)/r} $ units of time on a sequential machine. For $r$=2, memoryless and well-parallelisable
algorithms are known. The current paper describes memory-efficient and parallelisable algorithms for $r \ge 3$. The main results are: (1)~A sequential algorithm for 3-collisions, roughly using memory
$N^\alpha$ and time $N^{1-\alpha}$ for $\alpha\le1/3$. I.e., given $N^{1/3}$ units of storage, on can find 3-collisions in time $N^{2/3}$. Note that there is a time-memory tradeoff which allows to
reduce the memory consumption. (2)~A parallelisation of this algorithm using $N^{1/3}$ processors running in time $N^{1/3}$. Each single processor only needs a constant amount of memory. (3)~An
generalisation of this second approach to $r$-collisions for $r \ge3$: given $N^s$ parallel processors, on can generate $r$-collisions roughly in time $N^{((r-1)/r)-s}$, using memory $N^{((r-2)/r)-s}
$ on every processor. Category / Keywords: foundations / multicollision, random mapsDate: received 23 Jun 2009Contact author: Antoine Joux at m4x orgAvailable format(s): PDF | BibTeX Citation
Version: 20090624:072015 (All versions of this report) Discussion forum: Show discussion | Start new discussion[ Cryptology ePrint archive ] | {"url":"http://eprint.iacr.org/2009/305","timestamp":"2014-04-21T04:36:00Z","content_type":null,"content_length":"2754","record_id":"<urn:uuid:433f2407-d350-4038-b6bc-7b686202e010>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00421-ip-10-147-4-33.ec2.internal.warc.gz"} |
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If 4! = 1x2x3x4 and is called 4 factorial, what is the sum of the numbers e.g. 4(?)= (1+2+3+4) called?
I know how to easily calculate the answer. Is there an operator name for it?
6 factorial is 6! = 1×2x3×4x5×6 = 720
6 summit is 6| = 1+2+3+4+5+6 = 21
Obviously I made up the name “summit” and symbol ”|”
Is there a commonly used term and symbol for this? I can use the summation symbol but that is unwieldy.
Using Fluther
Using Email | {"url":"http://www.fluther.com/154922/if-4-1x2x3x4-and-is-called-4-factorial-what-is/send/","timestamp":"2014-04-18T22:22:58Z","content_type":null,"content_length":"19181","record_id":"<urn:uuid:b2aa9b49-8137-484d-8897-eb06f8d4ac79>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00429-ip-10-147-4-33.ec2.internal.warc.gz"} |
41A15 Spline approximation
5 search hits
Multiscale Modeling of CHAMP-Data (2003)
W. Freeden V. Michel
The following three papers present recent developments in multiscale gravitational field modeling by the use of CHAMP or CHAMP-related data. Part A - The Model SWITCH-03: Observed orbit
perturbations of the near-Earth orbiting satellite CHAMP are analyzed to recover the long-wavelength features of the Earth's gravitational potential. More precisely, by tracking the low-flying
satellite CHAMP by the high-flying satellites of the Global Positioning System (GPS) a kinematic orbit of CHAMP is obtainable from GPS tracking observations, i.e. the ephemeris in cartesian
coordinates in an Earth-fixed coordinate frame (WGS84) becomes available. In this study we are concerned with two tasks: First we present new methods for preprocessing, modelling and analyzing
the emerging tracking data. Then, in a first step we demonstrate the strength of our approach by applying it to simulated CHAMP orbit data. In a second step we present results obtained by
operating on a data set derived from real CHAMP data. The modelling is mainly based on a connection between non-bandlimited spherical splines and least square adjustment techniques to take into
account the non-sphericity of the trajectory. Furthermore, harmonic regularization wavelets for solving the underlying Satellite-to-Satellite Tracking (SST) problem are used within the framework
of multiscale recovery of the Earth's gravitational potential leading to SWITCH-03 (Spline and Wavelet Inverse Tikhonov regularized CHamp data). Further it is shown how regularization parameters
can be adapted adequately to a specific region improving a globally resolved model. Finally we give a comparison of the developed model to the EGM96 model, the model UCPH2002_02_0.5 from the
University of Copenhagen and the GFZ models EIGEN-1s and EIGEN-2. Part B - Multiscale Solutions from CHAMP: CHAMP orbits and accelerometer data are used to recover the long- to medium- wavelength
features of the Earth's gravitational potential. In this study we are concerned with analyzing preprocessed data in a framework of multiscale recovery of the Earth's gravitational potential,
allowing both global and regional solutions. The energy conservation approach has been used to convert orbits and accelerometer data into in-situ potential. Our modelling is spacewise, based on
(1) non-bandlimited least square adjustment splines to take into account the true (non-spherical) shape of the trajectory (2) harmonic regularization wavelets for solving the underlying inverse
problem of downward continuation. Furthermore we can show that by adapting regularization parameters to specific regions local solutions can improve considerably on global ones. We apply this
concept to kinematic CHAMP orbits, and, for test purposes, to dynamic orbits. Finally we compare our recovered model to the EGM96 model, and the GFZ models EIGEN-2 and EIGEN-GRACE01s. Part C -
Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, EGM96: Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been
successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical
harmonic models of the Earth's gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise and
these shall be made available to other interested groups. These wavelet coefficients allow the reconstruction of the wavelet approximations. Different types of wavelets are considered:
bandlimited wavelets (here: Shannon and Cubic Polynomial (CP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances
are given. The data format of the wavelet coefficients is also included.
Harmonic Spline-Wavelets on the 3-dimensional Ball and their Application to the Reconstruction of the Earth´s Density Distribution from Gravitational Data at Arbitrarily Shaped Satellite Orbits
Martin J. Fengler Dominik Michel Volker Michel
We introduce splines for the approximation of harmonic functions on a 3-dimensional ball. Those splines are combined with a multiresolution concept. More precisely, at each step of improving the
approximation we add more data and, at the same time, reduce the hat-width of the used spline basis functions. Finally, a convergence theorem is proved. One possible application, that is
discussed in detail, is the reconstruction of the Earth´s density distribution from gravitational data obtained at a satellite orbit. This is an exponentially ill-posed problem where only the
harmonic part of the density can be recovered since its orthogonal complement has the potential 0. Whereas classical approaches use a truncated singular value decomposition (TSVD) with the
well-known disadvantages like the non-localizing character of the used spherical harmonics and the bandlimitedness of the solution, modern regularization techniques use wavelets allowing a
localized reconstruction via convolutions with kernels that are only essentially large in the region of interest. The essential remaining drawback of a TSVD and the wavelet approaches is that the
integrals (i.e. the inner product in case of a TSVD and the convolution in case of wavelets) are calculated on a spherical orbit, which is not given in reality. Thus, simplifying modelling
assumptions, that certainly include a modelling error, have to be made. The splines introduced here have the important advantage, that the given data need not be located on a sphere but may be
(almost) arbitrarily distributed in the outer space of the Earth. This includes, in particular, the possibility to mix data from different satellite missions (different orbits, different
derivatives of the gravitational potential) in the calculation of the Earth´s density distribution. Moreover, the approximating splines can be calculated at varying resolution scales, where the
differences for increasing the resolution can be computed with the introduced spline-wavelet technique.
Time-Dependent Cauchy-Navier Splines and their Application to Seismic Wave Front Propagation (2006)
Paula Kammann Volker Michel
In this paper a known orthonormal system of time- and space-dependent functions, that were derived out of the Cauchy-Navier equation for elastodynamic phenomena, is used to construct reproducing
kernel Hilbert spaces. After choosing one of the spaces the corresponding kernel is used to define a function system that serves as a basis for a spline space. We show that under certain
conditions there exists a unique interpolating or approximating, respectively, spline in this space with respect to given samples of an unknown function. The name "spline" here refers to its
property of minimising a norm among all interpolating functions. Moreover, a convergence theorem and an error estimate relative to the point grid density are derived. As numerical example we
investigate the propagation of seismic waves.
Splines on the 3-dimensional Ball and their Application to Seismic Body Wave Tomography (2007)
Abel Amirbekyan Volker Michel
In this paper we construct spline functions based on a reproducing kernel Hilbert space to interpolate/approximate the velocity field of earthquake waves inside the Earth based on traveltime data
for an inhomogeneous grid of sources (hypocenters) and receivers (seismic stations). Theoretical aspects including error estimates and convergence results as well as numerical results are
The Application of Reproducing Kernel Based Spline Approximation to Seismic Surface and Body Wave Tomography: Theoretical Aspects and Numerical Results (2007)
Abel Amirbekyan
The main aim of this work was to obtain an approximate solution of the seismic traveltime tomography problems with the help of splines based on reproducing kernel Sobolev spaces. In order to be
able to apply the spline approximation concept to surface wave as well as to body wave tomography problems, the spherical spline approximation concept was extended for the case where the domain
of the function to be approximated is an arbitrary compact set in R^n and a finite number of discontinuity points is allowed. We present applications of such spline method to seismic surface wave
as well as body wave tomography, and discuss the theoretical and numerical aspects of such applications. Moreover, we run numerous numerical tests that justify the theoretical considerations. | {"url":"https://kluedo.ub.uni-kl.de/solrsearch/index/search/searchtype/collection/id/10689","timestamp":"2014-04-18T01:14:21Z","content_type":null,"content_length":"36417","record_id":"<urn:uuid:189dc683-48cb-4db8-a7e6-86dfe021b2a6>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00210-ip-10-147-4-33.ec2.internal.warc.gz"} |
Find the minimum cost Sum Of Prods and Product of Sum forms for the function
September 24th 2013, 09:30 AM #1
Sep 2013
Ramallah, Palestine
Find the minimum cost Sum Of Prods and Product of Sum forms for the function
Find the minimum-cost Sum of Products and Product of Sums forms for the function: f(x1,x2,x3( = $\sum$m(1,2,3,5)
I know how to get the truth table etc. But they want me to use a karnaugh map to get the answer. How?
Last edited by shamieh; September 24th 2013 at 09:32 AM.
Re: Find the minimum cost Sum Of Prods and Product of Sum forms for the function
How do you measure the cost of a Boolean expression? There are several ways to do it. Also, there is no way around learning how to use Karnaugh maps. If you need a reference or have concrete
questions about how they work, feel free to ask, but we can't learn Karnaugh maps for you.
September 24th 2013, 10:17 AM #2
MHF Contributor
Oct 2009 | {"url":"http://mathhelpforum.com/discrete-math/222246-find-minimum-cost-sum-prods-product-sum-forms-function.html","timestamp":"2014-04-21T12:19:29Z","content_type":null,"content_length":"33436","record_id":"<urn:uuid:eb374d07-bda9-4df6-8ba2-7fa95280d7c5>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00064-ip-10-147-4-33.ec2.internal.warc.gz"} |
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|dw:1360298829035:dw| find this vector
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your x axis coordinate is incorrect
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it should be (4,0)
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yes , thanks
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That is a beautiful triangle you have there.
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then use mid point formula your gonna get, (2,1) as coordinates of bisector n then your vector is , V= 2i +j
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modphys doesn't want the mid point. He want the vector that's perpendicular.
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well thats perpendicular bisector that means dividing equally that means mid point
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What I would do, is write the line in the form \(y=-x/2+2\). The slope of the intersecting line will be \[-\frac{1}{-\frac{1}{2}}=2.\] So you have a line that looks like \(y=2x\) as the line that
intersects. Set them equal to each other to get \(2x=-x/2+2\implies 5x/2=2\implies x=1\).
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Plug that in again, \(x=1\implies y=2(1)=2\), so the point of intersection is \((1,2)\).
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Which in turn means the vector is \[\vec{i}+2\vec{j}\]
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I got same answer but here what my book did
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If \(x=.8\), then the vector is \[.8\vec{i}+1.4\vec{j}\] Taking the dot product of that with the vector \[4\vec{i}-2\vec{j},\] we get \[.8\cdot4-2\cdot1.4=0.4\neq0\]so that solution is wrong.
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|dw:1360299909016:dw| just check the magnitude of vector A : (4*2)/sqrt(4^2+2^2) = 8/sqrt(20) = 8/2sqrt(5) = 4/5 * sqrt(5) 0.8 and 1.6 has the same magnitude
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Just as in your book, we can say the direction vector B (from pt (0,2) to (4,0) is (4,-2) let vector A be perpendicular to B. A dot B = 0 (x,y) dot (4,-2) =0 we get 4x = 2y pick a nice number for
x, like 1. then y =2 vector A = (1,2) is perpendicular to vector B make A unit length, by dividing by its length. |A|= sqrt(5) Au = (1,2)/sqrt(5) now to get the length to scale Au so it reaches
the line between (0,2) and (4,0), we notice that the vector (4,0) projected onto Au has the correct length Au dot (4,0) = 4/sqrt(5) The vector we want is 4/sqrt(5) * (1,2)/sqrt(5) = 4/5 * (1,2)
or (0.8, 1.6)
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is replying to Can someone tell me what button the professor is hitting...
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Transport properties of two finite armchair graphene nanoribbons
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Nanoscale Res Lett. 2013; 8(1): 1.
Transport properties of two finite armchair graphene nanoribbons
In this work, we present a theoretical study of the transport properties of two finite and parallel armchair graphene nanoribbons connected to two semi-infinite leads of the same material. Using a
single Π-band tight binding Hamiltonian and based on Green’s function formalisms within a real space renormalization techniques, we have calculated the density of states and the conductance of these
systems considering the effects of the geometric confinement and the presence of a uniform magnetic field applied perpendicularly to the heterostructure. Our results exhibit a resonant tunneling
behaviour and periodic modulations of the transport properties as a function of the geometry of the considered conductors and as a function of the magnetic flux that crosses the heterostructure. We
have observed Aharonov-Bohm type of interference representing by periodic metal-semiconductor transitions in the DOS and conductance curves of the nanostructures.
Keywords: Graphene nanostructures, Transport properties, Magnetic field effects
Graphene is a single layer of carbon atoms ordered in a two-dimensional hexagonal lattice. In the literature, it is possible to find different experimental techniques in order to obtain graphene such
as mechanical peeling, epitaxial growth or assembled by atomic manipulation of carbon monoxide molecules over a conventional two-dimensional electron system at a copper surface [1-4]. The physical
properties of this crystal have been studied over the last 70 years; however, the recent experimental breakthroughs have revealed that there are still a lot of open questions, such as time-dependent
transport properties of graphene-based heterostructures, the thermoelectric and thermal transport properties of graphene-based systems in the presence of external perturbations, the thermal transport
properties of graphene under time-dependent gradients of temperatures, etc.
On the other hand, graphene nanoribbons (GNRs) are quasi one-dimensional systems based on graphene which can be obtained by different experimental techniques [5-8]. The electronic behaviour of these
nanostructures is determined by their geometric confinement which allows the observation of quantum effects. The controlled manipulation of these effects, by applying external perturbations to the
nanostructures or by modifying the geometrical confinement [9-13], could be used to develop new technological applications, such as graphene-based composite materials [14], molecular sensor devices [
15-17] and nanotransistors [18].
One important aspect of the transport properties of these quasi one-dimensional systems is the resonant tunneling behaviour which, for certain configurations of conductors or external perturbations,
appears into the system. It is has been reported that in S- and U-shaped ribbons, and due to quasi-bound states present in the heterostructure, it is possible to obtain a rich structure of resonant
tunneling peaks by tuning through the modification of the geometrical confinement of the heterostructure [19]. Another way to obtain resonant tunneling in graphene is considering a nanoring structure
in the presence of external magnetic field. It has been reported that these annular structures present resonance in the conductance at defined energies, which can be tuned by gate potentials, the
intensity of the magnetic field or by modifying their geometry [20]. From the experimental side, the literature shows the possibility of modulating the transport response as a function of the
intensity of the external magnetic field. In some configuration of gate potential applied to the rings, it has been observed that the Aharonov-Bohm oscillations have good resolution [21-23].
In this context, in this work, we present a theoretical study of the transport properties of GNR-based conductors composed of two finite and parallel armchair nanoribbons (A-GNRs) of widths N[d] and
N[u], and length L (measured in unit cell units), connected to two semi-infinite contacts of width N made of the same material. We have thought this system as two parallel ‘wires’ connected to the
same reservoirs, whether the the leads are made of graphene or another material. This consideration allows us to study the transport of a hypothetical circuit made of graphene ‘wires’ in different
scenarios. A schematic view of a considered system is shown in Figure Figure1.1. We have focused our analysis on the electronic transport modulations due to the geometric confinement and the
presence of an external magnetic field. In this sense, we have studied the transport response due to variations of the length and widths of the central ribbons, considering symmetric and asymmetric
configurations. We have obtained interference effects at low energies due to the extra spatial confinement, which is manifested by the apparition of resonant states at this energy range, and
consequently, a resonant tunneling behaviour in the conductance curves. On the other hand, we have considered the interaction of electrons with a uniform external magnetic field applied perpendicular
to the heterostructure. We have observed periodic modulations of the transport properties as function of the external field, obtaining metal-semiconductor transitions as function of the magnetic
Schematic view of the conductor. Two finite armchair graphene ribbons (red lines). The length L of the conductor is measured in unitary cell units.
All considered systems have been described using a single Π-band tight binding Hamiltonian, taking into account only the nearest neighbour interactions with a hopping γ[0 ]= 2.75eV[24]. We have
described the heterostructures using surface Green’s function formalism within a renormalization scheme [16,17,25]. In the linear response approach, the conductance is calculated using the Landauer
formula. In terms of the conductor Green’s function, it can be written as [26]:
where $T¯E$, is the transmission function of an electron crossing the conductor region, $ΓL/R=i[ΣL/R−ΣL/R†]$ is the coupling between the conductor and the respective lead, given in terms of the
self-energy of each lead: Σ[L/R]= V[C,L/R]g[L/R]V[L/R,C]. Here, V[C,L/R] are the coupling matrix elements and g[L/R ]is the surface Green’s function of the corresponding lead [16]. The retarded
(advanced) conductor Green’s function is determined by [26]: $GCR,A=[E−HC−ΣLR,A−ΣRR,A]−1$, where H[C] is the hamiltonian of the conductor. Finally, the magnetic field is included by the Peierls phase
approximation [27-31]. In this scheme, the magnetic field changes the unperturbed hopping integral $γn,m0$ to $γn,mB=γn,m0e2ΠiΔΦn,m$, where the phase factor is determined by a line integral of the
vector potential A by:
Using the vectors exhibited in Figure Figure1,1, R[1 ]= (1,0)a, $R2=−1,3a/2$ and $R3=−1,3a/2$, where a = |R[n,m]| = 1.42 Å, the phase factors for the armchair configuration in the Landau gauge A =
(0,Bx) are given by:
where y[n ]is the carbon atom position in the transverse direction of the ribbons. In what follows, the Fermi energy is taken as the zero energy level, and all energies are written in units of γ[0].
Results and discussion
Unperturbed systems
Let us begin the analysis by considering the effects of the geometrical confinement. In Figure Figure2,2, we present results of (a) Local density of sates (LDOS) and (b) conductance for a conductor
composed of two A-GNRs of widths N[d ]= N[u ]= 5 connected to two leads of width N = 17 for different conductor lengths (L = 5,10 and 20 unit cells). The most evident result is reflected in the LDOS
curves at energies near the Fermi level. There are several sharp states at defined energies, which increase in number and intensity as the conductor length L is increased. These states that appear in
the energy range corresponding to the gap of a pristine N = 5 A-GNRs [24,32] correspond to a constructive interference of the electron wavefunctions inside the heterostructure, which can travel forth
and back generating stationary (well-like) states. In this sense, the finite length of the central ribbons imposes an extra spatial confinement to electrons, as analogy of what happens in open
quantum dot systems [16,17,19,33,34]. Independently of their sharp line shape, these discrete levels behave as resonances in the system allowing the conduction of electrons at these energies, as it
is shown in the corresponding conductance curves of Figure Figure2b.2b. It is clear that as the conductor length is increased, the number of conductance peaks around the Fermi level is also
increased, tending to form a plateau of one quantum of conductance (G[0 ]= 2e^2/h) at this energy range. These conductance peaks could be modulated by the external perturbations, as we will show
further in this work.
LDOS and conductance for different geometries. (a) LDOS (black line) and (b) conductance of two A-GRNs (red line) of widths N[d ]= N[u ]= 5, connected to two leads of widths N = 17 for different
conductor lengths: L = 5,10,20 u.c. (c) Conductance of a system ...
At higher energies, the conductance plateaus appear each as 2G[0], which is explained by the definition of the transmission probability T(E) of an electron passing through the conductor. In these
types of heterostructures, if the conductor is symmetric (N[u ]=N[d]), the number of allowed transverse channels are duplicated; therefore, electrons can be conduced with the same probability through
both finite ribbons. On the other hand, in Figure Figure2c,2c, we present results of conductance for a conductor of length L = 15 and composed of two A-GNRs of widths N[d ]= 5 and N[u ]= 7,
connected to two leads of widths N = 17. As a comparison, we have included the corresponding pristine cases. As it is expected, the conductance for an asymmetric configuration (red curve) reflects
the exact addition of the transverse channels of the constituent ribbons, with the consequent enhancement of the conductance of the systems. Nevertheless, there is still only one quantum of
conductance near the Fermi energy due to the resonant states of the finite system, whether the constituent ribbons are semiconductor or semimetal. We have obtained these behaviours for different
configurations of conductor, considering variations in length and widths of the finite ribbons and leads.
Magnetic field effects
In what follows, we will include the interaction of a uniform external magnetic field applied perpendicularly to the conductor region. We have considered in our calculations that the magnetic field
could affect the ends of the leads, forming an effective ring of conductor. The results of LDOS and conductance as a function of the Fermi energy and the normalized magnetic flux (ϕ/ϕ[0]) for three
different conductor configurations are displayed in the contour plots of Figure Figure3.3. The left panels correspond to a symmetric system composed of two metallic A-GNRs of widths N[u ]=N[d ]= 5.
The central panels correspond to an asymmetric conductor composed of two A-GNRs of widths N[d ]= 5 (metallic) and N[u ]= 7 (semiconductor). The right panels correspond to a symmetric system composed
of two semiconductor A-GNRs of widths N[u ]=N[d ]= 7. All configurations have been considered of the same length L = 10 and connected to the same leads of widths N = 17. Finally, we have included as
a reference, the plots of LDOS versus Fermi energy for the three configurations.
Magnetic field effects on LDOS and conductance. Contour plots of LDOS (lower panels) and conductance (upper panels) as a function of the Fermi energy and the magnetic flux crossing the hexagonal
lattice for three different configurations of conductor. ...
From the observation of these plots, it is clear that the magnetic field strongly affects the electronic and transport properties of the considered heterostructures, defining and modelling the
electrical response of the conductor. In this sense, we have observed that in all considered systems, periodic metal-semiconductor electronic transitions for different values of magnetic flux ratio ϕ
/ϕ[0], which are qualitatively in agreement with the experimental reports of similar heterosructures [21-23]. Although the periodic electronic transitions are more evident in symmetric
heterostructures (left and right panels), it is possible to obtain a similar effect in the asymmetric configurations. These behaviours are direct consequences of the quantum interference of the
electronic wave function inside this kind of annular conductors, which in general present an Aharonov-Bohm period as a function of the magnetic flux.
The evolution of the electronic levels of the system, depending of their energy, exhibits a rich variety of behaviours as a function of the external field. In all considered cases, the LDOS curves
exhibit electronic states pinned at the Fermi Level, at certain magnetic flux values. This state corresponds to a non-dispersive band, equivalent with the supersymmetric Landau level of the infinite
two-dimensional graphene crystal [30,35]. At low energy region and for low magnetic field, it is possible to observe the typical square-root evolution of the relativistic Landau levels [36]. The
electronic levels at highest energies of the system evolve linearly with the magnetic flux, like regular Landau levels. This kind of evolution is originated by the massive bands in graphene, which is
expected for these kinds of states in graphene-based systems [37,38].
By comparing the LDOS curves and the corresponding conductance curves, it is possible to understand and define which states contribute to the transport of the systems (resonant tunneling peaks), and
which ones only evolve with the magnetic flux but remain as localized states (quasi-bond states) of the conductor. These kind of behaviour has been reported before in similar systems [19,20]. This
fact is more evident in the symmetric cases, where there are several states in the ranges ϕ/ϕ[0]E(γ[0])
In this work, we have analysed the electronic and transport properties of a conductor composed of two parallel and finite A-GNRs, connected to two semi-infinite lead, in the presence of an external
perturbation. We have thought these systems as two parallel wires of an hypothetical circuit made of graphene, and we have studied the transport properties as a function of the separation and the
geometry of these ‘wires’, considering the isolated case and the presence of an external magnetic field applied to the system. We have observed resonant tunneling behaviour as a function of the
geometrical confinement and a complete Aharonov-Bohm type of modulation as a function of the magnetic flux. These two behaviours are observed even when the two A-GNRs have different widths, and
consequently, different transverse electronic states. Besides, the magnetic field generates a periodic metal-semiconductor transition of the conductor, which can be used in electronics applications.
We want to note that our results are valid only in low temperature limits and in the absence of strong disorder into the systems. In the case of non-zero temperature, it is expected that the
resonances in the conductance curves will become broad and will gradually vanish at room temperature [20].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LR and JWG have worked equally in all results presented in this paper. Both authors read and approved the final manuscript.
Authors’ information
LR is a professor at the Physics Department, Technical University Federico Santa Maria, Valparaiso, Chile. JWG is a postdoctoral researcher at the International Iberian Nanotechnology Laboratory,
Braga, Portugal.
Authors acknowledge the financial support of FONDECYT (grant no.: 11090212) and USM-DGIP (grant no.: 11.12.17).
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Learning and Teaching Math
VEX Robots can be more competitive when they have addressed several drive motor control challenges:
1. Stopping a motor completely when the joystick is released. Joysticks often do not output a value of “zero” when released, which can cause motors to continue turning slowly instead of stopping.
2. Starting to move gradually, not suddenly, after being stopped. When a robot is carrying game objects more than 12 inches or so above the playing field, a sudden start can cause the robot to tip
3. Having motor speeds be less sensitive to small joystick movements at slow speeds. Divers seeking to position the robot precisely during competition need “finer” control over slow motor speeds
than fast motor speeds.
These challenges can be solved using one or more “if” statements in the code controlling the robot, however using a single polynomial function can often solve all of these challenges in one step. A
graph can help illustrate the challenges and their solution:
A system of linear equations consists of multiple linear equations. The solution to a linear system, if one exists, is usually the point that all of the equations have in common. Occasionally, the
solution will be a set of points.
There are four commonly used tools for solving linear systems: graphing, substitution, linear combination, and matrices. Each has its own advantages and disadvantages in various situations, however I
often wondered about why the linear combination approach works. My earlier post explains why it works from an algebraic perspective. This post will try to explain why it works from a graphical
Consider the linear system:
which, when graphed, looks like: Read more…
As a parent, I look for two categories of attributes when choosing a school for my child:
– Ones which benefit my child directly
– Ones which benefit my child indirectly, by helping others (teachers, parents) do their jobs more effectively
Schools that satisfy more of the attributes in both categories are likely to have happier parents and more successful students.
The Administration and Teachers Should Help My Child
Directly By:
• Being aware of history. Before the start of each school year, my child’s current teacher(s) should have reviewed all of
– last years’ teacher comments for my child
– my child’s transcript (all courses, all years at the school)
• Helping my child to both pursue existing Read more…
When working with quantitative relationships, three concepts help “set the stage” in your thinking as you seek to understand the relationship’s behavior: domain, range, and co-domain.
The “domain” of a function or relation is:
• the set of all values for which it can be evaluated
• the set of allowable “input” values
• the values along the horizontal axis for which a point can be plotted along the vertical axis
For example, the following functions can be evaluated for any value of “x”:
therefore their domains will be “the set of all real numbers”.
The following functions cannot be evaluated for all values of “x”, leading to restrictions on their Domains – as listed to the right of each one:
$h(x)=\dfrac{1}{x}~~~~~~~~~\text{x cannot be zero}\\*~\\*j(x)=\dfrac{1}{(x-2)(x+4)}~~~~~~\text{x cannot be 2 or -4}\\*~\\*k(x)=3x+2,~1<x<10~~~~\text{only values between -1 and 10 may be used for x}$
The values for which a function or relation cannot be Read more…
Although addition and multiplication are commutative, exponentiation is not: swapping the value in the base with the value in the exponent will produce a different result (unless, of course, they are
the same value):
$2^3 e 3^2$
Therefore, two different inverse functions are needed to solve equations that involve exponential expressions:
- roots, to undo exponents
- logarithms, to undo bases
Just as there are many versions of the addition function (one for each number you might wish to add), and many versions of the “logarithm” function (each with a different base), there are many
versions of the “root” function: one for each exponent value to be undone.
The symbol for a root is $\sqrt{~~~~}$, and is referred to as a “radical“. It consists of a sort of check mark on the left, followed by a horizontal line, called a “vinculum”, that serves as a
grouping symbol (like parentheses) to Read more…
Long assessments can waste precious class time unless there is much material to be assessed, but shorter assessments (with few questions) can cause small errors to have too big an impact on a
student’s grade.
For example, consider the following assessment lengths where each question is worth 4 points, and the student has a total of two points subtracted from their score for errors:
│# Questions │Points │% │% Grade│
│1 │2 / 4 │50%│ F / F│
│2 │6 / 8 │75%│ D / C│
│3 │10 / 12│83%│ C+/ B│
│4 │14 / 16│88%│ B / B+│
│5 │18 / 20│90%│ B+/ A-│
The “% Grade” in the table above reflects a 7-point / 10-point per letter grade approach. A one question quiz is risky for students: they could get a failing grade for losing two points on the only
question. Two question quizzes are only slightly less risky. Only with three or more questions does this scenario start to minimize the risk of actively discouraging a student who loses several
Should quizzes therefore only have three or more questions? What if I don’t want the class to spend that much time on an assessment, or don’t have Read more…
Unlike the two most “friendly” arithmetic operations, addition and multiplication, exponentiation is not commutative. You will get a different result if you swap the value in the base with the one in
the exponent (unless, of course, they are the same value):
$3^2 e 2^3$
The most significant impact of this lack of commutativity arises when you need to solve an equation that involves exponentiation: two different inverse functions are needed, one to undo the exponent
(a root), and a different one to undo the base (a logarithm).
Just as there are many versions of the addition function (adding 2, adding 5, adding 7.23, etc.), and many versions of the “root” function (square roots, cube roots, etc.), there are also many
versions of the “logarithm” function. Each version has a “base”, which corresponds to the base of its inverse exponential expression.
Inverse Functions: Logarithms & Exponentials
Logarithms are labelled with a number that corresponds to the base of the exponential that they undo. For example, the Read more…
If a student makes four errors in the course of answering ten questions, what is an appropriate grade? Presumably, it would depend on the severity of the errors and the nature of the questions.
Consider how your approach to grading might vary if students had been asked to:
- match ten vocabulary words to a word bank, or
- define each of ten words, then use each appropriately in a sentence
- complete ten 2-digit multiplication problems, or
- solve ten multi-step algebra problems, each requiring a unique sequence of steps
- answer ten questions similar to what they have seen for homework or in class, or
- answer ten questions unlike ones they have been asked before
Would you label each answer as right or wrong, then use percentage right as the grade?
Would you assign a number of points to each answer (if so, out of how many points per question)?
Would you assign a letter grade to each answer (whole letters only, or with +/-)?
Would your answers depend on whether you had created the assessment yourself, or were using someone else’s questions?
Many math/science teachers seem to use a percentage approach (based on total points earned or number correct) more often than any other, particularly when their school defines its letter grades using
a 0 – 100 scale. Teachers of other subjects also use this scale often, but less so for “free-response” questions. While a percentage approach can work well for some assessments, it can have
unintended consequences for others.
Similar Right/Wrong Questions
When asking a series of similar questions, such as Read more…
What is a “system” of linear equations?
A “system of linear equations” means two or more linear equations that must all be true at the same time.
When represented symbolically, a system of equations will usually have some sort of grouping symbol to one side of them, such as the curly brace below, which is intended to convey that the set of
equations should be considered all at once. For example:
When graphed, all of the equations in a system will be shown on the same set of axes, so that they can be compared to one another easily:
What is “a solution” to a linear system?
A solution to a system of linear equations is Read more…
The phrase “Flipped Classroom” is appearing with increasing frequency in publications and blog postings. Yet, it seems to mean different things to different people. Many of the references I see to
flipped classrooms are made by people or organizations who have a vested interest in selling goods or services, which probably affects their view of the issues.
As proposed by Salman Khan in his TED Lecture, flipping the classroom involves using internet-based video to move “lecture” out of the classroom to some other place and time of a student’s choosing.
Class time can then be used for student problem solving and group work. Dan Meyer and others have critiqued aspects of Salman Khan’s approach, with some such as Michael Pershan offering constructive
ideas for improvements.
Eric Mazur, a physics professor at Harvard, has also been advocating a “flipped” approach - and for considerably longer than Salman Khan. His conception of “flipping” focuses on getting students to
Read more…
What is the difference between a Problem and a Project? While it is difficult to draw a definitive line that separates one from the other, the attributes of each and their differences as I see them
• Require less student time to complete (usually less than an hour)
• Focus on a single task, with fewer than 10 questions relating to it
• Can involve open-ended questions, but more often does not
• Are often one of a series of problems relating to a topic
• Look similar to many exam questions
• Can be used to introduce new concepts (Exeter Math)
• Can be used as practice on previously introduced concepts (most math texts)
• Require more student time to complete (hours to weeks)
• Focus on a theme, but with many tasks and questions to complete
• Provide an opportunity to acquire and demonstrate mastery
• Ask students to demonstrate a greater depth of understanding
• Ask students to reach and defend a conclusion, to connect ideas or procedures
• Can introduce new ideas or situations in a more scaffolded manner
Why Use Problems?
• Convenience
- Short time to completion makes it easier to fit into a class plan
- Multiple problems allow Read more…
Once a set of learning objectives have been settled on for an activity, problem, or project, what should the problem’s context be? Since linear equations model situations where there is a constant
rate of change, common contexts for linear equation projects often include the following:
• Steepness, height, angle
Examples: road grade, hillside, roof, skateboard park element, tide height over the two weeks before (or after) a full moon, sun angle at noon over a six month period
• Estimating time to complete a task (setup plus completion)
Examples: mowing a lawn, painting a wall, writing a research paper
• Purchase and delivery costs of bulk materials
Examples: mulch, gravel, lumber
• Purchasing a service that charges by consumption
Examples: cell phone, electricity, water, movie rental, etc.
• Total earnings over time from differing wage and bonus plan structures
Examples: hiring bonuses, longevity bonuses
• Energy use over time
Examples: calories burned, electricity, heating oil, gasoline
• Game points accumulated over time
Examples: by a professional athlete, a team, a video game player
• Pollutant levels over time Read more…
A number of historically “good” math students seem to reach a point during their High School years where their feeling of mastery seems to be slipping away. While teachers usually expect more from a
student with each passing year, student frustration usually arises from more than just increasing teacher expectations. I believe it arises because a tried and true study habit, memorization, is no
longer enough to assure mastery.
My experience
I used to read a math or science textbook in pretty much the same way I read anything: as quickly as I could. In fact, for math I often skipped the reading entirely as I had been shown how to do the
new types of problems in class, so all I had to do was sit down and follow the procedure I had been shown – no need for all the verbiage.
However, this approach stopped working when I got to college. If my notes from lecture did not help me figure out how to solve a problem, I had to rely on the text part of the textbook for almost the
first time. I learned that “believing I understood everything that happened in class” was a very different thing from being able to solve the problems assigned for homework.
After skimming through my math text, I often found that
The lesson plans I find most interesting, both to read and to teach from, have both “public” and “hidden” learning objectives. The public objectives focus student attention and help interest
students in the problem: they need to be short, to the point, and tightly related to the problem or project at hand.
The “hidden” objectives are the focus of teacher attention. They reflect the skills and concepts that the teacher hopes to see students grappling with, discussing with peers, and mastering over time
while working on successive problems. If students are informed about a teacher’s list of objectives in assigning a task, students are likely to use only that list in their work. By not publicizing
the teacher’s objective list, students are more likely to try a wider variety of approaches to solving a problem. I think the problem solving process starts with determining which concepts and skills
seem relevant to the problem, therefore keeping the teacher’s objective list hidden helps students become better problem solvers.
The list below covers topics typically taught over a large percentage of the school year, so not all objectives are appropriate at any given point in the year. However, by the end of the year
hopefully most of the following objectives will have been mastered by most students in a class: | {"url":"https://mathmaine.wordpress.com/","timestamp":"2014-04-18T03:22:49Z","content_type":null,"content_length":"83771","record_id":"<urn:uuid:b271f35e-015e-41e7-be5d-630d8e919b81>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00181-ip-10-147-4-33.ec2.internal.warc.gz"} |
DOCUMENTA MATHEMATICA, Extra Vol. ICM II (1998), 691-702
DOCUMENTA MATHEMATICA
, Extra Volume ICM II (1998), 691-702
Stefan Müller and Vladimir Sverák
Title: Unexpected Solutions of First and Second Order Partial Differential Equations
This note discusses a general approach to construct Lipschitz solutions of $Du \in K$, where $u: \Omega \subset {\mathbb R}^n \to {\mathbb R}^m$ and where $K$ is a given set of $m\times n$ matrices.
The approach is an extension of Gromov's method of convex integration. One application concerns variational problems that arise in models of microstructure in solid-solid phase transitions. Another
application is the systematic construction of singular solutions of elliptic systems. In particular, there exists a $2 \times 2$ (variational) second order strongly elliptic system $ \mbox{\rm div}
\, \sigma(Du) = 0 $ that admits a Lipschitz solution which is nowhere $C^1$.
1991 Mathematics Subject Classification: 35F30, 35J55, 73G05
Keywords and Phrases: Partial differential equations, elliptic systems, regularity, variational problems, microstructure, convex integration
Full text: dvi.gz 22 k, dvi 48 k, ps.gz 77 k.
Home Page of DOCUMENTA MATHEMATICA | {"url":"http://www.math.uni-bielefeld.de/documenta/xvol-icm/08/MullerS.MAN.html","timestamp":"2014-04-20T00:44:06Z","content_type":null,"content_length":"1961","record_id":"<urn:uuid:b2157f89-ff3a-4358-b317-261d506ec17c>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00388-ip-10-147-4-33.ec2.internal.warc.gz"} |
proving sin^2(x)<=|sin(x)|
September 22nd 2010, 12:00 PM #1
Sep 2009
proving sin^2(x)<=|sin(x)|
For all x, prove that $sin^{2}(x)\leq|sin(x)|$ given that x is in the set of real numbers.
I think i need to break this into a couple different cases, specifically: (0, $\pi$),( $\pi$, $2\pi$), but am not sure. Any help would be appreciated.
Thanks, I was definitely over-thinking that.
September 22nd 2010, 12:13 PM #2
September 22nd 2010, 12:14 PM #3
Sep 2009 | {"url":"http://mathhelpforum.com/trigonometry/157085-proving-sin-2-x-sin-x.html","timestamp":"2014-04-17T20:58:18Z","content_type":null,"content_length":"36797","record_id":"<urn:uuid:760c3289-0175-482d-b8fd-4f1295c4db8a>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00623-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Life Expectancy: The number of years a person in the united states is expected to live can be approximated by the equation y=0.189x=70.8,where x is the number of years since 1970.solve this
equation for x. use this new equation to determine in which year the approximate life expectancy will be 80.2 years.
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What's supposed to be between 0.189x and 70.8? Is it a + sign?
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thank you!
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i have one more ?
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Some doctors use the formula ND=1.08T to relate the variables N (the number of patient appointments the doctor schedules in one day), D (the duration of each patient appointment), and T(the total
number of minutes the doctor can use to see patients in one day). Solve the formula for N. Use this result to find the number of patient appointments N a doctor should make if she has 6 hours
available for patients and each appointment is 15 minutes long. ( Round your answer to the nearest whole number).
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Some doctors use the formula ND=1.08T to relate the variables N (the number of patient appointments the doctor schedules in one day), D (the duration of each patient appointment), and T(the total
number of minutes the doctor can use to see patients in one day). A. )Solve the formula for N. B.)Use this result to find the number of patient appointments N a doctor should make if she has 6
hours available for patients and each appointment is 15 minutes long. ( Round your answer to the nearest whole number).
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A. To solve for N, just divide both sides by D
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im not sure how to work it out :/ i suck at math
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For part B., you need to use T = 6 hours, but you must convert this to minutes. D = 15
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is that the answer
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I have another question
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whats the second part to this: The number of years a person in the united states is expected to live can be approximated by the equation y=0.189x+70.8,where x is the number of years since
1970.solve this equation for x. use this new equation to determine in which year the approximate life expectancy will be 80.2 years.
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this is what i have already y=0.189x+70.8 Subtract 70.80 from both sides y- 70.8=0.189x Divide both sides by 0.189x y-70.8/0.189=x
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Cryptology ePrint Archive: Report 2009/405
Generic Attacks on Misty Schemes -5 rounds is not enough-Valerie Nachef and Jacques Patarin and Joana TregerAbstract: Misty schemes are classic cryptographic schemes used to construct pseudo-random
permutations from $2n$ bits to $2n$ bits by using $d$ pseudo-random permutations from $n$ bits to $n$ bits. These $d$ permutations will be called the ``internal'' permutations, and $d$ is the number
of rounds of the Misty scheme. Misty schemes are important from a practical point of view since for example, the Kasumi algorithm based on Misty schemes has been adopted as the standard blockcipher
in the third generation mobile systems. In this paper we describe the best known ``generic'' attacks on Misty schemes, i.e. attacks when the internal permutations do not have special properties, or
are randomly chosen. We describe known plaintext attacks (KPA), non-adaptive chosen plaintext attacks (CPA-1) and adaptive chosen plaintext and ciphertext attacks (CPCA-2) against these schemes. Some
of these attacks were previously known, some are new. One important result of this paper is that we will show that when $d=5$ rounds, there exist such attacks with a complexity strictly less than $2^
{2n}$. Consequently, at least 6 rounds are necessary to avoid these generic attacks on Misty schemes. When $d \geq 6$ we also describe some attacks on Misty generators, i.e. attacks where more than
one Misty permutation is required.
Category / Keywords: secret-key cryptography / Misty permutations, pseudo-random permutations, generic attacks on encryption schemes, Block ciphers.Date: received 19 Aug 2009Contact author: valerie
nachef at u-cergy frAvailable format(s): PDF | BibTeX Citation Version: 20090824:144113 (All versions of this report) Discussion forum: Show discussion | Start new discussion[ Cryptology ePrint
archive ] | {"url":"http://eprint.iacr.org/2009/405","timestamp":"2014-04-20T03:15:14Z","content_type":null,"content_length":"3081","record_id":"<urn:uuid:e7ac5e5a-3287-4e69-bca0-fb343f57cb03>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Re: [Discuss-gnuradio] OFDM demodulation problem / example video
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Re: [Discuss-gnuradio] OFDM demodulation problem / example video
From: Jens Elsner
Subject: Re: [Discuss-gnuradio] OFDM demodulation problem / example video
Date: Thu, 6 Apr 2006 12:20:28 +0200
User-agent: Mutt/1.3.28i
thanks for your comments. I am beginning to like this problem, as it has
extended my knowlegde of OFDM tremendously - thanks everybody.
Nevertheless I'd like to see it solved. We have ruled out pretty much
every effect - I'm beginning to suspect that the transmitter actually sends
shifted symbols. The remaindre of this week I'll be pretty busy, but I'll come
back to
the problem and get data from other radio stations to verify.
> >Frequency offset results in ICI.
> > This is "white noise" interference.
> Correct: v0 will result in a time offset of N*v0
> which will manifest itself as ICI.
> There is MORE to it.
> It ALSO introduces an unknown phase offest (constant in each subchannel):
> Consider the i-th component of consequtive OFDM symbols.
> They are effectively rotated by multiples of 2*pi*(N+G)*v0.
> This is the phase accumulated over an OFDM symbol duration.
> Although the linear phase increase will be compensated by D-QPSK, there
> will be a constant phase rotation equal to 2*pi*(N+G)*v0 in all
> demodulated QPSK symbols.
