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convert 33 hectares to acres
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Multiplying Fractions
To find the product of multiplying fractions, you have to do the following:
Lets say the question is 1/4 x 1/3.
It is as simple as this. Multiply numerator x numerator, and denominator x denominator.
So the answer would be 2/12.
Reduced to 1/6. Its that simple.
If it was a fraction x a whole number then all you have to do is put a 1 under the whole number.
Lets say it was 4 x 1/4 .
So we change it to 4/1 x 1/4.
And the answer to that would be 1.
Multiplying with Mixed Numbers:
If the question is 2 1/3 x 3 1/4 we turn them into improper fractions.
So, 7/3 x 13/4.
Then simply again we multiply numerator x numerator and denominator x denominator.
13 x 7 = 91, and 3 x 4 = 12.
The answer is 91/12 reduced to 7 1/2. | {"url":"http://fractionproject123.webs.com/multiplyingfractions.htm","timestamp":"2014-04-16T04:11:56Z","content_type":null,"content_length":"13772","record_id":"<urn:uuid:cec7e06b-0c56-458c-b27e-2da0e0f5dc1d>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00497-ip-10-147-4-33.ec2.internal.warc.gz"} |
Power: Putting Charges to Work
Electric circuits are designed to serve a useful function. The mere movement of charge from terminal to short circuit. With charge flowing rapidly between terminals, the rate at which energy would be
consumed would be high. Such a circuit would heat the wires to a high temperature and drain the battery of its energy rather quickly. When a circuit is equipped with a light bulb, beeper, or motor,
the electrical energy supplied to the charge by the battery is transformed into other forms in the electrical device. A light bulb, beeper and motor are generally referred to as a load. In a light
bulb, electrical energy is transformed into useful light energy (and some non-useful thermal energy). In a beeper, electrical energy is transformed into sound energy. And in a motor, electrical
energy is transformed into mechanical energy.
An electrical circuit is simply an energy transformation tool. Energy is provided to the circuit by an electrochemical cell, battery, generator or other electrical energy source. And energy is
delivered by the circuit to the load at the location of the load. The rate at which this energy transformation occurs is of great importance to those who design electrical circuits for useful
functions. Power - the rate at which mechanical work is done - was introduced in Unit 5 of the Physics Classroom. Here, we will discuss power in electrical terms; while the context has changed, the
essential meaning of the concept of power will remain the same. Power is the rate at which electrical energy is supplied to a circuit or consumed by a load. The electrical energy is supplied to the
load by an energy source such as an electrochemical cell. Recall from Lesson 1 that a cell does work upon a charge to move it from the low energy to the high energy terminal. The work done on the
charge is equivalent to the electrical potential energy change of the charge. Thus, electrical power, like mechanical power, is the rate at which work is done. Like current, power is a rate quantity.
Its mathematical formula is expressed on a per time basis.
Whether the focus is the energy gained by the charge at the energy source or the energy lost by the charge at the load, electrical power refers to the rate at which the charge changes its watt,
abbreviated W. (Quite obviously, it is important that the symbol W as the unit of power not be confused with the symbol W for the quantity of work done upon a charge by the energy source.) A watt of
power is equivalent to the delivery of 1 joule of energy every second. In other words:
1 watt = 1 joule / second
When it is observed that a light bulb is rated at 60 watts, then there are 60 joules of energy delivered to the light bulb every second. A 120-watt light bulbs draws 120 joules of energy every
second. The ratio of the energy delivered or expended by the device to time is equal to the wattage of the device.
The kilowatt-hour
Electrical utility companies who provide energy for homes provide a monthly bill charging those homes for the electrical energy that they used. A typical bill can be very complicated with a number of
line items indicating charges for various aspects of the utility service. But somewhere on the bill will be a charge for the number of kilowatt-hours of electricity that were consumed. Exactly what
is a kilowatt-hour? Is it a unit of power? time? energy? or some other quantity? And when we pay for the electricity that we use, what exactly is it that we are paying for?
A careful inspection of the unit kilowatt-hour reveals the answers to these questions. A kilowatt is a unit of power and an hour is a unit of time. So a
It is a common misconception that the utility company provides electricity in the form of charge carriers or electrons. The fact is that the mobile electrons that are in the wires of our homes would
be there whether there was a utility company or not. The electrons come with the atoms that make up the wires of our household circuits. The utility company simply provides the energy that causes the
motion of the charge carriers within the household circuits. And when they charge us for a few hundred kilowatt-hours of electricity, they are providing us with an energy bill.
The electrical potential difference across the two inserts of a household electrical outlet varies with the country. Use the Household Voltages widget below to find out the household voltage values
for various countries (e.g., United States, Canada, Japan, China, South Africa, etc.).
Calculating Power
The rate at which energy is delivered to a light bulb by a circuit is related to the electric potential difference established across the ends of the circuit (i.e., the voltage rating of the energy
source) and the current flowing through the circuit. The relationship between power, current and electric potential difference can be derived by combining the mathematical definitions of power,
electric potential difference and current. Power is the rate at which energy is added to or removed from a circuit by a battery or a load. Current is the rate at which charge moves past a point on a
circuit. And the electric potential difference across the two ends of a circuit is the potential energy difference per charge between those two points. In equation form, these definitions can be
stated as
Equation 3 above can be rearranged to show that the energy change across the two ends of a circuit is the product of the electric potential difference and the charge - ΔV • Q. Substituting this
expression for energy change into Equation 1 will yield the following equation:
In the equation above, there is a Q in the numerator and a t in the denominator. This is simply the current; and as such, the equation can be rewritten as
The electrical power is simply the product of the electric potential difference and the current. To determine the power of a battery or other energy source (i.e., the rate at which it delivers energy
to the circuit), one simply takes the electric potential difference that it establishes across the external circuit and multiplies it by the current in the circuit. To determine the power of an
electrical device or a load, one simply takes the electric potential difference across the device (sometimes referred to as the voltage drop) and multiplies it by the current in the device.
As discussed above, the power delivered to an electrical device in a circuit is related to the current in the device and the electrical potential difference (i.e., voltage) impressed across the
device. Use the Electric Power widget below to investigate the effect of varying current and voltage upon the power.
Check Your Understanding
1. The purpose of every circuit is to supply the energy to operate various electrical devices. These devices are constructed to convert the energy of flowing charge into other forms of energy (e.g.,
light, thermal, sound, mechanical, etc.). Use complete sentences to describe the energy conversions that occur in the following devices.
a. Windshield wipers on a car
b. Defrosting circuit on a car
c. Hair dryer
2. Determine the ...
a. ... current in a 60-watt bulb plugged into a 120-volt outlet.
b. ... current in a 120-watt bulb plugged into a 120-volt outlet.
c. ... power of a saw that draws 12 amps of current when plugged into a 120-volt outlet.
d. ... power of a toaster that draws 6 amps of current when plugged into a 120-volt outlet.
e. ... current in a 1000-watt microwave when plugged into a 120-volt outlet.
3. Your 60-watt light bulb is plugged into a 110-volt household outlet and left on for 3 hours. The utility company charges you $0.11 per kiloWatt•hr. Explain how you can calculate the cost of such a
4. Alfredo deDarke often leaves household appliances on for no good reason (at least according to his parents). The deDarke family pays 10¢/kilowatt-hour (i.e., $.10/kW•hr) for their electrical
energy. Express your understanding of the relationship between power, electrical energy, time, and costs by filling in the table below.
│ Power Rating │Time │ Energy Used │ Costs │Costs │
│ │ │ │ │ │
│ (Watt) │(hrs)│(kilowatt-hour) │(cents)│ ($) │
│ 60 Watt Bulb │ 1 │ 0.060 kW•hr │ 0.6 ¢ │$0.006│
│ 60 Watt Bulb │ 4 │ │ │ │
│120 Watt Bulb │ 2 │ │ │ │
│100 Watt Bulb │ │ 10 kW-hr │ │ │
│ 60 Watt Bulb │ │ │1000 ¢ │ $10 │
│ │ 100 │ 60 kW-hr │ │ │ | {"url":"http://www.physicsclassroom.com/Class/circuits/U9L2d.cfm","timestamp":"2014-04-17T01:05:19Z","content_type":null,"content_length":"71894","record_id":"<urn:uuid:f3df2060-c137-499f-83e0-0ccfcace9390>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00216-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: [SI-LIST] : Trace impedance
Dennis Tomlinson (det@tellabs.com)
Mon, 09 Nov 1998 13:40:39 -0600
C.C. Chiu wrote:
> hi! anyone who knows how to calculate the PCB trace width which flow with 1A,2A.......
> thxs!!
Hi Mr. Chiu,
Since the currents you refer to are 1A, 2A, .. I assume you are asking a
question about DC trace resistance and not characteristic impedance. That
is the question I will answer. My apologies if I am wrong.