Yes, agreed. This effect can be seen nicely if you demodulate a symbol
and change the fractional frequency offset in small steps. The
constellation pulsates (white interference) and rotates (phase rotation).
> BTW, If you make sure that you estimate v0 with an accuracy of 1e-6
> then the time offset is on the order of 1e-3 (negligible) and the
> frequency offset on the order of 1 degree (negligible).
I think I hit the right frequency by +- 1 Hz, subcarrier spacing being
1KHz. Should be alright.
> BTW, because of the way you have coded your frequency correction (in
> your Matlab code) even perfect estimation of v0 will still result in the
> above phenomenon.
> If you correct for v0 INDIVIDUALLY in each OFDM symbol (as you do in
> your code), then--assuming perfect estimation of v0--there will be two
> effects:
> 1) within each OFDM symbol there will be no phase variation since you
> have corrected it already
> 2) each consequtive sample in the i-th subchannel will be advanced by
> 2*pi*(N+G)*v0 radians. This is the phase accumulated over an OFDM symbol
> duration.
> This is why it is important to derotate your entire frame at once.
Ok. I changed that in my code.
> Having said that, it seems that even if this is done, the problem you
> observe is still there!
> >I am clueless.
> I attach a piece of Matlab code that simulates both the Tx and Rx.
> Everything mentioned above can be demonstrated in this controled experiment.
Great - nice way to learn about OFDM.
> I still do not understand why the real-data application does not work.
Neither do I - I'll get other data next week.
[Prev in Thread] Current Thread [Next in Thread]
Re: [Discuss-gnuradio] OFDM demodulation problem / example video, Jens Elsner, 2006/04/04
Re: [Discuss-gnuradio] OFDM demodulation problem / example video, Achilleas Anastasopoulos, 2006/04/04
Re: [Discuss-gnuradio] OFDM demodulation problem / example video, Achilleas Anastasopoulos, 2006/04/05
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Cell-Oriented Modeling of Angiogenesis
Volume 11 (2011), Pages 1735-1748
Review Article
Cell-Oriented Modeling of Angiogenesis
Department of Human Anatomy and Physiology, University of Padova Medical School, via Gabelli 65, 35121 Padova, Italy
Received 9 August 2011; Accepted 12 September 2011
Academic Editor: David Tannahill
Copyright © 2011 Diego Guidolin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Due to its significant involvement in various physiological and pathological conditions, angiogenesis (the development of new blood vessels from an existing vasculature) represents an important area
of the actual biological research and a field in which mathematical modeling proved particularly useful in supporting the experimental work. In this paper, we focus on a specific modeling strategy,
known as “cell-centered” approach. This type of mathematical models work at a “mesoscopic scale,” assuming the cell as the natural level of abstraction for computational modeling of development. They
treat cells phenomenologically, considering their essential behaviors to study how tissue structure and organization emerge from the collective dynamics of multiple cells. The main contributions of
the cell-oriented approach to the study of the angiogenic process will be described. From one side, they have generated “basic science understanding” about the process of capillary assembly during
development, growth, and pathology. On the other side, models were also developed supporting “applied biomedical research” for the purpose of identifying new therapeutic targets and clinically
relevant approaches for either inhibiting or stimulating angiogenesis.
Computational models and simulations play many roles in science [1]. They are used to make precise and accurate predictions and to summarize data. They are used as heuristic approaches for designing
experiments or to demonstrate surprising and counterintuitive consequences of particular forms of systematic organization. As far as biological systems are concerned, attempts have been made at
modeling many different processes. In this respect morphogenesis represented a quite common target of modeling efforts. They addressed situations ranging from the formation of bacterial [2] and
mesenchymal cell [3] aggregation patterns and Dictyostelium morphogenesis (see [4]) to tumor growth [5–7], limb patterning [8], and gastrulation [9].
Due to its significant involvement in various physiological and pathological conditions [10], angiogenesis (the development of new blood vessels from an existing vasculature) represents an important
area of the actual biological research and a field in which mathematical modeling proved particularly useful. Perhaps the first mathematical analyses of vascular networks can be found in the seminal
work of Wilhelm Roux (see [11]) and in the classic work of Thompson [12] where he studies “…a number of interesting points in connection with the form and structure of blood vessels.” However, is
within the past two decades that the application of mathematical and computational models has significantly supplemented experimental approaches in this field and enhanced our understanding of the
main factors regulating the vascular pattern formation. One way to categorize the existing set of published models is according to the spatial scale they were developed to encompass [13, 14].
Some computational studies focused on the “molecular level,” building models of the intracellular dynamics (see [15, 16]), such as signaling phenomena and gene expression. The coupling of many
detailed single-cell models was suggested by some authors [17] as a possible modeling strategy to reproduce multicellular phenomena. However, very accurate models of a single-cell (see [18]) can, at
best, treat clusters formed by a quite low number of cells.
On the other side several models have reproduced vessel-like patterns consistent with those observed in vitro [19–24] or in vivo (see for instance [25–27]) by following a “tissue level” approach (see
[28]), in which the system is treated as a continuous substance, and the involved cells are described in terms of densities (using partial differential equations). Continuum models of this type
average out the behavior of the individual elements and are capable of efficiently capturing features of angiogenesis at a “macroscale” (such as average sprout density, network expansion rates,
etc.). They, however, are unable to provide detailed information at a “microscale” concerning the actual structure and morphology of the capillary network. In fact, the self-organization of the
endothelial cells (EC) leading to the formation of new capillary branches is mainly the result of several intimately linked single-cell behaviors [29].
Thus, working at too coarse or fine a level of detail makes quite hard an accurate modeling of the complex process of angiogenesis. For this reason, “cell-centered” approaches, working at a
“mesoscopic scale” and treating the cell as the fundamental module of development, have been devised [30]. They also proved quite useful to build multiscale models of the process, providing a sort of
natural interface between “molecular level” and “tissue level” modeling.
This specific modeling strategy and the role it can play in the study of the angiogenic process are the focus of the present paper.
The underlying principles of the “cell-centered” approach to modeling have been extensively discussed by Merks and Glazier [30], and its main characteristics will be only briefly recalled below.
The key concept on which cell-centered models are based is to assume the cell as the natural level of abstraction for mathematical and computational modeling of development. Thus, to a first
approximation, the cell’s internal properties (i.e., the details of the intracellular processes) are not explicitly taken into account and only its essential behaviors (such as movement, division,
death, differentiation, adhesion, and secretion of chemicals) are considered. A significant advantage of this strategy is the relative simplicity of the models it generates. Systems composed by a
quite large number of cells (up to 10^5-10^6 cells) can be simulated, opening a concrete possibility to study how tissue-level processes could emerge from the collective dynamics of multiple
interacting cells. It follows that cell-centered methods appear particularly suitable to investigate morphogenesis as also illustrated by very recent studies [31, 32] showing how cell shape, most
likely sensed by the mitotic spindle, serves as a major determinant of future cell and tissue development.
To achieve this goal, some methodological steps are required, in which cell-centered simulations are compared with experimental observations to identify the minimal set of single cell behaviors
needed to produce certain tissue-level patterns. A typical flow-chart for this protocol of computational prediction and experimental validation is provided in Figure 1.
As far as the mathematical modeling and simulation techniques are concerned, several cell-centered computational strategies have been proposed to study morphogenesis.
Some of them were focused on tissue processes in which cells keep a fixed position with respect to each other [33]; others considered mobile cells and the physics of the adhesive forces between cells
and the extracellular matrix (ECM) to simulate aggregates of thousand of cells (see [34]). In the Lagrangian Monte-Carlo method proposed by Drasdo et al. [35], for instance, attraction, compression,
and bending energies determine movements of spheroidal cells to simulate cleavage and gastrulation [9] and avascular tumor growth [5], while a statistical mechanics-based approach was developed by
Newman and Grima [36] to model chemotactic cell-cell interactions and to study cell ensembles analytically.
However, a quite popular strategy to model the self-organization of mobile cells is using cellular automata [37–39]. For this reason this computational technique will be here described in more
detail. Cellular automata (CA) consist of discrete particles that occupy some or all sites of a regular lattice [40]. Each particle is characterized by one or more internal state variables and a set
of rules describing the evolution of their state and position. Both the movement and change of state depend on the current state of the particle and those of neighboring particles. Again, the
evolution rules may either be discrete or continuous, deterministic or probabilistic, and usually they are applied in time steps, following a synchronous or stochastic updating scheme.
Philosophically, CA are attractive because they show some analogy with biological systems. In fact, their large-scale behaviors are completely self-organized [41, 42]. An individual cell does not
carry a road map, it can only respond to signals in its local environment. Furthermore, they need not privilege any single cell as pacemaker or director: all cells are fundamentally equivalent.
A relatively simple type of CA models is the so-called lattice-gas-based CA (LGCA) [39]. In LGCA individual particles on a discrete grid represent cells, each characterized by a velocity determined
by the local interactions the cell experiments. At each time step, each cell will move to a neighboring site according to the velocity that cell had. Thus, in their biological applications LGCA treat
cells as point-like objects with an internal state but no spatial structure. As a consequence LGCA models can be convenient and efficient for reproducing qualitative patterning in colonies where
cells retain simple shapes during migration. Eukaryotic cells, however, often move by remodeling their cytoskeleton and changing their shapes. Since in some cases shape change significantly
influences patterning, a modeling approach that takes into account cell shape is required. In this respect, a more efficient and complex CA is the Cellular Potts Model (CPM), in which a cell consists
of a domain of lattice sites, thus describing cell volume and shape more realistically. Originally it was developed by Graner and Glazier [43] to simulate the cell rearrangement resulting from cell
adhesion, in order to quantitatively simulate cell-sorting experiments. However, a number of cell behaviors can be quite easily implemented in this computational framework, and improvements to the
CPM included the possibility to model cell growth, cell division, apoptosis and cell differentiation, chemotaxis, extracellular materials, and cell polarity (see [30]). The basic characteristic of
the CPM is to represent the cell behaviors of interest in the form of terms within a generalized energy function which also includes the interactions with the ECM and parameters constraining
individual cell behavior. As an example, a simple form of CPM is illustrated in Figure 2. Cells are represented on a rectangular numerical grid as patches of lattice sites, , with identical nonzero
indices , while an index value 0 identifies the sites corresponding to the extracellular space. Grid points at patch interfaces represent cell surfaces, and the interaction between cell surfaces is
modeled by defining coupling constants representing the adhesion energy involved in the specified interaction. Each cell also has a set of attributes, including a “target” area and elongation, which
poses some constraint on the possible cell shape changes. Thus, the following “energy function” can be defined for this system: where represents the eight neighbors of , and represent resistances to
changes in size and elongation; respectively, and are the “target” values for cell area and length, and are the actual cell area and length values, and the Kronecker delta is .
The parameters involved in (2.1) can be estimated by specific experiments or based on biological considerations, and the dynamics of such an energy formalism can be solved using a variety of
minimization methods, as, for instance, the well-known Metropolis or Kawasaki algorithms (see [30, 39] for reviews).
CA-based approaches for modeling morphogenesis recently applied to studies of tumor growth [5–7, 44–46]. LGCA-based models proved useful to describe the ripple formation in myxomycetes [47] and
germinal center dynamics [48], and a quite wide range of biological problems (see [49–53]) were addressed with the CPM. In particular, as detailed in the next section, this type of cell-oriented
strategy to modeling played a significant role in studies focused on angiogenesis.
Cell-oriented computational models of angiogenesis can be categorized around some key questions they have been developed to answer. From one side, they have generated “basic science understanding”
about the process of capillary assembly during development, growth, and pathology. On the other side, models were also developed with the intention of supporting “applied biomedical research” for the
purpose of identifying new therapeutic targets and clinically relevant approaches for either inhibiting or stimulating angiogenesis. These two targets motivating the development of the models will be
the central thread of the present section.
One of the first, and most frequently cited, cell-centered models of angiogenesis has been developed by Stokes and Lauffenburger [54], who simulated individual cell movements by considering cell
motility and chemotaxis as partially stochastic events. They used the model to assess microvascular endothelial cells migration in the presence or absence of acidic fibroblast growth factor (aFGF),
and realistic capillary network structures were generated by incorporating rules for sprout branching and anastomosis. Although the model incorporated random motility and chemotaxis as mechanisms for
cell migration, no account was taken of the interactions between the endothelial cells and the ECM. To account for this specific point, more recent models of cell migration improved the accuracy of
the simulations by using force-based dynamics approaches to simulate internally generated forces and external traction forces, as well as matrix compliance and ECM stiffness [55].
Key morphological events involved in new vessel formation can be experimentally investigated by in vitro studies analyzing the endothelial cell self-organization in vitro [29, 56]. In this context an
important supporting tool for the interpretation of the observed patterns is represented by the CPM-based two-dimensional model by Merks et al. [57] simulating the process of in vitro vasculogenesis
or the assembly of human umbilical vein endothelial cells (HUVEC) into networks of connected cells in a Matrigel environment (see [58]). The model considered a set of single cell behavior, including
cell adhesion (between cells and with the ECM), chemotaxis, and cytoskeleton rearrangement. In this CPM cells are represented on a rectangular numerical grid as shown in Figure 2. By repeatedly
replacing a value at cell interface by a neighboring grid point’s value, it is possible to mimic active, random extension of filopodia and lamellipodia. If the resulting variation in effective energy
is negative, then the cell change will be accepted, otherwise it will be accepted with the Boltzmann-weighted probability. The preferential extension of filopodia in the direction of chemoattractant
gradients was implemented by including in (2.1) an extra reduction of energy whenever the cell protrudes into an area with a higher concentration of chemoattractant: where is the neighbor into which
site moves (i.e., copies its value), is the strength of the chemotactic response, and is the local concentration of the chemoattractant.
At each time instant, the concentrations were estimated from the following diffusion partial differential equation (PDE): where is the rate at which the cells release chemoattractant, is the
clearance rate of the chemoattractant, and its diffusion coefficient. The Kronecker delta simply indicates that the release occurs at the cell locations, while the factor is cleared in the
extracellular space.
As shown in Figure 3, with a proper choice of the parameters, the model generates cell patterns in close agreement (from both a qualitative and quantitative point of view) with those generated in
vitro by unstimulated HUVEC, suggesting that the three considered single-cell behaviors are essential for correct spatiotemporal vasculogenesis in vitro.
The same modeling approach was used by our group to analyze and interpret the results of in vitro angiogenesis experiments in conditions involving cell stimulation with proangiogenic factors or
performed with nonendothelial cells potentially able to differentiate towards an endothelial phenotype. In the first study [59], a cell-centered mathematical modeling approach was used to determine
essential cellular behaviors for pattern formation when human saphenous vein endothelial cells are stimulated by the pro-angiogenic factor adrenomedullin (AM) [60]. Cell culture measurements provided
the key parameters to customize the model. When put to the test, the simulated pattern and morphometric parameters closely matched that of untreated EC, confirming that cell elongation, in
conjunction with autocrine secretion of a chemoattractant, results in a cell-shape-dependent motility representing the key factor driving the formation of vascular-like morphologies by EC in vitro.
However, the model failed to predict patterns of EC cultured with AM, revealing that it lacked input from an important cell behavior. Hypothesizing that the missing ingredient was cell proliferation,
the model was extended to include it and called upon to estimate the percentage increase in cell number that would yield observed patterns. Remarkably, the proliferation rates predicted by the model
showed consistency with bromodeoxyuridine incorporation experiments performed to verify such a prediction. In another study [61], aimed at investigating the vasculogenic potential of bone marrow
macrophages from patients with multiple myeloma (MMMA), when seeded on Matrigel, a number of MMMA rapidly changed their morphology and developed an elongated shape. After 18 hours, the formation of a
pattern consisting of cord- and tubular-like structures was observed, sometimes arranged to form closed meshes. When biophysical parameters consistent with the available experimental evidence were
used to customize the model, it was quite accurate in quantitatively reproducing the observed in vitro patterns, provided that the possibility for cell differentiation was included in the model. In
particular, it indicated that about 30% of the seeded MMMA were differentiated towards an endothelial phenotype, suggesting that in multiple myeloma a quite high number of MMMA could become involved
in the process of capillarization by converting into a cell type at least similar to the endothelial one.
Following the above-mentioned CPM-based modeling, Merks et al. [62] also showed that including VE-cadherin-mediated contact inhibition of chemotaxis in the simulation causes randomly distributed
cells to organize into networks and cell aggregates to sprout, hence, reproducing aspects of both de novo and sprouting blood-vessel growth. This study, therefore, further confirmed the CPM as a
potentially very helpful tool to investigate the whole spectrum of patterns formed during angiogenesis.
As far as in vivo studies are concerned, a popular experimental setup for studying the sprouting of new vessels is the corneal pocket model, in which exogenous growth factors can be supplied in a
controlled manner to induce reproducible angiogenic sprouts from the limbic vessels. This experimental model has recently been the subject of a cell-based mathematical model [63] allowing for a
detailed study of the relative roles of EC migration, proliferation, and maturation in sprouts development. It showed that cell elasticity and cell-to-cell adhesion allow only limited sprout
extension in the absence of proliferation, and the maturation process combined with bioavailability of VEGF can explain the localization of proliferation to the leading edge, consistently with
experimental observations.
The vascularized phase of tumor growth has dominated as the most common context in which to develop mathematical and computational models of angiogenesis. In this respect, the above-mentioned
cell-centered model by Stokes and Lauffenburger [54] predicted, for the first time, that chemotaxis is needed to orient vascular growth toward the tumor. More complex cell-based models including a
number of key events of angiogenesis (such as the migratory response of endothelial cells to cytokines secreted by a solid tumor, endothelial cell proliferation, endothelial cell interactions with
ECM macromolecules, such as fibronectin, and capillary sprout branching and anastomosis) were proposed (see [13, 64]). They provided capillary networks with a very realistic structure and morphology,
capturing the early formation of loops, the essential dendritic structure of a capillary network, and the formation of the experimentally observed “brush border.”
A sophisticated cell-centered model of tumor angiogenesis was developed by Bauer et al. [65]. It describes diffusion, uptake, and decay of proangiogenic factors secreted by tumor cells and was used
to understand the role of cell-cell and cell-matrix dynamics in regulating tumor angiogenesis. The model incorporates both discrete and continuous approaches (see Figure 4): a PDE describes diffusion
and decay of tumor-secreted VEGF, while a CPM is used to describe EC migration, growth, division, and adhesion, as well as ECM degradation. Notably, this model is the first to capture anastomosis and
branching without needing to predefine rules for these events: these properties emerged from the independent behaviors of the individual simulated EC. Furthermore, the model proved useful to address
several questions, including the impact of ECM-binding affinity of VEGF on capillary morphology, the rate of capillary sprout elongation, and to what extent the composition of the stroma (ECM density
and anisotropy) influences angiogenesis.
In almost all the above-mentioned models, the intracellular events are not modeled explicitly, and the information concerning the intracellular dynamics is embedded in the model parameters.
Cell-centered approaches, however, can be extended to generate more realistic multiscale models of the complex process of angiogenesis. A recent example was provided by Scianna et al. [67]. The model
spans three fundamental biological levels: at the extracellular level a continuous model describes secretion, diffusion, uptake, and decay of the autocrine VEGF, at the cellular level a CPM
reproduces cell dynamics (migration, adhesion, chemotaxis), and at the subcellular level a set of reaction-diffusion equations describes a simplified VEGF-induced calcium-dependent intracellular
pathway. The results agree with the known interplay between calcium signals and VEGF dynamics and with their role in malignant vasculogenesis.
Moving from the basic science of angiogenesis to the applied biomedical research, a number of cell-oriented models were developed to support the search for therapies and/or technologies aimed at
favouring or inhibiting angiogenesis.
The development of tissue-engineered constructs greater than about 1 mm^3 is limited by the necessity to overcome oxygen diffusion limitations. Thus, the development of novel approaches for
engineering microvascular networks ex vivo or inducing their ingrowth upon implantation of the construct is imperative. Jabbarzadeh and Abrams [68] developed a model of VEGF-mediated EC chemotaxis
through a porous membrane in response to three different VEGF presentation strategies in order to assess which one could lead to the most extensive vascular coverage of the construct.
As far as the inhibition of angiogenesis is concerned, the early stages of tumor angiogenesis, in which EC escape the parent vessel and invade the ECM, were the focus of a cell-based mathematical
model by Plank and Sleeman [69] with the aim to study the antiangiogenic potential of pharmacological strategies based on angiostatin.
A high fidelity simulation model of angiogenesis induced by solid tumors was developed by Mahoney et al. [66] as an evolution of the above-mentioned model by Bauer et al. [65]. The aim was to
identify specific medically relevant intervention targets. The simulation system integrates (see Figure 4) the following: (a) a CPM that captures mechanisms of endothelial cell growth, cell adhesion,
ECM fiber adhesion and degradation, and tip cell chemotaxis and haptotaxis, (b) a continuous model of VEGF secretion from the tumor, diffusion through the stroma (host tissue), and endothelial cell
uptake and activation, (c) a flow model that estimates blood flow through the irregular network of vessels that emerge during angiogenesis, and (d) a continuous model of oxygen secretion from vessel
loops, diffusion through the stroma, and uptake by the tumor. This model captures behaviors such as vessel branching, loop formation (anastomosis), progression and termination of tip movement, and
activation and growth of new vessels. All these complex behaviors emerge from interactions among the simpler, biologically relevant component mechanisms of the model. The results of the simulations
showed the effectiveness of this computational method in finding interventions that are currently in use (such as those aimed at disrupting VEGF) and, more interestingly, suggesting some new
approaches that are counterintuitive yet potentially effective (such as those targeting the ECM to decrease the probability of the growing vessels forming loops).
Developmental biology classically aims to understand how gene regulation leads to the development and morphogenesis of multicellular organisms. In this respect it has to be observed that genetic
information influences the morphology and physiology of multicellular systems only indirectly. In fact, the gene network steers the biophysical properties of the cell by tuning the production of
regulatory RNA sequences and proteins which in turn determine the behavior of the cell and its response to signals from its environment. In many aspects of biological development; therefore, what
really matters are just these properties at the cell level (the type of signals released and the response to extracellular stimuli), and the cell’s inner workings can be neglected. Thus, as proposed
by some authors (see [30] for instance), two separate questions can be considered: the first one concerns how genetics drives cell behavior and the second one how cell behavior drives morphogenesis.
As far as the second question is concerned, the cell-centered modeling approach is certainly a significant tool to generate and test hypotheses in developmental biology, helping to understand which
cell behaviors are essential to structure tissues. The studies on the angiogenic process represent a significant example of this concept. In fact, cell-oriented computational screening of the
parameter dependence of patterning significantly helped to identify regulators of vascular development and suggest new hypotheses. Consistent with continuum models [20, 22], the cell-centered
approach confirmed the key role of chemotaxis in driving vascular formation both in vitro and in vivo [62, 65]. Furthermore, it suggested that EC adhesion is essential to form stable vascular
networks and that cell extension also strongly affects the multicellular patterns [57, 59]. The understanding of the role of EC proliferation and of their interaction with the ECM on the formation of
capillary sprouts in vivo was also greatly enhanced by cell-centered modeling efforts [65, 66]. In this respect, continuous models have great difficulty assessing the role of these parameters. Thus,
cell-oriented strategy to modeling angiogenesis represented a better tool to direct specific experiments, and recently a number of experimental validations of proposed models have been obtained [57,
59]. As demonstrated by some of the studies here reviewed, this modeling approach can also assist in the identification of cell properties representing potential targets to improve tissue engineering
constructs [68] or therapies [66, 69] against angiogenesis-dependent pathologies.
To assist developmental biologists in the investigation of the question concerning which molecular processes are responsible for the essential cell behaviors leading to specific tissue-level or
organism-level phenotypes, the cell-centered approximation will require extensions to both larger- and smaller-length scales [70]. The integration of models of the intracellular activity with
cell-centered models of cell behavior seems to be possible in two fashions. The simpler strategy is likely to use microscopic models to provide estimates of the parameters controlling the
cell-centered model. Alternatively, true hybrid models could be devised in which simulations of the inner cell processes function as components within cell-centered models (as in [71] and in the
example provided by Scianna et al. [67]). Similarly, the cell-centered models can be interfaced with macroscale models of tissue or organ behavior either by providing estimates of tissue properties
for continuum models or interacting directly with them in a hybrid model (an example is provided in [72]). As pointed out by Merks and Glazier [30], in this effort the key advantage of starting from
a mesoscopic standpoint, such as a cell-oriented approach, is that we often need to introduce only a minimal additional algorithmic complexity and computation time to achieve results consistent with
existing experimental data. Thus, this modeling strategy could also be a convenient tool to devise more complex models aimed to reach a better insight on the links between different levels of
biological organization. Such a research effort may represent an important target for future work in the field of modeling angiogenic processes. In fact, it has to be observed that the development of
multiscale models with capabilities to integrate processes spanning different spatial and temporal scales appears of particular relevance to complement experimental studies and address important
questions concerning the vascular system, whose developmental and remodeling aspects seem intimately characterized by a complex multiscale logic [73].
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How do you calculate the cost basis for a mutual fund over an extended time period?
Investors must pay taxes on any investment gains they realize. Subsequently, any
capital gain
realized by an investor over the course of a year must be identified when they file their income taxes. For this reason, being able to accurately calculate the
cost basis
of an investment, particularly one in a
mutual fund
, becomes extremely important.
The cost basis represents the original value of an
that has been adjusted for stock splits, dividends and capital distributions. It is important for tax purposes because the value of the cost basis will determine the size of the capital gain that is
taxed. The calculation of cost basis becomes confusing when dealing with mutual funds because they often pay
and capital gains distributions usually are reinvested in the fund.
For example, assume that you currently own 120 units of a fund, purchased in the past at a price of $8 per share, for a total cost of $960. The fund pays a dividend of $0.40 per share, so you are due
to receive $48, but you have already decided to reinvest the dividends in the fund. The current price of the fund is $12, so you are able to purchase four more units with the dividends. Your cost
basis now becomes $8.1290 ($1008/124 shares owned).
When shares of a fund are sold, the investor has a few different options as to which cost basis to use to calculate the capital gain or loss on the sale. The
first in, first out method
(FIFO) simply states that the first shares purchased are also the ones to be sold first. Subsequently, each investment in the fund has its own cost basis. The average cost single category method
calculates the cost basis by taking the total investments made, including dividends and capital gains, and dividing the total by the number of shares held. This single cost basis then is used
whenever shares are sold. The average cost double category basis requires the separation of the total pool of investments into two classifications:
short term
long term
. The average cost is then calculated for each specific time grouping. When the shares are sold, the investor can decide which category to use. Each method will generate different capital gains
values used to calculate the
tax liability
. Subsequently, investors should choose the method that provides them with the best tax benefit.
To learn more, see
Selling Losing Securities For A Tax Advantage
Using Tax Lots: A Way To Minimize Taxes
Mutual Fund Basics Tutorial
comments powered by Disqus | {"url":"http://www.investopedia.com/ask/answers/06/mutualfundcostbasis.asp","timestamp":"2014-04-17T04:36:31Z","content_type":null,"content_length":"72446","record_id":"<urn:uuid:5eba964e-e073-44ca-a485-beecf4fa6f4a>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00174-ip-10-147-4-33.ec2.internal.warc.gz"} |
Edges Connection Graph [Archive] - OpenGL Discussion and Help Forums
05-04-2010, 05:08 AM
Hi All,
Does an efficient implementation to extract edges from a mesh of triangles without duplicates?
I mean:
1) Vertices are a list of 3D points
2) Triangles are a list of 3 indices pointing to vertices above
We want to find very quickly:
1) A list of 2 indices representing the triangle sides but *without* duplicates (of course, you always get two with opposite direction) | {"url":"http://www.opengl.org/discussion_boards/archive/index.php/t-170887.html","timestamp":"2014-04-19T04:28:50Z","content_type":null,"content_length":"12910","record_id":"<urn:uuid:f83f978f-139c-4ab4-b1d8-d7de81d998f4>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00206-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Plus Advent Calendar Door #14: Euler's polyhedron formula
A polyhedron is the 3D version of a polygon. It's a solid object whose surface is made up of a number of polygonal faces. Two faces meet in an edge and the corners of a polyhedron are called
Euler's polyhedron formula, named after Leonhard Euler, is a pretty amazing equation relating the number
As an example, think of a cube. It's got 6 faces, 8 vertices and 12 edges:
as required.
The amazing thing is that this formula holds for all polyhedra, except for those that have holes running through them.
Using this formula, you can figure out quickly that there is no simple polyhedron (that is one without holes) with exactly seven edges. Similarly, there is no simple polyhedron with ten faces and
seventeen vertices.
The formula also makes it possible to prove one of the most beautiful results in geometry, that there are only five Platonic solids.
Figure 7: The Platonic solids. From left to right we have the tetrahedon with four faces, the cube with six faces, the octahedron with eight faces, the dodecahedron with twelve faces, and the
icosahedron with twenty faces.
Euler's polyhedron formula applies to solids that, in a topological sense, are equivalent to the sphere: you can turn each simple polyhedron into a sphere by smoothing out the edges and corners and
making it round. But you can also look at the number
You can find out more about Euler's polyhedron formula, including a proof, in this Plus article. And there's an interesting application to designing footballs in A fly walks around a football. | {"url":"http://plus.maths.org/content/plus-advent-calendar-door-14-eulers-polyhedron-formula","timestamp":"2014-04-16T16:26:54Z","content_type":null,"content_length":"26376","record_id":"<urn:uuid:2584b245-d0f7-4934-8b46-8988bb8a8e81>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00049-ip-10-147-4-33.ec2.internal.warc.gz"} |
Frustrum rate of change
March 5th 2007, 12:50 AM #1
Junior Member
Mar 2006
Frustrum rate of change
A shock wave is advancing in the shape of a frustrum of a right circular cone. The upper radius, x of the frustrum is increasing at the rate of 2m per sec, the lower radius, y is increasing at
the rate of 3m per sec, and the height, z is increasing at the rate of 1.5m per sec. Using the chain rule find the rate of increase of the volume, V of the frustrum at the moment when x = 10m, y
= 16m and z = 8m.
Hello, Shaun!
A shock wave is advancing in the shape of a frustrum of a right circular cone.
The upper radius, x of the frustrum is increasing at the rate of 2m/sec,
the lower radius, y is increasing at the rate of 3m/sec,
and the height, z is increasing at the rate of 1.5m/sec.
Using the chain rule find the rate of increase of the volume, V of the frustrum
at the moment when x = 10m, y = 16m and z = 8m.
The volume of a frustum of a cone (or pyramid) is given by:
. . . . . . . . . . . . . . . _____
. . V .= .(h/3)(B1 + √B1·B2 + B2)
where h is the height and B1, B2 are the areas of the two bases.
Our problem has: .B1 = πx², B2 = πy², h = z.
. . . . . . . . . . . . . . . . . . .______
Then: .V .= .(z/3)(πx² + √πx²·πy² + πy²)
. .or: . V .= .(π/3)z(x² + xy + y²)
Can you finish it now?
Differentiate with respect to time (product rule and chain rule)
. . then plug in the given values.
A shock wave is advancing in the shape of a frustrum of a right circular cone. The upper radius, x of the frustrum is increasing at the rate of 2m per sec, the lower radius, y is increasing at
the rate of 3m per sec, and the height, z is increasing at the rate of 1.5m per sec. Using the chain rule find the rate of increase of the volume, V of the frustrum at the moment when x = 10m, y
= 16m and z = 8m.
The volume V of the frustrum is
V = (pi/3)z[x^2 +y^2 +xy]
Differentiate both sides with respect to time t,
dV/dt = (pi/3){(dz/dt)[x^2 +y^2 +xy] +z[2x(dx/dt) +2y(dy/dt) +x(dy/dt) +y(dx/dt)}
Substituting all the givens,
dV/dt = (pi/3){(1.5)[10^2 +16^2 +10*16] +(8)[2(10)(2) +2(16)(3) +10(3) +16(2)]}
dV/dt = (pi/3){774 +1584}
dV/dt = 786pi
Therefore, at that instant, the volume is increasing at the rate of 786pi cubic meters per second. -------------answer.
March 5th 2007, 04:17 AM #2
Super Member
May 2006
Lexington, MA (USA)
March 5th 2007, 04:29 AM #3
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/calculus/12184-frustrum-rate-change.html","timestamp":"2014-04-18T00:26:00Z","content_type":null,"content_length":"37755","record_id":"<urn:uuid:c7fea36d-724a-4c23-a61c-e76dbc61f72a>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00656-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Sidereal day and solar day
A sidereal day is the amount of time it takes for the Earth (or, in general, any planet) to make precisely one revolution, relative to the Celestial Sphere. Distant stars make good approximations of
fixed points on the celestial sphere, so a sidereal day can be considered to be the time between a star reaching zenith (or it's cloest approach to zenith) and it's reaching zenith again.
A solar day is the time it takes for a planet to make precisely one revolution, relative to the Sun (or whatever star it is orbiting). In other words, this is the time from noon (sundial time) to
noon (sundial time).
It is difficult to imagine the difference between the two without a diagram, so here's one. Imagine a planet where the (sidereal) year is only eight solar days (and for simplicity imagine the planet
has no axial inclination).
\ B
A \
| star at zenith for B8
/ A8B - - - - - - - - - - - - -
sun at / A
zenith 7
for A9 / |B
/ 6
/ | A
/ B|
/ B4A
/ B|
/ |A
/ 2
/ |B
/ A|
sun at zenith for A0 | star at zenith for B0
* - - - - - - - - - - - - - - - - - A0B - - - - - - - - - - - - -
Imagine two observers on opposite sides of the planet at some time 0, so that it's noon for observer A (and consequently midnight for observer B). As time passes, the planet spins and goes around in
it's orbit. (In this diagram, the spin and the orbit are both in the same direction - anticlockwise. The Earth's orbit and spin looks the same when viewed from some point to the North of the ecliptic
, except of course it spins much faster.) At time 8, the planet has made one complete revolution (relative to the celestial sphere), so the same star that was at zenith for B at time 0 is at zenith
once more. The sidereal day is 8 units long. However, as observed from the planet, the position of the sun has changed relative to the celestial sphere, so observer B doesn't see the sun at zenith
until time 9. The solar day is 1/8 longer than the sidereal day.
It is no coincidence that 8 is the number of solar days in the year, and 1/8 is the factor by which the solar day differs from the sidereal day. Over the course of a year, the number of sidereal days
is precisely one more than the number of solar days. Thus, there are about 365.25 solar days in the year, but there are about 366.25 sidereal days. This difference of 1 can be conceptually grasped by
considering a planet that is in tidal lock with the sun - imagine observer A always seeing the sun at zenith in the diagram above, and observer B always on the opposite side. There are no solar days
at all, but when the planet makes one revolution in one year, observer B sees the same star at zenith as a year before - so there's one sidereal day where there are no solar days. (If the planet were
spinning in the direction opposite the orbit, with more than 1 solar day per year, there would be one fewer sidereal days than solar days. If the planet were spinning in the opposite direction, with
less than 1 solar day per year, the sum of the fraction of the solar day per year and the fraction of the sidereal day per year would be 1.)
Earthlings most easily recognise the solar day because the sun is so significant in the sky; it lights up the whole sky by atmospheric scattering and obscures all the other stars, so the solar day is
the most significant type of day on Earth. On Neptune or Pluto, however, the sun (at first glance) looks like it's not much more than just another star - brighter than usual, but it does not have
such a glare as to make all the other stars invisible. On distant, airless planets, the sidereal day would be more recognisable than the solar day. Neptunians would be more likely to make their
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CALCULUS - need help!
Number of results: 83,537
Calculus. Please!! i need help so bad!!!
Post a New Question | Current Questions | Chat With Live Tutors Homework Help: Calculus. Please!! i need help so bad!!! Posted by heather on Tuesday, April 24, 2012 at 7:16pm. 1. Graph y =sec(1/
2O-2pi)– 3. the O after the 2 has a slash through it 2. Write an equation for ...
Tuesday, April 24, 2012 at 7:16pm by Anonymous
What is the simplest solution to the Brachistochrone problem and the Tautochrone problem involving calculus? (I know that the cycloid is the solution but I need a simple calculus proof as to why this
is the case)
Thursday, May 27, 2010 at 12:13am by Sam
this is part of calculus cause i need it to find critical points of a graph
Tuesday, November 18, 2008 at 5:39am by alice
I need help with this calculus problem. H(S)=S/(1+S) If F 1/2, -1/2 & (a+1)? I can't figure out what this mean /?
Monday, May 30, 2011 at 12:46pm by Tammy
First of all, it is spelled calculus. Next, show your work so that we know where you need help. Okay?
Sunday, February 17, 2008 at 7:11am by Guido
I think you need a real tutor. You are asking questions that are fundamental to understanding calculus.
Tuesday, June 28, 2011 at 11:26am by bobpursley
Calculus, bobpursley!
This has to do with my last Calculus problem. I need help on it..The part I replied to after you answered the question.
Tuesday, September 21, 2010 at 8:01pm by Leanna
Urgent help calculus
you have a rectangle with height 5 and width 8-x. So, 5(8-a) = 20 8-a = 4 a = 4 No need for calculus here!
Wednesday, November 28, 2012 at 11:40am by Steve
By the way there is no need to use calculus for these since you can complete the square to find the vertex of a parabola with algebra 2.
Sunday, July 15, 2012 at 1:51pm by Damon
I need to take a college calculus class. Can someone direct me to a site which might help me get a good start? Thanks in advance.
Thursday, July 10, 2008 at 3:38pm by jules
Calculus - Urgent!!!
In case this looks familiar, yes, it is a repost. I really need help. Find the anti-derivitive of f(x)= x^3(x-2)^2. Multiple choice gives me the options: A: 1/6 x^6 - 4/5 X^5 + x^4 + C B: 1/6 x^6 - 4
/5 X^5 + 1/4 x^4 + C C: x^6 - X^5 + x^4 + C D: x^5 - 4x^4 + 4x^3 + C I got as ...
Thursday, April 26, 2012 at 5:18pm by In Need Of Help!
Calculus AP need help!
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. If f(x)= ∫(4,x^3)√(t^2+10) dt then f '(x).
Tuesday, April 17, 2012 at 4:08pm by dania
Did you follow suggestion and sketch the lines? I see a triangle with vertices (0,0), (6,3) and (6,-6) You certainly would not need Calculus for this base = 9, height = 6 area = (1/2)(9)(6) = 27 If
you have to use Calculus .... effective height = x/2 - (-x) = (3/2)x area = ...
Thursday, April 12, 2012 at 10:32pm by Reiny
You do not need Calculus for this question, since this results in a trapezoid. The two parallel sides have lengths of 2 and 18 and the distance between them is 2 Area = 2(2+18)/2 = 20 if you insist
on Calculus Area = [integral](8x+2)dx from 0 to 2 = |4x^2 + 2x| from 0 to 2 = ...
Sunday, December 5, 2010 at 4:33pm by Reiny
I need help with integrals and I need help with the problem. Integral of 4/sqrt(x)dx
Monday, April 11, 2011 at 9:30pm by Leanna
Math: Calculus
I REALLY NEED HELP ANSWERING THIS. I mange to get b and c but i'm stuck on the other two
Wednesday, November 3, 2010 at 11:14pm by NEED HELP!!!!
Please help. I need ( with details) how to differentiate with the chain rule f(x)=(x^2-6x+23)^3/2. I need to
Sunday, April 10, 2011 at 1:58pm by ben
Need an answer to this Calculus Question. I keep getting this type of question wrong. Thanks. Differentiate the Function: y = (3x-1)^2(2-x^4)^2 y =
Monday, April 19, 2010 at 5:04pm by Dave
I need help with calculus...
Tuesday, October 28, 2008 at 12:46pm by mohamed
pre calculus
You need to write out the question. I have no idea what "pre calculus" this could be, maybe zeroes, however, that is standard fare for an algebra course.
Tuesday, December 10, 2013 at 6:26pm by bobpursley
pre calculus
You need to write out the question. I have no idea what "pre calculus" this could be, maybe zeroes, however, that is standard fare for an algebra course.
Tuesday, December 10, 2013 at 6:26pm by bobpursley
6 - 2x is a straight line with a negative slope. In the interval [-1,2], it is highest when x is smallest (-1) and lowest when x is highest (2). You don't need to use calculus to answer this
Monday, November 16, 2009 at 5:51am by drwls
Grade 12 Calculus
you don't need calculus for this. It can just confirm your answers. these are just ordinary parabolas. Find the vertex, and check the values at the ends of the intervals.
Tuesday, April 16, 2013 at 12:15pm by Steve
I need to find the critical number of 4x^3-36x^2+96x-64 by factoring. Basically I need to know the zeros.
Monday, November 9, 2009 at 1:25pm by Matt
Sorry, I need more help! I am reallly struggling with Euler's method, and here's a problem I need help on: y'=2y + sin(x), y(0)=0, dx=0.1, the interval is 0 to 1, and f(x)= [e^(2x)-2sin(x)-cos(x)]/5
Monday, December 18, 2006 at 10:44pm by Raine
i need to find the derivative using chain rule and show steps: y= (x) / ((x^2)-1)^(-1/2) and is it possible to have free tutoring online like through a chatroom? i am in need of help
Friday, January 10, 2014 at 5:44pm by eric
Calculus - Integrals
Well, I think you just need to look at a simpler case. The algebra may sometimes obscue things. Also, you don't need to understand Lioville's theorem. Also, you don't need to stick to what you've
been taught in class. All you need to do is pick up a piece of paper and try to ...
Monday, March 24, 2008 at 12:53pm by Count Iblis
Hi there, I need help with composition of functions. I need to find fog, gof, gog, and fof and their domains for the following: f(x) = square root of 2x +3 g(x) = x^2 + 1 if someone can help me asap
that would be so great!
Wednesday, January 21, 2009 at 1:55pm by Lola
Applied Calculus
I know how to solve it to get .21, but I need to write it using positive exponents only. I know this isn't technically Calculus, but it is what we are reviewing currently. I am a bit rusty with this
one, can't quite remember how to do it. I am trying some different things, but...
Thursday, March 21, 2013 at 1:33am by Jacob
college algebra--need help Please!!!
Reiny, It seems like Calculus, but it actually falls under college algebra. Because I can get some calculus tutoring and the answers fit. Thank you for the help, it is greatlt appreciated!!!
Monday, November 12, 2012 at 10:19pm by ladybug
I need help simplifying this problem. f(x)=x^4/5(x-4)^2 f'(x)=x^4/5*2(x-4)+(x-4)^2*4/5x^-1/5 it is suposed to end up as this: 1/5x^-1/5(x-4)[5*x*2+(x-4)*4] but how do i get it to there? I need to see
all the steps and how to get them. Thanks
Tuesday, March 31, 2009 at 10:52am by Riley
drwls is right no need for calculus here but if youd like to use it f(x)=2(3-x) [-1,2] f'(x)=-2 no critical values therefore only test endpoints f(-1)=8 <--- abs. max (-1,8) f(2)=2 <---- abs. min
Monday, November 16, 2009 at 5:51am by Anonymous
Your question is not clear at all. When you say "under" the graph of f(x) = 6x+1, the region would be infinitely large. You will need to close it. Is it bounded by the x-axis? If so according to your
interval part would be below the x-axis and part would be above it. Make a ...
Wednesday, December 8, 2010 at 12:32pm by Reiny
Can't you answer any of these? If not, you need private tutoring. That is not what we provide here. Show us what you can do so we can see where you most need help, and someone will provide help with
those areas.
Sunday, December 6, 2009 at 5:41pm by drwls
CALCULUS!!! tangent lines
If f(2)=4 and f'(2)=3, use the line tangent to the graph of f at x=2 to find a linear approximation for f(1.992) HELP! The only one I need, I need a Genius!!!!
Tuesday, March 23, 2010 at 10:12pm by Anonymous
A vector perpendicular to both m and n is the cross product. After that, you only need to normalize it to make a unit vector. Post if you need more help.