First, let me define a square trace. A trace which is of width w and length
l = w is said to be a one square trace. A trace of width w and length l = 10*w
is then a 10 square trace. As an illustration, consider the following:
PCB trace segment rated as a 5 square trace
_________________________________________________ ___
| | | | | | |
| | | | | | |
| | | | | | w
| | | | | | |
| | | | | | |
|_________|_________|_________|_________|_________| __|_
In tabular form, some numerical examples would look like:
Trace width (mils) Number of squares per routed inch
------------------ ---------------------------------
30 33.333
w 1000/w
With this concept, the DC resistance of routed trace can then
be calculated from the following table based on copper weight:
Cu wt. in ounces /in^2 thickness (in.) milli-Ohms/square
---------------------- ---------------- -----------------
1/2 0.0007 1
1 0.0014 0.5
2 0.0028 0.25
These values are based on the conductivity of copper, and are nearly
exact at room temperature. The process employed in the manufacture of
PCB's often cause thickness to be slightly less than above. A simple
proportion can be used to re-scale the mill-Ohms/square appropriately.
As an example of how to use the above tables, assume I have 6 inches
of 20 mil wide trace routed with 1 oz/in^2 Cu. The number of squares in
this trace is:
(6 in)*(50 squares/in) = 300 squares.
The total trace resistance is given by:
(300 squares)*(0.5 milli-Ohms/square) = 150 milli-Ohms
With this information, augmented by requirements from the design
such as:
1. How much DC voltage drop you can afford,
2. How much routed trace length is expected, and
3. What Cu weights can or will be used,
you can choose trace width appropriately.
On the other hand, if I have misunderstood your question, I should
now be very embarrassed!
ps. Thanks to Tom Stamm who taught this to me when I was a fledgling
engineer, more concerned with which end of the soldering
iron to hang onto.
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www.qsl.net/wb6tpu/si-list **** | {"url":"http://www.qsl.net/wb6tpu/si-list2/pre99/1974.html","timestamp":"2014-04-19T04:21:19Z","content_type":null,"content_length":"5797","record_id":"<urn:uuid:4c079179-c192-414c-bbe7-a630a3fd04ea>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00177-ip-10-147-4-33.ec2.internal.warc.gz"} |
DOCUMENTA MATHEMATICA, Vol. 16 (2011), 1-31
DOCUMENTA MATHEMATICA
, Vol. 16 (2011), 1-31
Richard Hill and David Loeffler
Emerton's Jacquet Functors for Non-Borel Parabolic Subgroups
This paper studies Emerton's Jacquet module functor for locally analytic representations of $p$-adic reductive groups, introduced in \cite{emerton-jacquet}. When $P$ is a parabolic subgroup whose
Levi factor $M$ is not commutative, we show that passing to an isotypical subspace for the derived subgroup of $M$ gives rise to essentially admissible locally analytic representations of the torus
$Z(M)$, which have a natural interpretation in terms of rigid geometry. We use this to extend the construction in of eigenvarieties in \cite{emerton-interpolation} by constructing eigenvarieties
interpolating automorphic representations whose local components at $p$ are not necessarily principal series.
2010 Mathematics Subject Classification: 11F75, 22E50, 11F70
Keywords and Phrases: Eigenvarieties, $p$-adic automorphic forms, completed cohomology
Full text: dvi.gz 65 k, dvi 154 k, ps.gz 420 k, pdf 327 k.
Home Page of DOCUMENTA MATHEMATICA | {"url":"http://www.kurims.kyoto-u.ac.jp/EMIS/journals/DMJDMV/vol-16/01.html","timestamp":"2014-04-21T12:15:13Z","content_type":null,"content_length":"1919","record_id":"<urn:uuid:bee0223c-315b-4e7b-a6a4-cfed217007df>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00223-ip-10-147-4-33.ec2.internal.warc.gz"} |
cross sectional area
If spacetime is built out of quantum bits, does the shape of space depend on how the bits are entangled? The ER=EPR conjecture relates the entanglement entropy of a collection of black holes to the
cross sectional area of Einstein-Rosen (ER) bridges (or wormholes) connecting them. We show that the geometrical entropy of classical ER bridges satisfies the subadditivity, triangle, strong
subadditivity, and CLW inequalities. These are nontrivial properties of entanglement entropy, so this is evidence for ER=EPR. We further show that the entanglement entropy associated to classical ER
bridges has nonpositive interaction information. This is not a property of entanglement entropy, in general. For example, the entangled four qubit pure state |GHZ_4>=(|0000>+|1111>)/\sqrt{2} has
positive interaction information, so this state cannot be described by a classical ER bridge. Large black holes with massive amounts of entanglement between them can fail to have a classical ER
bridge if they are built out of |GHZ_4> states. States with nonpositive interaction information are called monogamous. We conclude that classical ER bridges require monogamous EPR correlations.
Recent Comments
• susanstewart_aa on The Unified Astronomy Thesaurus
On standing sausage waves in photospheric magnetic waveguides
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(1 vote from 1 institution)
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I. Dorotovic, R. Erdelyi^1†, N. Freij, V. Karlovsky, I. Marquez^2†
^1University of Sheffield, ^2Instituto de Astrofisica de Canarias
^†Listed affiliation is based on previous publications and was not specified in this preprint.
ArXiv #: 1210.6476 (PDF, PS, ADS, Papers, Other)
Comments: 7 Pages and 7 figures
Originally posted 10/24/2012
By focusing on the oscillations of the cross-sectional area and the intensity of magnetic waveguides located in the lower solar atmosphere, we aim to detect and identify magnetohydrodynamic (MHD)
sausage waves. Capturing several series of high-resolution images of pores and sunspots and employing wavelet analysis in conjunction with empirical mode decomposition (EMD) makes the MHD wave
analysis possible. For this paper, two sunspots and one pore (with a light bridge) were chosen as representative examples of MHD waveguides in the lower solar atmosphere. The sunspots and pore
display a range of periods from 4 to 65 minutes. The sunspots support longer periods than the pore – generally enabling a doubling or quadrupling of the maximum pore oscillatory period. All of these
structures display area oscillations indicative of MHD sausage modes and in-phase behaviour between the area and intensity, presenting mounting evidence for the presence of the slow sausage mode
within these waveguides. The presence of fast and slow MHD sausage waves has been detected in three different magnetic waveguides in the lower solar photosphere. Furthermore, these oscillations are
potentially standing harmonics supported in the waveguides which are sandwiched vertically between the temperature minimum in the lower solar atmosphere and the transition region. Standing harmonic
oscillations, by means of solar magneto-seismology, may allow insight into the sub-resolution structure of photospheric MHD waveguides.
On standing sausage waves in photospheric magnetic waveguides [Replacement]
0 votes @Harvard-Smithsonian CfA
(0 votes over all institutions)
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I. Dorotovic, R. Erdelyi^1†, N. Freij, V. Karlovsky, I. Marquez^2†
^1University of Sheffield, ^2Instituto de Astrofisica de Canarias
^†Listed affiliation is based on previous publications and was not specified in this preprint.
ArXiv #: 1210.6476 (PDF, PS, ADS, Papers, Other)
Comments: 7 Pages, 7 figures and submitted to A&A
Originally posted 10/25/2012
By focusing on the oscillations of the cross-sectional area and the intensity of magnetic waveguides located in the lower solar atmosphere, we aim to detect and identify magnetohydrodynamic (MHD)
sausage waves. Capturing several series of high-resolution images of pores and sunspots and employing wavelet analysis in conjunction with empirical mode decomposition (EMD) makes the MHD wave
analysis possible. For this paper, two sunspots and one pore (with a light bridge) were chosen as representative examples of MHD waveguides in the lower solar atmosphere. The sunspots and pore
display a range of periods from 4 to 65 minutes. The sunspots support longer periods than the pore – generally enabling a doubling or quadrupling of the maximum pore oscillatory period. All of these
structures display area oscillations indicative of MHD sausage modes and in-phase behaviour between the area and intensity, presenting mounting evidence for the presence of the slow sausage mode
within these waveguides. The presence of fast and slow MHD sausage waves has been detected in three different magnetic waveguides in the lower solar photosphere. Furthermore, these oscillations are
potentially standing harmonics supported in the waveguides which are sandwiched vertically between the temperature minimum in the lower solar atmosphere and the transition region. Standing harmonic
oscillations, by means of solar magneto-seismology, may allow insight into the sub-resolution structure of photospheric MHD waveguides.
On standing sausage waves in photospheric magnetic waveguides [Replacement]
0 votes @Harvard-Smithsonian CfA
(0 votes over all institutions)
Please log in or create an account to vote!
I. Dorotovic, R. Erdelyi^1†, N. Freij, V. Karlovsky, I. Marquez^2†
^1University of Sheffield, ^2Instituto de Astrofisica de Canarias
^†Listed affiliation is based on previous publications and was not specified in this preprint.