Thursday, May 24, 2012 at 10:51pm by MathMate
Using Calculus: Area = ∫(x+3) dx from 8 to 15 = [x^2/2 + 3x] from 8 to 15 = 225/2 + 45 - 64/2 - 24 = 203/2 or 101.5 of course we don't need Calculus to do this, since the shape is a simple trapezoid
at x=8, height = 11 at x = 15 , height = 18 distance between = 15-8 = 7 ...
Saturday, October 1, 2011 at 3:51pm by Reiny
Marlyn, I do not know if you have gone back to look at my detailed answer to your post on Dec. 5. http://www.jiskha.com/display.cgi?id=1260035879 If after reading the answer, you still have
difficulties doing this current question, I must conclude that drwls is right. You need...
Sunday, December 6, 2009 at 5:41pm by MathMate
CALCULUS. Check my answers, please! :)
Thank you!! Unfortunately I couldn't wait for your response, but I went over my answers before checking to see if they were right, and I got them all right! Thank you so much for your help! It means
the world to me, and will count on you if I need help on more topics in ...
Saturday, October 5, 2013 at 12:24pm by Samantha
convert to the alternate form to exponential form. a) 5^4=625 b) e^x=y c) log.001=x I really need help on these problems i have no idea how to do it? pls help me i need to study for it on my exam.
Friday, December 6, 2013 at 1:25am by Lane
I need help with these two review calculus problems for my final exam. What is the derivative of a). (5x^2+9x-7)^5 b). Lne^x^3 For a. I got 50x+45(5x^5+9x-7)^4 For b. I got 3x^2 Can you verify
Monday, May 21, 2012 at 11:58am by Sam
You have to specify your domain. For log(x-1) you need x>1 For log(1-x) you need x<1 May times you will find it written that ∫ dx/x = log |x| + C just for this reason.
Wednesday, January 8, 2014 at 4:04am by Steve
college algebra--need help Please!!!
You can complete the square .... y = -(x^2 + 6x +9 - 9 = -( (x+3)^2 - 9) = -(x+3)^2 + 9 vertex is (-3, 9) alternate method: for y = ax^2 + bx + c the x of the vertex is -b/2a for for yours the x of
the vertex is -(-6)/-2 = -3 sub into the function ... y = -9 + 18 = 9 so the ...
Monday, November 12, 2012 at 10:19pm by Reiny
You must be kidding. You need to provide numbers in to get numbers out. If you want a formula, you need to define the symbols for the variables. The least you could do is provide the complete
Friday, October 21, 2011 at 9:29am by drwls
Need help with the following proof: prove that if lim x->c 1/f(x)= 0 then lim x->c f(x) does not exist.I think i need to use the delta epsilon definition i am not sure how to set it up.
Sunday, July 26, 2009 at 4:32pm by jeff
In an interview of 50 math majors, 12 liked calculus and geometry 18 liked calculus but not algebra 4 liked calculus, algebra and geometry 25 liked calculus 15 liked geometry 10 liked algebra but
neither calculus nor geometry 2 liked geometry and algebra but not calculus. Of ...
Monday, January 11, 2010 at 8:45pm by Anita
Calculus 2
Duplicate post, already answered. I gave you the formula for the total volume. If you want the volume at different heights, you have to say which level heights you want. That is where the need for
calculus comes in. You must have mistyped something. You can't have two minor ...
Tuesday, March 16, 2010 at 3:20am by drwls
y=a(x-1500)^2+b is the form of the equation. you need to solve for a, b. Well, you know immediately that at x=1500, b has to be 25 Now, at x=0, you know y=450. Solve for a. Now, with a,b solve for y
when x=750 What is the calculus in this?
Friday, November 5, 2010 at 10:36am by bobpursley
Calculus Graph help!!!
I need help with a calulus graph and figuring out its equation but it wont let me post the picture of the graphs here. so will someone who understands Calculus and their graphs super well please
email me k m k underscore horse lover at hot mail dot com. no spaces and an actual...
Wednesday, May 2, 2012 at 9:44pm by kelsey
I need to repost
Sunday, December 2, 2007 at 1:20pm by kelly
So what do you need? the derivative?
Friday, April 25, 2008 at 4:18pm by Damon
I need help. i am so lost
Tuesday, September 14, 2010 at 9:25pm by Preston
please i need an example !
Friday, April 15, 2011 at 2:32am by E.T
need derivative
Friday, October 14, 2011 at 2:20pm by Beth
Calculus AP need help!
Tuesday, February 28, 2012 at 11:12pm by josephine
Calculus AP need help!
You're welcome! :)
Tuesday, February 28, 2012 at 11:12pm by MathMate
I really need help on this!!!!
Monday, October 21, 2013 at 8:25pm by anon
Calculus: drwls, please
Hi, You need to draw f(x) =1/x first and then transform it to f(x) = (1/x) - 4 To draw f(x) = 1/x, all you need to do is find several points, example: let x=1, then f(1)=1, you get one point (1,1)
let x = 0.5, f(0.5)=1/0.5=2, another point (0.5,2). Try to get 5-10 points and ...
Friday, April 4, 2008 at 9:31pm by Qun
come on. even without calculus you know that the vertex is at (3,-1). With calculus, f' = 2x-6, and f is increasing where f' > 0
Sunday, February 12, 2012 at 10:31pm by Anonymous
8. In an interview of 50 math majors, 12 liked calculus and geometry 18 liked calculus but not algebra 4 liked calculus, algebra, and geometry 25 liked calculus 15 liked geometry 10 liked algebra but
neither calculus nor geometry 2 liked geometry and algebra but not calculus. ...
Monday, November 30, 2009 at 9:02pm by poo
You need to review your calculus. pdw ydy is the operand of the integral. INT pdw ydy= pdw int ydy= pdwy^2/2 evaluated over the limits.
Saturday, October 24, 2009 at 11:12pm by bobpursley
I don't see why you need calculus for this, nor why they are calling the height 7h and the radius 4r. The differential volume of any cylindrical shell of radius r, height h and thickness dr is 2 pi r
h dr Integrating that from r = 0 to the outside radius R will give you the ...
Thursday, September 9, 2010 at 11:18am by drwls
really need help asap
Saturday, September 26, 2009 at 9:05pm by kim
I don't get it..Do I need to plug in anything for x or no?
Thursday, September 9, 2010 at 11:19pm by Anonymous
Math: Calculus
Wednesday, November 3, 2010 at 12:03am by REALLY NEED HELP!!!!
h(r)=1-r^2/2-r^3 find h'(1)
Tuesday, January 10, 2012 at 12:07pm by scarlette. . .need help asap
Need help for calcul: ( 1/2 ( -1 + i√3 )) ^2 thanks for your help
Saturday, January 14, 2012 at 6:50am by Sébastien
We need the given interval
Friday, March 2, 2012 at 12:23pm by Dean
I did that and got 19. Is this right?
Thursday, April 26, 2012 at 3:29pm by I Need Help
Calculus I
This post was also placed at freemathhelp, on the calculus board with titled, "Calculus I". No work is shown at either location. Sad.
Tuesday, September 2, 2008 at 4:53pm by Mark
CALCULUS - need help!
I will be happy to critique your thinking on these.
Monday, October 1, 2007 at 11:00pm by bobpursley
pre Calculus
need the points not the equation
Sunday, September 7, 2008 at 9:04pm by bill
pre Calculus
need the points not the equation
Sunday, September 7, 2008 at 9:04pm by bill
Calculus II
i need help with long divison
Wednesday, March 24, 2010 at 12:57pm by jessica
Math: Calculus
I don't know why but the answer is wrong.
Wednesday, November 10, 2010 at 1:00am by REALLY NEED HELP!!!!
Verify this equation, y = 4^x-1 You need parentheses. Is it, y = 4^(x - 1)? Or, y = 4^(x) - 1? Or ?
Sunday, January 30, 2011 at 1:44pm by helper
Calculus (pleas help I really need to check this)
a) -6 b) -1/4 c) 1/2
Thursday, February 17, 2011 at 7:24pm by Reiny
yes I know that but I need to steps for how you find the derivative.
Sunday, October 2, 2011 at 7:20pm by Allison
Calculus. I need help!
my worksheet say sin^5 x. well thanks
Tuesday, February 28, 2012 at 2:00am by danny
Calculus. I need help!
oops my bad.. 26(x^2-y^2)
Friday, March 2, 2012 at 10:17pm by MILY
Calculus AP need help!
f' = 3/x^2 f = -3/x + C f(1) = 0 = -3 + C f(-1) = 0 = 3 + C Typo somewhere?
Sunday, March 25, 2012 at 7:08pm by Steve
I need to integrate ((e^-x)^2 - (-ex)^2)dx Can someone help?
Sunday, May 6, 2012 at 6:36pm by Kira
What is the use of Calculus? How is it use in jobs? What jobs use Calculus? Calculus is used in engineering, economics, any physical science, and in business (e.g., actuary studies and statistics).
Saturday, May 5, 2007 at 9:27pm by Anonymous
Calculus (Optimization, Still Need Help)
I just wanted to correct something for my equation, it should be: V = (14 - 2x)(8 - 3x)(x), which simplifies to V = 112x - 44x^2 - 4x^3. Take the derivative: V' = 112 - 88x - 12x^2 Now all I need are
the roots, any help? I think I found one around 1.10594, possibly?
Friday, December 16, 2011 at 8:20pm by Mishaka
This is not calculus,it is beginning algebra. What is the point of labeling it calculus?
Saturday, January 26, 2008 at 10:14pm by bobpursley
Integral Calculus
A jeepney's velocity over time in ft/sec as it moves along the x-axis is governed by the function v(t)=3t^2-10t+15. If the jeep's position at t= 1 second is 8ft, what is the total displacement made
by the jeepney from t=0 to t=5sec? please I need help, i just need the idea on ...
Sunday, January 12, 2014 at 7:01pm by Lisa
I am too embarassed to ask this Calculus (really pre-calculus) question in tutoring, because I know I should know. Is the inverse of f(x)=3x-1 actually f(x)=1/3x+1? How do I find it? What if it asks
the same equation replaced with f to the -1 power (x)? I think I know how the ...
Tuesday, July 1, 2008 at 10:38pm by Molly
f(x)=(y+a)and b=the limit as y aproches 2+f=1000. I need to know what y and a equal.
Wednesday, November 28, 2007 at 9:26pm by Roseliandystarajestarahjaesracritreah
Calculus-Optimal Form: need help... please
thnx! :)
Tuesday, November 4, 2008 at 7:46pm by Stephanie
ap calculus
yes i need help can you solve it out then i will ask you questions from then
Saturday, September 26, 2009 at 8:20pm by kate
f(x)=ln(x^4+27) find the inflection points?
Monday, April 5, 2010 at 10:24pm by NEED Help
I just need help getting started with this one.
Tuesday, September 14, 2010 at 9:25pm by Preston
Oh wait I made a mistake..still need help! Thanks.
Thursday, October 28, 2010 at 8:24pm by Anonymous
Math: Calculus
This website doesn't help me in solving my question.
Thursday, November 4, 2010 at 4:28pm by REALLY NEED HELP!!!!
Math: Calculus
This website doesn't help me in solving my question.
Thursday, November 4, 2010 at 4:32pm by REALLY NEED HELP!!!!
Math: Calculus
Never mind i got it.
Thursday, November 11, 2010 at 2:42pm by REALLY NEED HELP!!!!
i need a real life application on parabola.
Monday, May 16, 2011 at 6:25pm by Reem
I need a better more indepth answer
Friday, October 26, 2007 at 12:14am by fool
Hi i need help for calcul with a Moivre's theorem: ( 1/2 ( -1 - i√3 )) ^3 thanks for your help
Tuesday, January 24, 2012 at 12:44pm by Sébastien
I need help finding the derivative of (x^2+sinx)secx
Tuesday, February 21, 2012 at 5:19pm by jholm
Pages: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Next>> | {"url":"http://www.jiskha.com/search/index.cgi?query=CALCULUS+-+need+help!","timestamp":"2014-04-16T05:35:13Z","content_type":null,"content_length":"33658","record_id":"<urn:uuid:5d1e4515-12c5-42f2-bf42-52bd3bf326c6>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00308-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hummingbird Graphics Processor, Galaxy S
Well, this came as a complete surprise to me. The Galaxy S will be the strongest handset as far as graphics and gaming goes when it is released (unless something stronger is in the making). Once you
pair this outstanding GPU with its beautiful SUPER AMOLED display and powerful processor, you will have a handset that truly rivals today’s mobile gaming platforms.
Here are the numbers of some Android Phones compared with consoles:
• Motorola Droid: TI OMAP3430 with PowerVR SGX530 = 7 million(?) triangles/sec
• Nexus One: Qualcomm QSD8×50 with Adreno 200 = 22 million triangles/sec
• iPhone 3G S: 600 MHz Cortex-A8 with PowerVR SGX535 = 28 million triangles/sec
• Samsung Galaxy S: S5PC110 with PowerVR SGX540 = 90 million triangles/sec
• PS3: 250 million triangles/sec
• Xbox 360: 500 million triangles/sec
For some odd reason Samsung did not discuss this in their official press release for the device. You would think that this would be one of the top features highlighted in the device specs. I
personally have no idea what million triangles/sec means but once I seen the comparison above, it’s pretty clear to me that it means something cool. Androidguys dug up the following definition on
A polygon in a computer graphics (image generation) system is a two-dimensional shape that is modelled and stored within its database. A polygon can be coloured, shaded and textured, and its position
in the database is defined by the co-ordinates of its vertices (corners).
Naming conventions differ from those of mathematicians:
* A simple polygon does not cross itself.
* a concave polygon is a simple polygon having at least one interior angle greater than 180°.
* A complex polygon does cross itself.
Use of Polygons in Real-time imagery. The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to
the display system (screen, TV monitors etc) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of the processed
data to the display system. Although polygons are two dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation so that as the viewing
point moves through the scene, it is perceived in 3D.
Morphing. To avoid artificial effects at polygon boundaries where the planes of contiguous polygons are at different angle, so called “Morphing Algorithms” are used. These blend, soften or smooth the
polygon edges so that the scene looks less artificial and more like the real world.
Meshed Polygons. The number of meshed polygons (”meshed” is like a fish net) can be up to twice that of free-standing unmeshed polygons, particularly if the polygons are contiguous. If a square mesh
has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square. There are (n+1) 2/2n2 vertices per triangle. Where n
is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines).
Polygon Count. Since a polygon can have many sides and need many points to define it, in order to compare one imaging system with another, “polygon count” is generally taken as a triangle. When
analysing the characteristics of a particular imaging system, the exact definition of polygon count should be obtained as it applies to that system as there is some flexibility in processing which
causes comparisons to become non-trivial. Vertex Count. Although using this metric appears to be closer to reality it still must be taken with some salt. Since each vertex can be augmented with other
attributes (such as color or normal) the amount of processing involved cannot be trivially inferred. Furthermore, the applied vertex transform is to be accounted, as well topology information
specific to the system being evaluated as post-transform caching can introduce consistent variations in the expected results.
Point in polygon test. In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x0,y0) lies inside a simple polygon given by a sequence of line
segments. It is known as the Point in polygon test.
Looks like the Galaxy S has more up its sleeves then we thought. | {"url":"http://androidcommunity.com/hummingbird-graphics-processor-galaxy-s-20100327/","timestamp":"2014-04-21T12:48:39Z","content_type":null,"content_length":"40117","record_id":"<urn:uuid:3a6ca7b8-2b6d-4395-be6b-aa33d0ca23e9>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00104-ip-10-147-4-33.ec2.internal.warc.gz"} |
2 short article vs. a long one
up vote 3 down vote favorite
I am a beginning mathematician and I need some publishing advice. I have some results obtained during my PhD research and I was wondering about the best way to publish them.
My initial intuition was to write one long article containing all my results (because for me it is all part of the same research). On the other hand, a more experienced mathematician suggested to
publish two medium articles, with the following rationalizations: a short article is more likely to be read/published, the results can be presented independently and as a beginning mathematician it
is better that I have a reasonable number of short articles instead of very few long articles (this may sound a bit cynical).
This sounded reasonable enough, but when I wrote the two articles, I found out that a quit large part which I introduce the setting is practically the same in both articles (the results and main
theorems are obviously different).
I am looking for advice – how reasonable is it for a mathematician to publish two articles which has a very similar beginning and themes? (I should stress – the results are different – I am not
trying to publish the same result twice!)
5 I don't think there's generic advice that works for all people in your situation. Why not talk to your dissertation advisor, or some other senior mathematician wherever you acquired your Ph.D? –
Ryan Budney Feb 19 '12 at 8:57
2 Please don't use the username "mathoverflow". There is a computer controlled account of almost exactly this name which does housekeeping work-- it's very confusing if now a real person is also
using this name... – Matthew Daws Feb 19 '12 at 9:22
2 what about a splitted version, with the same title, into "part I" and "part II", to be submitted possibly (but not necessarily) to the same journal? – Pietro Majer Feb 19 '12 at 9:50
1 It might help to know in which field you work (the average article length and count varies a lot among sub-fields), and how many pages you consider "long" to be. – Federico Poloni Feb 19 '12 at
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closed as not constructive by Ryan Budney, Will Jagy, Yemon Choi, Alain Valette, quid Feb 19 '12 at 11:14
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate,
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4 Answers
active oldest votes
It depends! I am not giving any advice but I am only recalling an anti-measure theoretic quote in this context:
“The whole is greater than the sum of its parts.” - Aristotle
up vote 1 down vote
It may help to think in these lines. All the best!
add comment
I think this a constant question for everyone, e.g. right now my co-authors have been debating about this for months and they are experienced researchers. I would follow the advice given
by your experienced friend, as probably he knows better, plus I also agree with what he said in general. You should not feel bad about repeating some definitions twice, people do it all
up vote 0 the time.
down vote
add comment
I would say that writing two articles with the same introductory body is not the problem. Rather, are the results of both articles going in the same direction? If yes, then it should be
up vote -1 better writing a single article, or a two-part article.; If, on the other hand the results of the two articles point in different directions then maybe you should split the article.
down vote
add comment
I think if you have sure-shot audiences/readers then only you should come up with a one long article publication otherwise the best thing is to give your articles in parts(depending upon
how long it is). Repitition of a a few results is not always bad. It may help people link and remember the points that you have discussed in your Part 1 and the best thing in coming with
two parts is you can assure some pre-available readers for your Article.2
Why I said this is Ramanujam prepared three notebooks with only results and didn't send all of his work at once to the mathematicians of his time. Think if he would have sent his complete
up vote -3 set of notebooks along with the solution and derivations then may be he won't have been what he is today.
down vote
Only a small piece is enough for people to see the spark in you and keep subscribed to your upcoming works which you can mention (in very brief ) along with your Part 1.
But this is purely a piece of **my personal thinking**. *Do whatever you feel suits you the best for nothing in this world is predictable.* Anyways good luck in advance.
add comment
Not the answer you're looking for? Browse other questions tagged advice or ask your own question. | {"url":"http://mathoverflow.net/questions/88910/2-short-article-vs-a-long-one/88918","timestamp":"2014-04-19T00:03:03Z","content_type":null,"content_length":"59732","record_id":"<urn:uuid:d6bd5ad0-6231-4c63-abe3-fbec2f3d59fb>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00097-ip-10-147-4-33.ec2.internal.warc.gz"} |
Storing very small numbers - but only order of magnitude
February 9th, 2013, 10:50 PM
Storing very small numbers - but only order of magnitude
I want to store a very small number - too small for even a long double. However, I don't need the number to be stored very precisely - in fact, all I really need is the order of magnitude of the
number. For example, the number 5.205849034425 X 10^-381 can just be stored as an integer of -381 for my purposes. I don't really care about the precision of the 5.205849034425 part.
The reason I get this very small number, is because I am multiplying together thousands of numbers, all of which are around 0.0001 in value. So, one way to do this would be to store all these
individual values as simply their order of magnitude (e.g. -4) and then instead of multiplying the numbers, I would just add up their orders of magnitude. However, I am wondering whether there is
a way in C++ to do this without me having to change the way I store my data - I still would like to keep the original values if possible.
February 10th, 2013, 04:17 AM
Re: Storing very small numbers - but only order of magnitude
without more specific requirements, I'd simply compute the product as the exponential of the sum of the logarithms ...
February 11th, 2013, 07:43 AM
Re: Storing very small numbers - but only order of magnitude
if your end number doesn't fit into a double.... and 10^-381 doesn't (double can be 10^-307 at best)...
Then you'll need a special numbers class just to handle all the multiplications reliably. Whatever this class is and how it works... how you extract the exponent from that class will be dependant
on what class you're using.
It could be a double-like type that allows for larger exponents and/or larger mantisa's, or it could be a class that "more or less" behaves in the way you want for your results.
if you want to know how to extract the exponent from a double: use frexp() from math.h
If you aren't interested at all in the mantissa and don't care about any effect of the mantissa on the exponent, then you can extract all the exponents and just add them up.
note that 0.5 * 0.5 = 0.25 = 5*10^-1 * 5*10^-1 = 2.5*10-1
so adding the exponents would have given you -2 when the end result is -1 because of the scaling effect of mantissa's.
if you need correct mantissa and exponent evaluation.
then after each multiplication, scale up by factors of 10 the resulting double, and accumulate the exponents.
0.5 * 0.5 = 0.25 = 2.5*10^-1
extract the exponent, and accumulate (exponent_accumulation is now -1), scale the double up to be noted in *10^0 (= multiply by 10^-extracted_exponent = *10)
we now have result=2.5*10^0 and exponent_accumulation=-1
multiply by 0.001 = 2.5*10-3;
extract exponent (-3) and accumulate (exponent_accumulation is now -4), scale the double up to be noted in *10^0 (= multiply by 10^-extracted_exponent = *1000)
we now have result=2.5*10^0 and exponent_accumulation=-4
multiply by 0.033 = 8.25*10-2
extract exponent (-2) and accumulate (exponent_accumulation is now -6), scale the double up to be noted in *10^0 (= multiply by 10^-extracted_exponent = *100)
we now have result=8.25*10^0 and exponent_accumulation=-6
... etc
you'll still have an exponent that ends up being "off" by a bit for very large number of multiplications because you're loosing some precision in the mantissa each time, and you're introducing
errors each time you're rescaling the double. Whether this is or isn't an issue for your problem, I can't tell, it depends how accurate you need things to be. It'll need some error analysis if
you need an actual grip on how much of an error you are accumulating in each step.
It will be "close", but it can't ever be "accurate", floating point and accurate don't go hand in hand. | {"url":"http://forums.codeguru.com/printthread.php?t=534467&pp=15&page=1","timestamp":"2014-04-17T13:51:11Z","content_type":null,"content_length":"8798","record_id":"<urn:uuid:ec9ab8ca-7da1-4e38-b4a8-417d1130611e>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00285-ip-10-147-4-33.ec2.internal.warc.gz"} |
Multidimensional Calculus
MATH 2057 Multidimensional Calculus Fall 2010
│ Section 4 │ Instructor: Dan Cohen │ Phone: 578-1576 │
│ T Th 7:40-9:00 am │ Office Hours: T Th 9:30-11:00 am │ E-mail: │
│ │ and by appointment │ │
│ 112 Lockett │ Office: 372 Lockett │ URL: www.math.lsu.edu/~cohen │
Text: Rogawski, Calculus (Early Transcendentals Version). We will cover (portions of) chapters 14 though 17.
Prerequisites: MATH 1552 (Calculus II).
Description: In the third semester of the Calculus sequence, we extend the basic notions from first-year Calculus, differentiation and integration, and their interpretations and applications
(tangents, optimization, area, volume...) to functions of several variables. We also study Vector Calculus, generalizing integration to line and surface integrals. These integrals arise in a number
of applications (work, fluid flow, electricity and magnetism...), and are related to their predecessors by analogues of the Fundamental Theorem of Calculus.
Catalog Description: Three-dimensional analytic geometry, partial derivatives, multiple integrals.
The departmental syllabus may be found at www.math.lsu.edu/courses/syllabi/2057.html.
Homework: This section of Calculus III will use the computer package WeBWorK for the assigning and grading of homework.
Your username is your PAWS email name, and your initial password is your 89 ID number (with no spaces or hyphens). You may change your password if you wish. WeBWorK also stores your PAWS email
address. If you prefer another email address, you may change this as well.
WeBWorK assignments will be given throughout the semester. The first assignments are available now. You should make sure you can successfully log in, and begin working on it immediately. In total,
WeBWorK homework will be worth 100 points.
I will assign other homework problems from the text (to be done by hand for practice) essentially every class. These homework assignments will be announced in class
Homework, both from WeBWorK and the text, will occassionally be discussed in class as necessary.
Exams: There will be two in-class exams, each worth 100 points. The exams will tentatively take place in the weeks of October 4 and November 15. Exact exam dates will be announced in class. No
make-up exams, except in extreme cases. If you must miss an exam, I will expect you to notify me before the exam takes place.
Final: There will be a comprehensive final exam worth 150 points. The final exam is scheduled for Saturday, December 11, 12:30 - 2:30 pm.
Grade: Your course grade will be out of the 450 possible points outlined above. I may curve course grades. In any case, 90-100% is assured an A, 80-89% a B, and so on.
Important Dates: August 30 is the last day to drop; September 1 is the last day to add; Labor Day is September 6; Fall Holiday is October 21-22; November 5 is the last day to withdraw; Thanskgiving
is November 24-26.
Notes: Calculators, computers, books, notes, etc. may not be used on the in-class exams or the final exam.
Bear in mind that you are taking this course under the guidelines of the Code of Student Conduct.
Dan Cohen Fall 2010 | {"url":"https://www.math.lsu.edu/~cohen/courses/PastSemesters/FALL10/M2057/M2057.html","timestamp":"2014-04-20T03:09:55Z","content_type":null,"content_length":"6668","record_id":"<urn:uuid:270e6361-f6b4-4e7c-827a-61985a8cbd59>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00470-ip-10-147-4-33.ec2.internal.warc.gz"} |
Profit Maximization
April 14th 2009, 01:18 PM
Profit Maximization
The profit (P) (in thousands of dollars) for a company spending an amount s (in thousands of dollars on advertising is: P = -1/10s^3 + 6s^2 + 400.
A.) The amount of money the company should spend on advertising in order to obtain a maximum profit.
B.) The maximum Profit.
I am stuck on this problem and I have to have it submitted soon. Can anyone help me????!?!?!?
April 14th 2009, 01:51 PM
The profit (P) (in thousands of dollars) for a company spending an amount s (in thousands of dollars on advertising is: P = -1/10s^3 + 6s^2 + 400.
A.) The amount of money the company should spend on advertising in order to obtain a maximum profit.
B.) The maximum Profit.
I am stuck on this problem and I have to have it submitted soon. Can anyone help me????!?!?!?
find the derivative of $P$ w/r to $s$ , then set $P'(s) = 0$ and solve for the critical values of $s$ ... finally, determine which (if any) of those critical values yields a maximum for $P(s)$ by
using the first or second derivative test for extrema. | {"url":"http://mathhelpforum.com/calculus/83725-profit-maximization-print.html","timestamp":"2014-04-19T15:14:21Z","content_type":null,"content_length":"5565","record_id":"<urn:uuid:f5b1c7ac-3c5f-4b56-a4f3-dbc23f16391d>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00642-ip-10-147-4-33.ec2.internal.warc.gz"} |
Properties of Triangles
This tutorial is designed to improve and deepen students understanding of the characteristics and properties of triangles.
An applet is used to explore the properties of triangles interactively.
Interactive Tutorial
1 - Press the button above to start the applet.
2 - A triangle with vertices A, B and C and the values of all its sides, angles (in degrees) and area are displayed. You can DRAG any vertex of the triangle to change its sides and angles.
3 - Add all three angles of the triangle and round your answer to the nearest degree. What is the answer? Do this activity for several angles. Conclusion.
3 - Explore the famous triangle inequality: "the sum of the lengths of two sides is always greater than the length of the third side". This triangle inequality is equivalent to : "The shortest
distance between two points is a straight line."
4 - Explore, for different triangles, that the longest side is opposite the largest angle.
5 - Drag the vertices of the triangle so that the triangle obtained is acute ie all its 3 angles are acute.
6 - Drag the vertices of the triangle so that the triangle obtained is acute ie all its 3 angles are acute.
7 - Drag the vertices of the triangle so that the triangle obtained is obtuse ie the triangle has an obtuse angle.
8 - Drag the vertices of the triangle so that the triangle obtained is an isosceles triangle ie 2 sides of the triangle are (approximately) equal. Check that the 2 angles adjacent to the two equal
sides are also (approximately) equal.
9 - Drag the vertices of the triangle so that the triangle obtained is an equilateral triangle ie all 3 sides of the triangle are equal. Check that all 3 angles are (approximately) equal to 60
10 - Drag the vertices of the triangle so that the triangle obtained is a right triangle ie one of its angles is (approximately) equal to 90 degrees. You may also want to verify Pythagora's theorem
for this triangle.
11 - Use different formulas to obtain the area of triangles and check with the displayed value.
12 - You may also check the sine and cosine laws.
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Re: st: RE: RE: Covariates in ANOVAs
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Re: st: RE: RE: Covariates in ANOVAs
From David Airey <david.airey@Vanderbilt.Edu>
To statalist@hsphsun2.harvard.edu
Subject Re: st: RE: RE: Covariates in ANOVAs
Date Fri, 15 May 2009 08:38:16 -0500
There are two ways to control age. One is as a continuous covariate where the age of each individual is used (eg ANCOVA). The other is as a blocking variable where age is grouped and variation
associated with each age bracket is modeled separate from the error term (eg randomized block). Both of these assumes the age is not changing across the design relative to the other variables.
On May 14, 2009, at 11:30 PM, Miss Gina Micke wrote:
Thanks for your input. Maybe I didn't make myself clear enough re "age range". Age range refers to actual age in days at each measurement. As all subjects were measured on the same day at each
measurement event, the difference between subject ages is the same at each measurement event, so age at first measurement would be the data in the "age range" variable. Do you still think this
should be inlcuded as a categorical variable?
From: owner-statalist@hsphsun2.harvard.edu [owner-statalist@hsphsun2.harvard.edu ] on behalf of Joseph Coveney [jcoveney@bigplanet.com]
Sent: Friday, 15 May 2009 11:48 AM
To: statalist@hsphsun2.harvard.edu
Subject: st: RE: Covariates in ANOVAs
Miss Gina Micke wrote:
I would like to know how to control for an independent continuous variable when using a repeated measures ANOVA. I have a 2x2 factorial design experiment with males and females in each of the
treatment groups (T1 & T2) measured at multiple "events", each subject assigned an "id". The subjects are all different ages so
I would like to be able to control for the effect of "age range".
If I didn't have "age range" as a covariate my model would be as follows:
anova x T1 T2 sex T1*T2 T1*sex T2*sex T1*T2*sex/id|T1*event T2*event T1*T2*event sex*event sex*T1*event sex*T2*event T1*T2*sex*event, repeated(event) partial
Do I use the ANCOVA and cont(age range) command? Is this output the effects of
my fixed factors taking into account the effect of age range?
First, the command line text shown doesn't look at all like correct syntax. Second, by the name of it, a variable called "age range" would be categorical and so you would not be using the
-continuous()- option for it in - anova-.
Unless you have balanced data (equal representation of age ranges within each factorial cell, as well as equal sex-by-treatment group cell sizes maintained throughout the observation period), try
something like the following to start
xi3 i.treatment_group*i.age_group*i.sex*i.observation_interval
xtmixed response _I* || patient_id: , nolrtest variance reml
You'll need to install the user-written command, -xi3-, from SSC if you haven't
Joseph Coveney
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Here's the question you clicked on:
solve dy/dx = e^{x+y} (x+y)^{-1} - 1 y ????
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solve what
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\[\frac{ dy }{ dx } = e^{x+y} (x+y)^{-1} - 1 \]
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dame i am not at a level like this one, i am sorry
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ok..., no problem @MikiaseKebede nevermind :)
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he is good
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i think you have to apply the product rule
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what did u get it??
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hmm i havent solved it just trying to help you with it
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or there is another easier way u =x+y du/dx = 1 du =dx
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solve dy/dx = e^{x+y}
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@mayankdevnani No, i mean \[\frac{ dy }{ dx } = e^{x+y} (x+y)^{-1} - 1 \]
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let \[z=x+y\]and u have\[\frac{\text{d}z}{\text{d}x}=1+\frac{\text{d}y}{\text{d}x}\]put this in the original equation\[\frac{\text{d}z}{\text{d}x}-1=\frac{e^z}{z}-1\]\[\frac{\text{d}z}{\text{d}x}
=\frac{e^z}{z}\]u have a separable differential equation :)
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ok @mukushla, then i got \[\int\limits z dz = \int\limits e^{z} dx\] \[\frac{ z^{2} }{ 2 } = e^{z} x\] \[\frac{ (x+y)^{2} }{ 2 } = e^{x+y} x\] Did I miss something?
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emm...there is a little mistake in ur separation\[ze^{-z}\text{d}z=\text{d}x\]
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\[\int\limits \frac{ z }{ e^{z} } dz = \int\limits dx \]
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\[\frac{ -1 + z }{ e^{z} } = x\] \[\frac{ -1 + x + y }{ e^{x+y} } = x\] \[\frac{ x+y-1 }{ x } = e^{x+y}\]
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integration by parts for z :)
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\[\int\limits ze^{-z} dz = \int\limits dx\] u = z, \(dv = e^{-z} dz\), then du = dz, and \(v = -e^{-z}\) So \[-ze^{-z} - \int\limits -e^{-z} dz = \int\limits dx\] \[-ze^{-z} - \frac{ e^{-z} }{ \
ln e } = x\] then ??
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ln e = 1 re substitute z as x+y
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\[-ze^{-z} - e^{-z} = x\] \[-(x+y)e^{-(x+y)} - e^{(x+y)} = x\] \[(-x-y - 1) e^{-(x+y)} = x\] \[(1+x+y) =- x e^{(x+y)}\]
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did i miss something?
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nopes, thats correct. :)
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so how to find y?
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i don't think, y can be isolated....
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what it means to "W is the product log function"??
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even idk that... not a standard function for sure....
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ok Thank you all so much! i really appreciate it :)
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welcome ^_^
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
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logarithm assistance
September 13th 2012, 10:10 AM #1
Sep 2012
spokane, wa
I'm working on some problems and I have hit a stumbling block. Everything I try comes up with the wrong answer. Hopefully someone can help and with the right answer I can work it backwards to
Thank you!
Rewrite the logarithm as an exponential equation. (Use capital letters for variables P and U.) P = log[b] U
Rewrite the logarithm as an exponential equation. (Use capital letters for variables L, P and R.) L = log[b](P + R)
Rewrite the exponential equation as a logarithm. (Use capital letters for variables F and G.) b^F + 1 = G
Rewrite the exponential equation as a logarithm. (Use capital letters for Q, F, S, and P.) b^QF = S − P
An October 2009 article in The Industry Standard states that "independent Twitter data shows exponential tweet growth." GigaTweet, an independent tweet-counting service, reports that the number
of tweets was 5.0 billion in October 2009. In April 2009, the number of tweets was 1.6 billion. (a) Develop the exponential growth model that fits with these data, where t is the number of months
after April 2009 and p is the number of tweets in billions. (Round the coefficient of t to seven decimal places.)
p(t) =
Re: logarithm assistance
$y = \log_b{x} \iff x = b^y$
Re: logarithm assistance
Don't just say, "Everything I try comes up with the wrong answer". Show us what you tried and we will be better able to point out your misunderstandings.
Rewrite the logarithm as an exponential equation. (Use capital letters for variables P and U.) P = log[b] U
Rewrite the logarithm as an exponential equation. (Use capital letters for variables L, P and R.) L = log[b](P + R)
Rewrite the exponential equation as a logarithm. (Use capital letters for variables F and G.) b^F + 1 = G
Rewrite the exponential equation as a logarithm. (Use capital letters for Q, F, S, and P.) b^QF = S − P
An October 2009 article in The Industry Standard states that "independent Twitter data shows exponential tweet growth." GigaTweet, an independent tweet-counting service, reports that the number
of tweets was 5.0 billion in October 2009. In April 2009, the number of tweets was 1.6 billion. (a) Develop the exponential growth model that fits with these data, where t is the number of months
after April 2009 and p is the number of tweets in billions. (Round the coefficient of t to seven decimal places.)
p(t) =
Re: logarithm assistance
Rewrite the logarithm as an exponential equation. (Use capital letters for variables L, P and R.) L = logb(P + R)
I put " P+R = L ^B"
Re: logarithm assistance
September 13th 2012, 10:48 AM #2
September 13th 2012, 10:50 AM #3
MHF Contributor
Apr 2005
September 13th 2012, 10:57 AM #4
Sep 2012
spokane, wa
September 13th 2012, 01:35 PM #5 | {"url":"http://mathhelpforum.com/business-math/203394-logarithm-assistance.html","timestamp":"2014-04-20T06:56:47Z","content_type":null,"content_length":"47538","record_id":"<urn:uuid:7b065ce3-a9b3-401a-b084-67a176511030>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00524-ip-10-147-4-33.ec2.internal.warc.gz"} |
Preconceived notions about place value
One thing that struck me about the math talks given at this weekend's New England Conference on the Gifted and Talented was the emphasis on manipulatives and the concerns about whether children
understand place value. Are these the most appropriate things to be focusing on when it comes to students who are gifted in math? The mathematically gifted kids I know grasp place value and other
aspects of arithmetic with only minimal exposure to manipulatives, and quickly advance to higher levels of abstraction by the time they hit first or second grade. But the education establishment
seems bent on convincing itself that children--however gifted--don't understand place value.
Why would you want to convince yourself of this? Because it gives you an excuse not to teach the standard algorithms of arithmetic. If children don't understand place value, then they can't
understand borrowing and carrying (regrouping), let alone column multiplication and long division. And unless they understand how these procedures work from the get-go, educators claim (though
mathematicians disagree
), using them will permanently
harm their mathematical development
So, given how nice it would be not to feel any pressure to teach the standard algorithms (because, let's admit it, they are rather a pain to teach), wouldn't it nice to convince ourselves that our
elementary school students, however gifted in math, don't understand place value?
But how do you convince yourself of this? As that ground-breaking math education theorist
Constance Kamii
has shown, it's child's play. All you have to do is ask a child the right sort of ill-formed question. Here's how it works:
1. Show the child a number like this:
2. Place your finger on the left-most digit and ask the child what number it is.
3. When the child answers "two" rather than "twenty," immediately conclude that he or she doesn't understand place value.
4. Banish from your mind any suspicion that a child who can read "27" as "twenty seven" might simultaneously (a) know that the "2" in "27" is what contributes to twenty seven the value of twenty and
(b) be assuming that you were asking about "2" as a number rather than about "2" as a digit.
11 comments:
Hmmm. You're making me feel less bad about not attending. At least, it sounds like I wouldn't have gained much guidance for how to deal with the math issues in this household.
I was surprised to find my son's fifth grade math class starting out the year with place value. I didn't expect to see an emphasis on it at that grade level, don't recall it from when my older
child was in that grade. The only rationale I could think of was that maybe last year's class had done poorly on that part of the state tests.
At least they didn't use manipulatives! (I think...)
Oh man! This is exactly what I've been noticing about the way reading gets taught these days. Reading teachers are convinced that there's tons of kids out there who read perfectly fluently but
don't "comprehend".
This is because they ask the kid a question like "what do you think will happen next?" and the kid says "I don't know." They think this proves the kid "doesn't comprehend".
ARGH! It's like they're just inventing problems out of thin air, as if nature doesn't provide us with enough.
On the other hand, when you complain that your bright child can no longer do simple calculations that she once understood, they say, "but she understands place value!"
I've never understood all this concern about teaching place value. It's not a particularly difficult concept and it is a crucial foundational concept that shouldn't be put off. I introduced my
daughter to tens and ones at the age of 4. She's not gifted but she understands it. She is now doing online school with the California Virtual Academy and they cover place value in a K-Grade 1
online math class.
If students really are not able to grasp place value, it can only be because schools aren't teaching it properly. If I'm not mistaken, most countries introduce place value in 1st grade. If their
students get it and ours don't, we have to seriously look at how math is being taught.
That is most curious. There are, of course, children who at 4th and 5th grade are still remarkably shaky on numbers in a way that place value work (yes, even with manipulatives to some extent) is
useful (I tutor some of those kids). But a gifted and talented conference? What? Those kids (almost) all have a great grasp on numbers (I'm a parent to some of those kids), and going back to
manipulatives would be a serious waste of their time. Talk about not knowing your audience!
The idea that you shouldn't teach an algorithm before children can developmentally understand a concept is really dangerous. Often teaching the algorithm first will actually make it far easier to
understand the concept later on. There is no harm in telling a child that these are the steps you need to solve a problem. Later on the why will make sense.
We've got this problem now that kids don't understand the concepts because we don't teach them the algorithms and we're not teaching the algorithms because they don't understand the concept. It's
like being stuck in an infinite loop.
I think an important idea touched on here is the connection between understanding math concepts and being able to verbalize those concepts. The connection is not necessarily very close. Some
people would argue that "If you can't say it, you don't know it". I don't agree with that. And the contrapositive, "If you know it, you can say it" is obviously just as untrue. So what is true
about knowing and saying? My perspective would be that saying is very valuable to knowing, and knowing is very valuable to saying. We should try to keep them together as much as we can. But we
always need to remember that they can be separate, and often are.
I have a few years experience teaching college freshman math. Here is something I have noticed again and again when trying to help a student in my office. I will have the student attempt a
problem, and I will offer as much explanation as I can. But time and again I will be struggling to put together a string of words that concisely expresses whatever idea or information the student
is missing, when the student will suddenly say, "Oh, I get it!" That's my cue to shut up.
I am not a constructivist in the usual meaning of the term these days, but the constructivists are very much right about the central idea of the learner constructing his or her own knowledge.
People who call themselves constructivists, however, don't seem to apply the really important meaning of constructivism. When a student comes to me for help, and after a bit of preliminary
groping to understand the situation, I say, "Let's try this problem here . . .". In doing so I am setting up some elements that the student may use to construct knowledge. The student takes the
problem and tries to assemble the elements of that problem into some sort of meaningful structure. I offer help as best I can, which typically includes verbalizing the essential math concepts
that are involved. But my verbalizations are imperfect. The student's understanding of what I am saying is imperfect. Understanding comes when the student manages to take the elements of the
problem and see how they are to be assembled together in a meaningful way as required by the problem.
I think a very common fallacy for teachers is to think, "I explained it. They understand it. Therefore my explaining caused the understanding." That's not totally fallacious, of course, in most
cases. A good explanation can be a powerful contributing cause to understanding, often even a necessary cause, but not necessarily a complete and sufficient cause. We don't just explain. We also
assign problems. Every problem is a set of elements that the student must assemble into some form as required by whatever mathematical idea is being taught. Students must indeed construct their
own learning. It's strange that adherents of constructivism seem to want to do anything but deliver to the student a carefully crafted "learning kit", which is what the combination of a good
textbook, a good explanation, and a well chosen assignment is.
All this points to the idea that understanding, or the lack of understanding, is not necessarily easy to diagnose. Suppose I ask a student, "How many thirds make a whole", and the student looks a
little puzzled. (And I have had a few occasions to suspect college students may not understand the basic meaning of a fraction.) Whether in third grade, seventh grade, or even college, a blank
look may mean the student is wondering just what I mean and where it is leading, or it may indeed mean the student really doesn't understand the very simple primitive meaning of a fraction. Or, I
may ask a student that same question and the immediate response is "three" with no hesitation at all. Can we take this to mean that of course that student understands the basic meaning of a
fraction? I don’t think so. We should not take this as full and definitive confirmation that the student has the basic understanding of what a fraction means that we want that student to have. It
is simple one bit of evidence
Understanding, I would argue, is never easy to assess. It normally must come from observations and analysis over time.
Unless you're an ideologue, of course. Then it's amazingly simple and easy, as examples given attest.
And, if I may mention it, I have elaborated on what I consider real constructivism at http://www.brianrude.com/constv.htm.
Thanks for all the great comments! Look for some of these later in my "favorite comments of 2010" series.