ArXiv #: 1210.6476 (PDF, PS, ADS, Papers, Other)
Comments: 7 Pages, 7 figures and submitted to A&A
Originally posted 01/07/2013
By focusing on the oscillations of the cross-sectional area and the intensity of magnetic waveguides located in the lower solar atmosphere, we aim to detect and identify magnetohydrodynamic (MHD)
sausage waves. Capturing several series of high-resolution images of pores and sunspots and employing wavelet analysis in conjunction with empirical mode decomposition (EMD) makes the MHD wave
analysis possible. For this paper, two sunspots and one pore (with a light bridge) were chosen as representative examples of MHD waveguides in the lower solar atmosphere. The sunspots and pore
display a range of periods from 4 to 65 minutes. The sunspots support longer periods than the pore – generally enabling a doubling or quadrupling of the maximum pore oscillatory period. All of these
structures display area oscillations indicative of MHD sausage modes and in-phase behaviour between the area and intensity, presenting mounting evidence for the presence of the slow sausage mode
within these waveguides. The presence of fast and slow MHD sausage waves has been detected in three different magnetic waveguides in the lower solar photosphere. Furthermore, these oscillations are
potentially standing harmonics supported in the waveguides which are sandwiched vertically between the temperature minimum in the lower solar atmosphere and the transition region. Standing harmonic
oscillations, by means of solar magneto-seismology, may allow insight into the sub-resolution structure of photospheric MHD waveguides.
Analysis and Modeling of Two Flare Loops Observed by AIA and EIS
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Y. Li^1†, J. Qiu^2†, M. D. Ding
^1Department of Physics, Institute of Astrophysics, & Center for Theoretical Sciences, National Taiwan University, ^2Mathematical and Computer Science, Colorado School of Mines, USA
^†Listed affiliation is based on previous publications and was not specified in this preprint.
ArXiv #: 1208.5440 (PDF, PS, ADS, Papers, Other)
Comments: Accepted for publication in ApJ
Originally posted 08/27/2012
We analyze and model an M1.0 flare observed by SDO/AIA and Hinode/EIS to investigate how flare loops are heated and evolve subsequently. The flare is composed of two distinctive loop systems observed
in EUV images. The UV 1600 \AA emission at the feet of these loops exhibits a rapid rise, followed by enhanced emission in different EUV channels observed by AIA and EIS. Such behavior is indicative
of impulsive energy deposit and the subsequent response in overlying coronal loops that evolve through different temperatures. Using the method we recently developed, we infer empirical heating
functions from the rapid rise of the UV light curves for the two loop systems, respectively, treating them as two big loops of cross-sectional area 5\arcsec by 5\arcsec, and compute the plasma
evolution in the loops using the EBTEL model (Klimchuk et al. 2008). We compute the synthetic EUV light curves, which, with the limitation of the model, reasonably agree with observed light curves
obtained in multiple AIA channels and EIS lines: they show the same evolution trend and their magnitudes are comparable by within a factor of two. Furthermore, we also compare the computed mean
enthalpy flow velocity with the Doppler shift measurements by EIS during the decay phase of the two loops. Our results suggest that the two different loops with different heating functions as
inferred from their footpoint UV emission, combined with their different lengths as measured from imaging observations, give rise to different coronal plasma evolution patterns captured both in the
model and observations.
Cryogenic microstripline-on-Kapton microwave interconnects
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A. I. Harris^1†, M. Sieth, J. M. Lau, S. E. Church, L. A. Samoska^2†, K. Cleary^3†
^1Maryland, ^2JPL, ^3Caltech
^†Listed affiliation is based on previous publications and was not specified in this preprint.
ArXiv #: 1206.1461 (PDF, PS, ADS, Papers, Other)
Comments: 3 pages, 3 figures, submitted to The Review of Scientific Instruments
Originally posted 06/07/2012
Simple broadband microwave interconnects are needed for increasing the size of focal plane heterodyne radiometer arrays. We have measured loss and cross-talk for arrays of microstrip transmission
lines in flex circuit technology at 297 and 77 K, finding good performance to at least 20 GHz. The dielectric constant of Kapton substrates changes very little from 297 to 77 K, and the electrical
loss drops. The small cross-sectional area of metal in a printed circuit structure yields overall thermal conductivities similar to stainless steel coaxial cable. Operationally, the main performance
tradeoffs are between crosstalk and thermal conductivity. We tested a patterned ground plane to reduce heat flux.
Very High Resolution Solar X-ray Imaging Using Diffractive Optics
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B. R. Dennis, G. K. Skinner^1†, M. J. Li, A. Y. Shih
^†Listed affiliation is based on previous publications and was not specified in this preprint.
ArXiv #: 1205.4762 (PDF, PS, ADS, Papers, Other)
Originally posted 05/22/2012
This paper describes the development of X-ray diffractive optics for imaging solar flares with better than 0.1 arcsec angular resolution. X-ray images with this resolution of the \geq10 MK plasma in
solar active regions and solar flares would allow the cross-sectional area of magnetic loops to be resolved and the coronal flare energy release region itself to be probed. The objective of this work
is to obtain X-ray images in the iron-line complex at 6.7 keV observed during solar flares with an angular resolution as fine as 0.1 arcsec – over an order of magnitude finer than is now possible.
This line emission is from highly ionized iron atoms, primarily Fe xxv, in the hottest flare plasma at temperatures in excess of \approx10 MK. It provides information on the flare morphology, the
iron abundance, and the distribution of the hot plasma. Studying how this plasma is heated to such high temperatures in such short times during solar flares is of critical importance in understanding
these powerful transient events, one of the major objectives of solar physics. We describe the design, fabrication, and testing of phase zone plate X-ray lenses with focal lengths of \approx100 m at
these energies that would be capable of achieving these objectives. We show how such lenses could be included on a two-spacecraft formation-flying mission with the lenses on the spacecraft closest to
the Sun and an X-ray imaging array on the second spacecraft in the focal plane \approx100 m away. High resolution X-ray images could be obtained when the two spacecraft are aligned with the region of
interest on the Sun. Requirements and constraints for the control of the two spacecraft are discussed together with the overall feasibility of such a formation-flying mission.
Longitudinal oscillations in density stratified and expanding solar waveguides
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M. Luna-Cardozo, G. Verth, R. Erdélyi
ArXiv #: 1204.4201 (PDF, PS, ADS, Papers, Other)
Comments: 10 pages, 5 figures, published in ApJ, uses emulateapj
Journal: Astrophys.J.748:110-119,2012
Originally posted 04/19/2012
Waves and oscillations can provide vital information about the internal structure of waveguides they propagate in. Here, we analytically investigate the effects of density and magnetic stratification
on linear longitudinal magnetohydrodynamic (MHD) waves. The focus of this paper is to study the eigenmodes of these oscillations. It is our specific aim is to understand what happens to these MHD
waves generated in flux tubes with non-constant (e.g., expanding or magnetic bottle) cross-sectional area and density variations. The governing equation of the longitudinal mode is derived and solved
analytically and numerically. In particular, the limit of the thin flux tube approximation is examined. The general solution describing the slow longitudinal MHD waves in an expanding magnetic flux
tube with constant density is found. Longitudinal MHD waves in density stratified loops with constant magnetic field are also analyzed. From analytical solutions, the frequency ratio of the first
overtone and fundamental mode is investigated in stratified waveguides. For small expansion, a linear dependence between the frequency ratio and the expansion factor is found. From numerical
calculations it was found that the frequency ratio strongly depends on the density profile chosen and, in general, the numerical results are in agreement with the analytical results. The relevance of
these results for solar magneto-seismology is discussed.
1D Modeling for Temperature-Dependent Upflow in the Dimming Region Observed by Hinode/EIS
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S. Imada, H. Hara, T. Watanabe, I. Murakami, L. K. Harra, T. Shimizu^1†, E. G. Zweibel^2†
^1Institute of Space and Astronautical Science, Sagamihara, Japan, ^2University of Wisconsin-Madison
^†Listed affiliation is based on previous publications and was not specified in this preprint.
ArXiv #: 1108.5031 (PDF, PS, ADS, Papers, Other)
Comments: accepted for publication in The Astrophysical Journal
Originally posted 08/25/2011
We have previously found a temperature-dependent upflow in the dimming region following a coronal mass ejection (CME) observed by the {\it Hinode} EUV Imaging Spectrometer (EIS). In this paper, we
reanalyzed the observations along with previous work on this event, and provided boundary conditions for modeling. We found that the intensity in the dimming region dramatically drops within 30
minutes from the flare onset, and the dimming region reaches the equilibrium stage after $\sim$1 hour later. The temperature-dependent upflows were observed during the equilibrium stage by EIS. The
cross sectional area of the fluxtube in the dimming region does not appear to expand significantly. From the observational constraints, we reconstructed the temperature-dependent upflow by using a
new method which considers the mass and momentum conservation law, and demonstrated the height variation of plasma conditions in the dimming region. We found that a super radial expansion of the
cross sectional area is required to satisfy the mass conservation and momentum equations. There is a steep temperature and velocity gradient of around 7 Mm from the solar surface. This result may
suggest that the strong heating occurred above 7 Mm from the solar surface in the dimming region. We also showed that the ionization equilibrium assumption in the dimming region is violated
especially in the higher temperature range.
Debris disk size distributions: steady state collisional evolution with P-R drag and other loss processes
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Mark C. Wyatt^1†, Cathie J. Clarke^2†, Mark Booth^2†
^1IoA, ^2IoA, Cambridge
^†Listed affiliation is based on previous publications and was not specified in this preprint.