"Discovery" or "constructing ones own knowledge" comes about through careful hints and prods as Brian Rude so accurately points out. In the examples he provides, what he is talking about is
"scaffolding". Start with something the student understands and use that as the springboard for the concept/procedure you are trying to get across.
If an assignment is constructed properly, sudents take away a good understanding of the material by the time they are done whether conducted in class or as homework Sometimes this manifests
itself in an an "aha" experience brought about by procedural fluency as Brian Rude pointed out with his "Oh, now I get it" example. In tutoring students, I've seen that the procedural fluency
resulting from the exercises helps clarify the concept, even if it wasn't fully understood before starting the problem set.
I also agree with Brian's characterization and thoughts on "understanding". (See http://www.educationnews.org/commentaries/opinions_on_education/93277.html
I recall walking along line of black cherry trees during first grade (1956-57) to pick up sticks. [Apparently we were too poor to buy "manipulatives", or our teachers knew when and when not to
spend precious resources on such things.] The sticks were brought into the classroom where some were kept as “ones”. Others were bundled into “tens” with rubber bands. Some of the tens bundles
were (without removing the tens rubber bands) grouped into ten and bundled with a bigger rubber band into “hundreds” bundles.
These bundles could be grouped and regrouped, (carried to and borrowed from) much more easily than can be done with those yellow cubes and sticks and flats that cost a lot of money. Let’s see
what happens when a student tries to “unbundle” (break) a stick or flat.
But I digress. We learned place value both on the blackboard and with physical models that showed what we were doing on the blackboard. We combined rote with logic and visualization. It worked.
Why do we have so much trouble remembering what works?
Mark Bohland | {"url":"http://oilf.blogspot.com/2010/10/preconceived-notions-about-place-value.html","timestamp":"2014-04-19T06:51:45Z","content_type":null,"content_length":"123104","record_id":"<urn:uuid:e06734bd-9e62-4b74-8b9f-b26e07c7c457>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00416-ip-10-147-4-33.ec2.internal.warc.gz"} |
More Numbers in the Ring
Copyright © University of Cambridge. All rights reserved.
'More Numbers in the Ring' printed from http://nrich.maths.org/
Vidhya from Kensri School in India sent in a very well reasoned solution:
When we subtract an even number from an odd number, or vice versa, the difference is always odd. So if we fill up odd numbers and even numbers alternately, if there are an even number of squares, the
differences will all be odd. But there is no solution
(in other words the differences cannot all be odd)
if there is an odd number of squares. | {"url":"http://nrich.maths.org/2783/solution?nomenu=1","timestamp":"2014-04-17T15:31:54Z","content_type":null,"content_length":"3532","record_id":"<urn:uuid:da9e6839-e624-4416-9f67-31c78a2b748e>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00446-ip-10-147-4-33.ec2.internal.warc.gz"} |
An Algorithm to Determine a probable profit
up vote 0 down vote favorite
I 'm searching for an algorithm (and except the naive brute force solution had no luck) that efficiently ($O(n^2)$preferably) does the following:
Supposing I’m playing a game and in this game I’ll have to answer n questions (each question from a different category). For each category $i$, $i=1,...,n$ I’ve calculated the probability $p_i$ to
give a correct answer.
For each consecutive k correct answers I’m getting $k^4$ points. What is the expected average profit?
I will clarify what I mean by expected profit in the following example:
In the case n=3 and $p_1=0.2,p_2=0.3,p_3=0.4$
The expected profit is
$ EP=\left(0.2\cdot 0.3\cdot 0.4\right)3^4+$ (I get all 3 answers correct)
$+\left(0.2\cdot 0.3\cdot 0.6\right)2^4+\left(0.8\cdot 0.3\cdot 0.4\right)2^4+\left(0.2\cdot 0.7\cdot 0.4 \right)2+$ (2 answers correct)
$+\left(0.2\cdot 0.7\cdot 0.6\right) +\left(0.8\cdot 0.3\cdot 0.6\right)+\left(0.8\cdot 0.7\cdot 0.4\right)$ (1 answer correct)
clearly for each possible outcome I'm calculating the probability and multiply it with the points gained. And then get the sum off all those.
Any ideas? I'm only interested in the sum itself.
Thank you!
This seems like a simple exercise rather than on the level of research, so it might fit some of the other sites in the FAQ better. If you really mean "consecutive" (contrary to your example) then
there is an $O(n^2)$ algorithm by considering the events that a streak of length $i$ ends in position $j$. If you don't mean consecutive then there is an even faster algorithm from expressing the
probabilities as a convolution. – Douglas Zare Jan 5 '13 at 4:55
@Douglas Zare If you have 5 questions then a possible answer could be CWCCC where C=correct and W=wrong. This answer will get $(1+3^4)p_1⋅(1−p_2)⋅p_3⋅p_4⋅p_5$ points. So consecutive answers get
more points that's what I meant. And I'm only interested on the sum of the points. Can you be more precise? – Vertical Jan 5 '13 at 12:07
add comment
1 Answer
active oldest votes
It's best, I think, to think of the questions as being asked in backwards order: $n$ first, then $n-1$, etc., down to question $1$. And just for fun, let's tack on one final question
for which the probability of answering correctly is $p_0=0$. The relevant recursion then is
\begin{array}{rcl} E(p_0,p_1,\ldots,p_n) &=& p_np_{n-1}\cdots p_1(1-p_0)(n-0)^2 \cr && +\ p_np_{n-1}\cdots p_2(1-p_1)\left((n-1)^2 +E(p_0)\right) \cr && +\ p_np_{n-1}\cdots p_3(1-p_2)\
left((n-2)^2+E(p_0,p_1)\right)\cr && +\ p_np_{n-1}\cdots p_4(1-p_3)\left((n-3)^2 + E(p_0,p_1,p_2)\right)\cr && +\ \ldots \cr && +\ (1-p_n)\left((n-n)^2 + E(p_0,p_1,\ldots p_{n-1})\
up vote 0 right)\cr &=& p_np_{n-1}\cdots p_1\left( n^2 + \sum_{k=1}^n{ (1-p_k)\over p_1\cdots p_k}\left((n-k)^2+E(p_0,\ldots p_{k-1}) \right)\right) \end{array}
down vote
accepted Starting from $E(p_0)=0$, one can compute (and store) values for $E(p_0,p_1)$, $E(p_0,p_1,p_2)$, etc. The complexity is probably something like $O(n^3)$.
To repeat, I've reordered the questions from last to first, for notational convenience -- I found it best to think in terms of the number of questions that remain to be answered. I hope
this doesn't cause too much confusion.
Thank you! Answer accepted :) – Vertical Jan 5 '13 at 20:32
add comment
Not the answer you're looking for? Browse other questions tagged algorithms or ask your own question. | {"url":"http://mathoverflow.net/questions/118095/an-algorithm-to-determine-a-probable-profit?sort=votes","timestamp":"2014-04-21T04:51:03Z","content_type":null,"content_length":"54079","record_id":"<urn:uuid:2ff1d7ee-06c0-42ba-bbdf-4bc0cdc7e44e>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00405-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: Q > rare matrix implementation?
Replies: 1 Last Post: Apr 18, 1997 11:34 AM
Re: Q > rare matrix implementation?
Posted: Apr 18, 1997 11:34 AM
In article <5il26l$8e3$10@halon.vggas.com>, JYoungman@vggas.com (James Youngman) writes:
>In article <ptcg1wzfc3y.fsf@hammer.thor.cam.ac.uk>, ey200@thor.cam.ac.uk
>>I'm afraid I'm not 100% sure what you mean. I assume you want to know
>>how to implement sparse matrices (ones where only a few elements are
>>non-zero). I'd suggest you look in a numerical analysis book, such as
>>"Numerical Recipes in C" [...]
>Looking at "NR in C" really isn't a bad idea. The bad idea is reading it, or
>worse, using its coding style :-)
Are there any books using Pascal (or even Borland's Delphi) to code
numerical analysis algorithms? I realize Pascal is not usually thought
of in this vein (over Fortran, C/C++, etc), but rather as a "learning"
language. That is actually how it would be used here. The person
using it would be a gifted high school student in an AP Computer Class.
Yes, AP Exam uses Pascal for one more year (next year) then moves to
C. Personally, I was hoping for Smaltalk, but that is another story
If you can cite any such sources, please respond to me directly since
I do not read this newsgroup.
John Neubert | {"url":"http://mathforum.org/kb/thread.jspa?threadID=529120&messageID=1604241","timestamp":"2014-04-19T17:44:39Z","content_type":null,"content_length":"15323","record_id":"<urn:uuid:9b1b08d4-8580-4bfe-ac40-b3ad58ea94a4>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00560-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Wednesday, July 3, 2013 at 10:18am
curriculum content
which one of the following schools is specifically intended for gifted and college-prep students ?
Tuesday, July 2, 2013 at 5:11pm
College Algebra
x is at lest 9 what is x
Tuesday, July 2, 2013 at 4:04pm
The incomes in a certain lg. population of college teachers have a normal distribution with a mean of $35,000 and st. dev. of $5000. Four teachers are randomly selected. What is the probability their
average salary will exceed $40,000?
Tuesday, July 2, 2013 at 2:59pm
A social psychologist asked 15 college students how many times they fell in love before they were 11 years old. The numbers of times were as follows: 2, 0, 6, 0, 3, 1, 0, 4, 9, 0, 5, 6, 1, 0, 2
Make (a) a frequency table and (b) a histogram. Then (c) describe the ...
Monday, July 1, 2013 at 3:07pm
college algebra: probability and statistics
Hmm. The way you asked it, the probability is 1. You have to roll an odd number or an even number!
Monday, July 1, 2013 at 3:37am
College Physics
A)2 b)5.7 C)87.7 D) 76.4 E)4.5
Sunday, June 30, 2013 at 7:13pm
As I reflected back<~~"back" is repetitive; the word "reflect" means to look back, so delete "back;" also you need to add a comma after "reflected" I realized that my friends got better grades than I
did. (Otherwise, you're trying...
Sunday, June 30, 2013 at 6:05pm
college algebra: probability and statistics
Sunday, June 30, 2013 at 4:00pm
Pages: <<Prev | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | Next>> | {"url":"http://www.jiskha.com/college/?page=12","timestamp":"2014-04-16T11:30:10Z","content_type":null,"content_length":"37617","record_id":"<urn:uuid:42e3110c-8f33-484e-b121-dfd83df6cc52>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00412-ip-10-147-4-33.ec2.internal.warc.gz"} |
Shifted Dirichlet series
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If $\sum_{n=1}^\infty \frac{a_n}{n^s} $ converges, does $\sum_{n=1}^\infty \frac{a_n}{(n+1)^s} $ also converge?
add comment
Yes, because $(n+1)^{-s} = n^{-s} + sn^{-s-1} + O(|s|^2n^{-\sigma - 2})$. The first series necessarily converges in the open half-plane strictly to the right of s, and converges absolutely in the
half-plane strictly to the right of s + 1. I hope I am not doing homework from a course in analytic number theory here.
add comment | {"url":"http://mathoverflow.net/questions/6450/shifted-dirichlet-series?sort=newest","timestamp":"2014-04-20T11:39:31Z","content_type":null,"content_length":"48966","record_id":"<urn:uuid:cd46c0ce-be28-4b53-afe2-f5b1baa34cd4>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00051-ip-10-147-4-33.ec2.internal.warc.gz"} |
Help solving an equation for x
1. March 22nd 2011, 12:04 PM #1
2. March 22nd 2011, 12:16 PM #2
This is a nasty equation and will be difficult to solve for x under normal conditions, what methods do you know?
This looks like a Wolfram output, what was the solution given?
3. March 22nd 2011, 12:33 PM #3
This is a Wolfram output, and it did not give any solutions !
I don't know any methods to solve that, I am actually in Calc II and I don't see how I could solve it..
If anybody knows how to solve it, I would really appreciate your help
4. March 23rd 2011, 04:50 AM #4
5. March 23rd 2011, 05:12 AM #5
This is the Prandtl Meyer equation. I need it for a project I am currently doing...
Similar Math Help Forum Discussions
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Cable Size For 3 Phase Motors - Motor Starting
Hi all,
I'm trying to commonise the cables used for few sizes of 3-phase motors used in my work.
Here's a summary of what we have
1. 2hp & 3hp with DOL start (FLA 3.5A & 5A)
2. 7.5hp, 10hp & 15hp coupled with centrifugal pump with DOL start (FLA 11A, 14A &21A)
3. 15hp & 20hp coupled with radiator fan with Wye-Delta start (FLA 21A & 28A)
4. 50hp air compressor with Wye-Delta start (FLA 66A)
5. 75hp motor coupled with a centrifugal pump with Solid State soft start (FLA 100A)
And the cable size we are currently using for the above
1. 4 x 4mm2 (36A cable current rating)
2. 4 x 10mm2 (64A cable current rating)
3. 4 x 10mm2 (64A cable current rating)
4. 4 x 25mm2 (114A cable current rating)
5. 4 x 50mm2 (170A cable current rating)
From what I read in this forum, we can safely size the cable to 125% of the motor FLA assuming the cable is not too long to effect voltage drop.
And the new size of cable to use as below,
For group 1,2 & 3, use the same cable (for more durability)
For group 4, use 4x16mm2 cable with 86A current rating.
For group 5, use 4 x 35mm2 cable with 140A current rating.
These motors needs to be mobilised every now and then, therefore using the smallest size is most preferable for ease of moving from site to site.
The maximum cable length that we use is 30m the most.
Are these cable sized correctly?
Is there any effect between different type of starting and application on these selection?
Thanks in advance. | {"url":"http://www.lmpforum.com/forum/topic/4131-cable-size-for-3-phase-motors/","timestamp":"2014-04-18T03:07:50Z","content_type":null,"content_length":"155238","record_id":"<urn:uuid:370f01aa-21df-4779-8fae-f92be1957b33>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00375-ip-10-147-4-33.ec2.internal.warc.gz"} |
Patent US8000917 - Method and system for S-parameter capture
This application is a continuation of, and hereby claims priority under 35 U.S.C. §120 to, U.S. patent application Ser. No. 10/930,291, entitled “Method and Apparatus For Integrated Channel
Characterization,” by the same inventors as the instant application, and filed on 31 Aug. 2004.
This application is related to the following U.S. patent application that is herein incorporated by reference in its entirety:
Title: “Method and Apparatus For Integrated Undersampling”;
Filing date: same as the present application;
Inventor: Jeffrey Lee Sonntag;
Docket number: 10/930,292; and
Express Mail No.: EU893-957-354US.
The present invention relates to the collection of discrete time digital representations of analog signals, and more particularly to undersampling of such signals.
During the design or maintenance of systems involving high-speed (e.g., 6.25 Giga-bit/sec) data transfer, it can be desirable to measure signals, as received by certain receiver circuits, as analog
waveforms. Analog waveform measurement can be approximated by capturing a series of discrete time samples of a signal and converting the samples into digital values. Analog waveform measurement can
be accomplished with digital sampling oscilloscopes (or DSO's).
DSO's, however, can pose several disadvantages. First, the physical packaging, of the system whose receiver circuits are to be monitored, can be very dense and not admit of the physical insertion of
a DSO probe. Second, for high speed data transfer, the DSO probe itself can significantly change the characteristics of the channel and therefore the analog waveform to be measured.
It would therefore be desirable to provide an alternative to the DSO that can be connected in dense packaging environments and/or will not significantly change channel characteristics.
The frequency response of a channel (the manner in which a channel attenuates signals at each frequency) is also known as the S[21 ]measurement of the set of possible S-parameter measurements.
One approach to obtaining this information is a network analyzer. At each point in time, a network analyzer drives a sinusoidal signal of a particular frequency into an input port of the channel. At
an output port of the channel, the network analyzer measures the amplitude attenuation and phase shift. Thus, in a time-serial fashion, frequency by frequency, the S[21 ]of the channel is
In addition to being a temporally serial approach to exploring a spectrum, a network analyzer relies on an ability to output a highly-accurate sinusoid at any frequency of interest, and on a receiver
having the ability to very accurately measure the amplitude and phase of the sinusoid received.
It would therefore be desirable to capture S[21 ]measurements in a non-serial manner and with equipment that does not rely upon precision sinusoidal generation or reception.
Summary of Undersampling Overview
An advantage of undersampling, over full sampling, is that the sampling frequency can be arbitrarily low. To obtain this advantage, however, the input signal and sample clocks must be synchronous to
a reference clock.
Because the input signal and sample clocks are synchronous to the same reference clock, the sample clocks have a certain offset phase, referred to herein as a “symbol offset phase,” with respect to
each symbol of the input signal.
A symbol stream can be either periodic or non-periodic, and it will still have a certain symbol offset phase. To have a pattern offset phase, however, in addition to having a symbol offset phase, a
symbol stream needs to be periodic.
Sample clocks can have a pattern offset phase, with respect to the content of a periodic symbol stream, if the time period between sample clocks is an integer multiple of the pattern period.
Sample clocks that are spaced apart integer multiples of a pattern period can be referred to as performing “coherent” undersampling of a symbol stream. Coherent undersampling can be characterized by
its ability to reconstruct a pattern, or a portion of a pattern, as a signal.
Sample clocks that sample a non-periodic symbol stream can be referred to as performing “incoherent” undersampling of the symbol stream.
For a given phase (symbol and, possibly, pattern) of sample clocks relative to an input signal, the input signal is sampled until a sufficient number of samples are collected. Such samples can be
stored in a one dimensional histogram array associated with the current phase of the sample clocks relative to the input signal.
The phase of the sample clocks, relative to the input signal, can then be shifted. Such phase shift can be accomplished by a component referred to herein as a “phase mixer” (discussed further below).
The phase shift can be small enough to provide sufficient resolution in a composite sampled image of the input signal.
For each phase increment, samples can be taken and stored in a one dimensional histogram associated with the current phase of the sample clock relative to the input signal. When a sufficiently
temporally wide segment of the input signal is assembled, as a temporally-ordered array of one dimensional histograms, the undersampling process can be stopped.
The phase mixer can operate by shifting sample clocks, relative to the input signal, in accordance with an amount of phase shift indicated at a “PS” input to the phase mixer. The PS input to the
phase mixer can be specified as an n-bit value, with a range from zero to 2^n−1, corresponding to phase shifts from zero to two symbol intervals. The range of phase shift available with a phase
mixer, resulting from varying its phase shift input from zero to 2^n−1, can be measured in degrees as 360 degrees.
For coherent undersampling, where the desired output is a function suitable for further processing, from the temporally ordered array of one dimensional histograms, a mean value can be computed from
each one dimensional histogram, resulting in a vector of mean values versus time.
Summary of Further Considerations for an Undersampling System
In general, the maximum temporal resolution of an undersampling system can depend on the amount of phase shift, measured in symbol intervals, per unit increment of the phase mixer's PS input.
If a visual representation is desired, for a two dimensional histogram, it can be desirable to increase the signal-level resolution such that visually smooth and continuous waveforms are presented.
In an embodiment of an undersampling system, 512 levels of signal-level resolution, can be used.
For purposes of generating a visual representation, the number of samples per bin can be converted into an appropriate color with a color map.
To be able to distinguish, visually, among those bins containing relatively few samples, log-based color mapping can be used.
Summary of Undersampling Implementation
For an example undersampling system, a suitable Data Processing System 1600 can be any stored-program computer capable of receiving data values, as a result of undersampling, and producing a
histogram thereof.
An undersampling system can further comprise an IC 1611 that has integrated, on the same physically contiguous IC, both an undersampler circuit and a receiver circuit whose input is to be monitored
by undersampling. A connection 1615, that carries a differential signal to be undersampled, can couple a transmitter 1614 to the receiver on IC 1611.
In order to use undersampling, transmitter 1614 and the undersampler of IC 1611 must be synchronized to the same reference clock. A single reference clock generator 1616 can distribute, via a
connection 1620, a reference clock to both transmitter 1614 and the undersampler of IC 1611.
FIG. 4A depicts an example apparatus for IC 1611.
The example receiver circuit, whose input is to be monitored by undersampling, is shown on IC 1611 as a receiver block 455. IC 1611 contains an internal controller 402 that interfaces with a control
bus input, such as JTAG bus 1612, to produce or receive control signals for accomplishing the sampling. Also on IC 1611 are two instances, US 430 and US 431, of an undersampler of type 403.
A functional overview of an undersampler of type 403 follows. The node to be sampled is connected to input 438, while input 436 controls whether input 438 is “tracked” or “held.” Tracking of input
438 occurs while input 436 is high. While tracking of input 438 occurs, on a sample capacitor of US 403, a signal level previously transferred to a hold capacitor in US 403 can be driven onto its
output 424 by asserting output enable 426.
The undersamplers can be controlled by a trigger unit 432. Output 434 of trigger unit 432 produces the sample clocks referred to in the above “Summary of Undersampling Overview.” Trigger unit 432 is
an instance of a trigger unit of type 404. A functional overview of trigger unit type 404 follows.
In its steady state, trigger unit 404 outputs a high signal at 434 causing the undersamplers to track. Trigger unit 404 can change the level at 434 to low, causing the undersamplers to hold, when the
following two events occur in sequence: i) trigger enable input 435 is asserted and ii) a sufficient number of clock edges occur at input 433. As part of determining a sufficient number of clock
edges at input 433, trigger unit 404 can divide clock input 433 by N. The value of N can be set to accomplish coherent undersampling.
Clock input 433 of trigger unit 432 is clocked by an output of APLL 405 of FIG. 4A. APLL 405 can be an instance of type APLL 100 (APLL 100 is discussed below in “Summary of APLL 100”). The
functionality of APLL 405 is comprised of two parts. First, to produce a clock 453 synchronized with reference clock 1620. Second, to provide a phase mixer, as described above, for shifting the phase
relationship between clock 453 and differential input 476. APLL 405 has a control bus 415 for control of the phase shift input to APLL 405's phase mixer.
Acquisition of a sample value can proceed as follows.
DPS 1600 can assert, through the JTAG connection 1612 to internal controller 402, a read ADC signal 454. The value read from ADC 452 (actually, the value read is the last value produced by ADC 452,
as stored in a buffer 459) is available to DPS 1600, for loading into a histogram array, via an ADC out bus 451. In addition, a positive edge on read ADC signal 454 can start a sequencer 458.
Sequencer 458 can first assert a trigger enable line 422. As discussed above, assertion of trigger enable line 422 can cause trigger unit 432 to produce a sample clock (or hold pulse), at its output
434, such that undersamplers 430 and 431 capture a sample of the signal at differential input 476.
Second, if ABUS enable line 427 enables the undersamplers to drive differential to single converter 448, sequencer 458 can assert a hold line 456 such that a sample is taken of a single-ended
conversion of the signal sample.
Third, sequencer 458 can assert an ADC convert line 457 that causes ADC 452 to produce a digital value for the sampled single-ended signal.
Summary of APLL 100
APLL 100 (see FIG. 1) contains a feedback loop, referred to herein as a “reference clock loop,” that acts to phase and frequency lock a VCO 103 with a reference clock 111.
Each output 120 to 127 of VCO 103 provides the same frequency, but at a different phase. Expressed in degrees, there is an ordering of the eight VCO 103 outputs such that each output differs, from a
succeeding and a preceding output, by 360/8 or 45 degrees.
While any one of outputs 120 to 127 could be used, VCO clock 121 (also referred to as clock 453 in FIG. 4A) is shown as the output that causes trigger unit 432 to produce sample clock edges at output
In general, a phase mixer takes N clock inputs of the same frequency but different phases and, in response to a phase shift control signal, interpolates between two of the input phases. The phase
mixer outputs the interpolated phase.
The phase mixer design of APLL 100 has eight clock inputs (labeled 130 to 137), a 10 bit wide phase shift control signal “PS_139” and the interpolated phase is at output 138. The three most
significant bits of PS_139 (e.g., PS_139[9:7]) select one of the eight pairs of clock inputs that differ by 45 degrees. The seven least significant bits of PS_139 (e.g., PS_139[6:0]) set a position
of phase interpolation between the selected pair of inputs.
Because of the reference clock loop, in the absence of changes at input PS_139, the outputs of VCO 103 are phase and frequency locked with reference clock 111. Input PS_139 can be loaded with any
value, under the program control of DPS 1600, through JTAG connection 1612.
In FIG. 4A, reference clock 111, of the APLL 405, is connected to a reference clock via connection 1620. Since transmitter 1614 is sending data to IC 1611 in synchronization with reference clock 1620
, it can be seen that clock 453, produced by APLL 405, is synchronized with the signal at differential input 476.
Increments to the value input to phase mixer 105, at PS_139, cause the phase relationship, between any output of VCO 103 (e.g., output 121 that produces clock 453) and the signal at differential
input 476, to be shifted.
Summary of S-Parameter Capture
A broadband signal can be used to ascertain a channel's S[2], measurement. The broadband signal, sent by the transmitter, is designed to stimulate the channel across a spectrum of interest. The
response of the channel to such a broadband signal, if the broadband signal is periodic, can be measured from coherent undersampled data captured at the receiver location.
A function of received periodic content can be constructed by concatenating the mean value for each one dimensional histogram (one such histogram for each temporal location). The Fourier transform of
the broadband signal as received after transmission through the channel (such received broadband signal determined from the constructed function of received periodic content), divided by the Fourier
transform of the transmitted broadband signal, constitutes the S[21 ]of the channel.
An example apparatus for determining a channel's S[21 ]measurement can be comprised of a transmitter board 902, a backplane board 901 and a receiver board 903.
Transmitter board 902 can comprise a physically contiguous transmitter integrated circuit 924. Transmitter IC 924 can comprise a transmitter 922 that transmits data from a multiplexer 921.
Multiplexer 921 can select either a typical source of data 920 or a pattern generator 923. The typical source of data 920 can be data as typically transmitted, from transmitter board 902 to receiver
board 903, when the S[21 ]characteristic is not being measured.
Pattern generator 923 can generate a broadband pattern that is injected into the channel and is received by the receiver board 903. In particular, receiver board 903 can receive the injected pattern
with an IC 1611.
Pattern generator 923 need not be on transmitter IC 924, however it can be desirable to integrate pattern generator 923 on the same transmitter IC 924.
Receiver board 903 can comprise an IC 1611 as discussed above. Specifically, on the same physically contiguous IC 1611 can be integrated both a receiver circuit, at which an S[21 ]is to be measured,
along with an undersampler for taking the needed measurements of the received broadband signal.
The S[21 ]of a channel (also known as the “through transmission”), referred to herein as C[f](n), is the Fourier transform of the signal received via the channel, referred to herein as R[f](n),
divided by the Fourier transform of the transmitted signal, referred to herein as T[f](n), where T[f](n) is the signal that produced R[f](n). Expressed in equation form:
$C f ( n ) = R f ( n ) T f ( n )$
where T[t](n) and R[t](n) are, respectively, the time domain versions of T[f](n) and R[f](n). T[f](n) can be found from T[t](n), and R[f](n) can be found from R[t](n), by applying the Discrete
Fourier Transform (DFT).
Since T[t](n) has periodic content of finite length, the space of potential patterns for T[t](n) can be searched for those patterns of content that maximize a particular metric. A metric that can be
used, discussed further below, is maximizing the minimum power level, across the spectrum of interest. Based upon the highest frequency of interest, and the highest frequency which can reasonably be
expected to be delivered to the receiver, a minimum effective sample rate at which to collect R[t](n) can be determined which avoids aliasing high frequency components down into the range of
frequencies of interest.
C[f](n) is a vector of complex numbers comprising both phase and magnitude information. In many applications, only the magnitude information is used.
The frequency resolution of C[f](n) can be increased by increasing the time duration of the pattern injected into the channel.
In general, over the period of T[t](n), noise energy in the channel can be approximated as constant and uniform across the spectrum of interest. Different patterns for T[t](n), however, can inject
different amounts of signal energy into the channel at different frequencies (i.e. different patterns have different spectra).
In general, to maximize the signal to noise ratio, a pattern can be selected for T[t](n) that maximizes the minimum power across the spectrum of interest. This constraint can be utilized by
constructing a pattern generator that generates all permissible patterns, and weighting the desirability of each such pattern according to the metric of its minimum power level across the relevant
portion of spectrum.
The accompanying drawings, that are incorporated in and constitute a part of this specification, illustrate several embodiments of the invention and, together with the description, serve to explain
the principles of the invention:
FIG. 1 depicts an APLL unit.
FIG. 2 depicts a timing diagram of eight outputs of a VCO relative to an example signal.
FIG. 3 depicts symbol offset phase for a symbol stream and pattern offset phase for a periodic symbol stream.
FIG. 4A depicts an example apparatus for an IC 1611 that includes both a receiver and undersampler.
FIG. 4B depicts internal details of a trigger unit of type 404.
FIG. 4C depicts the internals of an undersampler.
FIG. 5A depicts an example phase mixer design.
FIG. 5B depicts an example phase circle.
FIG. 5C shows how the “Gain Even” (or GE) and “Gain Odd” (or GO) vary with location on the phase circle.
FIGS. 6A and 6B depict an example two dimensional histogram resulting from incoherent undersampling.
FIGS. 7A and 7B depict an example two dimensional histogram resulting from coherent undersampling.
FIG. 8A depicts DPS 1600 as a type of data processing system known as a personal computer.
FIG. 8B depicts an example overall equipment arrangement in which undersampling can be used.
FIG. 9 depicts an example apparatus for determining a channel's S[21 ]measurement.
FIG. 10A depicts an example T[t](n) and R[t](n), while FIG. 10B depicts corresponding examples for T[f](n) and R[f](n).
FIG. 10C shows the example |C[f](n)| resulting from a division of R[f](n) by T[f](n).
FIG. 11 depicts an example procedure that searches for a pattern for T[t](n).
Reference will now be made in detail to preferred embodiments of the invention, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numbers will be
used throughout the drawings to refer to the same or like parts.
Table of Contents to Detailed Description
1. Undersampling Overview
2. Further Considerations For An Undersampling System
3. Undersampling Implementation
3.1. Undersampler 403, Internal Operation
3.2. Trigger Unit Type 404, Internal Operation
3.3. Example Computing Environment
4. APLL 100
4.1. Reference Clock Loop
4.2. VCO 103
4.3. Phase Mixer 105
4.3.1. Overview
4.3.2. Further Details
4.4. APLL 405 of IC 1611
5. S-Parameter Capture
5.1. DFT Determination
5.2. Signal to Noise Considerations
5.3. AC Coupled Receivers
5.4. Example Pattern
5.5. Example Search Procedure
6. Glossary of Selected Terms
The present invention comprises techniques or undersampling, with the purpose of producing a histogram output, where the undersampler, and the receiver circuit whose input is to be undersampled, are
integrated on the same physically contiguous integrated circuit.
1. Undersampling Overview
In full sampling, an input signal can be sampled without need for synchronization between a sample clock and the input signal. The sample frequency, however, must be high enough to permit the highest
frequency component of interest, in the input signal, to be captured. Specifically, the frequency of the sample clock must be at least twice the frequency of the highest frequency component to be
An advantage of undersampling is that the sampling frequency can be arbitrarily low. To obtain this advantage, however, the input signal and sample clocks must be synchronous to a reference clock.
Because the input signal and sample clocks are synchronous to the same reference clock, the sample clocks have a certain offset phase, referred to herein as a “symbol offset phase,” with respect to
each symbol of the input signal. Symbol offset phase is illustrated in FIG. 3: FIG. 3 shows an input signal as a stream of symbols 310, where the symbols are sequentially labeled. Several sample
clock edges (SCEs) are also shown in FIG. 3: SCE[n], SCE[n+1 ]and SCE[n+2]. SCE[n ]and SCE[n+1 ]are shown as having a symbol offset phase such that they occur halfway through the temporal extent of,
respectively, symbols 1 and 9.
A symbol stream, such as 310, can be either periodic or non-periodic, and it will still have a certain symbol offset phase. To have a pattern offset phase, however, in addition to having a symbol
offset phase, a symbol stream needs to be periodic.
Sample clocks can have a pattern offset phase, with respect to the content of a periodic symbol stream, if the time period between sample clocks is an integer multiple of the pattern period. The
example coherent symbol stream 311 of FIG. 3 is comprised of a four-symbol pattern, with the symbols of the pattern labeled 0 to 3. Sample clocks SCE[n], SCE[n+1 ]and SCE[n+2 ]are shown occurring
halfway through symbol 1 of the pattern.
Sample clocks that are spaced apart integer multiples of a pattern period can be referred to as performing “coherent” undersampling of a symbol stream. Coherent undersampling can be characterized by
its ability to reconstruct a pattern, or a portion of a pattern, as a signal.
Sample clocks that sample a non-periodic symbol stream can be referred to as performing “incoherent” undersampling of the symbol stream. Also, sample clocks of a periodic sample stream, where the
spacings between the sample clocks are prime relative to the periodicity in the data content to be sampled, can be referred to as performing “incoherent” undersampling of the symbol stream.
Incoherent undersampling can be characterized by its ability to capture a “data eye” in which the various types of transitions that occur, over a symbol interval, are overlaid.
For a given phase (symbol and, possibly, pattern) of sample clocks relative to an input signal, the input signal is sampled until a sufficient number of samples are collected. Such samples can be
stored in a one dimensional histogram array associated with the current phase of the sample clocks relative to the input signal.
The phase of the sample clocks, relative to the input signal, can then be shifted. Such phase shift can be accomplished by a component referred to herein as a “phase mixer” (discussed further below).
The phase shift can be in accordance with a fixed increment, in accordance with a fixed direction, and small enough to provide sufficient resolution in a composite sampled image of the input signal.
For each phase increment, samples can be taken and stored in a one dimensional histogram associated with the current phase of the sample clock relative to the input signal.
When a sufficient number of phase increments have been sampled, such that a sufficiently temporally wide segment of the input signal is assembled, as a two dimensional histogram that is a
temporally-ordered array of one dimensional histograms, the undersampling process can be stopped.
The phase mixer can operate by shifting sample clocks, relative to the input signal, in accordance with an amount of phase shift indicated at a “PS” input to the phase mixer. The PS input to the
phase mixer can be specified as an n-bit value, with a range from zero to 2^n−1, corresponding to phase shifts from zero to two symbol intervals. The range of phase shift available with a phase
mixer, resulting from varying its phase shift input from zero to 2^n−1, can be measured in degrees as 360 degrees.
While the PS input to the phase mixer has a finite range, the range of phases that can be selected, by repeated application of the phase mixer, is not limited. Repeated application of the PS input is
analogous the setting of time with a mechanical clock by controlling the minute hand: each complete revolution of the minute hand changes the hour hand by one hour, but continued wrapping-around by
the minute hand suffices to continue advancing the hour hand through multiple hours. Similarly, the PS control to the phase mixer has a circular range: advancing the control through an interval
corresponding to it's nominal range (0 to 2^n−1) plus one more bit returns the PS control to the same value; however the phase of the input signal has been advanced two entire symbols.
FIGS. 6A and 6B depict an example of incoherent undersampling.
FIGS. 6A and 6B depict an array of 10 one dimensional histograms (numbered 0 to 9), each histogram having 8 bins (numbered 0 to 7).
Also depicted in FIGS. 6A and 6B are the four types of transitions (labeled 610, 611, 612 and 613 in FIG. 6A and unlabeled in FIG. 6B) for a binary encoded signal that are sampled by incoherent
undersampling, where the temporal extent of the 10 histograms corresponds to the temporal extent of one bit period. The symbol offset phase, between the input signal and the sample clocks, is such
that histogram 0 corresponds to halfway through a bit while histogram 8 corresponds to halfway through a following bit. Signal transitions 610, 611, 612 and 613 are shown in ideal form, without
accounting for such real world effects as noise or variation in the generated signal.
FIG. 6B depicts an example result for the array of one dimensional histograms, when undersampling transitions 610, 611, 612 and 613 with variations due to real world effects. Each one dimensional
histogram of FIG. 6B has a total of 100 samples. The bins with the greatest number of hits are those that lie directly on the ideal paths of the transition types. However, the histograms also show
the spreading of hits, beyond the ideal signal paths, due to real world signal variation.
FIGS. 7A and 7B correspond, respectively, to FIGS. 6A and 6B, but depict a coherent undersampling example where each one dimensional histogram corresponds to a particular pattern offset phase.
For coherent undersampling, where the desired output is a function suitable for further processing, from the temporally ordered array of one dimensional histograms, a mean value can be computed from
each one dimensional histogram, resulting in a vector of mean values versus time.
2. Further Considerations for an Undersampling System
In general, the maximum temporal resolution of an undersampling system can depend on the amount of phase shift, measured in symbol intervals, per unit increment of the phase mixer's PS input. An
example undersampling system is presented herein where each 360 degree phase-shift, specified at the PS input, is divided into 1024 increments (i.e., the phase mixer's phase shift control input is 10
bits) and corresponds to two symbol intervals of the undersampled differential signal. Thus, each unit increment of the phase mixer corresponds to a phase shift of
$1 512$
a symbol interval.
In general, the maximum signal-level resolution of an undersampling system can depend upon the resolution of its ADC. An example undersampling system is presented herein where the signal-level
resolution comprises 2^13 signal levels as a result of using a 13-bit ADC.
In general, acquisition time, to obtain an undersampled image of a signal, remains constant as the signal-level resolution is increased, if the number of samples collected per one dimensional
histogram is held constant.
If a visual representation is desired, for a two dimensional histogram, it can be desirable to increase the signal-level resolution such that visually smooth and continuous waveforms are presented.
In an embodiment of an undersampling system, presented further below, 512 levels, of signal-level resolution, can be used. A suitable unit of signal level (e.g., an amount of volts or amperes), per
unit of signal-level resolution, can be determined from a peak-to-peak measurement of the signal to be displayed. Such peak-to-peak measurement can be compressed into less than a full-scale
signal-level representation (e.g., into 80% of a full-scale signal-level representation) such that succeeding peaks or troughs of the input signal, that may be outside the range of a particular
peak-to-peak measurement used to establish the unit amount of signal-level resolution, do not exceed the full-scale of representable signal-level.
For purposes of generating a visual representation, the number of samples per bin can be converted into an appropriate color with a color map. Prior to applying the number of samples per bin to a
color map, it can be desirable to normalize the number of samples per bin. An example normalization technique is dividing the value in each bin by the value of the bin with the largest number of
samples, thereby converting the value in every bin to be within the range of zero to one. The normalized values can then be multiplied by a number equal to the maximum input value of the color map
and then input to the color map.
In general, if the total number of samples collected per one dimensional histogram is held constant, as the number of bins per one dimensional histogram is increased, the maximum number of samples
collected, in any one bin, decreases. If the ratio, between the number of bins and the total number of samples collected, becomes too great, the maximum number of samples per bin can become too close
to the minimum number of samples per bin to provide for a suitable level of visual differentiation when the bin contents are color mapped. Such a result can also occur if the acquisition time is too
short, but fast acquisition time can be an advantage and the collected signal representation can be sufficient for certain purposes.
Even if there is a large range, between those bins with a maximum number of samples and those bins with relatively few samples, it can still be desirable to be able to distinguish, visually, among
those bins containing relatively few samples. Log-based color mapping can be used to accomplish this. In one approach, the log is found for each normalized bin value. Such log value is then scaled
and shifted into the range of the color map. Scaling can be accomplished by multiplying the log by a scaling factor. Shifting converts the logs into values that are greater than, or equal to, zero,
and can be accomplished by adding an offset.
Another approach is to design the color map itself to logarithmically map from changes in input number to changes in color, such that color change, per unit change of the input number, is greater
towards the lower value end of color map input. For example, bins with 1 and 10 samples can be mapped to colors providing greater contrast than the colors for bins with 1000 and 1100 samples.
To emphasize the difference between those bins containing relatively few samples and those bins containing zero samples, a color map can be used wherein there is a visually pronounced difference
between the color assigned to zero samples per bin and the colors assigned to those bins with relatively few samples.
3. Undersampling Implementation
FIG. 8B depicts an example overall equipment arrangement in which undersampling can be used. Data Processing System 1600 can be any stored-program computer suitable for receiving data values, as a
result of undersampling, and producing a histogram thereof. FIG. 8A depicts DPS 1600 as a type of data processing system known as a personal computer.
FIG. 8B depicts the following equipment. An example “USB” (i.e., Universal Serial Bus) connection 1613 between DPS 1600 and a USB-to-JTAG interface unit 1610 (where “JTAG” refers to the “Joint Test
Action Group” that developed the IEEE 1149.1 boundary-scan standard). An example JTAG connection 1612 couples interface unit 1610 and an integrated circuit 1611. IC 1611 has integrated, on the same
physically contiguous IC, both an undersampler circuit and a receiver circuit whose input is to be monitored by undersampling. A connection 1615, that carries the differential signal to be
undersampled, couples transmitter 1614 to the receiver on IC 1611.
In order to use undersampling, transmitter 1614 and the undersampler of IC 1611 must be synchronized to the same reference clock. In FIG. 8B, a single reference clock generator 1616 distributes, via
connection 1620, a reference clock to both transmitter 1614 and the undersampler of IC 1611.
FIG. 4A depicts an example apparatus for IC 1611. The example receiver circuit, whose input is to be monitored by undersampling, is shown on IC 1611 as a receiver block 455. Receiver block 455
receives its input from IC 1611's differential input 476. An example data rate, for a differential signal to be applied to differential input 476, is 6.25 Giga-baud/sec and an example data encoding
is binary. APLL 405 can be an instance of type APLL 100 (APLL 100 is discussed in section “APLL 100”) and has a control bus 415 for control of the phase shift input to APLL 405's phase mixer. IC 1611
contains an internal controller 402 that interfaces with a control bus input, such as JTAG bus 1612, to produce or receive control signals for accomplishing the sampling. Also on IC 1611 are two
instances, US 430 and US 431, of an undersampler of type 403 (type 403 to be discussed below and in section “Undersampler 403, Internal Operation”). US 430 and 431 sample, respectively, inputs VIN[P
]and VIN[N ]of differential input 476.
An example contents, for the physically contiguous extent of IC 1611, is indicated in FIG. 4A by the dashed outline. While FIG. 4A shows IC 1611 as containing more than just a receiver block 455 and
undersamplers 430 and 431, another example embodiment of the invention can just include a receiver circuit (such as receiver block 455) and an undersampler or undersamplers (such as undersamplers 430
and 431) on the same physically contiguous integrated circuit.
A functional overview of an undersampler of type 403 follows, while internal details of its operation are covered in a below section. The node to be sampled is connected to input 438, while input 436
controls whether input 438 is “tracked” or “held.” For the example of FIG. 4C, tracking of input 438, onto sample capacitor 460, occurs while input 436 is high. While tracking input 438, a signal
level (e.g., a voltage level) previously transferred to a hold capacitor 463 in US 403 can be driven onto its output 424 (and therefore onto an analog bus, called “ABUSP 440” or “ABUSM 441”) by
asserting output enable 426.
The undersamplers can be controlled by a trigger unit 432. Output 434 of trigger unit 432 produces the sample clocks referred to in the above section “Undersampling Overview.” Trigger unit 432 is an
instance of a trigger unit of type 404, whose internal details are shown in FIG. 4B. A functional overview of trigger unit type 404 follows, while a discussion of its internal details of operation
are covered in a below section. In its steady state, trigger unit 404 outputs a high signal at 434 causing the undersamplers to track. Trigger unit 404 can change the level at 434 to low, causing the
undersamplers to hold, when trigger enable input 435 is asserted. In addition to enable input 435 being asserted, the condition needed, for the level at 434 to go low, is a sufficient number of clock
edges (in this example, positive edges) at input 433. Trigger unit 404 can comprise a divide-by-N counter 474, of clock input 433, where N can be adjustable by an input 437. The value of N can be set
to accomplish coherent undersampling. Undersampling of a signal with periodic content is coherent when the period of the sample clock, produced by the counter 474, is an integer multiple of the
signal-content periodicity.
Where the pattern for coherent sampling is of fixed length (e.g., 80 bits) counter 474 can always divide by the same amount (e.g., 40, where two bits of the pattern occur for each clock edge at input
433). A fixed value for counter 474, set to accomplish coherent sampling for a particular pattern, will not have a detrimental effect on incoherent sampling, so long as the sampling period set by
counter 474 is prime relative to any periodicity in the data content to be incoherently sampled.