ArXiv #: 1103.5499 (PDF, PS, ADS, Papers, Other)
Comments: Accepted for publication by Celestial Mechanics and Dynamical Astronomy (special issue on EXOPLANETS)
Originally posted 03/29/2011
We present a new scheme for determining the shape of the size distribution, and its evolution, for collisional cascades of planetesimals undergoing destructive collisions and loss processes like
Poynting-Robertson drag. The scheme treats the steady state portion of the cascade by equating mass loss and gain in each size bin; the smallest particles are expected to reach steady state on their
collision timescale, while larger particles retain their primordial distribution. For collision-dominated disks, steady state means that mass loss rates in logarithmic size bins are independent of
size. This prescription reproduces the expected two phase size distribution, with ripples above the blow-out size, and above the transition to gravity-dominated planetesimal strength. The scheme also
reproduces the expected evolution of disk mass, and of dust mass, but is computationally much faster than evolving distributions forward in time. For low-mass disks, P-R drag causes a turnover at
small sizes to a size distribution that is set by the redistribution function (the mass distribution of fragments produced in collisions). Thus information about the redistribution function may be
recovered by measuring the size distribution of particles undergoing loss by P-R drag, such as that traced by particles accreted onto Earth. Although cross-sectional area drops with 1/age^2 in the
PR-dominated regime, dust mass falls as 1/age^2.8, underlining the importance of understanding which particle sizes contribute to an observation when considering how disk detectability evolves. Other
loss processes are readily incorporated; we also discuss generalised power law loss rates, dynamical depletion, realistic radiation forces and stellar wind drag.
Hysteresis of Backflow Imprinted in Collimated Jets
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Akira Mizuta, Motoki Kino, Hiroki Nagakura
ArXiv #: 0912.3662 (PDF, PS, ADS, Papers, Other)
Comments: 9 pages, 5 figures, accepted for publication in ApJL
Originally posted 12/20/2009
We report two different types of backflow from jets by performing 2D special relativistic hydrodynamical simulations. One is anti-parallel and quasi-straight to the main jet (quasi-straight
backflow), and the other is bent path of the backflow (bent backflow). We find that the former appears when the head advance speed is comparable to or higher than the local sound speed at the hotspot
while the latter appears when the head advance speed is slower than the sound speed bat the hotspot. Bent backflow collides with the unshocked jet and laterally squeezes the jet. At the same time, a
pair of new oblique shocks are formed at the tip of the jet and new bent fast backflows are generated via these oblique shocks. The hysteresis of backflow collisions is thus imprinted in the jet as a
node and anti-node structure. This process also promotes broadening of the jet cross sectional area and it also causes a decrease in the head advance velocity. This hydrodynamic process may be tested
by observations of compact young jets.
On the origin of Fanaroff-Riley classification of radio galaxies: Deceleration of supersonic radio lobes
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Nozomu Kawakatu, Motoki Kino, Hiroshi Nagai
ArXiv #: 0904.4752 (PDF, PS, ADS, Papers, Other)
Comments: 5 pages, 1 figure, accepted for publication in ApJ Letters
Originally posted 05/03/2009
We argue that the origin of "FRI/FRI{-.1em}I dichotomy" — the division between Fanaroff-Riley class I (FRI) with subsonic lobes and class I{-.1em}I (FRI{-.1em}I) radio sources with supersonic lobes
is sharp in the radio-optical luminosity plane (Owen-White diagram) — can be explained by the deceleration of advancing radio lobes. The deceleration is caused by the growth of the effective
cross-sectional area of radio lobes. We derive the condition in which an initially supersonic lobe turns into a subsonic lobe, combining the ram-pressure equilibrium between the hot spots and the
ambient medium with the relation between "the hot spot radius" and "the linear size of radio sources" obtained from the radio observations. We find that the dividing line between the supersonic lobes
and subsonic ones is determined by the ratio of the jet power $L_{\rm j}$ to the number density of the ambient matter at the core radius of the host galaxy $\bar{n}_{\rm a}$. It is also found that
there exists the maximal ratio of $(L_{\rm j}/\bar{n}_{\rm a})$ and its value resides in $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}\approx 10^{44-47} {\rm erg} {\rm s}^{-1} {\rm cm}^{3}$, taking account
of considerable uncertainties. This suggests that the maximal value $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ separates between FRIs and FRI{-.1em}Is.
Fate of baby radio galaxies: Dead or Alive ?