Clock input 433 of trigger unit 432 is clocked by an output of APLL 405 of FIG. 4A. The functionality of APLL 405 is comprised of two parts. First, to produce a clock 453 synchronized with reference
clock 1620. Second, to provide a phase mixer, as described above, for shifting the phase relationship between clock 453 and differential input 476.
As explained below, in the section “APLL 100,” APLL 405 can comprise an instance of APLL 100 along with a register for setting values of the phase mixer's phase shift input PS 139.
Acquisition of a sample value can proceed as follows.
DPS 1600 can assert, through USB to JTAG interface 1610, a read ADC signal 454. ADC 452 can output a value from a previous request that ADC 452 perform an analog to digital conversion. A positive
edge on read ADC signal 454, therefore, can load a previous conversion value from ADC 452 into a buffer 459. The value of buffer 459 is available to DPS 1600, for loading into a histogram array, via
an ADC out bus 451. In addition, a positive edge on read ADC signal 454 can start a sequencer 458.
Sequencer 458 can first assert a trigger enable line 422. As discussed above, assertion of trigger enable line 422 can cause trigger unit 432 to produce a sample clock (or hold pulse), at its output
434, such that undersamplers 430 and 431 capture a sample of the signal at differential input 476. If ABUS enable line 427 is asserted, the captured sample can be driven by undersamplers 430 and 431
onto the analog bus. Differential to single converter 448 can convert the differential signal, on the analog bus, into a single-ended signal at its output 446. The single-ended signal at output 446
can be sampled by input 443 of sample and hold unit 447.
Second, sequencer 458 can assert a hold line 456 such that a sample is taken of the single-ended signal at output 446. The sample taken by sample and hold unit 447 is a (single-ended) discrete time
analog value, available to ADC 452 for conversion into a discrete time digital value.
Third, while still asserting hold line 456, sequencer 458 can assert an ADC convert line 457 that causes ADC 452 to produce a digital value for the sampled signal at ADC signal input 450.
Fourth, once ADC 452 has produced a digital value, sequencer 458 can de-assert hold line 456.
3.1. Undersampler 403, Internal Operation
The internals of an undersampler of type 403 are shown in FIG. 4C. As can be seen, the node to be sampled is connected to input 438, while input 436 controls whether input 438 is “tracked” (by sample
capacitor 460) or “held.” In the example of FIG. 4C, input 438 is tracked when input 436 is high.
When input 436 is low, the voltage of sample capacitor 460 is held by turning off tracking transistor 461. Also, when 436 is low, holding capacitor 463 is charged according to a predetermined
relationship with sample capacitor 460. More specifically, when input 436 is low, transistor 462 transfers to holding capacitor 463 the voltage which is required to produce, at V[OUT ] 464, the
voltage of sample capacitor 460. Such transfer from the sample to hold capacitors can be accomplished by a unity-gain operational amplifier, indicated by outlined region 467. Operational amplifier
467 drives the voltage at V[OUT ] 464 to equal the voltage across sample capacitor 460.
Holding capacitor 463 can be much larger than sample capacitor 460. For example, holding capacitor 463 can be large enough to hold its charge until an analog-to-digital conversion of the charge has
been completed. By having a separate holding capacitor 463, sampling capacitor 460 can be made small enough to permit a sufficiently high bandwidth data signal to be received by receiver circuit 455.
However, sampling capacitor 460 still needs to be large enough to permit a charge to be established on holding capacitor 463.
When input 436 returns high, the operational amplifier output voltage at V[OUT ] 464, caused by the charge trapped on holding capacitor 463, can be driven onto an analog bus (e.g., “ABUSP 440” or
“ABUSM 441”) by enabling transmission gate 466. When transmission gate 466 is initially enabled, the voltage at V[OUT ] 464 can be expected to change due to the capacitance of the analog bus.
However, operational amplifier 467 eventually charges the analog bus to the voltage caused by the charge trapped on hold capacitor 463.
Input 436 is depicted as directly coupled to the following three components of FIG. 4C: tracking transistor 461, transistor 462 and AND gate 468. However, input 436 can be connected to these three
components with non-overlapping clocks. Non-overlapping clocks can be used to ensure that a switch, through which a capacitor has been charged, is opened prior to enabling or disabling other
circuits. The time difference, between opening such capacitor-charging switch and enabling or disabling other circuits, can be equal to the following: several times the delay associated with an
inverter in the relevant technology.
In the example of FIG. 4C, when input 436 transitions to a low signal value, non-overlapping clocks can be used to ensure that tracking transistor 461 turns off before both of the following occur:
transistor 462 turns on and transmission gate 466 is disabled. When input 436 transitions to a high signal value, non-overlapping clocks can be used to ensure that transistor 462 turns off before
both of the following occur: tracking transistor 461 turns on and transmission gate 466 is enabled.
3.2. Trigger Unit Type 404, Internal Operation
Each time counter 474 produces a positive edge, it clocks D-type flip-flops 469 to 472 simultaneously. If a logical one is present at trigger enable input 435, this logical one is clocked, by
positive edges from counter 474, into flip-flops 469 to 471. When a logical one is present at the Q output of flip-flop 471 and a logic zero is present at the Q output of flip-flop 472, AND gate 473
transitions to a logic zero at its 434 output. Due to the asynchronous relationship between trigger enable input 435 and the edges of counter 474, it can take two or three edges of counter 474 before
a logic one is at the Q output of flip-flop 471. Due to the asynchronous relationship between trigger enable input 435 and the edges of counter 474, there can be metastability problems at the output
of flip-flop 469 that can be addressed by adding flip-flops 470 and 471.
With trigger enable 435 still high, when the next positive edge is produced by counter 474, a logic one is also clocked into flip-flop 472, causing output 434 to go high.
When trigger enable 435 is returned low, over the next three to four positive edges from counter 474, a logic zero is clocked into flip-flops 469 to 472. With a logical zero is present at the Q
output of flip-flop 471, output 434 remains high until trigger enable 435 is once again brought high.
3.3. Example Computing Environment
The data processing of samples, captured by the present invention, can be accomplished by a computing environment (or data processing system) such as that of FIG. 8A. FIG. 8A depicts a personal
computer 1600 comprising a Central Processing Unit (CPU) 1601 (or other appropriate processor or processors) and a memory 1602. Memory 1602 has a portion of its memory 1603 in which are stored the
software tools (or computer programs) and data of the present invention. While memory 1603 is depicted as a single region, those of ordinary skill in the art will appreciate that, in fact, such
software and data may be distributed over several memory regions or several computers. Furthermore, depending upon the computer's memory organization (such as virtual memory), memory 1602 may
comprise several types of memory (including cache, random access memory, hard disk and networked file server). Computer 1600 can be equipped with a display monitor 1605, a mouse pointing device 1604
and a keyboard 1606 to provide interactivity between the software of the present invention and the equipment designer or maintainer. Computer 1600 also includes a way of reading computer readable
instructions from a computer readable medium 1607, via a medium reader 1608, into the memory 1602. Computer 1600 also includes a way of reading computer readable instructions via the Internet (or
other network) through network interface 1609.
In some embodiments, computer programs embodying the present invention are stored in a computer readable medium, e.g. CD-ROM or DVD. In other embodiments, the computer programs are embodied in an
electromagnetic carrier wave. For example, the electromagnetic carrier wave may include the programs being accessed over a network.
4. APLL 100 4.1. Reference Clock Loop
APLL 100 (see FIG. 1) contains a feedback loop, referred to herein as a “reference clock loop,” that acts to phase and frequency lock VCO 103 with a reference clock 111. An example frequency for
reference clock 111 is 625 MHz. Specifically, the loop acts to vary the frequency of VCO 103 such that the output of divider 104 is phase locked with reference clock 111. When the signal to be
undersampled, at differential input 476 (see FIG. 4A), encodes 6.25 Giga-baud/sec, divider 104 can divide by five such that the reference clock loop acts to frequency lock VCO 103 to 3.125 GHz.
The reference clock loop, of APLL 100, operates as follows.
Phase differences, between reference clock 111 and the output of divider 104, are detected by phase detector 101. The output of phase detector 101 is filtered by Charge Pump/Low Pass Filter 102 to
produce a signal for controlling VCO 103. While VCO 103 can provide eight clock outputs, labeled in FIG. 1 as outputs 120 to 127, for purposes of the reference clock loop, VCO 103 can be regarded as
having a single frequency-controlled output. Also, for purposes of the reference clock loop, phase mixer 105 can be regarded as simply a wire that couples a single frequency-controlled output of VCO
103 to the input of divider 104.
4.2. VCO 103
Each output 120 to 127 of VCO 103 provides the same frequency, but at a different phase. The phase differences can be illustrated with a “phase circle” diagram, shown in FIG. 5B.
FIG. 5B represents a complete cycle, of any output of VCO 103, as a complete circle (or 360 degrees). In general, the phases of the outputs of VCO 103 are such that, when placed upon a phase circle
diagram, they divide the circumference of the circle into equal-sized segments of arc. Expressed in terms of degrees, this means that, for an N output VCO 103, there is an ordering of the N outputs
such that each output differs, from a succeeding and a preceding output, by 360/N degrees. In the case where VCO 103 has eight outputs, the case shown in FIG. 5B, the phases of the outputs divide the
circumference of the phase circle into eight equal sized segments. The eight equal sized segments are numbered in binary, in FIG. 5B, from 000 to 111. Expressed in degrees, there is an ordering of
the eight VCO 103 outputs such that each output differs, from a succeeding and a preceding output, by 360/8 or 45 degrees.
FIG. 2 depicts a timing diagram of these eight outputs relative to an example differential signal at input 476 of FIG. 4A. Each of the eight selected outputs of VCO 103 differs from a succeeding and
a preceding output by 45 degrees.
FIG. 2 also shows an edge of VCO clock 121 (also referred to as clock 453 in FIG. 4A) as the edge that causes trigger unit 432 to produce a sample clock edge n (SCE[n]) at output 434. An example bit
phase offset, for such a sample clock, is also shown in FIG. 2.
4.3. Phase Mixer 105
4.3.1. Overview
In general, a phase mixer takes N clock inputs of the same frequency but different phases and, in response to a phase shift control signal, interpolates between two of the input phases. The phase
mixer outputs the interpolated phase.
An example phase mixer design is shown in FIG. 5A, and FIG. 1 shows an instance 105 of such design. The phase mixer design of FIG. 5A has eight clock inputs (labeled 130 to 137), a 10 bit wide phase
shift control signal “PS_139” and the interpolated phase is at output 138. The three most significant bits of PS_139 (e.g., PS_139[9:7]) select one of the eight arcs depicted in the phase circle of
FIG. 5B (i.e., they select a pair of clock inputs, one from multiplexer 512 and the other from multiplexer 513, that differ by 45 degrees). The seven least significant bits of PS_139 (e.g., PS_139
[6:0]) set a position of phase interpolation along the selected arc (i.e., they set a position of phase interpolation between the selected pair of clock inputs).
For example, in FIG. 5B, when PS_139[9:7] is equal to 111, the pair of inputs 137 and 130 are selected. For PS_139[6:0] equal to 0000000, the phase of input 137 is sent to output 138, while for PS_
139[6:0] equal to 1111111, the phase of input 130 is sent to output 138. For values of PS_139[6:0] that are between 0000000 and 1111111, an interpolation, between the phases of 130 and 137, is sent
to output 138. Details, of how phase mixer 105 can accomplish such interpolation, are presented below.
4.3.2. Further Details
A pair of points on the phase circle of FIG. 5B, that are 45 degrees apart, is selected by applying the three most significant bits of the phase shift control 139 (PS_139 [9:7]) to multiplexers 512
and 513. Multiplexers 512 and 513 are each depicted in FIG. 5A as 8-input multiplexers whose inputs have been connected together into four pairs. For multiplexer 512, inputs 7 or 0 select phase 130,
inputs 1 or 2 select phase 132, inputs 3 or 4 select phase 134 and inputs 5 or 6 select phase 136. For multiplexer 513, inputs 0 or 1 select phase 131, inputs 2 or 3 select phase 133, inputs 4 or 5
select phase 135 and inputs 6 or 7 select phase 137.
In summary, multiplexer 512 selects one of the four even-numbered inputs, located at 0, 90, 180 and 270 degrees on the phase circle. Multiplexer 513 selects one of the four odd-numbered inputs,
located at 45, 135, 225 and 315 degrees on the phase circle.
The length of the chord (or the linear distance) between each pair of phases separated by 45 degrees (i.e., a pair of phases comprising one odd-numbered phase and one even-numbered phase, the phases
being consecutively numbered) is normalized to a value of one. Along any such unit-distance, the seven least significant phase shift control bits (i.e., PS_139 [6:0]) select the interpolation point.
This can be accomplished by applying the selected even-numbered phase input to an adjustable gain amplifier 514 and by applying the selected odd-numbered phase input to an adjustable gain amplifier
515. The gains of even-phase amplifier 514 and odd-phase amplifier 515 are adjusted such that they always sum to a constant value. The outputs of amplifiers 514 and 515 are input to summer 516. The
result of summer 516 is filtered by low pass filter 517 and converted back to full logic levels by slicer 518.
Consider the phase indicated by arrow 530 of FIG. 5B, located within the 45 degree section selected by PS_139 [9:7]=000. The closer phase 530 is to 0 degrees, the greater the gain of 514 and the less
the gain of 515. This is illustrated in FIG. 5C, that shows how the gain for 514 (also referred to as “Gain Even” or GE) and for 515 (also referred to as “Gain Odd” or GO) vary with location on the
phase circle.
FIG. 5C is divided into eight sections, numbered 000 to 111, in accordance with the value for PS_139 [9:7]. For each of these regions, PS_139 [6:0] increases from 0 to 127, from left to right. Within
the region for PS_139 [9:7]=000, it can be seen that, as PS_139 [6:0] is increased, from 0 to 127, GE decreases while GO increases. The net result, at any point across FIG. 5C, is that the sum of GE
and GO is always the normalized distance of one. FIG. 5B also shows, graphically, how the functions for GE and GO of FIG. 5C, for region PS_139 [9:7]=000, accurately model the position of phase 530
on the chord between 0 and 45 degrees. Specifically, FIG. 5B shows how phase 530 divides the chord into two sections, where the length of one section is proportional to GE and the length of the other
section is proportional to GO.
Referring back to FIG. 5C, within the region for PS_139 [9:7]=001, it can be seen that, as PS_139 [6:0] is increased, from 0 to 127, the change in GE and GO becomes opposite to that for the region
PS_139 [9:7]=000. Specifically, for PS_139 [9:7]=001, as GE increases GO decreases. FIG. 5B shows, graphically, how the functions for GE and GO of FIG. 5C, for region PS_139 [9:7]=001, accurately
model the position of phase 531 on the chord between 45 and 90 degrees. Specifically, FIG. 5B shows how phase 531 divides the chord into two sections, where the length of one section is proportional
to GE and the length of the other section is proportional to GO.
FIG. 5A shows an example embodiment for controlling the gain of 514 and 515. A digital to analog converter (DAC) 520 is depicted, that produces a GE control output 510 and a GO control output 511.
For the four regions of the phase circle of FIG. 5B where PS_139 [7]=0, as PS_139 [6:0] is increased from 0 to 127, input bus 529 to DAC 520 (as shown in FIG. 5C) also increases from 0 to 127.
Therefore, DAC output 510 increases GE while DAC output 511 decreases GO. For the four regions of the phase circle of FIG. 5B where PS_139 [7]=1, due to the seven XOR gates labeled 519 in FIG. 5A, as
PS_139 [6:0] is increased from 0 to 127, input bus 529 to DAC 520 (as shown in FIG. 5C) decreases from 127 to 0. Therefore, DAC output 510 decreases GE while DAC output 511 increases GO.
DAC 520 is designed to provide complementary outputs 510 and 511. DAC 520 can be designed such that, if the sum of the signal levels at its outputs 510 and 511 is determined, such sum is a constant
4.4. APLL 405 of IC 1611
To produce clock 453 of APLL 405, any one of the eight outputs of VCO 103 can be used. For purposes of example, FIG. 4A assumes VCO output 121 is used.
Because of the reference clock loop, in the absence of changes at input PS_139, the outputs of VCO 103 are phase and frequency locked with reference clock 111. Input PS_139 can be loaded with any
value, under the program control of DPS 1600, through JTAG connection 1612.
In FIG. 4A, reference clock 111, of the APLL 405, is connected to a reference clock via connection 1620. Since transmitter 1614 is sending data to IC 1611 in synchronization with reference clock 1620
, it can be seen that clock 453, produced by APLL 405, is synchronized with the signal at differential input 476.
Increments to the value input to phase mixer 105, at PS_139, cause the phase relationship, between any output of VCO 103 (e.g., output 121 that produces clock 453) and the signal at differential
input 476, to be shifted.
5. S-Parameter Capture
This section describes a use of coherent undersampling to perform S-Parameter capture.
When a transmitter transmits into a channel from a particular location and a receiver receives from the same channel at a particular location, the characteristics of the channel can be expressed as
an S[21 ]measurement.
A broadband signal can be used to ascertain the channel's S[21 ]measurement. The broadband signal, sent by the transmitter, is designed to stimulate the channel across a spectrum of interest. The
response of the channel to such a broadband signal, if the broadband signal is periodic, can be measured from coherent undersampled data captured at the receiver location. As discussed above, due to
such factors as signal noise, even the coherent undersampling process captures a variety of signal values at each temporal location of a periodic pattern. The variation is captured by the use of a
one dimensional histogram for each temporal location. A function of received periodic content can be constructed by concatenating the mean value for each one dimensional histogram (one such histogram
for each temporal location). The Fourier transform of the broadband signal as received after transmission through the channel (such received broadband signal determined from the constructed function
of received periodic content), divided by the Fourier transform of the transmitted broadband signal, constitutes the S[21], of the channel.
The “channel” depends upon what is defined to be the transmitted broadband signal. If the transmitted signal is defined to be the digital representation of the pattern to be transmitted, then every
limitation of the translation of the digital representation to analog form, as well as the limitations of the wiring connecting the transmitter to the receiver, is included in the channel. If,
however, the transmitted signal is a filtered version of the digital representation, then the channel will not include those transmitter limitations approximated by such filtering.
An example apparatus for determining a channel's S[2], measurement is depicted in FIG. 9. The apparatus can be comprised of a transmitter board 902, a backplane board 901 and a receiver board 903.
Transmitter board 902 can connect to backplane 901 via a connector 910. Likewise, receiver board 903 can connect to backplane 901 via a connector 911. Connectors 910 and 911 can be coupled via
conductors 912.
Transmitter board 902 can comprise a physically contiguous transmitter integrated circuit indicated by outline 924. Transmitter IC 924 can comprise a transmitter 922 that transmits data from a
multiplexer 921. Multiplexer 921 can select either a typical source of data 920 or a pattern generator 923. The typical source of data 920 can be data as typically transmitted, from transmitter board
902 to receiver board 903, when the S[21 ]characteristic is not being measured. Pattern generator 923 can generate a broadband pattern that is injected into the channel and is received by the
receiver on IC 1611. An example pattern generator generates a binary pattern of 80 bits. Pattern generator 923 need not be on transmitter IC 924, however it can be desirable to integrate pattern
generator 923 on the same transmitter IC 924. This desirability can be due to two factors. First, the physical packaging of the apparatus, whose channel is to be measured, can be dense and not
readily admit the connection of probes through which a test pattern can be sent. Second, for channels that carry very high data rates (e.g., 6.25 Giga-bits/sec), non-integrated probes, for injecting
the test pattern, can introduce significant bandwidth limitations and spectral shaping in the injected test pattern due to impedance discontinuities caused by the connection of the probes to the
channel. The quality of the signal delivered from an integrated pattern generator 923 is higher due to a lack of such impedance discontinuities. Therefore, an integrated test pattern generator 923
can obtain an essentially in situ measurement of a channel's S[21 ]characteristic.
Receiver board 903 can comprise an IC 1611 as discussed above. Specifically, on the same physically contiguous IC 1611 can be integrated both a receiver circuit, at which an S[21 ]is to be measured,
along with an undersampler for taking the needed measurements of the received broadband signal.
The S[21 ]of a channel (also known as the “through transmission”), referred to herein as C[f](n), is the Fourier transform of the signal received via the channel, referred to herein as R[f](n),
divided by the Fourier transform of the transmitted signal, referred to herein as T[f](n), where T[f](n) is the signal that produced R[f](n). Expressed in equation form:
$C f ( n ) = R f ( n ) T f ( n )$
where T[t](n) and R[t](n) are, respectively, the time domain versions of T[f](n) and R[f](n). T[f](n) can be found by applying the Discrete Fourier Transform (DFT) to T[t](n) and R[f](n) can be found
by applying the Discrete Fourier Transform (DFT) to R[t](n).
Since T[t](n) has periodic content of finite length, the space of potential patterns for T[t](n) can be searched for those patterns of content that maximize a particular metric. A metric that can be
used, discussed further below, is maximizing the minimum power level, across the spectrum of interest. Based upon the highest frequency of interest, and the highest frequency which can reasonably be
expected to be delivered to the receiver, a minimum effective sample rate at which to collect R[t](n) can be determined which avoids aliasing high frequency components down into the range of
frequencies of interest.
In an example embodiment, T[t](n) can comprise 80 binary symbols transmitted at 6.25 Giga-bits/second. For this example T[t](n), R[t](n) can comprise 640 samples, or 8 samples per binary symbol. A
sample frequency of 8 samples/binary-symbol indicates the highest frequency that can be captured is 25 GHz. The highest frequency of interest is less than 6 GHz, and only frequencies higher than 44
GHz in the received signal would be folded down to interfering frequencies less than 6 GHz; however, the bandlimited nature of the transmit driver, backplane channel 901, and undersampler provide
large attenuation at frequencies which could alias down below 6 GHz.
Since T[t](n) is a generated signal (i.e. it is a representation produced by convolving an assumed trapezoidal pulse shape with the chosen broadband data pattern), it can be “sampled” at any sample
rate needed to match the sample rate used to acquire the received signal.
5.1. DFT Determination
A general formula for discrete Fourier transform (or DFT) determination is as follows:
$X f ( n ) = ∑ k = 0 N - 1 x t ( k ) ⅇ - j N 2 π nk$ $n = { 0 , 1 , … N - 1 }$
where x[t](k) is an array of N samples of a time domain signal. The result X[f](n) ranges from the result for the zero frequency (or “DC”) at X[f](0) to the result for the Nyquist frequency at either
X[f](N/2), for N being even, or X[f](└N/2┘), for N being odd.
Thus, a formula for determination of C[t](n) is:
$C f ( n ) = R f ( n ) T f ( n ) = ∑ k = 0 N - 1 R t ( k ) ⅇ - j N 2 π nk ∑ k = 0 N - 1 T t ( k ) ⅇ - j N 2 π nk$
where C[f](n) is a vector of complex numbers comprising both phase and magnitude information. In many applications, only the magnitude information is used. The above DFTs can be solved according to
the fast Fourier transform (FFT) approach that can be particularly efficient.
A phase response can be analyzed as having two components: a component that is a linear ramp with frequency and a residual “nonlinear” component. The linear component corresponds to channel delay,
while the nonlinear component accounts for channel dispersion (i.e., how the channel delays different frequencies by different amounts). C[f](n) provides the information about the nonlinear phase
component only; the actual channel delay cannot be determined by the techniques herein described.
The resolution of C[f](n) can be determined as follows:
$res_C f = 1 pattern_duration$
For example, an 80 bit pattern, at 6.25 Giga-bits/sec, has a pattern_duration of 12.8×10^−9 sec. Using the above formula, res_C[f ]is 78.125×10^6 Hz. Thus, the frequency resolution of the resulting C
[f](n) can be increased by increasing the time duration of the pattern injected into the channel.
5.2. Signal to Noise Considerations
In general, over the period of T[t](n), noise energy in the channel can be approximated as constant and uniform across the spectrum of interest. Different patterns for T[t](n), however, can inject
different amounts of signal energy into the channel at different frequencies (i.e. different patterns have different spectra).
A pattern for T[t](n), where the transmitted signal encodes binary symbols, can be a binary one followed by all zeros. For the example discussed above, comprised of 80 binary symbols, a pattern can
be comprised of one binary one followed by 79 zeros. This pattern shall be referred to herein as a “pulsie” pattern. A pulse pattern distributes its energy evenly across frequencies, but the total
amount of energy injected is relatively low. Therefore, its R[t](n) can have a relatively poor signal to noise ratio.
In general, to maximize the signal to noise ratio, a pattern can be selected for T[t](n) that maximizes the minimum power across the spectrum of interest. This constraint can be utilized by
constructing a pattern generator that generates all permissible patterns, and weighting the desirability of each such pattern according to the metric of its minimum power level across the relevant
portion of spectrum.
5.3. AC Coupled Receivers
For systems where the receiver is AC coupled to the channel, R[f](0) is zero and therefore C[f](0) is zero. To obtain information on the channel's S[21 ]characteristic for zero frequency,
extrapolation can be used. Specifically, linear extrapolation to C[f](0) can be determined from C[f](1) and C[f](2).
5.4. Example Pattern
This section presents an example S[21 ]measurement where T[t](n) comprises 40 binary symbols transmitted at 3.125 Giga-bits/second. For this example T[t](n), R[t](n) can comprise 320 samples, or 8
samples/binary-symbol. A sample frequency of 8 samples/binary-symbol indicates the highest frequency that can be captured (i.e., the Nyquist frequency) is 12.5 Giga-hertz.
The resolution of C[f](n) can be determined as follows:
$res_C f = 1 pattern_duration$
For a 40 bit pattern, at 3.125 Giga-bits/sec, the pattern_duration is 12.8×10^−9 sec. Using the above formula, res_C[f ]is 78.125×10^6 Hz.
The frequency resolution of C[f](n) can also be determined by dividing the Nyquist frequency by one less than the number of values of C[f](n), from zero to Nyquist frequency. For this example, the
number of values of C[f](n), from C[f](0) to the Nyquist frequency at C[f](160), is 161. Dividing 12.5×10^9 Hz by 1.6×10^2 equals 78.125×10^6 Hz.
To maximize the signal to noise ratio, a pattern can be selected for T[t](n) that maximizes the metric of the minimum power across the spectrum of interest. This metric can be utilized by a search
procedure that includes the following operations: generating permissible patterns and weighting such patterns according to the metric of its minimum power level across the relevant portion of
spectrum. An example search procedure is discussed in the following section. A non-exhaustive search, of a total search space of 2^40 possible patterns, can result in the following value for T[t](n):
bits 0-9: 1011010000
bits 10-19: 1011011001
bits 20-29: 0111100110
bits 30-39: 1010001010
FIG. 10A depicts the above listed T[t](n) as a signal 1010. Each bit of the above-listed pattern has a corresponding portion of signal 1010, in accordance with the bit-position indicated by the
horizontal axis of FIG. 10A. For example, bits 6 through 9 of the above-listed pattern are zero. Examining signal 1010, for the portion corresponding to bits 6 through 9 on the horizontal axis (this
portion of signal 1010 is also indicated by arrow 1020), it can be seen that signal 1010 is at −0.23 volts (where −0.23 volts corresponds to a logic zero).
Signal 1011 of FIG. 10A shows an example R[t](n), received as a result of transmitting signal 1010 into a channel. Unlike signal 1010, signal 1011 is not synchronized with the horizontal axis of FIG.
10A. For example, portion 1021 of signal 1011 is a result of portion 1020 being transmitted into the channel, yet portion 1021 of signal 1011 is located, with respect to the horizontal axis, at bits
28 to 31.
Converting signal 1010 of FIG. 10A into its corresponding T[f](n) yields spectrum 1012 of FIG. 10B. Since spectrum 1012 has only the magnitude information, it is actually a representation of |T[f](n)
|. Similarly, converting signal 1011 of FIG. 10A into its corresponding |R[f](n)| yields spectrum 1013 of FIG. 10B. Since the receiver for signal 1011 is AC coupled its spectrum 1013 has no DC
component. The first unit of resolution, after DC (or zero Hertz), is at 78.125 MHz and this is the lowest frequency point of spectrum 1013 that has a non-zero value.
The suitability of the example pattern can be analyzed by examining the spectrum of interest for spectrum 1012. Assuming a data transmission system with ideal equalization available, this can be from
zero Hertz up to one half the data transmission speed. Herein, one half the data transmission speed is referred to as the “Nyquist frequency of the transmitted data.” In the case of this example, one
half the data transmission speed of 3.125 Giga-bits/sec yields an upper end, for the spectrum of interest, of 1.5625 GHz. Assuming non-ideal equalization, however, the upper end of the spectrum of
interest can be 1.3 or 1.4 times the Nyquist frequency of the transmitted data. In the case of this example, this yields an upper end, for the spectrum of interest, of about 2.0 GHz.
As can be seen in FIG. 10B, spectrum 1012 is relatively flat, or “white,” from zero to about 2.0 GHz. This means that signal 1010 provides the broadband stimulus needed for S-Parameter capture.
|C[f](n)| is found by dividing |R[f](n)| by |T[f](n)|. FIG. 10C shows the |C[f](n)| resulting from a division of 1013 by 1012. As discussed above, extrapolation can be used to obtain a value at DC
for |C[f](0)|.
5.5. Example Search Procedure
FIG. 11 depicts an example procedure that searches for a pattern for T[t](n). The procedure seeks to find a pattern whose T[f](n), for the spectrum of interest, has the greatest minimum power level.
A variable for keeping track of the best pattern found, “BEST_PAT,” is initialized to null (step 1110). A variable for keeping track of the minimum power level of the best pattern found,
“MIN_PWR_BEST_PAT,” is initialized to zero (step 1111).
A current pattern to be tested is generated and assigned to the variable “CURR_PAT” (step 1112). The spectrum of the current pattern to be tested is found (step 1113) and its frequency domain power
spectrum density (frequency domain PSD) is determined (step 1114). Step 1113 can be accomplished by an FFT, and such FFT can be squared in order to produce the result of step 1114.
The minimum power level of the PSD is found and assigned to the variable “MIN_PWR” (step 1115).
If the minimum power level of the current pattern is greater than the minimum power level of the best pattern found thus far (“yes” branch of step 1116) then the best pattern found thus far is
updated (steps 1118 and 1119). After the update, if a limit on the number of patterns to be tested has not been reached (step 1117), the procedure loops back (“no” branch of step 1117) and another
pattern is generated (by step 1112).
If the minimum power level of the current pattern is not greater than the minimum power level of the best pattern found thus far (the “no” branch of 1116), if a limit on the number of patterns to be
tested has not been reached (step 1117), the procedure loops back (“no” branch of step 1117) and another pattern is generated (by step 1112).
6. Glossary of Selected Terms
APLL 100: Analog Phase-Locked Loop 100.
ADC: Analog to Digital Converter.
CP/LPF 102: Charge Pump/Low-Pass Filter 102.
DIS 112: Differential input signal 112.
DIV 104: Frequency divider 104.
DLPF 106: Digital Low-Pass Filter 106 DRU 115: Data Recovery Unit 115.
High signal level: corresponds to logic one.
Low signal level: corresponds to a logic zero.
PD 101: Phase Detector 101. Outputs pulses of width proportional to phase difference between its two inputs. PD 101 can be a frequency detector in addition to being a phase detector. Such frequency
detection can help with startup transients.
US 403: undersampler type 403.
US 430: an instance 430 of an undersampler of type 403.
US 431: an instance 431 of an undersampler of type 403.
VCO 103: Voltage Controlled Oscillator 103.
While the invention has been described in conjunction with specific embodiments, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art in
light of the foregoing description. Accordingly, it is intended to embrace all such alternatives, modifications and variations as fall within the spirit and scope of the appended claims and | {"url":"http://www.google.com/patents/US8000917?dq=5,815,488","timestamp":"2014-04-18T21:39:51Z","content_type":null,"content_length":"155521","record_id":"<urn:uuid:80728709-af6b-4630-a736-fe092e543e71>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00162-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Protosimplex] [Examples and illustrations]
last update 06-03-21
Examples and illustrations
Playing with the corrected gravitation law
Integrating field mass into gravitation results in a corrected gravitation law, which deviates for very large distances from Newton's description.
Furthermore its area of validity now has an upper (R[0]) and a lower limit (r[0]), called reality barriers.
Any mass which is situated in the range between the upper border distance R[0] and ρ must overcome a very weak repulsion force, if it wants to approach the source of field. Since this effect occurs
only for very large distances, it is practically not observable. However the value of astronomical red shift can be acknowledged as result of this repulsion by computating this with Heim's corrected
gravition law (see below).
For distances smaller than ρ the field now corresponds in good approximation to Newton's approximation. All empirical processes in space we are able to observe take place within this area.
This corrected gravitation law plays a key role in Heim's mathematical calculations, because Heim says “Gravitation is the only physical background phenomenon which accompanies all physical effects.”
(This is a result of equivalence of gravitation and inertia and a second equivalence of mass and energy. Therefore all energy phenomena can be expressed by matter-field-quanta).
1. Final geometrical unit
Here is another fundamental thought of Burkhard Heim. He says “If gravitation is the general background phenomenon of physical world, then there cannot be a world outside of the boundaries of
gravitation!” Therefore with this law it is possible now to determine both the smallest thing in the world (fundamental geometrical unit) and the largest thing existing – the diameter of the whole
Now we will see, how this things are done.
First Heim was examining whether there would be a final geometrical value, which remains still existing even if all masses are disappearing in empty space. In fact such a value is calculable while
transitioning mass against 0, if you form a product between the lower reality border of gravitation and Kompton's wave length of this infinitesimal mass.
Interesting enough, this final geometrical unit then will be a surface. It's size is approximately the square of Planck's length. (Heim named it τ, because this letter existed coincidentally on his
The exact value of this “metrons” (6.15 * 10^-70 m^2) describes the final geometrical unit of empty space, where no mass exists. Real physical space – in contrast to this – always is curved, whereby
this elementary surfaces become more or less compressed (“condensed”) depending upon densities of all fields existing in this space.
2. Diameter of the physical world
Just as the smallest thing of our world Heim also derives the largest thing from the boundaries of gravitation law – the maximum diameter of our physical world.
If you calculate an upper boundary R[0] of the gravitation law for the smallest mass conceivable (elementary mass), you will receive the largest diameter for which gravitation law really exists.
3. Cosmic red shift
Finally Heim found that cosmic red shift too is a result of the corrected gravitation law. Therefore each particle of this world must approach primarily against the repulsive gravitation component of
almost the whole remaining world. (This corresponds to the field curve between ρ and R[0].) This is using energy whereby each photon becomes longer in it's wavelength during this journey.
Heim inserted estimated middle mass density of universe into his formula and than he received as result Hubble radius, which is the radius of our visible world. Each photon coming from further on
behind this radius has lost all of his energy.
Even observed exceptions in red shift are plausible now with this model. They are only a result of inhomogenous mass density in space. | {"url":"http://www.engon.de/protosimplex/posdzech/px_g_gravi1e.htm","timestamp":"2014-04-18T13:07:23Z","content_type":null,"content_length":"6156","record_id":"<urn:uuid:af75a423-dcea-41bb-8db7-c6d2d60400fb>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00580-ip-10-147-4-33.ec2.internal.warc.gz"} |
Solids of Revolution
Do you have to use the washer method? I think it would be easier to use cylindrical shells.
Mr. Snookums
Find the volume formed by rotating the area contained by y=sqrt(6x+4), the y-axis and the line y=2x about the y-axis. Set up, but do not evaluate the integral.
First I graphed it, then did the "washer" method of finding the area of the circle formed, and found that the radius is (y/2-sqrt[(y^2-4)/6]. Is this right?
Wrong. First of all, if your solving [itex] y = \sqrt{6x + 4}[/itex] for x, the answer is not [itex] x = \sqrt{\frac{y^2 -4}{6}}[/itex], (where did the square root come from?)
Secondly, there is no one "radius." The volume for a washer is given by,
[tex]V = \pi[(\mbox{outer radius})^2 - (\mbox{inner radius})^2)] * (\mbox{thickness}) [/tex]
You want to find functions for the "outer radius" and the "inner radius." (Be careful, these might not be the same over your whole interval of integration)
I then found the height, which would be sqrt(y/2-sqrt[(y^2-4)/6], wouldn't it?
Where do you get "height" from. The formula for the volume of a washer is
[tex]V = \pi[(\mbox{outer radius})^2 - (\mbox{inner radius})^2)] * (\mbox{thickness}) [/tex]
I think it would help if you went over some examples from your text book.
Here is an example of the washer method when rotated about the x-axis.
Go to example 5 under heading "washers"
washer example
your book should have a better example (hopefully) | {"url":"http://www.physicsforums.com/showthread.php?t=118436","timestamp":"2014-04-18T18:16:12Z","content_type":null,"content_length":"28596","record_id":"<urn:uuid:ea7a9629-1fa6-4905-8431-0ff355fbc847>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00611-ip-10-147-4-33.ec2.internal.warc.gz"} |
the 4 written maths methods
ADD TO TES PRO
the 4 written maths methods
Last updated 12 June 2013, created 12 June 2013, viewed 118
4 A4 posters showing the 4 maths methods I use at school (chunking up for division, number line for subtraction, partitioning for addition and grid method for multiplication) I printed them onto A3
(resized) and backed on coloured paper and laminated for display. Really useful for the kids to refer More…to. Also included an A4 sheet with all 4 on, which I printed off and stuck in the front of
their homework books for them to reference at home, and to help parents know what methods we use too! Please leave a comment and let me know if you find it useful too!
Adapted from a resource contributed to TES Connect by clangercrazy
Downloads and web links
Reviews (10)
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Awesome! Thank you! :) (report comment)
Thank you so much for sharing, great for my display and brilliant for their folders. (report comment)
Thanks!! Excellent resource both for display board and books :)) (report comment)
Brilliant - just what I was looking for! (report comment)
great exactly what i have been looking for! :) (report comment)
More reviews… Report a problem with this resource | {"url":"http://www.tesaustralia.com/teaching-resource/the-4-written-maths-methods-6035154/","timestamp":"2014-04-20T01:26:56Z","content_type":null,"content_length":"61403","record_id":"<urn:uuid:575c07bd-b0a3-4e6a-b93f-5570a928def6>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00266-ip-10-147-4-33.ec2.internal.warc.gz"} |
unbiased estimator
September 13th 2008, 06:36 AM #1
Nov 2006
unbiased estimator
I want to show that the following estimator is unbiased:
s^2 = SSE/(n-2) = [sum from i = 1 to n of (yi - yi*)^2]/(n-2) = [sum from i = 1 to n of (yi - b0 - b1xi)^2]/(n-2)
For it to be unbiased E(s^2) must equal sigma^2.
I know E(yi) = beta0 + beta1*xi, Var(yi) = sigma^2, E(b1) = beta1, E(b0) = beta0
I've tried working on this (it's difficult for me to write out all my work), but I get lost trying to calculate E(b0b1) and other E's for instance.
Can someone show me how to do this?
Thanks in advance for any help.
I want to show that the following estimator is unbiased:
s^2 = SSE/(n-2) = [sum from i = 1 to n of (yi - yi*)^2]/(n-2) = [sum from i = 1 to n of (yi - b0 - b1xi)^2]/(n-2)
For it to be unbiased E(s^2) must equal sigma^2.
I know E(yi) = beta0 + beta1*xi, Var(yi) = sigma^2, E(b1) = beta1, E(b0) = beta0
I've tried working on this (it's difficult for me to write out all my work), but I get lost trying to calculate E(b0b1) and other E's for instance.
Can someone show me how to do this?
Thanks in advance for any help.
You need to show that $E[SSE] = (n-2) \sigma^2$. I'll take you most of the way but there'll be a point where I become latex intolerant and that's where you'll need to finish it off .....
The following standard and well known results (0, 1, 4 and 5 are standard linear regression formulae) will be used:
0. $b_0 = \bar{y} - b_1 \bar{x}$.
1. $\sum_{i=1}^{n} (x_i - \bar{x}) (y_i - \bar{y}) = b_1 \, \sum_{i=1}^{n} (x_i - \bar{x})^2$.
2. $\sum_{i=1}^{n} (y_i - \bar{y})^2 = \left( \sum_{i=1}^{n} y_i^2 \right) - n \bar{y}^2$.
3. $Var[W] = E[W^2] - (E[W])^2$ where $W$ is any random variable. (I would've used $U$ here but it looks like there's some sort of latex glitch which keeps turning $U$ into $u$ inside Var - quite
4. $E[b_1] = \beta_1$.
5. $Var[b_1] = \frac{\sigma^2}{\sum_{i=1}^{n} (x_i - \bar{x})^2}$.
$E[SSE] = E\left[ \sum_{i=1}^{n} (y_i - b_0 - b_1 x_i)^2\right]$
Substitute standard and well known result 0:
$= E\left[ \sum_{i=1}^{n} (y_i - ( \bar{y} - b_1 \bar{x} ) - b_1 x_i)^2\right] = E\left[ \sum_{i=1}^{n} (y_i - \bar{y} + b_1 \bar{x} - b_1 x_i)^2\right]$
$= E\left[ \sum_{i=1}^{n} ( [y_i - \bar{y}] - b_1 [x_i - \bar{x}])^2\right]$
$= E\left[ \sum_{i=1}^{n} [y_i - \bar{y}]^2 + b_1^2 \sum_{i=1}^{n} [x_i - \bar{x}]^2 - 2 b_1 \sum_{i=1}^{n} (x_i - \bar{x}) (y_i - \bar{y}) \right]$
Substitute from standard and well known results 1. and 2:
$= E\left[ \left( \sum_{i=1}^{n} y_i^2 \right) - n \bar{y}^2 - b_1^2 \sum_{i=1}^{n} (x_i - \bar{x})^2 \right]$
$= \left( \sum_{i=1}^{n} E\left[ y_i^2 \right] \right) - n E\left[\bar{y}^2\right] - \sum_{i=1}^{n} (x_i - \bar{x})^2 E\left[ b_1^2 \right]$
Use standard and well known result 3:
$= \left( \sum_{i=1}^{n} Var[y_i] + \left( E\left[ y_i \right]\right)^2 \right) - n \left( Var [\bar{y}] + \left( E \left[ \bar{y}\right] \right)^2 \right) - \sum_{i=1}^{n} (x_i - \bar{x})^2 \
left( Var[ b_1] + \left( E[b_1]\right)^2 \right)$
$= n \sigma^2 + \sum_{i=1}^{n} (\beta_0 + \beta_1 x_i)^2 - n \left( \frac{\sigma^2}{n} + (\beta_0 + \beta_1 \bar{x})^2 \right) - \sum_{i=1}^{n} (x_i - \bar{x})^2 \left( \frac{\sigma^2}{\sum_{i=1}
^{n} (x_i - \bar{x})^2} + \beta_1^2\right)$
where I've substituted standard and well known results 4. and 5. Note that $E[y_i] = E[\beta_0 + \beta_1 x_i] = \beta_0 + \beta_1 x_i$.
And at this point I've become quite latex intolerant. You should be able to show that all this simplifies to $(n-2) \sigma^2 \, ....$
Last edited by mr fantastic; September 14th 2008 at 02:46 AM. Reason: Slight re-organisation, content unchanged
Re: unbiased estimator
Hey, I managed to get all the way to the last step on my own but i cant figure out the silliest thing! X_X
Could you please show all your steps regarding the last step (1/(n-2))(sigma^2)
What happens with the Sxx(b1^2)??
September 13th 2008, 09:39 PM #2
October 24th 2012, 01:12 PM #3
Oct 2012
South Africa | {"url":"http://mathhelpforum.com/advanced-statistics/48878-unbiased-estimator.html","timestamp":"2014-04-20T16:31:28Z","content_type":null,"content_length":"44123","record_id":"<urn:uuid:d4feb2f9-2307-4f74-b69b-4af5e79377d5>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00071-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: C Style Strings
Chris Smith wrote:
> Andrew Poelstra <(E-Mail Removed)> wrote:
> > On 2006-06-02, Noah Roberts <(E-Mail Removed)> wrote:
> > >
(E-Mail Removed)
> > >> function call. In C++ you can hope your compiler can figure it out; if
> > >> not it will use new/delete which eventually falls back to malloc/free
> > >> which is hundreds of times slower.
> > >
> > > That statement about C++ is simply incorrect; I can't even imagine
> > > where it is coming from.
> > >
> >
> > I imagine that it comes from a basic understanding of stack-based memory.
> I don't believe the complaint was about stack memory. It was about the
> incorrect statement regarding C++. The same statement may be considered
> valid concerning Java, C#, VB, or C++/CLI, for example; but those are
> different languages from C++. (The word "valid" should be taken lightly
> there; I haven't verified the hundreds of times.)
> C++ perfectly well allows programmers to allocate any "objects" (not
> quite, really, since they don't own their identity so they are a sort of
> 2/3-object... but in C++ vocab they are objects) on the stack, with all
> the accompanying performance benefits.
Indeed, I have again made the mistake of calling C++ what I mean to
call C++ but not using the C subset/paradigms. Of course you can do
this in C++ because you can just do it using the C-like subset (where
the resulting data types are still considered "objects".) My point was
just that C++ does not have a blanket advantage over C, since falling
back to ordinary C may still be the best way to do things.
Paul Hsieh
http://www.pobox.com/~qed/ http://bstring.sf.net/ | {"url":"http://www.velocityreviews.com/forums/t442953-p3-re-c-style-strings.html","timestamp":"2014-04-20T16:07:03Z","content_type":null,"content_length":"69944","record_id":"<urn:uuid:9d72b8be-5dca-4f46-9a05-07c0b73f0d31>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00072-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Primer on Learning in Bayesian Networks for Computational Biology
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More information
PLoS Comput Biol. Aug 2007; 3(8): e129.
A Primer on Learning in Bayesian Networks for Computational Biology
Fran Lewitter, Editor^
Bayesian networks (BNs) provide a neat and compact representation for expressing joint probability distributions (JPDs) and for inference. They are becoming increasingly important in the biological
sciences for the tasks of inferring cellular networks [1], modelling protein signalling pathways [2], systems biology, data integration [3], classification [4], and genetic data analysis [5]. The
representation and use of probability theory makes BNs suitable for combining domain knowledge and data, expressing causal relationships, avoiding overfitting a model to training data, and learning
from incomplete datasets. The probabilistic formalism provides a natural treatment for the stochastic nature of biological systems and measurements. This primer aims to introduce BNs to the
computational biologist, focusing on the concepts behind methods for learning the parameters and structure of models, at a time when they are becoming the machine learning method of choice.
There are many applications in biology where we wish to classify data; for example, gene function prediction. To solve such problems, a set of rules are required that can be used for prediction, but
often such knowledge is unavailable, or in practice there turn out to be many exceptions to the rules or so many rules that this approach produces poor results.
Machine learning approaches often produce better results, where a large number of examples (the training set) is used to adapt the parameters of a model that can then be used for performing
predictions or classifications on data. There are many different types of models that may be required and many different approaches to training the models, each with its pros and cons. An excellent
overview of the topic can be found in [6] and [7]. Neural networks, for example, are often able to learn a model from training data, but it is often difficult to extract information about the model,
which with other methods can provide valuable insights into the data or problem being solved. A common problem in machine learning is overfitting, where the learned model is too complex and
generalises poorly to unseen data. Increasing the size of the training dataset may reduce this; however, this assumes more training data is readily available, which is often not the case. In
addition, often it is important to determine the uncertainty in the learned model parameters or even in the choice of model. This primer focuses on the use of BNs, which offer a solution to these
issues. The use of Bayesian probability theory provides mechanisms for describing uncertainty and for adapting the number of parameters to the size of the data. Using a graphical representation
provides a simple way to visualise the structure of a model. Inspection of models can provide valuable insights into the properties of the data and allow new models to be produced.
Bayesian Networks
In a graphical model representation, variables are represented by nodes that are connected together by edges representing relationships between variables. Figure 1 provides an example of a BN
describing a gene regulation network. The expression of each gene is represented by one variable of a JPD that describes how the genes are regulated by each other. Such a JPD may be complex even for
just five variables; however, the graphical representation makes it clear where the regulatory relationships exist between the genes.
An Example: Gene Regulatory Networks
For BNs, the edges of the graph must form a directed acyclic graph (DAG)—a graph with no cyclic paths (no loops). This allows for efficient inference and learning. JPDs can be expressed in a compact
way, reducing model size through exploiting conditional independence relationships—two variables are conditionally independent if they are independent given the state of a third variable. A benefit
of BNs is that they may be interpreted as a causal model which generated the data. Thus, arrows (directed edges) in the DAG can represent causal relations/dependencies between variables. However, it
must be noted that to learn a causal model from data needs more than association data, and this is discussed toward the end of this primer under the heading Causality.
Bioinformatics applications of BNs have included gene clustering and the inference of cellular networks [1], since they are well-suited to modelling stochastic complex biological systems, and the
resulting networks can be easily understood. An excellent example of combining data and domain knowledge in the bioinformatics field is the MAGIC BN which has been designed using expert knowledge for
combining information from diverse heterogeneous data sources for the classification task of gene function prediction [3].
Conditional probability distributions (model parameters).
The relationships between variables are encoded by conditional probability distributions (CPDs) of the form p(B|A)—the probability of B given A. For discrete variables, probability distributions are
expressed as conditional probability tables (CPTs) containing probabilities that are the model parameters (see Figure 7 and related text for examples). For each node, the probability that the
variable will be in each possible state given its parents' states can be calculated based on the frequency observed in a set of training data. It is often useful/necessary to use a prior distribution
for the model parameters, as, without a prior, a possible configuration that was not seen in the training examples would be incorrectly assigned a zero probability of ever being observed. (Equally
well, these probabilities may be estimated by an expert and used alongside those learned from data).
Naïve Bayes Classifier with Model Parameters in the Form of CPTs
For BNs, which use continuous variables, conditional probability densities are used in a similar way to CPTs. Figure 2 presents a simple BN which introduces the concept of using continuous variables.
The usual notation is to use squares for discrete nodes and circles for continuous nodes. A continuous node, B, with a discrete parent, A, (say, a variable with k = 3 states) leads to a model of the
continuous data using k Gaussian distributions. Thus, given that A is in state a[i], the likelihood of a value of B may be inferred, or, alternatively, given a value b for variable B, the probability
that variable A is in state a[i] may be inferred. Parameters for the Gaussians (or other distributions) can be learned from training data. θ[B] is the parameter set that encodes the model for B in
terms of three Gaussians, one for each of the three possible states of A. A mean μ[i] and standard deviation σ[i] are the parameters for the Gaussian distribution which models p(b|a[i]).
Illustration of Model Parameters for Two-Node Bayesian Network
In a similar way, regression models for CPDs of continuous variables with continuous parents may be used. In this case, θ[B] = P(B|A) ~ N(c + ma, σ^2). i.e., the CPD for B is a Gaussian distribution
with a mean dependent on the value of A = a, with constants m and c determined by regression of B on A.
Joint probability distributions.
It is the JPD over all the variables that is of great interest. However, the number of model parameters needed to define the JPD grows rapidly with the number of variables. Through exploiting
conditional independence between variables, the models may be represented in a compact manner, with orders of magnitude fewer parameters.
Relationships between variables are captured in a BN structure S defined by a DAG (as in the gene regulatory network example in Figure 1). This enables the JPD to be expressed in terms of a product
of CPDs, describing each variable in terms of its parents, i.e., those variables it depends upon. Thus:
where x = { x[1], … , x[n] } are the variables (nodes in the BN), and θ = { θ[1] , … , θ[n] } denotes the model parameters, where θ[i] is the set of parameters describing the distribution for the ith
variable x[i], and pa(x[i]) denotes the parents of x[i]. Each parameter set θ[i] may take a number of forms—commonly a CPT is used for discrete variables, and CPDs (such as Gaussian distributions)
are used for continuous variables. Classification/regression models can be used to learn the parameters for each node in the network.
Inference in Bayesian networks.
For the known BN structure (gene regulatory network) in Figure 1 and a CPD for each node (modelling gene interactions), given evidence about the expression levels of some genes, inferences about the
likely values of other genes can be made. For example, the value of G1 may be inferred from the values of the other genes, i.e., p(G1|G2, G3, G4, G5). More generally, inferences of the values of a
set of variables may be made given evidence of another set of variables, by marginalising over unknown variables. (Marginalising means considering all possible values the unknown variables may take,
and averaging over them.) Simple inference examples are illustrated in the next section.
Conceptually, inference is straightforward, p(x|y) is calculated as a product of relevant CPDs, using Bayes rule [p(a|b) = p(b|a)p(a)/p(b)] to calculate any posterior probabilities. Computationally,
the calculation of inference in this way is hard and inefficient. A number of methods exist that exploit the structure of the graph to derive efficient exact inference algorithms such as the
sum–product and max–sum algorithms. For many problems, however, exact inference is not feasible, and, therefore, the use of approximation methods such as variational methods and sampling approaches
are required.
Conditional independence.
Two variables are conditionally independent if they are independent given the state of a third variable. Mathematically, a and b are conditionally independent given c if:
Conditional independence relationships are encoded in the structure of the network, as illustrated in the three cases below. Regulation of three genes x, y, and z is taken as an example. In each
case, the situation is described, along with a BN diagram, an equation for the JPD, and an equation for inference of p(z|x).
Serial connection. For example, when gene x promotes gene y, and gene y promotes gene z (Figure 3). In this case, evidence is transmitted unless the state of the variable in the connection is known:
if the expression level of gene y is unknown, then evidence of the level of x effects the expected level of z; if y is known, then the level of z depends only on the expression level of y. z is
conditionally independent from x.
Diverging connection. For example, when a transcription factor y turns on two genes x and z (Figure 4). As with a serial connection, evidence is transmitted unless the variable in the connection is
instantiated: if the expression level of y is unknown, then evidence of the level of x effects the level of z (since they are co-regulated—if x is highly expressed, then the likely level of y may be
inferred, which in turn would influence the expression level of z); if y is known, then the level of z depends only on the expression level of y. z is conditionally independent from x.
Converging connection. For example, when two genes x and z both promote gene y (Figure 5). Evidence is transmitted only if the variable in the connection or one of its children receives evidence: if
y is unknown, then evidence of the expression level of gene x does not help to infer the expression level of z—x and z are independent; however, if y is known, then the level of x does help to infer
the expression level of z. Importantly, at the v-structure in the network, the CPD for y encodes the dependency of y on both x and z. Note in this case that p(x,z|y) ≠ p(x|y)p(z|y).
In the case of a converging connection, it is also worthwhile noting that when the value of y is known as well as x, then this evidence helps to infer the value of z, and x and z are no longer
independent variables:
Thus, the structure of the model captures/encodes the dependencies between the variables and leads to a different causal model.
An example: Naïve Bayes classifier for interaction site prediction.
As a simple example, consider the task of predicting interaction sites on protein surfaces from measures of conservation and hydrophobicity of surface patches. This gives three variables: I, whether
the patch is an interaction site or not; C, conservation score for the patch; and H, the hydrophobicity of the patch. I is a discrete class variable. Both C and H are continuous variables (though may
be quantised to form discrete data). Conservation and hydrophobicity are both reasonably good predictors of interaction sites, and the information from these independent predictions may be combined
in a naïve Bayes classifier to improve performance. The structure of the model for a naïve Bayes classifier has a class node (the one to be inferred from the other observed variables) as a parent to
all other independent variables and is illustrated in Figure 7. Such a model structure is excellent for integrating information, and for maintaining a small model size. [For a set of n binary
variables, a completely connected DAG has 2^n − 1 free parameters, an inverted naïve Bayes classifier (where the class node depends on all other variables) has 2^n^−1 + n + 1 free parameters, whereas
a naïve Bayes classifier has only 2n + 1 free parameters! For a model with 100 binary variables, this is more than 2^90 times smaller!]. In the next section of this primer, the learning of parameters
for this simple example is illustrated. This example is inspired by [4] in which a naïve Bayes classifier is used within a classification scheme to predict protein–protein interaction sites using a
number of predictive variables.
Parameter Learning
The simplest approach to learn the parameters of a network is to find the parameter set that maximises the likelihood that the observed data came from the model.
In essence, a BN is used to model a probability distribution X. A set of model parameters θ may be learned from the data in such a way that maximises the likelihood that the data came from X. Given a
set of observed training data, D = { x[1], … , x[N] } consisting of N examples, it is useful to consider the likelihood of a model, L(θ), as the likelihood of seeing the data, given a model:
It should be noted here that x[i] is the ith training example and that the likelihood of D being generated from model θ is the product of the probabilities of each example, given the model.
Maximum likelihood.
The learning paradigm which aims to maximise L(θ) is called maximum likelihood (ML). This approximates the probability of a new example x given the training data D as p(x|D) ≈ p(x|θ[ML]) where θ[ML]
is the maximum (log) likelihood model which aims to maximise ln p(D|θ), i.e., θ[ML] = arg max[θ] ln p(D|θ). This amounts to maximising the likelihood of the “data given model.” ML does not assume any
prior. Using negative log likelihood is equivalent to minimising an error function.
Maximum posterior.
In order to consider a prior distribution, a maximum a posteriori (MAP) model can be used. This approximates the probability of a new example x given the training data D as p(x|D) ≈ p(x|θ[MAP]) where
θ[MAP] is the MAP probability (likelihood of the “model given data”) which aims to maximise ln p(θ|D), i.e., θ[MAP] = arg max[θ] ln p(θ|D). This takes into account the prior, since through Bayes'
theorem: p(θ|D) = p(D|θ)p(θ)/p(D).
Often ML and MAP estimates are good enough for the application in hand, and produce good predictive models. The numerical example at the end of this section illustrates the effects of ML and MAP
estimates with different strength priors and training set sizes. Both ML and MAP produce a point estimate for θ. Point estimates are a single snapshot of parameters (though confidence intervals on
their values can be calculated).
Marginal likelihood.
For a full Bayesian model, the uncertainty in the values of the parameters is modelled as a probability distribution over the parameters. The parameters are considered to be latent variables, and the
key idea is to marginalise over these unknown parameters, rather than to make point estimates. This is known as marginal likelihood. The computation of a full posterior distribution, or model
averaging, avoids severe overfitting and allows direct model comparison. In [8], Eddy introduces Bayesian statistics with a simple example, and integrates over all possible parameter values,
illustrating a more rigorous approach to handling uncertainty. Formulating Bayesian learning as an inference problem, the training examples in D can be considered as N independent observations of the
distribution X. Figure 6 shows a graphical model where the shaded nodes x[i] represent the observed independent training data and x the incomplete example observation for which the missing values are
to be inferred, all of which are dependent upon the model θ.
Graphical Model Illustrating Bayesian Inference
The joint probability of the training data, the model, and a new observation x is:
where p(θ) is the prior. Applying the sum rule [p(a) = ∫p(a,b)db]:
Applying the product rule [p(a,b) = p(a|b)p(b)] to the left-hand side, and substituting (4) for the joint probability on the right-hand side, then dividing both sides by p(D), gives the predictive
distribution for x:
This is computing a full Bayesian posterior. In order to do this, a prior distribution, p(θ), for the model parameters needs to be specified. There are many types of priors that may be used, and
there is much debate about the choice of prior [9]. Often the calculation of the full posterior is intractable, and approximate methods must be used, such as point estimates or sampling techniques.
Marginal likelihood fully takes into account uncertainty by averaging over all possible values.
Learning from incomplete data.
The parameters for BNs may be learned even when the training dataset is incomplete, i.e., the values of some variables in some cases are unknown. Commonly, the Expectation–Maximisation (EM) algorithm
is used, which estimates the missing values by computing the expected values and updating parameters using these expected values as if they were observed values.
EM is used to find local maxima for MAP or ML configurations. EM begins with a particular parameter configuration (possibly random) and iteratively applies the expectation and maximisation steps,
until convergence.
E-step. The expected values of the missing data are inferred to form D[C]—the most likely complete dataset given the current model parameter configuration.
M-step. The configuration of which maximises p(|D[C]) is found (for MAP).
Using EM to find a point estimate for the model parameters can be efficient to calculate and gives good results when learning from incomplete data or for network structures with hidden nodes (those
for which there is no observed data).With large sample sizes, the effect of the prior p(θ) becomes small, and ML is often used instead of MAP in order to simplify the calculation. More sophisticated
(and computationally expensive) sampling methods such as those mentioned below may also be applied to incomplete data. One advantage of these methods is that they avoid one of the possible drawbacks
of EM—becoming trapped in local optima.
There may be cases of hidden nodes in gene regulatory networks, where the network is known, but experiments have not provided expression levels for all genes in the network—model parameters can still
be learned. The ability to handle incomplete data is an important one, particularly when considering that expression data may come from different laboratories, each looking at different parts of a
gene regulatory network, with overlap of some genes whilst others are missing. In this case, all the collected data can be used.
Sampling methods.
A number of sampling methods have been used to estimate the (full) posterior distribution of the model parameters p(θ|D). Monte Carlo methods, such as Gibbs sampling, are extremely accurate, but
computationally expensive, often taking a long time to converge, and become intractable as the sample size grows. Gaussian approximation is often accurate for relatively large samples, and is more
efficient than Monte Carlo methods. It is based on the fact that the posterior distribution p(θ|D) which is proportional to p(D|θ) × p(θ) can often be approximated as a Gaussian distribution. With
more training data, the Gaussian peak becomes sharper, and tends to the MAP configuration θ[MAP].
Parameter learning numerical example.
In this numerical example, we illustrate the approaches described in the text for learning Bayesian network parameters, using the simple example of a naïve Bayes classifier to predict protein
interaction sites (I) using information on conservation (C) and hydrophobicity (H). Each variable has two possible values: I = yes/no; H = high/low and C = high/low. The conditional probability
tables defining the network are shown in Figure 7, and the learning problem is to determine values for the associated probabilities p[1–5].
To illustrate the different methods, we will focus on parameter p[2], the probability that conservation is high (C = high), given that this is a protein interaction site (I = yes). The value of p[2]
is to be estimated from count data; in this case, we assume that for N interaction sites, n have high conservation and N − n have low conservation.
Figure 8 describes a number of possible scenarios. In the Figure 8A–8D graphs, the red dashed line shows the likelihood, p(data|model). In this case, it is derived from the binomial distribution, and
represents the probability of observing n high conservation sites in N trials, as a function of the binomial parameter p[2]. The other graph curves are the prior p(model) (dotted green curve), giving
a prior distribution for the value of p[2], and the posterior p(model|data) (solid blue curve). Here we have used the beta distribution as the prior. This is a very flexible distribution on the
interval [0,1]; it has two parameters B(n,m), with B(1,1) representing the uniform distribution and other shapes being obtained with larger and different values of n and m. An advantage of the beta
distribution in this case is that when used as a prior with the binomial it yields a posterior that is also a beta distribution (but with different parameters). The beta distribution is the conjugate
prior of the binomial. In fact, the n and m parameters of the beta distribution can be viewed as pseudocounts, which are added to the observed counts to reflect prior knowledge.
The Effects of Different Strength Priors and Training Set Sizes
The Bayesian approach of calculating marginal likelihood does not involve making a point estimate of the parameter; instead, the posterior distribution is averaged over in order to fully take into
account the uncertainty in the data.
Structure Learning
Particularly in the domain of biology, the inference of network structures is the most interesting aspect; for example, the elucidation of regulatory and signalling networks from data. This involves
identifying real dependencies between measured variables; distinguishing them from simple correlations. The learning of model structures, and particularly causal models, is difficult, and often
requires careful experimental design, but can lead to the learning of unknown relationships and excellent predictive models.
Full Bayesian posterior.
So far, only the learning of parameters of a BN of known structure has been considered. Sometimes the structure of the network may be unknown and this may also be learned from data. The equation
describing the marginal likelihood over structure hypotheses S^h as well as model parameters is an extension of Equation 7; the predictive distribution is:
However, the computation of a full posterior distribution over the parameter space and the model structure space is intractable for all practical applications (those with more than a handful of
Sampling methods.
Even for a relatively small number of variables, there are an enormous number of possible network structures, and the computation of a full posterior probability distribution is difficult. There are
several approaches to this problem, including Markov chain Monte Carlo (MCMC) methods (such as the Metropolis–Hastings algorithm), which are used to obtain a set of “good” sample networks from the
posterior distribution p(S^h,θ[S]|D), where S^h is a possible model structure. This is particularly useful in the bioinformatics domain, where data D may be sparse and the posterior distribution p(S^
h,θ[S]|D) diffuse, and therefore much better represented as averaged over a set of model structures than through choosing a single model structure.
Variational methods.
A faster alternative to MCMC is to use variational methods for certain classes of model. By approximating parameters' posterior distributions (which are difficult to sample from) by simpler ones, a
lower bound on the marginal likelihood can be found which can then be used for model selection.
Structure learning algorithms.
The two key components of a structure learning algorithm are searching for “good” structures and scoring these structures. Since the number of model structures is large (super-exponential), a search
method is needed to decide which structures to score. Even with few nodes, there are too many possible networks to exhaustively score each one. Efficient structure learning algorithm design is an
active research area. A greedy search may be done by starting with an initial network (possibly with no (or full) connectivity) and iteratively adding, deleting, or reversing an edge, measuring the
accuracy of the resulting network at each stage, until a local maxima is found. Alternatively, a method such as simulated annealing should guide the search to the global maximum.
There are two common approaches used to decide on a “good” structure. The first is to test whether the conditional independence assertions implied by the structure of the network are satisfied by the
data. The second approach is to assess the degree to which the resulting structure explains the data (as described for learning the parameters of the network). This is done using a score function.
Ideally, the full posterior distribution of the parameters for the model structure is computed (marginal likelihood); however, approximations such as the Laplace approximation or the Bayesian
Information Criterion (BIC) score functions are often used, as they are more efficient (though approximate, and therefore less accurate). The BIC score approximates ln p(D|S^h) as , where is an
estimate of the model parameters for the structure, d is the number of model parameters, and N is the size of the dataset. For large N, the learned model often has parameters like θ[ML]. The BIC
score has a measure of how well the model fits the data, and a penalty term to penalise model complexity. This is an example of Occam's Razor in action; preferring the simplest of equally good
models. ML is not used as a score function here, as without a penalty function it would produce a completely connected network, implying no simplification of the factors.
In the case of gene regulatory networks, these structure learning algorithms may be used to identify the most probable structure to give an influence diagram for a gene regulatory network learned
from data. Imoto et al. [10] derive gene networks based on BNs from microarray gene expression data, and use biological knowledge such as protein–protein interaction data, binding site information,
and existing literature to effectively limit the number of structures considered to be the most biologically relevant. The fitness of each model to the microarray data is first measured using
marginal likelihood, then biological knowledge is input in the form of a prior probability for structures. The posterior probability for the proposed gene network is then simply the product of the
marginal likelihood of the parameters and the prior probability of the structure.
Often the really interesting problems involve the learning of causal relationships [11], such as protein signalling networks [2] and gene regulatory interactions. In order to discover the underlying
causal model, more than just structure learning is needed, because the available data may be insufficient to distinguish different network structures that imply the same conditional independences
(Markov equivalence) and have the same score. One way to determine the directionality of the causal relations is to use intervention data, where the value of one variable is held fixed. Consider two
correlated variables, X and Y, subjected to interventions (these may be expression levels of two genes, and interventions are gene knockouts). If inhibiting X leads to a limited range of observed
values of Y, whereas inhibiting Y leads to a full range of X values, then it can be determined that X influences Y, but Y doesn't influence X. This implies there is a causal relationship X → Y.
Sachs et al. [2] model a protein signalling network from flow cytometry data. Simultaneous observations of multiple signalling molecules in many thousands of cells in the presence of stimulatory cues
and inhibitory interventions (perturbations) and careful experimental design allow for identifying causal networks, which are potentially useful for understanding complex drug actions and
dysfunctional signalling in diseased cells.
Dynamic Bayesian networks.
An essential feature of many biological systems is feedback. BNs are perfectly suited to modelling time series and feedback loops. When BNs are used to model time series and feedback loops, the
variables are indexed by time and replicated in the BN—such networks are known as dynamic Bayesian networks (DBNs) [12] and include as special cases hidden Markov models (HMMs) and linear dynamical
systems. The creation of experimental time series measurements is particularly important for modelling biological networks.
As an example, if in the earlier gene regulatory network example, gene G5 regulated G1, then a feedback loop (cyclic graph) would be formed. In order to perform efficient inference, BNs require a DAG
to define joint probabilities in terms of the product of conditional probabilities. For probabilistic graphical models with loops, as described, either iterative methods such as loopy belief
propagation must be used, or the cyclic graph must be transformed into a DAG. Assuming a (first-order) Markov process governs gene regulation, the network may be rolled out in time, to create a DBN.
Generally, DBNs contain two time slices, with an instance of each variable in each time slice (t and t + Δt). Directed edges are added from nodes at time t to the nodes they influence at t + Δt. HMMs
are a special case of DBNs, where there is a hidden set of nodes (normally discrete states), a set of observed variables, and the slices need not be time; often HMMs are used for sequence analysis
and t is the transition from one base to the next. DBNs have been used for inferring genetic regulatory interactions from microarray data. Data from a few dozen time points during a cell cycle is a
very small amount of data on which to train a DBN. Husmeier has recently used MCMC on simulated data of microarray experiments in order to access the network inference performance with different
training set size, priors, and sampling strategies [13]. Variational Bayesian methods have been used to approximate the marginal likelihood for gene regulatory network model selection with hidden
factors from gene expression time series data. The hidden factors capture the effects that cannot be directly measured, such as genes missing from the microarray, the levels of regulatory proteins
present, and the effects of mRNA, etc. [14].
Many applications in computational biology have taken advantage of BNs or, more generally, probabilistic graphical models. These include: protein modelling, systems biology; gene expression analysis,
biological data integration; protein–protein interaction and functional annotation; DNA sequence analysis; and genetics and phylogeny linkage analysis. However, perhaps the most interesting
application of BNs in the biological domain has been the modelling of networks and pathways. These analyses combine all the features of BNs: the ability to learn from incomplete noisy data, the
ability to combine both expert knowledge and data to derive a suitable network structure, and the ability to express causal relationships. Recent application of DBNs has allowed more sophisticated
relationships to be modeled; for example, systems which incorporate feedback. Furthermore, the marriage of improved experimental design with new data acquisition techniques promises to be a very
powerful approach in which causal relations of complex interactions may be elucidated.
Additional Reading
Heckerman has written an excellent mathematical tutorial on learning with BNs [9], whose notation has been adopted here. This is the suggested text to consult for statistical details and discussion
of the concepts introduced in this primer. Murphy's introduction [15], along with the guide to the software Bayes Net Toolkit for Matlab, BNT [16], provides an overview of algorithms for learning.
Tipping's tutorial [17] contains good illustrations of marginal likelihood, and Ghahramani's tutorial [18] contains a clear overview introducing structure learning and approximation methods.
Husmeier's bioinformatics text [13] is also an excellent resource.
Bayesian network
Bayesian information criterion
conditional probability distribution, CPT, conditional probability table
directed acyclic graph
dynamic Bayesian network
hidden Markov model
joint probability distribution
maximum a posteriori
Markov chain Monte Carlo
maximum likelihood
Chris J. Needham and Andrew J. Bulpitt are with the School of Computing, University of Leeds, Leeds, United Kingdom. James R. Bradford and David R. Westhead are with the Institute of Molecular and
Cellular Biology, University of Leeds, Leeds, United Kingdom.
Competing interests. The authors have declared that no competing interests exist.
Author contributions. CJN is the primary author of the tutorial. JRB and DRW have advised on the biological examples. AJB and DRW have contributed their pedagogical knowledge on the topic. All
authors have advised on the selection and presentation of the material.
Funding. The authors would like to thank the Biotechnology and Biological Sciences Research Council for funding on grant BBSB16585 during which this article was written.
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A flagstaff 17.5 metre high casts a shadow of length 40.25 metre. The height of building, which casts a shadow of length 28.75 metre under similar conditions will be :
Important English Words
Means: Show Off
Means: Of Imposing Height
Means: Give Instructions To Or Direct Somebody To Do Something With Authority
Means: Obtainment, Acquirement
Means: Active, Fast, Energetic
...important english words
Question Detail
A flagstaff 17.5 metre high casts a shadow of length 40.25 metre. The height of building, which casts a shadow of length 28.75 metre under similar conditions will be :
• 12 metre
• 12.5 metre
• 13.5 metre
• 14 metre
Answer: Option B
Less shadow, Less Height (Direct Proportion)
So, let height of building be x metre
40.25 : 17.5 :: 28.75 : x \\
=> x = \frac{17.5*28.75}{40.25} \\
=> x = 12.5
So height of building should be 12.5 metre
Similar Questions : Read more from - Chain Rule Questions Answers
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Calculating Percentages?
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Calculating Percentages?
Posted by FW on March 27, 2001 1:26 PM
I'm an Excel rookie who has been thrown into this with no training or manual. I'm trying to calculate some percentages. I've had a look at both Percentile and Percentrank, and can't get them to do
what I need.
I have columns of numbers, adding up to various amounts. I need to calculate the percentage each one of these numbers bears to the total of each column.
IE Column contains the numbers 3, 11, and 6. Total is
20. I therefore need to know the percentages that 3, 11 and 6 would be of 20.
Does anyone have any idea of what formula would be needed to extrapolate the required percentages? I hope its a fairly simple one, as I have a number of them to do on a monthly basis.
Sorry if this is simplistic, but I have exactly three days experience with Excel. Thank you.
Check out our Excel Resources
Re: Calculating Percentages?
Posted by Dave Hawley on March 27, 2001 1:33 PM
Hi FW
Here is one easy way:
=A1/SUM(A1:A10) Then format the cell as a percentage.
This will tell you the what percentage A1 is of the Total of A1:A10
OzGrid Business Applications
Re: Calculating Percentages?
Posted by Mark W. on March 27, 2001 1:37 PM
FW, if you modify Dave's formula as follows:
...you can copy it down a column adjacent to
your list of numbers and accurate results.
Re: Calculating Percentages?
Posted by mseyf on March 27, 2001 1:47 PM
a brief overview of calculating percentages
A1 3 =A1/SUM($A$1:$A$3)
A2 11 =A2/SUM($A$1:$A$3)
A3 6 =A3/SUM($A$1:$A$3)
A1 3 =A1/$A$4
A2 11 =A2/$A$4
A3 6 =A3/$A$4
A4 20
then format the calculations as percentages.
The "$" creates an 'Absolute' reference that doesn't change as you copy the calculation
Re: Calculating Percentages?
Posted by FW on March 28, 2001 9:32 AM
Thanks muchly, everyone. It now seems to be working.
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Wi-Fi Location-Based Services 4.1 Design Guide - Determining Approximate Roots using Maxima [Design Zone for Mobility]
Table Of Contents
Determining Approximate Roots using Maxima
Determining Approximate Roots using Maxima
In the circle-circle intersection equations below:
•d represents the inter-access point distance in feet
•Obg represents the percentage of overlap desired for 802.11bg
•Oa represents the percentage of overlap desired for 802.11a
•R represents the cell radius in feet
Note that zero overlap occurs when the distance between the centers of the two circles is equal to twice the radius (d = 2R). This relationship becomes invalid if d is allowed to exceed 2R, as an
area of intersection would be impossible to calculate. This is why the find_root function is limited to the closed interval from d/2 to d.
wxMaxima 0.7.3a http://wxmaxima.sourceforge.net
Maxima 5.13.0 http://maxima.sourceforge.net
Using Lisp GNU Common Lisp (GCL) GCL 2.6.8 (aka GCL)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
(%i4) find_root(Obg*%pi*R^2=2*R^2*acos(d/(2*R))-(1/2*d*(sqrt(4*R^2-d^2))),R,d/2,d);
(%i5) find_root(Oa*%pi*R^2=2*R^2*acos(d/(2*R))-(1/2*d*(sqrt(4*R^2-d^2))),R,d/2,d);
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West Bridgewater Math Tutor
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Transition to Turbulence in Fluid Flows
Supported in part by NSF CAREER Award CMMI-06-44793;
U of M Digital Technology Center's 2010 Digital Technology Initiative Seed Grant
The 2007 Annual Review of Fluid Mechanics article on “Nonmodal Stability Theory” by Peter J. Schmid has devoted two sections to my papers on “Componentwise energy amplification in channel flows”, J.
Fluid Mech. 2005, and “The spatio-temporal impulse response of the linearized Navier-Stokes equations”, 2001 American Control Conference. Even though we haven't written a journal version of the
impulse response paper, a significantly polished version of the original conference submission can be found in Chapter 10 of my PhD Thesis.
The Annual Review of Fluid Mechanics covers the significant developments in the field of fluid mechanics; its 2012 impact factor is 12.600.
Project synopsis
During my doctoral work, I initiated a system-theoretic approach to model and analyze shear flows of Newtonian fluids, such as air and water. Using this approach I obtained a detailed
characterization of the flow structures triggering the transition to turbulence. This is likely to have an impact in a variety of applications including design of fuel-efficient and
environmentally-friendly vehicles. Since joining the University of Minnesota, I used this early success as a motivation for a new line of research; this effort resulted in a set of well developed
theoretical and computational methods for analysis and control of both Newtonian and non-Newtonian fluids.
In a recent interdisciplinary collaborative effort with Prof. Satish Kumar of Chemical Engineering and Materials Science Department at the University of Minnesota, I have been studying transition to
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micro/nano-fluidic devices. Even though such flows are inherently stable when inertial effects are negligible, they often exhibit deviations from laminar profiles impairing quality of polymer
products. My research has demonstrated that these deviations are triggered by high flow sensitivity. To counter this sensitivity I have explicitly accounted for modeling imperfections by quantifying
their influence on transient and asymptotic dynamics of viscoelastic fluids. My work was the first to reveal previously unknown structural similarities between weakly-inertial flows of viscoelastic
fluids and strongly-inertial flows of Newtonian fluids (see figure below for an illustration) and is enhancing the understanding of the early stages of transition to elastic turbulence.
Figure 1: Block diagrams of the frequency response operators that map the wall-normal and spanwise forces to the streamwise velocity fluctuation in streamwise-constant (a) inertialess flows of
viscoelastic fluids; and (b) inertial flows of Newtonian fluids. In Newtonian fluids amplification originates from vortex tilting, i.e. the operator
Relevant Publications | {"url":"http://www.ece.umn.edu/users/mihailo/transition.html","timestamp":"2014-04-21T07:04:36Z","content_type":null,"content_length":"8747","record_id":"<urn:uuid:f3511c78-346a-4063-80cc-06ca183f6bdf>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00613-ip-10-147-4-33.ec2.internal.warc.gz"} |
American Mathematical Society
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. --- G. Hardy (from A Mathematician's Apology,
London 1941)
What Do Mathematicians Do?
Mathematicians are often asked by friends, family, colleagues in other fields, and strangers: "What do mathematicians do?" Here are some resources and facts that may help answer that question.
Many people are familiar with mathematicians in academia, but mathematicians also work in many other fields, including:
The diversity of fields that employ mathematicians is reflected in Mathematical Moments and Mathematics Awareness Month themes.
Mathematicians make it possible to send secure emails and buy things online. Mathematicians are essential to analyze data and design accurate models in fields as diverse as biology and finance.
Mathematicians enabled researchers to complete The Human Genome Project quickly. And because of the prevalence of the computer at work and at play, mathematicians will continue to touch everyone in
modern society.
"Mathematician" is named the best job in America today, according to a 2009 JobsRated.com report.
How Many Mathematicians Are There in the U.S.?
There are over 35,800 individual members of the four leading professional mathematical sciences societies in the U.S.---the AMS, the Association for Women in Mathematics, the Mathematical Association
of America, and the Society for Industrial and Applied Mathematics. Most would call themselves mathematicians; many received their doctoral degrees outside the U.S. There are at least 10,000 more
members of the societies who are graduate students or in other categories, and there are also mathematicians who are not members of any of these societies.
Although they have advanced degrees in mathematics, many of those employed in academia might call themselves professors instead of mathematicians, and similarly, those in industry and government may
not have "mathematician" in their job title. The job title doesn't tell the whole story, however: These people are doing mathematics and are indeed mathematicians. Furthermore, the number of
mathematicians is increasing. The number of new Ph.D.'s in the U.S. has gone up every year since 2002.
Who Are Mathematicians?
Mathematicians are people of all ages and from all over the world who enjoy the challenge of a problem, who see the beauty in a pattern, a shape, a proof, a concept. Some of the best young
mathematicians compete in math olympiads, state and national science fairs, or the fun Who Wants to Be a Mathematician game. Some high school mathematicians go to summer Math Camps to learn more and
work with teams on projects; undergraduates participate in Summer Research Experiences. Many carry on their research and teach at colleges and universities, while others apply their skills in all
kinds of professions. (See Early Career Profiles for job profiles of recent undergraduate math majors, and Sloan Career Cornerstone for profiles of mathematicians at all stages of their careers).
There's probably a bit of the "mathematician" in all of us and we don't even realize it. Keith Devlin poses this idea in his book, The Math Instinct: Why You're a Mathematical Genius (along with
Lobsters, Birds, Cats, and Dogs). In any case, those who are not "mathematicians" can appreciate the subject by reading reviews of books about mathematicians, how mathematicians think, breakthroughs
in mathematics, and current applications at Math in the Media.
Resources on the Mathematics Profession: | {"url":"http://cust-serv@ams.org/profession/career-info/math-work/math-work","timestamp":"2014-04-18T01:09:41Z","content_type":null,"content_length":"42085","record_id":"<urn:uuid:93c259e6-67f9-404a-9cca-da0c0574f1bd>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00435-ip-10-147-4-33.ec2.internal.warc.gz"} |
SETHEO: A High-Performace Theorem Prover
- AUTOMATED DEDUCTION -- CADE-12, VOLUME 814 OF LNAI , 1994
"... PROTEIN (PROver with a Theory Extension INterface) is a PTTPbased first order theorem prover over built-in theories. Besides various standardrefinements known for model elimination, PROTEIN also
offers a variant of model elimination for case-based reasoning and which does not need contrapositives. ..."
Cited by 41 (10 self)
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PROTEIN (PROver with a Theory Extension INterface) is a PTTPbased first order theorem prover over built-in theories. Besides various standardrefinements known for model elimination, PROTEIN also
offers a variant of model elimination for case-based reasoning and which does not need contrapositives.
- Proceedings of the 16-th German AI-Conference (GWAI-92 , 1992
"... this paper, we will show how to extend model elimination with theory reasoning. Technically, theory reasoning means to relieve a calculus from explicit reasoning in some domain (e.g. equality,
partial orders) by taking apart the domain knowledge and treating it by special inference rules. In an impl ..."
Cited by 17 (10 self)
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this paper, we will show how to extend model elimination with theory reasoning. Technically, theory reasoning means to relieve a calculus from explicit reasoning in some domain (e.g. equality,
partial orders) by taking apart the domain knowledge and treating it by special inference rules. In an implementation, this results in a universal "foreground" reasoner that calls a specialized
"background" reasoner for theory reasoning. Theory reasoning comes in two variants (Sti85) : total and
- JOURNAL OF SYMBOLIC COMPUTATION , 1994
"... In this paper, stepwise and nearly stepwise simulation results for a number of first-order proof calculi are presented and an overview is given that illustrates the relations between these
calculi. For this purpose, we modify the consolution calculus in such a way that it can be instantiated to reso ..."
Cited by 14 (10 self)
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In this paper, stepwise and nearly stepwise simulation results for a number of first-order proof calculi are presented and an overview is given that illustrates the relations between these calculi.
For this purpose, we modify the consolution calculus in such a way that it can be instantiated to resolution, tableaux model elimination, a connection method and Loveland's model elimination.
- University of Koblenz, Institute for Computer Science , 1994
"... Theory Reasoning means to build-in certain knowledge about a problem domain into a deduction system or calculus, which is in our case model elimination. Several versions of theory model
elimination (TME) calculi are presented and proven complete: on the one hand we have highly restricted versions of ..."
Cited by 8 (6 self)
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Theory Reasoning means to build-in certain knowledge about a problem domain into a deduction system or calculus, which is in our case model elimination. Several versions of theory model elimination
(TME) calculi are presented and proven complete: on the one hand we have highly restricted versions of total and partial TME. These restrictions allow (1) to keep fewer path literals in extension
steps than in related calculi, and (2) discard proof attempts with multiple occurrences of literals along a path (i.e. regularity holds). On the other hand, we obtain by small modifications to TME
versions which do not need contrapositives (a la Near-Horn Prolog). We show that regularity can be adapted for these versions. The independence of the goal computation rule holds for all variants.
Comparative runtime results for our PTTP-implementations are supplied. 1 Introduction The model elimination calculus (ME calculus) has been developed already in the early days of automated theorem
proving [Lovel... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=3292701","timestamp":"2014-04-24T20:49:45Z","content_type":null,"content_length":"20223","record_id":"<urn:uuid:145289f2-7d3d-452c-81f5-8b4b4b72bb29>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00233-ip-10-147-4-33.ec2.internal.warc.gz"} |
Homework Help
Posted by Robin on Saturday, January 12, 2013 at 7:00pm.
There is also a graph i can't move it.
Building Isochrons and Determining the Ages of Three Rocks
For each of the three rocks listed below follow these steps to determine the rock's age of formation.
1. Prepare an isochron by plotting the points that represent the amounts of isotopes Rb-87 and Sr-87. (The values are RATIOS of parent and daughter to the stable isotope, Sr-86. Mass spectrometers
measure ratios and not absolute amounts.) Plotting Instructions
2. Click "Verify Points" and then use the "regressor" to find the slope and intercept of the line that Best Fits the points. The process, called "Least Squares," will result in the best value for the
slope. Regressor Instructions
3. The age of the rock will be calculated for you. After you minimize the error term, click the "Verify Age" button, and if age is OK, transfer the rock's age to the table on the right. After
completing table, click the "Next" button to receive your CERTIFICATE of completion as a Geochronologist.
Age (Million years)
Granite Scottish Highlands
Gabbro Sudburry, Canada
Gneiss Greenland
• Geology - Anonymous, Monday, March 11, 2013 at 11:43am
• Geology - Anonymous, Monday, March 11, 2013 at 11:44am
• Geology - Anonymous, Friday, March 22, 2013 at 10:02pm
Plot the point, verify them, it will tell you if they are good, and for the age verificacion is quitte the same.
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st: third variable...
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st: third variable...
From Christopher Baum <baum@bc.edu>
To statalist@hsphsun2.harvard.edu
Subject st: third variable...
Date Wed, 26 May 2004 09:44:37 -0400
Clive wrote
My variables of interest are concerned with net votes. The dependent
variable is GENCH = change in a party's vote at the current general
election from the previous one. The key independent variable is MIDCH =
change in a party's midterm election vote from the previous general
election. I hypothesise that MIDCH has a significant effect on GENCH over
time (or, over several GEs). In the models I've fitted so far (using both
- -reg- and -xtgls-) I have found this to be the case for all parties. So
far, so good.
At a recent workshop, however, I was told both of these variables may be
influenced by a third variable (let's call it REGION) and, once controlled
for, there may be no relationship between GENCH and MIDCH. How should one
handle this in a time-series context? -ivreg- would appear to be ruled
out, since this deals with endogeneity. -xtgls- could be used regardless,
but this may be a dangerous option to take given the above. I'm not
entirely sure if simultaneous equation models would help here or not.
What I do know, however, is that the key variables in my models are all
differenced, and time trends have been fitted, which goes part of the way
to solving this problem (if, indeed, I have it). If anybody has any
comments, I'd be very grateful to you.
If region is a categorical variable, and these are xt data, then there are two possibilities: region modifies the constant term (in which some sort of fe or re model should be used) or region modifies the entire relationship (including the coeff on midch). In the latter case a set of interacted dummies would be used in a fe context, or one could use some sort of random-coefficients model (Hildreth-Houck).
I did not respond to the original enquiry since the answer seemed obvious: if there is a third variable that (one suspects) should be in the relationship, and it is measurable, the correct methodology is to include it. After having done so, one may test for its relevance. Techniques such as dealing with proxy issues would only arise if the variable in question is not quantifiable.
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2004-05/msg00924.html","timestamp":"2014-04-17T21:50:30Z","content_type":null,"content_length":"6794","record_id":"<urn:uuid:b6230a26-6808-497e-9cfe-49df7ae009fe>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00552-ip-10-147-4-33.ec2.internal.warc.gz"} |
University of Dayton, Ohio
Careers in Theoretical and Applied Mathematics
Theoretical mathematicians advance mathematical knowledge by developing new principles and recognizing previously unknown relationships between existing principles of mathematics. Although these
workers seek to increase basic knowledge without necessarily considering its practical use, such pure and abstract knowledge has been instrumental in producing or furthering many scientific and
engineering achievements. Many theoretical mathematicians are employed as university faculty, dividing their time between teaching and conducting research.
Applied mathematicians, on the other hand, use theories and techniques, such as mathematical modeling and computational methods, to formulate and solve practical problems in business, government,
engineering, and the physical, life, and social sciences. For example, they may analyze the most efficient way to schedule airline routes between cities, the effects and safety of new drugs, the
aerodynamic characteristics of an experimental automobile, or the cost-effectiveness of alternative manufacturing processes. Applied mathematicians working in industrial research and development may
develop or enhance mathematical methods when solving a difficult problem. Some mathematicians, called cryptanalysts, analyze and decipher encryption systems – codes – designed to transmit military,
political, financial, or law enforcement-related information. Applied mathematicians start with a practical problem, envision its separate elements, and then reduce the elements to mathematical
variables. They often use computers to analyze relationships among the variables and solve complex problems by developing models with alternative solutions.
Individuals with titles other than mathematician do much of the work in applied mathematics. In fact, because mathematics is the foundation on which so many other academic disciplines are built, the
number of workers using mathematical techniques is much greater than the number formally called mathematicians. For example, engineers, computer scientists, physicists, and economists are among those
who use mathematics extensively. Some professionals, including statisticians, actuaries, and operations research analysts, are actually specialists in a particular branch of mathematics. Applied
mathematicians are frequently required to collaborate with other workers in their organizations to find common solutions to problems. | {"url":"http://www.udayton.edu/artssciences/mathematics/careers/index.php","timestamp":"2014-04-17T12:56:58Z","content_type":null,"content_length":"12895","record_id":"<urn:uuid:56c661b5-88d4-4a5c-8609-3f9f9a6cc418>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00060-ip-10-147-4-33.ec2.internal.warc.gz"} |
intersection theory
up vote 2 down vote favorite
Assume that $X$ is a smooth 3-fold (over $\mathbb C$). Let $V$ be a smooth divisor on $X$ and let $S_1,S_2$ be prime divisors on $X$. Assume that given a curve $C$ on $X$ not contained in $V$, then
$a:=C\cdot V>0$ (intersection product of C and V) and assume that the multiplicity of $S_i$ along the generic point of C is $b>0$.
My question is: how to bound (from below) the intersection multiplicity of the free components of $S_{1|V}$ and $S_{2|V}$ at a given point of $C\cap V$ (here $S_{i|V}$ is the restriction of $S_i$ on
Since it is a local problem, we may assume that $C\cap V$ is a single point $p\in V$. If we assume that $S_1,S_2$ and $T$ meet exactly at $p$, then I believe that the answer is easy, as I would
expect that $$I_p(V;S_{1|V},S_{2|V})=I_p(X;S_1,S_2,V)\ge b^2 C\cdot V=a\cdot b^2$$
Assuming that this is correct, my question is, is there a way to get a similar bound assuming that $S_1$ and $S_2$ meet in a curve contained in $V$?
In other words, I am assuming that $S_{i|V}=B_i\cup B$ where $B$,$B_1$ and $B_2$ are curves on $V$ (not necessarily reduced or irreducible) such that $B_1\cap B_2$ is a finite set of points (we may
assume $B_1\cap B_2=p$).
Besides the obvious lower bounds, I am expecting that $$I_p(V;B_1,B_2)\ge a\cdot (b-c)^2$$ where $c$ is the multiplicity of the curve $B$ at the point $p$. This is the answer we get assuming that
$S_1$ and $S_2$ have a common component $T$ such that $T_{|V}=B$. So I guess my question is: is the inequality true in general?
intersection-theory ag.algebraic-geometry
What's the relationship of $C$ to the other objects? Also, when you first use $T$ it is not yet defined. Are we supposed to assume the later definition for $T$? – Sándor Kovács Mar 16 '11 at 23:32
Both $B$ and $C$ are curves contained in the intersection of $S_1$ and $S_2$, but $B$ is contained in $V$ and $C$ is not. Moreover, if $T$ is a (not nec. reduced) surface contained in the
intersection of $S_1$ and $S_2$ and such that $T_{|V}=B$ then I believe that the inequality I wrote is correct. In general, $T$ might not exist (as $S_1$ and $S_2$ might be irreducible). I was
wondering if the inequality still holds. I am not sure if this answer your questions. – user8229 Mar 17 '11 at 8:15
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Calculate average of multiple matrices by reading an m.file
I'm having difficulty in calculating the mean of multiple matrices. I know how to calculate the mean, I have tried to use mean(match,3) in 3 dimension , but it does not give me the mean of all my
matrices by columns If I calculate my mean in the while loop, it creates a mean for every matrix, and if I do it outside the while loop, it does it only for the last matrix.
For example while(reading file) ...
if(some conditions) continue
match= strcmp('Hello', out)//compares script with the word hello and outputs a logical array of 1s and 0s
match=[1 0 0 0 0 0] match=[0 0 1 0 0 0] match=[0 0 1 0 0 0]
I want the mean to be M=[1/3 0 2/3 0 0 0]
Do I need to use a for loop? Because it seems to be a much longer process to use a for loop to get the mean, but I do not see how else I could do this.
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1 Answer
Accepted answer
k = 0;
while (reading file) ...
k = k+1;
match(k,:) = strcmp('Hello', out)
M = mean(match)
10 Comments
Show 7 older comments
No, that is not your output. Your output is like this:
match=[1 0 0]
match=[0 1 1]
match=[1 1 0 1]
And yes, thats very possible the reason for your difficulties. To fill the empty space with zeros, you could use:
ml = strcmp('Hello', out); % match for the line
match(i,1:length(ml)) = ml; %#ok<SAGROW>
Sorry, I meant this is what I want my output to be like. Thank you so much, it works perfectly now! I have been trying to get this code to work for some time, thanks for taking the time to help me
out, much appreciated. :D
You're welcome. | {"url":"http://www.mathworks.com/matlabcentral/answers/69411","timestamp":"2014-04-21T12:30:10Z","content_type":null,"content_length":"36731","record_id":"<urn:uuid:a54285b9-9fbf-44ae-8335-16c930a5e7bf>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00246-ip-10-147-4-33.ec2.internal.warc.gz"} |
Australian Mathematical Society
The George Szekeres Medal of the Australian Mathematical Society
The George Szekeres Medal is awarded by the Australian Mathematical Society for work done over a fifteen year period. Most of the work must have been carried out in Australia. The first award was
made in 2002.
2002 Ian Sloan
2002 Alf van der Poorten
2004 Robert (Bob) Anderssen
2006 Anthony J Guttmann
2008 Hyam Rubinstein
MacTutor links:
Australian Mathematical Society
Australian Mathematical Society Prizes, etc:
Australian Mathematical Society Medal
Australian Mathematical Society Szekeres Medal
Australian Mathematical Society Mahler Lectureship
Other Web site:
Australian Mathematical Society Web site
JOC/EFR August 2009
The URL of this page is: | {"url":"http://www-gap.dcs.st-and.ac.uk/~history/Societies/AuMSSzekeresMedal.html","timestamp":"2014-04-18T00:19:08Z","content_type":null,"content_length":"3681","record_id":"<urn:uuid:d055f352-bebb-4dfc-beed-bfc0bd34a59b>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00599-ip-10-147-4-33.ec2.internal.warc.gz"} |
Nonlinear Eigenvalue Problems of Schr¨odinger Type Admitting Eigenfunctions with
"... We introduce a new variational method in order to derive results concerning existence and nodal properties of solutions to superlinear equations, and we focus on applications to the equation \
Gamma\Deltau = h(x; u) u 2 L ); ru 2 L ); N 3 where h is a Caratheodory function which is od ..."
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We introduce a new variational method in order to derive results concerning existence and nodal properties of solutions to superlinear equations, and we focus on applications to the equation \Gamma\
Deltau = h(x; u) u 2 L ); ru 2 L ); N 3 where h is a Caratheodory function which is odd in u. In the particular case where h is radially symmetric, we prove, for given n 2 N, the existence of a
solution having precisely n nodal domains, whereas some results also pertain to a nonsymmetric nonlinearity.
, 2009
"... It is well known that the concept of a point charge interacting with the electromagnetic (EM) field has a problem. To address that problem we introduce the concept of wave-corpuscle to describe
spinless elementary charges interacting with the classical EM field. Every charge interacts only with the ..."
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It is well known that the concept of a point charge interacting with the electromagnetic (EM) field has a problem. To address that problem we introduce the concept of wave-corpuscle to describe
spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over the 4-dimensional space time
continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient- a nonlinear self-interaction term providing for
a cohesive force assigned to every charge. An ideal wave-corpuscle is an exact solution to the Euler-Lagrange equations describing both free and accelerated motions. It carries explicitly features of
a point charge and the de Broglie wave. A system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton equations of
motion for point charges interacting with the Lorentz forces. In this regime the nonlinearities are ”stealthy ” and don’t show explicitly anywhere, but they provide for the binding forces that keep | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1884765","timestamp":"2014-04-24T14:20:04Z","content_type":null,"content_length":"14981","record_id":"<urn:uuid:dbc207e6-a8ab-4d85-9d79-f15c61e27d25>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00540-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fourier series of functions with points of discontinuity
What I was thinking is that in the L2 space there is an equivalence relation such that if the Lebesgue integral of the diference is 0, then they are equivalent. However, the functions in the
trigonometric basis of Fourier are contained in C[a,b], and because C[a,b] is closed under addition, the infinite linear combination with real coefficients will also be contained in C[a,b]. So the
Fourier series will converge to the continuous equivalent function in the L2 space. Is that right? | {"url":"http://www.physicsforums.com/showthread.php?p=4165937","timestamp":"2014-04-16T19:08:01Z","content_type":null,"content_length":"47421","record_id":"<urn:uuid:899d1670-5abc-44b6-b98c-eab750285a85>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00481-ip-10-147-4-33.ec2.internal.warc.gz"} |
Morita equivalence
Algebraic theories
Algebras and modules
Higher algebras
Model category presentations
Geometry on formal duals of algebras
Equality and Equivalence
• equality (definitional?, propositional, computational, judgemental, extensional, intensional, decidable)
• isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
• Examples.
Classical Morita theorem
Given rings $R$ and $S$, the following properties are equivalent
1. The categories of left $S$-modules and left $R$-modules are equivalent;
2. The categories of right $S$-modules and right $R$-modules are equivalent;
3. There are bimodules ${}_R M_S$ and ${}_S N_R$ such that $\otimes_R M$ and $\otimes_S N$ form an adjoint equivalence between the category of right $S$- and the category of right $R$-modules;
4. The ring $R$ is isomorphic to the endomorphism ring of a generator in the category of left (or right) $S$-modules;
5. The ring $S$ is isomorphic to the endomorphism ring of a generator in the category of left (or right) $R$-modules.
Dmitri Pavlov: Tsit-Yuen Lam in his book “Lectures on modules and rings” on pages 488 and 489 states the Morita equivalence theorem using progenerators (i.e., finitely generated projective
generators) instead of just generators. Are these two versions equivalent?
Dmitri Pavlov: I would like to state the Morita equivalence theorem as a 2-equivalence between two bicategories: The bicategory of rings, bimodules and their intertwiners and the bicategory of
abelian categories that are equivalent to the category of modules over some ring (i.e., abelian categories that have all small coproducts and a compact projective generator), Eilenberg-Watts functors
between these categories (i.e., right exact functors that commute with direct sums) and natural transformations. Is it possible to do this and what is the precise statement then?
In algebra
Two rings are Morita equivalent if the equivalent statements in Morita theorem above are true. A Morita equivalence is a weakly invertible 1-cell in the bicategory $\mathrm{Rng}$ of rings, bimodules
and morphisms of bimodules.
In homotopy theory
In any homotopy theory framework a Morita equivalence between objects $C$ and $D$ is a span
$C \lt \stackrel{\simeq}{\leftarrow} \hat C \stackrel{\simeq}{\to} \gt D$
where both legs are acyclic fibrations.
In particular, if the ambient homotopical category is a category of fibrant objects, then the factorization lemma (see there) ensures that every weak equivalence can be factored as a span of acyclic
fibrations as above.
Important fibrant objects are in particular infinity-groupoids (for instance Kan complexes are fibrant in the standard model structure on simplicial sets and omega-groupoids are fibrant with respect
to the Brown-Golasinski folk model structure). And indeed, Morita equivalences play an important role in the theory of groupoids with extra structure:
In Lie groupoid theory
A Morita morphism equivalence of Lie groupoids is an anafunctor that is invertible, equivalently an invertible Hilsum-Skandalis morphism/bibundle.
Lie groupoids up to Morita equivalence are equivalent to differentiable stacks. This relation between Lie groupoids and their stacks of torsors is analogous to the relation between algebras and their
categories of modules, which is probably the reason for the choice of terminology. | {"url":"http://ncatlab.org/nlab/show/Morita+equivalence","timestamp":"2014-04-17T12:37:09Z","content_type":null,"content_length":"42606","record_id":"<urn:uuid:724e6402-dba0-427d-9ce5-5d30825a0908>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00134-ip-10-147-4-33.ec2.internal.warc.gz"} |
function g(x)=(1-2x)^2(x-3)
March 23rd 2010, 09:42 PM #1
function g(x)=(1-2x)^2(x-3)
I need to state whether the function is odd or even and i said odd since there is a degree of 3, and the x and y intercepts i solved for and ended up with y=-2 and x=3,1/2. now i need to find the
first and second derivative, and i was wondering if i should use the chain rule and product rule ?
I tried solving for the first derivative and i used the product rule and the chain rule
g'(x)=(1-2x)^2 (1) + (2(-2)(1-2x)
=(1-2x)^2 + -4(1-2x)
=2x^2 +8x -3
I need to state whether the function is odd or even and i said odd since there is a degree of 3, and the x and y intercepts i solved for and ended up with y=-2 and x=3,1/2. now i need to find the
first and second derivative, and i was wondering if i should use the chain rule and product rule ?
It is even if and only if g(x)=g(-x) and odd if and only if g(x)=-g(-x).
So does g(1)=g(-1)?
or does g(1)=-g(-1)?
i'll solve it like that then, i didnt think of that, thats what you get for working on these problems at 2am. do my derivatives look right though?
how should i approach this function in order to get the derivatives? Since its factored im not really sure what i'm suppose to do.
Product rule followed by chain rule:
$\frac{d}{dx}\left[(1-2x)^2(x-3)\right]=\left[\frac{d}{dx}(1-2x)^2 \right] (x-3) + (1-2x)^2\left[ \frac{d}{dx}(x-3) \right]$
March 23rd 2010, 09:55 PM #2
March 23rd 2010, 11:22 PM #3
Grand Panjandrum
Nov 2005
March 24th 2010, 07:08 AM #4
March 24th 2010, 07:15 AM #5
Grand Panjandrum
Nov 2005
March 24th 2010, 07:16 AM #6
March 24th 2010, 07:30 AM #7
Grand Panjandrum
Nov 2005 | {"url":"http://mathhelpforum.com/pre-calculus/135353-function-g-x-1-2x-2-x-3-a.html","timestamp":"2014-04-18T16:24:50Z","content_type":null,"content_length":"49483","record_id":"<urn:uuid:2242aa34-4723-4680-9f4b-8a9206dba0d5>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00590-ip-10-147-4-33.ec2.internal.warc.gz"} |
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PRATHYUSHA BSN 7 Months ago I am very grateful to m4maths. It is a great site i have accidentally logged on when i was searching for an answer for a tricky maths puzzle. It heped me greatly and i am
very proud to say that I have cracked the written test of tech-mahindra with the help of this site. Thankyou sooo much to the admins of this site and also to all members who solve any tricky puzzle
very easily making people like us to be successful. Thanks a lotttt
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n u'll surely succeed :)
Sandhya Pallapu 1 year ago Hai friends! this site is very helpful....i prepared for TCS campus placements from this site...and today I m proud to say that I m part of TCS family now.....dis site
helped me a lot in achieving this...thanks to M4MATHS!
vivek singh 2 years ago I cracked my first campus TCS in November 2011...i convey my heartly thanks to all the members of m4maths community who directly or indirectly helped me to get through
TCS......special thanks to admin for creating such a superb community
Manish Raj 2 years ago this is important site for any one ,it changes my life...today i am part of tcs only because of M4ATHS.PUZZLE
Asif Neyaz 2 years ago Thanku M4maths..due to u only, imade to TCS :D test on sep 15.
Harini Reddy 2 years ago Big thanks to m4maths.com. I cracked TCS..The solutions given were very helpful!!!
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help of m4maths.......it's just an osum site as well as a sure shot guide to TCS apti......Pls let me know wt can be asked in the interview by MBA students.
Anusha Alva 3 years ago a big thnks to this site...got placed in TCS!!!!!!
Oindrila Majumder 3 years ago thanks a lot m4math.. placed in TCS
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". If in coded language
MAMTA = 8
SAUMYA = 80
MONIKA = 35
RASHMI = ?
Options 1) 16
2) 26
3) 42
4) 38
5) 14
6) 81
7) 39
8) 14
9) 27
10)None of these"
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"waht is the sum of the product and quotient of 8 and 8."
UnsolvedAsked In: Bank Exam
"mathematics is very useful for us.this is truely motivation subject.i really appreciate." Aman tiwari
"If there is a God, he's a great mathematician." Paul Dirac
"my math textbook is my role model. no matter how many problems it has it never sucide's." prabhpreet singh
"Language is remarkable, except under the extreme constraints of mathematics and logic, it never can talk only about what it's supposed to talk about but is always spreading around. " Raju Sharma
"Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning." John Locke
"You may be an engineer if your idea of good interpersonal communication means getting the decimal point in the right place." Unknown
"find the output is 24 u have to use only 8,8,3,3 except then that u cant even use 1 or etc. and u can use +.-,/,* in any times" UnsolvedAsked In: HR Interview
". If in coded language MAMTA = 8 SAUMYA = 80 MONIKA = 35 then RASHMI = ? Options 1) 16 2) 26 3) 42 4) 38 5) 14 6) 81 7) 39 8) 14 9) 27 10)None of these" UnsolvedAsked In: M4maths
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Universal constructions
Local presentation
$(\infty,1)Cat$ is the (∞,2)-category of all small (∞,1)-categories.
Its full subcategory on ∞-groupoids is ∞Grpd.
The $(\infty,2)$-category
As an $SSet$-category
One incarnation of (∞,2)-categories is given by quasi-category-enriched categories (see there for details). As such $(\infty,1)Cat$ is the full SSet-enriched subcategory of SSet on those simplicial
sets that are quasi-categories. By the fact described at (∞,1)-category of (∞,1)-functors this is indeed a quasi-category-enriched category.
As an enriched model category
The model category presenting this (∞,2)-category is the Joyal model structure for quasi-categories $sSet_{Joyal}$. Its full sSet-subcategory is the quasi-category enriched category of
quasi-categories from above.
The $(\infty,1)$-category
Sometimes it is useful to consider inside the full $(\infty,2)$-catgeory of $(\infty,1)$-categories just the maximal $(\infty,1)$-category and discarding all non-invertible 2-morphisms. This is the
(∞,1)-category of (∞,1)-categories.
As an $SSet$-category
As an SSet-enriched category the (∞,1)-category of (∞,1)-categories is obtained from the quasi-category-enriched version by picking in each hom-object simplicial set of $(\infty,1)Cat$ the maximal
Kan complex.
As an enriched model category
One model category structure presenting this is the model structure on marked simplicial sets. As a plain model category this is Quillen equivalent to $sSet_{Joyal}$, but as an enriched model
category it is $sSet_{Quillen}$ enriched, so that its full SSet-subcategory on fibrant-cofibrant objects presents the $(\infty,1)$-category of $(\infty,1)$-categories.
Limits and colimits in $(\infty,1)$Cat
Limits and colimits over a (∞,1)-functor with values in $(\infty,1)Cat$ may be reformulation in terms of the universal fibration of (infinity,1)-categories $Z \to \infty Grpd^{op}$
Then let $X$ be any (∞,1)-category and
$F : X \to (\infty,1)Cat$
an (∞,1)-functor. Recall that the coCartesian fibration $E_F \to X$ classified by $F$ is the pullback of the universal fibration of (∞,1)-categories $Z$ along F:
$\array{ E_F &\to& Z \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& (\infty,1)Cat }$
Let the assumptions be as above. Then:
• The colimit of $F$ is equivalent to $E_F$:
$E_F \simeq colim F$
• The limit of $F$ is equivalent to the (infinity,1)-category of cartesian section of $E_F \to X$
$\Gamma_X(E_F) \simeq lim F \,.$
The full subcategory of the (∞,1)-category of (∞,1)-categories $Func((\infty,1)Cat, (\infty,1)Cat)$ on those (∞,1)-functors that are equivalences is equivalent to $\{Id, op\}$: it contains only the
identity functor and the one that sends an $(\infty,1)$-category to its opposite (infinity,1)-category.
This is due to
• Bertrand Toen, Vers une axiomatisation de la théorie des catégories supérieures , K-theory 34 (2005), no. 3, 233-263.
It appears as HTT, theorem 5.2.9.1 (arxiv v4+ only)
First of all the statement is true for the ordinary category of posets. This is prop. 5.2.9.14.
From this the statement is deduced for $(\infty,1)$ -categories by observing that posets are characterized by the fact that two parallel functors into them that are objectwise equivalent are already
equivalent, prop. 5.2.9.11, which means that posets $C$ are characterized by the fact that
$\pi_0 (\infty,1)Cat(D,C) \to Hom_{Set}( \pi_0 (\infty,1)Cat(*,D) , \pi_0 (\infty,1)Cat(*,C) )$
is an injection for all $D \in (\infty,1)Cat$.
This is preserved under automorphisms of $(\infty,1)Cat$, hence any such automorphism preserves posets, hence restricts to an automorphism of the category of posets, hence must be either the identity
or $(-)^{op}$ there, by the above statement for posets.
Now finally the main point of the proof is to see that the linear posets $\Delta \subset (\infty,1)Cat$ are dense in $(\infty,1)Cat$, i.e. that the identity transformation of the inclusion functor $\
Delta \hookrightarrow (\infty,1)Cat$ exhibits $Id_{(\infty,1)Cat}$ as the left Kan extension
$\array{ \Delta &\hookrightarrow& (\infty,1)Cat \\ \downarrow & earrow_{Lan = \mathrlap{Id}} \\ (\infty,1)Cat } \,.$
The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third
table) and of $\mathcal{O}$-monoidal (∞,1)-categories (fourth table). | {"url":"http://ncatlab.org/nlab/show/(infinity%2C1)Cat","timestamp":"2014-04-25T03:32:33Z","content_type":null,"content_length":"55284","record_id":"<urn:uuid:384e93ff-c554-4a7b-bf7b-5d1631013f33>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00385-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> Categorical outcomes
Anonymous posted on Thursday, March 13, 2003 - 7:32 am
I have a data with a binary outcome and a binary mediating variable. My exogenous variables are a mixture of latent and observed variables. Can I use m-plus version 2 to analyze this data?
Linda K. Muthen posted on Thursday, March 13, 2003 - 9:09 am
Yes, it may be necessary to use our THETA parameterization for this model. If so, when you run the program, it will tell you.
Angela posted on Monday, April 21, 2003 - 11:41 am
I have estimated a simple path model (no latent) with two measured continuous variables and four measured categorical variables. A reviewer is asking me to specify what type of correlations
(tetrachoric, polychoric, Pearson) I am reporting in the correlation matrix. There's no indication on the Mplus output or in the manual what type of correlations are repoted in the " CORRELATION
MATRIX (WITH VARIANCES ON THE DIAGONAL)" section. Could you clear this up for me? Thank you!
Linda K. Muthen posted on Monday, April 21, 2003 - 12:50 pm
The type of correlation is determined by the scale of the two variables. If both are continuous, the correlation is a Pearson product moment correlation. If both are binary, it is a tetrachoric
correlation. If both are ordered polytomous, it is a polychoric correlation. If one is dichotomous and one is continuous, it is a biserial correlation. If one is ordered polytomous and one is
continuous, it is a polyserial correlation. I hope this answers your question.
Angela posted on Monday, April 21, 2003 - 1:34 pm
You answered my quesiton perfectly. I was writing my responses to the reviewer exactly the way you described them, but wanted to know for sure I was saying the correct thing. Thank you so much!!
One more quick question - is there any way to get Mplus to print statistical significance levels for these correlations?
Linda K. Muthen posted on Monday, April 21, 2003 - 4:29 pm
If you ask for TYPE=BASIC, you will get the correlations and also the standard deviations for each correlation. If you divide the correlation by its standard error, this is like a z-test.
Greg posted on Wednesday, July 30, 2003 - 9:12 am
I have a model (3 exogenous and 3 endogenous latent constructs) with a binary categorical outcome. The other variables are partly ordinal, partly continuous.
As my data is skewed and the sample size is large enough (n=1000) I would like to use the WLS estimator. Do I have to apply a biserial correlation (as computed in PRELIS) or do I have to compute
point-biserial correlations. If the latter is true, is it possible to compute the point-biserial correlations and the corresponding weight matrix with Mplus?
Or do I have to use latent class analysis instead?
Linda K. Muthen posted on Wednesday, July 30, 2003 - 9:23 am
You need to have raw data. Mplus will compute the appropriate correlations for the scale of the variable. For example, with a dichotomous variable and a continuous variable, a biserial correlation is
Anonymous posted on Friday, August 15, 2003 - 1:13 pm
I have two dichotomous outcome measures (Y1 and Y2), that are supposed to be correlated with each other. The two outcomes were modeled simultaneously using Mplus 2.14, being predicted by the same set
of Xs. With "TYPE=GENERAL," Mplus provides probit regression coefficients for the two outcomes. However, the model fit statistics/indexes are confusing: Chi-Square=51.841, d.f.=13, P-Value=0.0000;
CFI=1.000, TLI=1.000; RMSEA= 0.000; and WRMR= 0.000. When some Xs were removed from the model, CFI or TLI had a value greater than 1.0 sometimes. What appropriate model fit statistics/indexes should
I use for the simultaneous probit model?
In addition, Mplus provides the correlation between Y1* and Y2*, which are the underlying latent variables of Y1 and Y2, respectively. How should I interpret this correlation given the fact that
probit model has no residual term? Setting the correlation to 0.0 didn't change the coefficient estimates and their standard errors, but changed chi-square statistic and other fit indexes (e.g., CFI
changed from 1.000 to 0.234 TLI changed from 1.000 to -7.424(why negative TLI)). Was the correlation between Y1* and Y2* factored in the modeling?
Your help will be highly appreciated!
bmuthen posted on Saturday, August 16, 2003 - 4:51 pm
Please send the output from both of the models you mention to support@statmodel.com so that your full results can be studied.
Regarding the residual correlation between y1* and y2*, the probit model does have a residual term. Although the variance of the residual is standardized to one, it is possible to estimate residual
correlations with multivariate outcomes. If residual correlations are given in the Model Results section, they are estimated (see also Tech1 regarding which parameters are estimated).
Anonymous posted on Thursday, August 21, 2003 - 11:23 am
I am doing path analysis with several categorical outcome. the dependent variable and the mediating variable are both binary. how do I specify the 'THETA' parametriztion. Is ther an example?
bmuthen posted on Thursday, August 21, 2003 - 4:10 pm
Here is an example:
TITLE: this is an example of a Monte Carlo simulation study for a path analysis
DATA: FILE = firstcat.dat;
NAMES = y1-y3 x1-x3;
CATEGORICAL = y1-y3;
y1 y2 on x1-x3*.5;
y3 on y1-y2*.5 x1-x3*.3;
davood posted on Thursday, December 02, 2004 - 9:34 am
Dear Dr. Muthen,
I am doing a two group CFA using categorical indicators. The mplus generated this errors:
*** ERROR
Group 5 does not contain all values of categorical variable: DADCARE1
group 5 is puertoricans. When I ran the model for just puerto ricans I did not face this error. Also, I ran the frequency and probed the DADCARE1 for group 5. Here is the result:
Value Frequency
-999 156
1.00 2
2.00 2
3.00 11
4.00 36
5.00 191
Total 398
Linda K. Muthen posted on Thursday, December 02, 2004 - 10:46 am
It is likely because of listwise deletion, you don't have 398 observations and without all 398, you may not have all values on DADCARE1. Or you may be reading your data wrong in Mplus. If you can't
figure it out, send your output and data to support@statmodel.com.
empty logistic model posted on Friday, February 24, 2006 - 6:24 am
Dear Dr. Muthen,
I am having trouble replicating results given on an ucla website which uses mplus for examples from the Multilevel Book by snijders/bosker, chapter 14,
(it provides mplus syntax and data fo a multilevel binary logistic regression)
The website provides as a result:
COHAB 0.032 0.019 1.635
My own calculations (same data same syntax):
several serious problems and
COHAB 0.000
own data als yielded
several serious problems and
XYZ 0.000
Is there a problem in the mplus syntax for the "empty logistic model" ?:
Title: multilevel logistic regression
File is ch14_1.dat ;
Names are
respnr ltbm eltwm sex age edu relserv class reg cohab;
Missing are all (-9999) ;
usevar are cohab reg;
categorical is cohab;
cluster is reg;
Type = twolevel random ;
Thanks in advance
Linda K. Muthen posted on Friday, February 24, 2006 - 10:41 am
What version of Mplus are you using. What version did they use. You can see this by looking at their output if they provide the full output.
empty logistic model posted on Saturday, February 25, 2006 - 12:53 am
Thanks for the quick response,
ucla version is not given.
How would you do the model in mplus syntax?
(binary outcome, only the level 2 cluster as explanatory variable)
Linda K. Muthen posted on Saturday, February 25, 2006 - 7:39 am
The Mplus input is on the UCLA website. If you cannot get this to run, you need to send the input, data, output, and your license number to support@statmodel.com.
Nyankomo Wambura Marwa posted on Monday, July 31, 2006 - 3:41 am
Dear Prof Muthen
I am fitting latent class models in MPLUs.I have four categorical manifest variables and i am interested to estimate the true status of the disease given the four diagnostic test.There is high
likelihood that my manifest variables are correlated .Intitutively one latent class with two levels will be ideal.But when i try it the model does not fit.Only when i go up to level four is when the
model fits better.
My questions are follows.
1.How do we detect the dependence among the manifest variables in MPLUs
2.Given that there is dependence how to model it.I saw in in MPlus users guide for version 1 of 2000 page 13 that WITH statement is used for correlational among continuous varibles but is not
applicable to categorical variable.
Thanks in advance for your help
Linda K. Muthen posted on Monday, July 31, 2006 - 10:27 am
The manifest variables should correlate. Their correlations are assumed to be zero within each class, but the variables will correlate when the classes are mixed. You can look at TECH10 to check for
dependence among the manifest variables within calss. The WITH statement is used with catgorical observed variables in some models. With latent class analysis, you need to put a factor behind the two
indicators to specify a correlation. See Example 7.16.
Nyankomo Wambura Marwa posted on Monday, July 31, 2006 - 11:33 am
Dear Linda
Thanks for your quick response.But i have one more question.In your response u refered me to see Example 7.6 on how to use WITH statement for categorical variables.I am not clear it is example 7.6
from the user's manual or from where??.Just to remind you i have user's guide for MPLUS version 1.
See your response attached below for reference
Hoping to here from u soon.
With regards
Nyankomo marwa
Nyankomo Wambura Marwa posted on Monday, July 31, 2006 - 12:00 pm
Dear Linda
Here is my syntax ,but i am not quite sure how to include with statement in modelling dependence among A B D H.
I tried to use
A B WITH D H but it does not work.
Thanks once again for your estimeed support.
File is D:\mp.txt;
Names are A B D H count;
usevariables are A B D H count;
FREQWEIGHT is count ;
categorical are A B D H;
Missing are all (-9999) ;
classes = cl(2);
Type = mixture;
Linda K. Muthen posted on Monday, July 31, 2006 - 2:53 pm
I always refer to the most recent user's guide which is posted on the website. See Example 7.6 in this user's guide. Version 1 is very old. Your research will be improved by updating to Version 4.1.
There are many changes that have occurred since Version 1 that will help you in mixture modeling particularly the addition of random starts.
Peter Croy posted on Monday, May 14, 2007 - 7:24 am
I'm just starting to model a categorical outcome variable (a binary y/n DV) and have generated a model as the first step toward a chi sq difference test per instructions on page 314 of the guide.
Using the savedata: difftest is deriv.dat command saves a dat file which contains an error message, namely: The input file does not contain valid commands. What does this suggest?
Linda K. Muthen posted on Monday, May 14, 2007 - 7:54 am
It is difficult to say without more information. If you send your input, data, output, and license number to support@statmodel.com, I can take a look at it.
Yu Kyoum Kim posted on Friday, June 08, 2007 - 12:55 pm
Dear Dr. Muthen,
I am a new user of Mplus and I really like Mplus.
I have 4 continuous latent variables (IVs) which are all measured by continuous indicators.
However, dependent variable is one observed binary variable. I understand Mplus can incorporate this relationship but how?
1. What correlation matrix is used? Does matrix analyzed include both Pearson and biserial correlation even though I just have one binary variable in structural part?
2. What estimation method should I use?
3. The relationship in measurement part (between continuous latent variables and continuous indicators) is described by linear regression equation separately from structural part (between 4
continuous latent variable and one binary observed dependent variable) or all relationship is described by probit or logistic regression equation?
I am looking forward to hearing from you soon.
I truly appreciate your time and help.
Linda K. Muthen posted on Friday, June 08, 2007 - 2:57 pm
Let's say you have four factors, f1 through f4, and one dependent binary variable, u1. You would specify the model as:
f1 BY ...
f2 BY ...
f3 BY ...
f4 BY ....
u1 ON f1-f4;
1. If you are using weighted least squares, the correlation depends on the scale of the two dependent variables.
2. You can use weighted least squares which is the default or maximum likelihood.
3. The type of regression coefficient is determined by the scale of the dependent variable. Factor indicators are dependent variables. When they are continuous, the regression coefficient is a linear
regression coefficient. The regression of the categorical dependent variable on the continuous latent variables is probit or logistic depending on the estimator and link. It is probit for weighted
least squares. It is a logistic regression coefficient for maximum likelihood using the default logit link. It is probit using the probit link.
Omar Abdellatif posted on Monday, February 08, 2010 - 10:48 pm
Dear Dr. Muthen,
Kindly I'd like to know which product of Mplus 5.2 to purchase:
Mplus Base Program
Mplus Base Program and Mixture Add-On
Mplus Base Program and Multilevel Add-On
Mplus Base Program and Combination Add-On
to perform latent class analysis with 4 manfiset variables (diagnostic tests) in 400 animals to estimate the true disease status & estimate the sensitivity & specificity of each test.
My study is similar to Nyankomo Wambura Marwa's study posted in this wall.
also I'd like to ask if Mplus 3 can perform this type of test.
with Best regards
Linda K. Muthen posted on Tuesday, February 09, 2010 - 9:54 am
I think the Mplus Base Program and Mixture Add-On would be the correct choice.
harvey brewner posted on Monday, March 07, 2011 - 7:15 am
i conducted a efa and found two items (e6 and e12) that loaded strongly on the intended factor (E) as well as another factor (N). i then conducted a cfa without the cross-loadings and the model fit
very well. i then ran another cfa allowing the first cross-loading item (e6) to cross-load and the model was not identified. the same was true when i ran another cfa allowing e12 instead of e6 to
cross-load -- it was also not identified. in the first cfa, i.e. with no cross-loadings, there was a extremely high MI (e.g. >200) for e6 with e12.
1.) is there a connection between the strong cross-loadings of e6 and e12 in the efa and the strong correlation (i.e., large MI value) between e6 and e12 in the first cfa (i.e., where no
cross-loadings were allowed)? for example, does the strong correlation in the cfa manifest itself as a large cross-loading in the efa?
2.)how can e6 and e12 have large cross-loadings on another factor (N) in the efa but the models were not identified in the cfa when the items were allowed to cross-load, separately, on E and N?
any thoughts on either question?
Bengt O. Muthen posted on Monday, March 07, 2011 - 6:09 pm
1) If e6 and e12 have something in common not covered by your E and N factor, then EFA covers the extra correlation by a cross-loading because EFA does not allow residual correlations. You can look
at the MIs for the EFA which probably point to a res corr. You can use ESEM to correlate residuals. The CFA tells you that correlating the residuals is better - that again points to the two items
both being influenced by another factor.
2) You have to send the model to support for us to see the specific features of the CFA.
mpduser1 posted on Wednesday, August 17, 2011 - 5:07 pm
I want to verify that I'm interpreting Mplus output and Chapter 14 of the User's Guide correctly.
I'm estimating model where I have a dichotomous outcome, Y (0 = no, 1 = yes), a dichotomous predictor, X, and a latent class variable that I'm specifying as a predictor, L. L has three classes.
I obtain "logistic regression intercepts" for Y from the latent class portion of my model as follows:
Class 1: Y$1 = 1.32
Class 2: Y$1 = .80
Class 3: Y$1 = .20.
For X = 0, am I correct that this indicates that 52% of respondents who respond "yes" to Y are in Class 1; 30.9% who respond "yes" to Y are in Class 2; 16.99% who respond "yes" to Y are in Class 3?
I obtain this via: EXP(1.32) = 3.74; EXP(.80) = 2.22; EXP(.20) = 1.22, with proportion in Class 1 as: 3.74 / (3.74 + 2.22 + 1.22).
Thank you.
Bengt O. Muthen posted on Wednesday, August 17, 2011 - 6:16 pm
It is simpler than that (you are half-ways slipping into multinomial regression). With a binary Y you have:
P(Y=1 | Class k) = 1/[1+exp(-logit)],
where the logit = -threshold. So, for example for Class 1, you have exp(-logit) = exp(+threshold) = exp(+1.32) and get P = 0.21.
mpduser1 posted on Wednesday, August 17, 2011 - 6:55 pm
That was actually my original thinking.
In one model, however, I get thresholds along the lines of:
Class 1: 4.197
Class 2: 3.589
Class 3: 3.330.
So as I understand your calculation this would indicate that for respondents for whom X = 0 (approximately) only 1.5%, 2.7%, and 3.4% responded "yes" to Y.
These numbers seem small; especially given that approximately 20% of my total sample responded "yes" to item Y; and Class 1 is 24% of my sample, Class 2 is 47% of my sample, and Class 3 is 29% of my
Am I being thrown off here because I'm only looking at the "X=0 segment" of the sample in this particular application?
Bengt O. Muthen posted on Wednesday, August 17, 2011 - 8:43 pm
Maybe X=0 is below the X mean. And perhaps X also influences L.
mpduser1 posted on Thursday, August 18, 2011 - 4:43 am
If X does influence L, is the model better rendered as:
X --> L --> Y
X --> Y
Would this adjust the otherwise large intercepts?
Bengt O. Muthen posted on Thursday, August 18, 2011 - 7:57 am
That's hard to say - try it - but it will change the class proportions at X=0 (assuming X has an effect on L).
mpduser1 posted on Thursday, August 18, 2011 - 8:25 am
Running the model as a path model doesn't change anything.
In the problematic example I noted above (with the large intercepts), X is actually a vector of covariates. I'll have to reduce the complexity of the model. Seems latent class models of either type
can't be too rich. Not really sure why that is, but you can start to see it in the SEs.
I'll have to go back and build the model piece-wise.
Thank you.
Alfred Mbah posted on Tuesday, September 20, 2011 - 6:24 am
Dear Dr Muthen
How do you assign categorical outcome variables with 2 categories(e.g. u1, u2, u3) to a categorical latent variable (with 2 classes for example) in MPLUS? The Type = mixture command does not seem to
do this.
Linda K. Muthen posted on Tuesday, September 20, 2011 - 10:16 am
See Example 8.12 in particular page 222.
Lisa M. Yarnell posted on Thursday, September 20, 2012 - 5:43 pm
Hi Linda,
If I read in a polychoric correlation matrix for a set of dichotomous indicators (being told to do this by a professor), using TYPE = CORRELATION, is it possible to specify the correlations as
polychoric? If so, how?
Linda K. Muthen posted on Friday, September 21, 2012 - 8:11 am
You can do this using only the ULS estimator. You do not need to use the CATEGORICAL option. With any other estimator you would need a weight matrix for correct results. It is much better to analyze
raw data.
K. Tan posted on Friday, March 01, 2013 - 2:13 pm
It seems like MPLUS generally uses only Logit and Probit models for categorical dependent variables (using the MLR and WSLMV estimators respectively). Is there any way to run a Linear Probability
Model with robust standard errors? Is that the model used when I fail to declare my dependent variable as categorical?
Linda K. Muthen posted on Friday, March 01, 2013 - 6:00 pm
If you do not put a categorical variable on the CATEGORICAL list, a linear regression is estimated for the categorical outcome.
K. Tan posted on Friday, March 01, 2013 - 7:37 pm
Thank you! Can I assume that if I use the MLR estimator with this specification, the standard errors are robust to heteroskedasticity?
Bengt O. Muthen posted on Saturday, March 02, 2013 - 1:09 pm
In principle they are robust, but in practive I am not sure how robust the results would be given that the coefficients are biased. You would need to do a Monte Carlo simulation study which is easy
to do in Mplus.
Martin Van Boekel posted on Thursday, March 07, 2013 - 11:27 am
I am a new MPLUS user and attempting to run a SEM model with five exogenous, and eight endogenous variables, all of which are categorical. Of the two dependent latent variables, one is made up of
three categorical rating scale items, the other is a single binary outcome.
I have reviewed much of the documentation on how to set up this type of model and consulted with peers, however when I run the syntax the time between iterations is very slow. I have left the program
running for over 15 minutes and observed only 115 iterations.
If I remove the dichotmous outcome variable the model will converge so I do not believe that there is a problem with the code.
This is my first attempt at running an SEM model with categorical outcome variables, so any help would be appreaciated thank you.
Linda K. Muthen posted on Thursday, March 07, 2013 - 12:17 pm
Please send your input, data, and license number to support@statmodel.com.
Isaac Yrigoyen M. posted on Friday, August 09, 2013 - 5:03 pm
Dear Dr. Linda and Dr. Bengt,
I am new in your website, and I'm very interested in knowing whether the following model can be run in Mplus:
-My dependent variable is unordered categorical variable (not latent, just observed nominal variable with 4 categories).
-My predictors are: 4 continuous latent variables (they are reflective and have ordinal indicators), and 2 of them are mediators in the model.
According to your user guide, it would be something like that:
Variables u1- u16
Categorical are u2 – u16
Nominal u1
f1 by u2 – u4
f2 by u5 – u7
f3 by u8 – u10
f4 by u11 – u16
f1 ON f3
f2 ON f3 f4
u1 ON f1 f2 f4
u1 IND f2 f4
Can Mplus run this model?
Does Mplus run multinomial logistic regression for the most endogenous paths (I mean between u1 and f1 f2 f4)?
Is it possible to model some of those latent variables (e.g.: f3) as formative?
Thank you very much
Bengt O. Muthen posted on Friday, August 09, 2013 - 6:32 pm
This is possible, although Model Indirect is not available with a nominal distal outcome because indirect effects with a nominal distal outcome need special attention to what is meant by the indirect
effect - it is not simply of the "a*b" kind. For a proper handling of this, see the paper (with Mplus scripts):
Muthén, B. (2011). Applications of causally defined direct and indirect effects in mediation analysis using SEM in Mplus.
Formative factors are straightforward. Multinomial logistic regression is used where u1 is the DV.
Cecily Na posted on Thursday, September 12, 2013 - 7:38 am
Hello Professors,
I have several count variables (range 0-200) which I want to treat as continuous variables, but they are heavily skewed (skewness=50). Can I still treat them as continuous and use the MLR estimator?
I also tried to normalize their distributions by transformation based on their frequency percentage. I recode these variables into 5- or 6-point ordinal variables (0-1 recoded into 0, 2-10 recoded
into 1 and so on), and want to treat them as continuous in my analysis. However, I need to incorporate weights in my analysis, and the transformation is done regardless of the weights.
Which method is better (after I apply weights)?
Thank you very much for your advice!
Linda K. Muthen posted on Thursday, September 12, 2013 - 1:31 pm
You should not transform count variables. You should not treat them as continuous if they have a lot of zeroes. You can treat them as count variables or perhaps censored variables. I don't see any
advantage of collapsing values unless there is a substantive reason.
Cecily Na posted on Friday, September 13, 2013 - 6:40 am
Thank you, Linda, for your reply. The reason that I do not want to treat them as count variables is I suspect the measurement is pretty lousy (not real counts), and I would rather collapse them into
several (continuous)ranges. So the focus of my question this time is:
Would it be okay if the unweighted data are normal, and then I apply weights and use MLR for estimation?
Thanks a lot!
Linda K. Muthen posted on Friday, September 13, 2013 - 8:50 am
Lily Wang posted on Saturday, October 12, 2013 - 2:50 pm
Hello professors,
I am fitting a multilevel path model with a multinomial outcome (3 categories). So I have odds ratios for category 1 vs 3 and 2 vs 3. I was wondering if there are ways for me to obtain odds ratios of
outcomes 1 vs 2, WHILE knowing if the estimates are statistically significant. I look forward to your advice on this! Thanks.
Linda K. Muthen posted on Monday, October 14, 2013 - 10:04 am
You can use the DEFINE command to change the reference category of the nominal variable.
Lily Wang posted on Monday, October 14, 2013 - 10:24 am
Thank you so much Linda. It sounds like a great option--I just wanted to make sure if there are statistical/reporting implications of this. Would you say that it is appropriate to report odds ratio
of 1 vs 2 alongside with the other two pairs (1 vs 3) (2 vs 3) where the base category is the same and generated from the running the model with 3 as the base? Are there any statistical issues that I
need to be addressing in this situation? Thanks again for your guidance!
Linda K. Muthen posted on Monday, October 14, 2013 - 2:29 pm
I can't see any issues as long as you are clear on what you are reporting.
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In the majority of our public schools nine years are devoted to the study of arithmetic. Notwithstanding this fact, the majority of these students who have had this amount of training are unable to
add a column of figures rapidly and accurately. They are unable to perform mentally the simplest operations in relation to business. The business world has deplored this condition of things for many
decades. Is it not time for the teachers of this country to investigate the causes of these meager results? All who have given the question serious consideration agree that too much time is already
devoted to the subject; all agree that the fault lies in the method employed in teaching numbers; this view is in the main correct. After all, may there not be another reason for the poor results?
Every teacher who has had a few years' experience must have occasionally met the backward pupil, twelve or fifteen years of age, who has begun arithmetic, as hi called it, and in one year
accomplished as much as his seatmate who has studied numbers for four or five years. In some instances these backward students have surpassed those of their own age who have had for years the routine
drills of the school. If it is a fact that the majority of these backward students accomplish this work in so short a time, is it not conclusive evidence that we begin pure number work altogether too
early? May it not be true that the child's mind is not sufficiently developed for this kind of training when he begins school at the age of five or six years? It is not implied here that in
elementary science and language that numbers are to be ruled out, but that they are to be taught incidentally.
If pure number work could be delayed even a year or two and then be presented in a proper way, a great deal would be gained. The second reason for failure is also deserving of serious consideration.
All teachers agree that, in teaching little folks numbers, objects should be employed until these objects can be readily pictured in the mind. They agree that the combinations, whether in addition,
subtraction, multiplication or division, should be given by the child without conscious effort, in other words, the process of number teaching should secure for the child a storing up of concepts to
be used forever afterwards without hesitation and with unvarying accuracy. This result is not secured, for the simple reason that early in the child's work the graphic representation of number is
taught. For example, in learning the number four, under the pretext of busy work, the little fellow is asked to solve on his slate such examples as the following- 2 and 2 = ? ; 2 x 2 =?; 4-2=?.
If the child has had these combinations in relations to objects, if the child has committed to memory the results, how much time will the child consume in writing the results on his slate? The truth
of the matter is, he does not write the result without hesitation. Without any reference to the teacher, he introduces new mental factors into his work. These factors have no relation whatever to the
mental operations provided by the child in the presence of the objects. In this way two methods of computation are established in the child's mind.
As he progresses in the study of numbers he has less to do with the mental. When he enters the fifth or sixth grade, he is a slave to his written work and continues a slave during the remainder of
his school life.
Graphic work in numbers in the first, second, and third grades of our public schools, not only has no value in itself, but is positively detrimental. Again we say that there can be no objection to
using the language of number where it becomes a part of their written science or language exercises. Let the number work be purely oral during the first three years of the child's school life and
thereafter use no graphic work except where large numbers are involved in addition, subtraction, multiplication and division. The time that is usually wasted in slate work, in busy work, could be
devoted to reading, literature and elementary science. The operations in numbers would all be performed in the class under the guidance of the skillful teacher. The first three years of the child's
work in numbers would enable him to acquire a power in mental arithmetic that the average high school graduate rarely possesses. The work of learning arithmetic would be revolutionized.
Source: Newton, Roy, editor. Life and Works of Woodbridge N. Ferris. (Big Rapids, Michigan: n.p., 1960), 163-165. | {"url":"http://www.ferris.edu/library/SpecCollections/WNF/Writings/Numbers.html","timestamp":"2014-04-17T21:55:53Z","content_type":null,"content_length":"11108","record_id":"<urn:uuid:c3d241ca-3b96-4b68-bf1d-0840291d2967>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00200-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mathematics 9-12
Experimenting with Yeast Metabolism: Students will use multiple technologies to gain an understanding of yeast metabolism.
Graphing Linear Equations and Slopes: Students will explore the relationship between the slopes of perpendicular lines and parallel lines
Set Theory Review Using the TI Navigator: Students in a Math III class use wireless graphing calculators to enhance their understanding of Set Theory.
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In Einstein's theory of General Relativity gravity is understood as a manifestation of the curvature of spacetime. One consequence of this is that light rays should bend near any mass. In particular,
Einstein predicted that the light from a star should bend as it goes past a massive object like the sun. That prediction was first tested by Sir Arthur Eddington in 1919 during an total eclipse of
the sun. A star that would otherwise not be visible, because it was behind the sun, was observed close to the sun's outer edge (its limb) confirming the bending of light by gravity.
In 1937 Zwicky (Phys. Rev. 51, 1937, 290) noted that the bending of light by massive objects should lead to gravitational lensing: the focusing of light from a distant object by a mass that lies in
the line-of-sight between the distant object and the observer. At the time the prediction was nothing more than an interesting curiosity because the technical means to test it simply did not exist,
and would not exist for another sixty years! Today, gravitational lensing not only affords a beautiful test of Einstein's theory, but also provides a tool for investigating the halo of dark matter
thought to surround the Milky Way.
The basic idea, as pointed out by the astrophysicist Bohdan Paczynski (Princeton) in 1986, is that if the galactic halo contains dark objects---called massive compact halo objects (machos), with
masses between that of Jupiter (about 1/1000 of a solar mass) and very dim stars (e.g., brown dwarfs) with masses too small (less than 1/10 of a solar mass) to trigger thermonuclear ignition, then
once in a while these massive halo objects should cross the line-of-sight between the earth and a more distant luminous star. If the halo object gets close enough to the line-of-sight we should see a
noticeable lensing effect, characterized by a temporary brightening of the light from the distant star. This effect, which I describe in detail below, is called microlensing.
Searching for halo objects
Four experimental teams, EROS, MACHO, OGLE and DUO are now searching for halo objects using the stars in the Large Magellanic Cloud (LMC), a satellite galaxy of the Milky Way, as the back-drop
against which the halo objects move. The basic idea is illustrated in the figure below.
Figure 1.
Light from a star (the yellow dot in the figure) in the LMC is deflected, indeed focused, by a passing halo object (the brown dot). These objects act as gravitational lenses. In principle, by
measuring the distribution and rate of microlensing events we could learn something about the nature and distribution of these halo objects. In practice, this is very difficult because the accidental
line-ups occur infrequently. The EROS collaboration, for example, has measured a microlensing rate of only 7.3 x 10^-7 events/star/year (astro-ph/9511073). So, if you looked at about 10 million stars
continuously for one year you might expect to detect only one or two microlensing events! Clearly, this research is very challenging. The EROS team is able to monitor about 4 million stars,
simultaneously. In a period of three years they found two events that can be interpreted as microlensing events. The MACHO team has found several possible microlensing candidates.
Many stars are known to brighten and dim; some in a regular and repeating pattern, while others do so irregularly. These are the variable stars. The brightness versus time of a stellar object is
called its light curve. With so many variable stellar objects in the sky how can we possibly distinguish the light curve of microlensing events from that of, say, variable stars?
The answer is that general relativity predicts a precise form for the light curve of a microlensed star. This light curve may even be unique. Also, microlensing events are singular events: a star
appears to brighten then dim, and this sequence is not repeated. Another interesting signature of microlensing is that the effect is achromatic: that is, it does not depend on the color of the light
(provided the source and lens are compact). Therefore, if you look at a star through two filters, usually blue and red, the blue and red light curves should be identical. We know of no stellar
phenomena, other than microlensing, which would produce such behaviour. Here is the light curve of one of the possible microlensing events found by the MACHO team.
Figure 2.
Notice the perfectly symmetrical nature of the light curves and the fact that the red and blue light curves are indeed the same. This is very convincing evidence that this is, in fact, the light
curve of a microlensing event.
The Einstein Ring
We shall now work through the mathematics of microlensing. What we want to calculate is the angle between the star's image and the halo object. Happily, the mathematics is not so bad; honest! It
requires just a bit of basic geometry and one formula from general relativity. Oh---and a spot of calculus! Consider the diagram below.
Figure 3.
Let's suppose that the earth (the blue dot on the left), a halo object (the brown dot in the middle) and a light source, say a star (the white dot on the right), are almost lined up. In the absence
of gravitational lensing the light from the star would travel from the star directly along the line-of-sight to the earth. However, the presence of the halo object causes light from the star, which
would otherwise have missed the earth, to bend towards the earth. The light now appears to come from the direction of the yellow dot, which is an image of the star. As we shall see shortly there is
another image on the other side of the halo object. (As drawn, the second image would lie below the halo object.)
The distance between the earth and the light source is denoted by L---this would be 55 kpc for a star in the LMC. The distance between the earth and the halo object is R, maybe 10 kpc. The distance
between the image (the yellow dot) and the halo object (the brown dot) is d. The angle between the halo object and the star is denoted by b, while the angle between the halo object and the star's
image is denoted by q. Obviously, the angle between the star and its image is q-b. The light from the star bends by an angle a, which according to general relativity is given by
(1) a = 4GM/c^2d.
This is the only formula we shall need from general relativiy! Notice, the formula can be written in terms of the Schwarzchild radius, Rs = 2GM/c^2:
(2) a = 2R[S]/d.
For simplicity, even though this is not a good approximation for the Large Magellanic Cloud, we shall assume that the star is very much further away from earth than the halo object; that is, we shall
assume that L >> R. With this assumption we may write the approximate equation a = q-b. When we combine it with Eq. (2) we get
(3) q-b = 2R[S]/d.
Now we need to figure out how the distance d is related to q. For small angles this is easy to work out; the desired relationship is
(4) d = Rq,
which, when combined with Eq. (3) gives
(5) q-b = 2R[S]/Rq.
A slight re-arrangement of Eq. (5) leads to the quadratic equation
(6) q^2 - bq - 2R[S]/R = 0,
whose solutions q[+ ]and q[- ]are
(7a) q[+] = b/2 + (b^2/4 + 2R[S]/R )^1/2,
(7b) q[-] = b/2 - (b^2/4 + 2R[S]/R )^1/2[.]
So here are the formulas we wanted. Notice that the q[+] solution is slightly larger than the angle b, the angular separation between the star and the halo object. This solution corresponds to an
image that is displaced away from the line-of-sight (at the location of the yellow dot in the figure above). The q[- ]solution corresponds to a second image displaced away from the halo object. The
angle q[- ]is actually negative! All this means is that the second image is diametrically apposite the first image, but on the other side of the halo object. In the figure above the second image
would lie below the gravitational lens. The diagram below shows what this might look like on the sky. (For convenience, we have oriented the lens and the star so that a line drawn through their
centers is horizontal.)
Figure 4.
Imagine a halo object (the lens) moving across the sky, close to a star (the light source). At some instance the lens will be an angle b away from the star. An image of the star would form at an
angle q[+ ]to the left of the lens; a second image would form at an angle |q[-]| to the right of the lens. The vertical bars | | simply means that we just consider the size of the angle and forget
about its sign. Note, since the halo object is moving (upwards in the diagram) the size of the angle b will change, as will the relative orientation of the two images. The left image will move along
a downward arc, while the right image will move along an upward arc. These microlensing events can last for days; it would be quite spectacular to make a movie of them. Unfortunately, for halo
objects, the angles involved are, at present, too small to be resolved.
What happens when the distant star and the halo object (that is, the lens) are perfectly aligned? Perfect alignment means that the angle b between the star and the halo is zero. In that case, we find
from Eq. (7)
(8a) q[+](b=0) = + (2R[S]/R )^1/2,
(8b) q[-](b=0) = - (2R[S]/R )^1/2.
The images, not surprisingly, are displaced symmetrically about the lens. In fact, because of the rotational symmetry about the line-of-sight (you can rotate everything about the line-of-sight
without changing the relative orientation of the halo object and the star) there would, in fact, be a ring of images about the halo object. The halo object would have a halo! This ring is called an
Einstein Ring. The radius of the ring R[E], at the position of the halo object, is given by
(9) R[E] = Rq[+](b=0) = (2R[S]R)^1/2.
The radius R[E] is called the Einstein Radius.
Let's work out the Einstein radius for a halo object that is 10 kpc away, and which has a Schwarzchild radius of R[S ]= 1/10 km. This Schwarzchild radius corresponds to a halo object whose mass is
about 1/30 that of the sun. (The sun's Schwarzchild radius is about 3 km.) One parsec is equal to approximately 31 trillion kilometers, so 10 kiloparsecs is equal to 3.1 x 10^17 km. Therefore, the
Einstein radius is
R[E] = (2x10^-1x3.1x10^17)^1/2 = 249 million km.
This is about 1.7 AU; therefore, the ring would be slightly larger than the orbit of Mars. How large would the diameter of the ring appear in the sky in arcseconds? (One arcsecond = 1/3600 of a
degree.) This answer is
Angular diameter = (2R[E]/R)*(360/2p)*3600 arcseconds
= (2 * 2.49 x 10^8/3.1 x 10^17)*(360/2p)*3600,
= 3.2 x 10^-4 arcseconds.
That is, about 1/10 of a milliarcsecond! At present, such a tiny angle is utterly beyond our ability to resolve.
In the above discussion we considered a gravitational lens at a fixed angular separation b from a star. In reality, of course, the angular separation would always be changing because of the relative
motion of the lens and the star. As we noted above, this causes the total brightness of the two images to change in a very special way. In this section we shall derive the form of the microlensing
light curve. The amount of light we receive from a star is determined by the solid angle subtended by the star. The solid angle is just the apparent angular area of the star in the sky. The lensing
effect increases the solid angle over which we receive light, thereby increasing the amount of light we receive. So if we can calculate the solid angle of the star in the absence of lensing and the
solid angle with lensing the light magnification would be simply
magnification = total solid angle with lensing/solid angle without lensing.
To see how this works consider the figure below
Figure 5.
The brown dot represents the gravitational lens. Far behind the lens is a star. Consider a strip of width Du on the star's surface. Let u be the distance between the lens and the points A and B on
the strip. Light from these points travels out in all directions; however, the only rays that we need consider are the ones that travel to points A+, B+, A- and B-. In the absence of lensing the
light from the points A and B would come straight towards us. But because of lensing the light from point A appears to come from the points A+ and A-; the light from point B appears to come from the
points B+ and B-. The lensing effect therefore creates two image strips of width Dy[+] and Dy[-], to the left and right of the lens, as illustrated. Let the radial distance from the lens to the
points A+ and B+ be y[+] and let y[-] be the radial distance from the lens to the points A- and B-.
The amount of light E from each strip is just the flux f (the energy per unit area per second) times the area of the strip:
(10a) E = f *Area = f u[ ]f Du
(10a) E[+ ]= f *Area = f y[+]f Dy[+]
(10a) E[-] = f *Area = f y[-]f Dy[-],
[where ][f ][is the angle between the points A and B, relative to the lens. (We have used the fact that the length of an arc = its radius times its angular size.) Therefore, the magnification m is
(11) m = (E[+] + E[-])/E = (y[+]/u)(Dy[+] /Du) + (y[-]/u)Dy[-] /Du).
Notice that the flux f and angle f cancel out. So all we need do now is to relate u, y[+] and y[- ]to b, q[+] and q[-], respectively, and to calculate the ratios Dy[+] /Du and Dy[-] /Du. The first
set of relations are easy:
(12) u = Rb, y[+] = Rq[+ ], y[-] = Rq[-].
[The second set (the ratios) needs a bit of calculus! Here are the results]
(13a) Dy[+] /Du = 1/2[1 + b/(b^2 + 8R[S]/R )^1/2]
= 1/2[1 + u/(u^2 + 4R[E]^2)^1/2]
(13b) Dy[-] /Du = 1/2[b/(b^2 + 8R[S]/R )^1/2 - 1]
= 1/2[u/(u^2 + 4R[E]^2)^1/2- 1].
In Eq. (13) we have substituted u for b. From Eqs. (12) and (7) we can write
(14a) y[+] = Rq[+ ]= Rb/2 + R(b^2/4 + 2R[S]/R )^1/2
= 1/2[u + (u^2 + 4R[E]^2)^1/2]
(14b) y[-] = Rq[- ]= R(b^2/4 + 2R[S]/R )^1/2 - Rb/2
= 1/2[(u^2 + 4R[E]^2)^1/2 - u].
We now substitute Eqs. (13) and (14) into Eq. (11), and after a bit of algebra we get the formula for the magnification m as a function of the distance u of the lens from the line-of-sight to the
(15) m = (u^2 + 2R[E]^2)/[u(u^2 + 4R[E]^2)^1/2].
We can simplify this formula further if we measure u in units of the Einstein radius R[E]. To do that we just divide u by the Einstein radius in Eq. (15). This gives, finally, the simple formula
(16) m = (u^2 + 2)/[u(u^2 + 4)^1/2].
However, we are not yet quite done! As the lens moves across the sky, or if we keep the lens fixed and imagine the star moving, the distance u will change; it will be a function u(t) of the time t.
What is that function? We shall assume that the lens (or the star) moves in a straight line across the sky at a constant speed V. The distance traveled in time t is just D = Vt. At some time, let's
say t = 0, the star and lens will be as close in the sky as they can get. Let's denote this distance of closest approach by u0 = u(t=0). Note that this distance is at right-angles to the direction of
motion of the lens (or the star, if we consider the lens to be fixed). See Fig. 4. At any other time t, the distance u(t) will be the hypotenuse of a right-angled triangle; the other side is u0 and
the third is Vt/R[E]. We divide Vt by R[E ]because we are measuring lengths in units of the Einstein radius. From Pythagorus' theorem we have for any other time t
(17) u(t) = (u0^2 + (V/R[E])^2 t^2)^1/2.
The time Dt =R[E]/V to cross a distance equal to one Einstein radius is called the lensing time. This can be anywhere from hours to weeks. Usually, we write u(t) in terms of the lensing time:
(18) u(t) = (u0^2 + (t/Dt)^2)^1/2.
^The formulas given by Eqs. (16) and (18) provide a precise description of the microlensing light curve, examples of which are shown in Fig. 2.
Last updated April 6, 1998, Harrison B. Prosper | {"url":"http://www.physics.fsu.edu/Courses/spring98/AST3033/Micro/lensing.htm","timestamp":"2014-04-18T03:29:22Z","content_type":null,"content_length":"32265","record_id":"<urn:uuid:2881e55b-d79b-4f33-b88c-b269d96a9107>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00006-ip-10-147-4-33.ec2.internal.warc.gz"} |
FRM Fun 12. Find the mistake in a published LVaR question
This is a circulating question on liquidity-adjusted value at risk (LVaR) that several members have submitted over the years (image source: a member posted elsewhere in the general forum):
At least one mistake has been confirmed!
Question: what is the correct 95% LVaR answer (hint: it is not given as an option)?
ShaktiRathore Well-Known Member
The Liquidity Adjusted VaR is,
LVaR=VaR+Liquidity cost
LVaR=VaR+(mean of spread+1.96*volatality of spread)*V; there should be plus instead of minus sign in the formula above
Another Mistake above seems that mean and volatility of spread are taken in USD and when multiplied by V gives USD^2 as unit which is not valid. Hence we first need to convert mean and volatility
to % terms before calculating LVaR.Instead of taking mean and volatlity as .1USD and .3USD they should be corrected to .1% and .3% accordingly.
Thereby after correction,
LVaR=1 million+.5*(.1%+.588%)*1million
LVaR=1 million+.5*(.688%)*1million
LVaR=1 million+.344%*1million
LVaR=1 million+3440
Hi ShaktiRathore,
Thank you, I agree that question should instead read "spread of 0.1% with spread volatility of 0.3%." The answer appears to treat the spread as %; and, also, if the spread is really USD, it is
hard to know how to treat (surely that cannot be the spread on the whole position). So, IMO, I agree this counts as one mistake.
I also agree that another mistake, which follows Culp, is to employ the +mean - volatility (i.e., +.1 - .3). This cannot be correct because it implies an increase in mean will lower the cost of
illiquidity. This is an old error from Culp due to computing VaR = mean - volatility * spread and this error is why I prefer Dowd's:
□ VaR = -mean + volatility*sigma; this format seems to be more robust to pilot error because then LC is always a natural addition:
□ LVaR = -mean + volatility*sigma + LC; i.e., +LC increases a positive VaR
Finally, I think the third error is to use 1.96. This has been much discussed on this forum. My view is that the spread deviate should also be 1.645 per a one-tailed critical value: we are not
interested in the other tail, only the adverse tail where the spread moves against us. | {"url":"https://www.bionicturtle.com/forum/threads/frm-fun-12-find-the-mistake-in-a-published-lvar-question.6015/","timestamp":"2014-04-16T07:53:28Z","content_type":null,"content_length":"35641","record_id":"<urn:uuid:fc9a11eb-d847-450d-8f88-67cb7bd92965>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00658-ip-10-147-4-33.ec2.internal.warc.gz"} |
February 1st 2007, 06:44 AM
For the linear regression model
(B= beta and e = epsilon)
ei~N(0,s^2) (s = sigma)
i = 1,...,n are independant, this model does not contain an intercept form
(E = sum of , ^B = beta hat)
i found that ^B = Ei xiyi / Ei xi^2
now i need to show that ^B is an unbiased estimator of B and that its variance is s^2/ Ei xi^2
jus wondering what the proof is for a linear regression model that the expected or estimated value of ^B =B in this circumstance and how to prove that var(^B)=s^2(XtX)^-1
ive been working in matrix form, t=transpose as im sure u might have guessed | {"url":"http://mathhelpforum.com/advanced-statistics/10988-estimators-print.html","timestamp":"2014-04-17T20:12:02Z","content_type":null,"content_length":"3616","record_id":"<urn:uuid:3e570e10-07b2-407b-9792-7064a10550c0>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00573-ip-10-147-4-33.ec2.internal.warc.gz"} |
Matlab functions need conversion
September 12th, 2011, 08:34 PM #1
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Sep 2011
Matlab functions need conversion
I need help in converting a few functions from matlab to VC++. Here's the code that I have done partially in C++ and some are still in MATLAB.
#include "mbed.h"
#include <stddef.h>
#include <stdlib.h>
#include <stdio.h>
#include "iostream"
//#include "math.h"
void Norma::norma(float Dn)
float D;
int ni;
float Dn;
[~,ni] = sizeof(D);
if (ni == 1)
Dn = (D - min(D))./(max(D)-min(D)); //convert this equation in C++
vmaxD = max(D); vminD = min(D);
for (int i = 1; i<= ni; i++)
Dn(:,i) = (D(:,i) - vminD(i))./(vmaxD(i)-vminD(i));//convert this equation in C++
Re: Matlab functions need conversion
I need help in converting a few functions from matlab to VC++. Here's the code that I have done partially in C++ and some are still in MATLAB.
Instead of hoping that someone knows MATLAB in a C++ forum, why not tell us the name of the formula you're trying to implement?
The reason why is that it may not be as easy as translating line-by-line a math formula that works in MATLAB to C++. There are things to consider such as round-off error, possible overflow/
underflow, etc.
If we knew what you were trying to do, then we won't need MATLAB syntax to figure it out.
Paul McKenzie
Re: Matlab functions need conversion
I'm developing an API which that will be embedded to a ARM micro controller. I was have the simulation version which is in .m or Matlab file. But it seems that I have to convert it into C++ since
the micro controller compiler doesn't support calling other third party libraries.
The function that I need to convert is from the equation itself which are the min(D) and max(D).
Re: Matlab functions need conversion
From what I can tell, D and Dn are actually arrays or vectors even though you've defined them as a single float here. You'll need to use a loop to sequentially do an operation on each element.
However, the min() and max() functions can be replaced almost directly by std::min_element() and std::max_element(). The only difference is, those return an iterator which you will need to
dereference rather than the element itself.
Precompute the min and max values outside the loop, of course, for efficiency.
Re: Matlab functions need conversion
From what I can tell, D and Dn are actually arrays or vectors even though you've defined them as a single float here. You'll need to use a loop to sequentially do an operation on each element.
However, the min() and max() functions can be replaced almost directly by std::min_element() and std::max_element(). The only difference is, those return an iterator which you will need to
dereference rather than the element itself.
I don't use matlab, but look at the "vminD(i)" expression; it seems D is a matrix and min(D) is a vector of min values ... perhaps, the OP should look at a ublas C++ library instead ...
BTW, note that there's also the new minmax_element to get the min and max of a sequence simultaneously ...
Re: Matlab functions need conversion
I'm developing an API which that will be embedded to a ARM micro controller. I was have the simulation version which is in .m or Matlab file. But it seems that I have to convert it into C++ since
the micro controller compiler doesn't support calling other third party libraries.
What I'm asking is for you help us out here -- tell us what all those symbols mean, or at the very least, using mathematical nomenclature, what that code is doing.
For example, what is this?
float D;
This is perfectly legal syntax in C++. It declares a single float variable called D. Is this what you want, or is it something else (maybe an array)?
Then this:
[~,ni] = sizeof(D);
What is that tilde supposed to mean? And sizeof() is legal syntax in C++.
In other words, something like "I want to have an array of floating point values called D, vMin(D) means get the minimum value in an array called D,...". etc..
Paul McKenzie
Last edited by Paul McKenzie; September 13th, 2011 at 11:30 AM.
Re: Matlab functions need conversion
Paul McKenzie
Then this:
[~,ni] = sizeof(D);
What is that tilde supposed to mean? And sizeof() is legal syntax in C++.
The size() function in MATLAB returns the dimensions of a matrix. The tilde indicates that a particular returned value should be ignored rather than stored, for functions with multiple return
values. So the above code (if it were using size() rather than sizeof()) would store the number of columns in the matrix D while ignoring the number or rows.
Of course, sizeof() does not do anything even remotely similar in C++, so it is almost certainly not what you want there.
September 12th, 2011, 09:41 PM #2
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how to really start learning some three-dimensional geometry
up vote 1 down vote favorite
I am neither a mathematician nor a maths major student. More likely, I am a dumb person in mathematics especially in three- dimensional geometry. I can’t draw an acceptable diagram and apply theorem
to proof any questions in my homework. I tried to find a way to became an “average person”, but I failed. My question is : are there any books , videos o, classes in college or whatever start to talk
about some basic problems in three- dimentional geometry and develop the basic understanding of it? I don’t need any answer like “ I am sorry that’s so bad” or “there are some courses in college
talking about geometry, such as 304 or 415.” I am hearing these for years! The question seems quite weird posting here. Give me your suggestions if you don’t mind to reduce your maths taste.
To give advice, it would help me if I knew your motivation to learn math. The term "three-dimensional geometry" has many possible interpretations. And I think your difficulty in understanding
3d-geometry courses indicates you might start learning another (easier) part of math first. – Konrad Voelkel Nov 8 '09 at 22:25
This isn't an appropriate place for your question. Please see the FAQ for some alternatives. – Scott Morrison♦ Nov 8 '09 at 23:49
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closed as off topic by Scott Morrison♦ Nov 8 '09 at 23:48
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2 Answers
active oldest votes
I'm not sure what you mean with three-dimensional geometry, but if you mean geometry of curves and surfaces in $\mathbb{R}^3$ I would recommend you
up vote 1 down vote Differential Geometry of Curves and Surfaces, by Manfredo Do Carmo .
1 Judging from the rest of the post it's quite likely the OP means something more basic. – Qiaochu Yuan Nov 9 '09 at 4:00
add comment
I'm guessing wildly here, but it sounds like you are having difficulty translating between algebraic manifestations of a concept (like with equations) and geometric representations (such
as with pictures, or the general idea of objects in space). I don't have any silver-bullet answers, since I also had trouble picking up these sorts of questions.
There are some online resources listed at mathforum.org, and I would also recommend looking at Project MATHEMATICS and The Geometry Center. I don't know how appropriate these are for you,
up vote 0 since I'm unable to ascertain your math background. Try not to get too discouraged if something is pitched too high.
down vote
I have an off-the-wall suggestion: if you can take an art class that emphasizes perspective studies (perhaps while reading a math book on projective geometry), it might help your
visualization skills. It can be beneficial to focus intensively on exactly how the different ways of viewing an object will change the way it appears.
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"Square root" of Beta(a,b) distribution
up vote 3 down vote favorite
Under what conditions on a and b is there a distribution $f_{a,b}$ such that the product $XY$ of two independent realizations $X$ and $Y$ from $f_{a,b}$ has a Beta(a,b) distribution?
A standard result on deriving the distibution of the product of two variables indicates that the p.d.f. $f_{a,b}$ needs to satisfy: $\int_0^1 f_{a,b}(x) f_{a,b}(\frac{v}{x}) \frac{1}{x} dx = \frac{v^
{a-1} (1-v)^{b-1}}{B(a,b)}$, but I have no idea how to find such an $f_{a,b}$. (B denotes the beta function).
This article ( http://www.jstor.org/stable/2045709 ) might be relevant for the case a=b=1, but my background in the relevant maths is not strong enough to understand it at all.
st.statistics pr.probability probability-distributions
2 The problem might simplify if you look for the logs of $X$ and $Y$, which should sum to a log beta of course. Summing RV's means convolving distributions and convolution becomes product after
taking a Fourier transform. – Jeff Schenker Jul 21 '10 at 15:18
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4 Answers
active oldest votes
A partial answer is that $f_{a,b}$ exists for every positive integer $b$.
To see this, first recall that for every positive $s$ and $a$ the distribution Gamma$(s,a)$ has density proportional to $z^{s-1}e^{-az}$ on $z\ge0$ and that the sum of independent
Gamma$(s,a)$ and Gamma$(t,a)$ is Gamma$(s+t,a)$.
If $b=1$, choose $X=e^{-Z}$ and $Y=e^{-T}$ where $Z$ and $T$ are i.i.d. and Gamma$(1/2,a)$. Then $X$ and $Y$ are i.i.d. and $XY=e^{-(Z+T)}$ where $Z+T$ is Gamma$(1,a)$, that is,
up vote 3 exponential of parameter $a$, hence $XY$ is Beta$(a,1)$ and you are done if $b=1$.
down vote
accepted From there, recall that the product of two independent Beta$(a,c)$ and Beta$(a+c,b-c)$ is Beta$(a,b)$ for every $c$ in $(0,b)$. Assume that $b$ is a positive integer and choose $X=e^
{-Z_1-Z_2-\cdots-Z_b}$ and $Y=e^{-T_1-T_2-\cdots-T_b}$ where all the $Z_k$ and $T_i$ are independent and, for each $k$, $Z_k$ and $T_k$ are both Gamma$(1/2,a+k-1)$. Then, by the $b=1$
case, each $e^{-(Z_k+T_k)}$ is Beta$(a+k-1,1)$ hence $XY$ is the product of some independent Beta$(a,1)$, Beta$(a+1,1)$, ..., Beta$(a+b-1,1)$, hence it is Beta$(a,b)$ and you are done
for every positive integer $b$.
add comment
If you're only interested in the existence of a solution Jeff's suggestion of looking at the logarithms seems to be the correct approach. Let $Z$ have Beta$(a,b)$ distribution. Suppose that
there is a solution to the "square root" problem in this case, and let $X$ be a random variable with this distribution. Then, if $\phi(u) = E[e^{i u \log(Z)}] = E[Z^{i u}]$ and $\psi(u) = E[X
^{i u}]$ are the characteristic functions (or Fourier transforms) of $\log(Z)$ it must be the case that $\psi(u)^2 = \phi(u)$ for all $u\in\mathbb{R}$.
Because we know a lot about Beta distributions we can actually solve explicitly that $\phi(u) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)} \frac{\Gamma(\alpha+i u)}{\Gamma(\alpha+\beta+i u)}
$. Then we can prove the existence of a solution by defining $\psi(u) = \sqrt{\phi(u)}$ and then letting $X$ be a random variable with distribution $\psi(u)$.
There are a couple of issues that need to be smoothed over. First of all, $\phi(u)$ is a complex-valued function so we need to be careful with the square root. However, since $\phi(u)\neq 0$
for all $u\in\mathbb{R}$ we can take the square root in a continuous manner. The bigger issue is that we need to be able to prove that $\psi(u)$ is a characteristic function of some
up vote probability distribution. For this we need some Fourier inversion theorems.
2 down
vote Theorem 14 in A Modern Approach to Probability Theory by Fristedt and Gray gives sufficient conditions for a function to be the characteristic function of a real-valued random variable. There
are several conditions to check. First, that $\psi(0)=1$ and $\psi$ is continuous. These are obviously satisfied. Next, that $\psi(u)$ is "positive definite". That is, $\sum_{k=1}^n \sum_{j=
1}^n \psi(u_k-u_j)z_j\bar{z}_k \geq 0$ for any complex $n$-tuple $(z_1,\ldots, z_n)$ and real $n$-tuple $(u_1,\ldots, u_n)$. I don't know how to check to see if this is true. The final
condition is that $\int| \psi(u)| du < \infty$. I'm not an expert in the $\Gamma$ function, so I plotted $\psi(u)$ with Mathematica and it looks like $|\psi(u)|$ decays roughly like $|u|^{-b}
$ as $|u|\rightarrow \infty$ so that $\int |\psi(u)| du < \infty$ if $b>1$.
I still can't think of how to check the "positive definite" condition, but if one can check that the problem would be solved for $b>1$.
Didier informed me of a small mistake in my above explanation. I said that $\psi(u)$ decays like $|u|^-b$. However, what is true is that $\phi(u)$ decays roughly at that rate, and so for $
\psi$ to be in $L^1$ we would need $b>2$. Due to Didier's above proof in the cases where $b$ is a positive integer I believe that the "square root" distribution probably exists for any
$a,b>0$. I still don't know what to do about proving the "positive definite" condition, but the integrability assumption on $\psi$ can probably be removed by someone who knows inverse
Fourier transforms better than me. – Jon Peterson Jul 28 '10 at 17:47
Thank you for the feedback. I am also interested in existence, but I would love to be able to find the square root distribution if it exists, given a and b. Since you indicate that it is
likely they exist, and Didier's work indicates how to obtain it for integral b, the best way forward for me might be to attempt to extend that characterization to non-integral b. – Steve
Kroon Aug 16 '10 at 13:55
1 Yes. Call $B$ the set of parameters $b$ such that $f_{a,b}$ exists for every positive $a$. We know that $B$ is closed by addition and that it contains $1$, hence that $B$ contains every
positive integer. And $B$ might also be a closed subset of the real line. But it is not clear (to me) that $B$ should contain every positive $b$. In particular the positive definiteness
condition which Jon alluded to, might fail for small positive values of $b$. Who knows... :-) – Did Aug 17 '10 at 16:04
It turns out that I'm now looking for the square root of the Beta(0.5, 0.5) distribution, so non-integral b has suddenly become more important to me. – Steve Kroon Sep 7 '10 at 7:38
Hmm, $\beta(1/2,1/2)$ is also the distribution of the fraction of time spent on the positive quadrant by Brownian Motion started at the origin. Have you pursued ideas of this sort taking
two independent copies of this random variable ? – Ivan Dornic Sep 27 '10 at 20:19
show 1 more comment
I might have misunderstood the question but, if not, I suggest this.
Denoting, $Z = X \ Y$, take Mellin-Transform expectations of both-sides. Since $E[X^{s-1}] = B(a+s,b)$, this condition you demand amounts to find the solutions of $B(a+s,b)/B(a,b)=1$,
which can also be rewritten as:
up vote 0
down vote $\Gamma(a) \Gamma(a+s) = \Gamma(a+b) \Gamma(a+b+s).$
EDIT: I apologize for not having read in details Jon Petersen's 1st answer, which is far more superior to my feeble (though unwanted) rephrasing...
I'm not clear on what you mean by "Mellin-transform expectation". However, there seems to be something wrong with your solution, since Γ(a)Γ(a+s)=Γ(a+b)Γ(a+b+s) can not hold for b=1
(RHS will be strictly greater than the LHS), while the discussion in the previous comments have shown that b=1 and any a satisfy my requirements. – Steve Kroon Sep 29 '10 at 10:39
add comment
I'm relatively new to MathOverflow, so I'm sorry if the following is inappropriate as a response- the faq doesn't mention posting related questions. It doesn't seem worth starting a new
thread about, but I had a question:
Is it an accident that the integral $\int f_{a,b}(x) f_{a,b}(\frac{v}{x}) \frac{dx}{x}$ is convolution on the multiplicative group of reals with respect to the Haar measure?
up vote 0 If $X$ and $Y$ are independent, identically distributed elements of a locally compact topological group, does a similar formula hold for the distribution of $X*Y$ ? Wikipedia is
down vote uninformative, and I've never seen anything about this in a textbook.
Many thanks.
The equation above for distribution of a product can be established by transforming from the multiplicative case to the additive convolution case using logarithms. For topological
groups, this approach is probably not viable, so it would depend on whether a more direct result for convolution is available when the domain is not the positive reals. – Steve Kroon
Sep 29 '10 at 10:43
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CAIMS_SCMAI Abstracts
by Paul Muir
Saint Mary's University
Coauthors: Christina Christara, University of Toronto,
Mathematical models in a great majority of applications involve the numerical solution of partial differential equations (PDEs). Essentially all such models are sufficiently complicated that the
associated PDEs typically must be solved using sophisticated numerical algorithms. This mini-symposium will bring together researchers who will discuss recent developments in new computational
algorithms for the numerical solution of PDEs including high-order collocation methods in one and two spatial dimensions, domain decomposition algorithms, and symplectic finite-difference time-domain
schemes, applied to problems including modelling of intracellular signalling pathways (mathematical biology), pricing of American options (computational finance), and the numerical solution of
Maxwell's equations (computational electromagnetism).
Confirmed speakers:
Zhi Li, Saint Mary's University
B-spline collocation software for 2D Parabolic PDEs, with application to Cell Biology
In this talk, we describe new software, BACOL2D, for solving two dimensional time-dependent parabolic partial differential equations (PDEs), defined over a two dimensional rectangular region. The
numerical solution is represented as a bi-variate piecewise polynomial (using tensor product B-spline bases) with unknown time dependent coefficients. These coefficients are determined by
imposing collocation conditions, i.e., by requiring the numerical solution to satisfy the PDE at a number of points within the spatial domain. This leads to a large system of time-dependent
differential algebraic equations (DAEs), which we solve using the high quality DAE solver, DASPK 2.0. BACOL2D is natural extension of the one-dimensional PDE solver BACOL and many of the
algorithms employed in BACOL are applicable to BACOL2D. We also describe an algorithm for a fast block LU solver with modified alternate row and column elimination with partial pivoting for the
treatment of the almost block diagonal linear systems that arise during the numerical solution of the DAEs. We will also briefly consider numerical results to demonstrate convergence rates for
the collocation solution and, in particular, the existence of superconvergent points that may be useful for error estimation. We also demonstrate the use of BACOL2D to solve a 2D cell biology
Duy Minh Dang, University of Waterloo
Adaptive and high-order methods for pricing American options
We develop space-time adaptive and high-order methods for valuing American options using a partial differential equation (PDE) approach. The linear complementarity
problem arising due to the free boundary is handled by a penalty method. Both finite difference and finite element methods are considered for the space discretization of the PDE, while classical
finite differences, such as Crank-Nicolson, are used for the time discretization. The high-order discretization in space is based on an optimal finite element collocation method, the main
computational requirements of which are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are
fourth-order. To control the space error, we use adaptive gridpoint distribution based on an error equidistribution principle. A time stepsize selector is used to further increase the efficiency
of the methods. Numerical examples show that our methods converge fast and provide highly accurate options prices, Greeks, and early exercise boundaries.
Dong Liang, York University,
New Energy Conservation Properties and Energy-Conserved S-FDTD Schemes for Maxwell's Equations
Computational electromaganetics has been playing a more and more important role in many areas of electromagnetic industry, such as radio frequency, microwave, integrated optical circuits,
antennas, and wireless engineering, etc. It is of special importance to develop efficient high-order methods for effective and accurately simulating propagation of electric and magnetic waves in
large scale field and long time duration. However, most previous ADI or splitting schemes break the energy conservation of elctromegnetic waves. In this talk, I will first give new energy
conservation properties of electromagnetic waves in lossless medium, and then provide our newly developed energy-conserved S-FDTD schemes for Maxwell's equations. Both theoretical analysis and
numerical experiment will be presented to show the efficiency of the new schemes.
Hans De Sterck, University of Waterloo
Global approximation of singular capillary surfaces: asymptotic analysis meets numerical analysis
Singular capillary surfaces in domains with sharp corners or cusps are well studied and the asymptotic series approximation of the solution is known. However, the asymptotic series approximation
is only valid in a sufficiently small neighbourhood of the singularity, and we wish to obtain a global approximation of the solution through finite element approximation. Yet it is also known
that the singularity of the solution spoils the accuracy of standard finite element approximations, which cannot reproduce the singularity accurately. We show that an accurate numerical
approximation can be obtained through an appropriate change of variable combined with a change of coordinates, motivated by the known
asymptotic behaviour. Using this accurate numerical approximation methodology, we can numerically confirm the validity of the known asymptotic expansions in great detail, and we can make two
conjectures on asymptotic behaviour of singular capillary surfaces at a cusp for two open cases.
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equation of the tangent line
May 10th 2010, 01:43 PM #1
Jan 2010
equation of the tangent line
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
I've found the derivative of the function to be:
$(x (-4 x^2-4 y^2+25))/(y (4 x^2+4 y^2+25))$
To find the tangent line at (3,1), you just plug in the values to find the slope, correct? I've done this and can't seem to find the right tangent line at that point..
Assuming your derivative is correct, so we have:
Find $y'(3,1)$ to get the slope ..
then substitute this in the well-known equation of any line ..
May 10th 2010, 01:46 PM #2 | {"url":"http://mathhelpforum.com/calculus/144045-equation-tangent-line.html","timestamp":"2014-04-19T12:52:42Z","content_type":null,"content_length":"32284","record_id":"<urn:uuid:99a10cd6-71f4-470f-94c4-b8dfd79621ba>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00417-ip-10-147-4-33.ec2.internal.warc.gz"} |