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Nozomu Kawakatu, Hiroshi Nagai, Motoki Kino
ArXiv #: 0812.1353 (PDF, PS, ADS, Papers, Other)
Comments: 5 pages and 3 figures, accepted for publication in Astronomische Nachrichten (issue dedicated to the Proceedings of "The 4th Workshop on Compact Steep Spectrum and GHz-Peaked Spectrum Radio
Sources" held at Riccione, Italy, 26-29 May 2008)
Originally posted 12/08/2008
In order to reveal the long-term evolution of relativistic jets in active galactic nuclei (AGNs), we examine the dynamical evolution of variously-sized radio galaxies [i.e., compact symmetric objects
(CSOs), medium-size symmetric objects (MSOs), Fanaroff-Riley type II radio galaxies (FRIIs)]. By comparing the observed relation between the hot spot size and the linear size of radio source with a
coevolution model of hot spot and cocoon, we find that the advance speed of hot spots and lobes inevitably show the deceleration phase (CSO-MSO phase) and the acceleration phase (MSO-FRII phase). The
deceleration is caused by the growth of the cross-sectional area of the cocoon head. Moreover, by comparing the hot spot speed with the sound speed of the ambient medium, we predict that only CSOs
whose initial advance speed is higher than 0.3-0.5c can evolve into FRIIs. | {"url":"http://harvard.voxcharta.org/tag/cross-sectional-area/","timestamp":"2014-04-16T10:14:46Z","content_type":null,"content_length":"103057","record_id":"<urn:uuid:cf742267-f7fb-4ee0-9894-b4c8aff5f927>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00387-ip-10-147-4-33.ec2.internal.warc.gz"} |
Divisibility 13
Are you trying to prove the converse, namely $a^k\mid b^k\ \Rightarrow\ a\mid b\,?$
I don't know... But good question... $a^3|b^3 \Rightarrow a|b$ (Why?) ...write... http://www.mathhelpforum.com/math-help/number-theory/65693-divisibility-14-a.html I think...Maybe use binomial
I would suggest maybe using prime factorization. $<br /> a^k|b^k (\Rightarrow):$ I think; $(p_1 \times p_2 \times p_3 \times ....\times p_m,N)=1$ and $(p_1 , p_2 , p_3 , ... , p_m)=1$ $a^k=(p_1^{y_1}
\times p_2^{y_2}\times p_3^{y_3} \times .......p_m^{y_S})^k$ $=(p_1^{y_1})^k\times (p_2^{y_2})^k\times (p_3^{y_3})^k\times .......(p_m^{y_S})^k$ $<br /> =(p_1^{ky_1}\times p_2^{ky_2}\times p_3^{ky_3}
\times .......p_m^{ky_S})<br />$ $<br /> b^k=(p_1^{z_1}\times p_2^{z_2}\times p_3^{z_3} \times .......p_m^{z_S})^k \times N$ $<br /> =(p_1^{z_1})^k\times (p_2^{z_2})^k\times (p_3^{z_3})^k \times
.......(p_m^{z_S})^k \times N<br />$ $=p_1^{kz_1}\times p_2^{kz_2}\times p_3^{kz_3} \times .......p_m^{kz_S} \times N$ $a^k|b^k \Rightarrow p_i^{ky_i} | p_i^{kz_i}$ and $i=1,2,3,....,s$$\Rightarrow
ky_i \leq kz_i \Rightarrow y_i \leq z_i , k\in Z^+ \Rightarrow p_i^{y_i} | p_i^{z_i}$ I think...?
Last edited by Sea; December 25th 2008 at 11:34 PM.
If your'e trying to prove that a^k|b^k => a|b then a^k=nb^k (a/b)^k=n a/b=n^1/k Now a|b iff n=m^k for some integer m, the problem is to show that it is so. You can assume by contradiction that a!|b
which means there exists p and r such that: a=pb+r r<=b Now write a^k=(pb+r)^k and see that it contradicts our assumption. | {"url":"http://mathhelpforum.com/number-theory/65685-divisibility-13-a.html","timestamp":"2014-04-18T21:55:31Z","content_type":null,"content_length":"51433","record_id":"<urn:uuid:3075a5d5-6f62-4237-b048-c665a8579549>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00033-ip-10-147-4-33.ec2.internal.warc.gz"} |
Greenbelt Statistics Tutor
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A Course in Commutative Algebra
A Course in Commutative Algebra
A Course in Commutative Algebra
by Gregor Kemper
Published: 2010-12-10 | ISBN: 3642035442 | PDF | 257 pages | 3 MB
This textbook offers a thorough, modern introduction into commutative algebra. It is intented mainly to serve as a guide for a course of one or two semesters, or for self-study. The carefully
selected subject matter concentrates on the concepts and results at the center of the field. The book maintains a constant view on the natural geometric context, enabling the reader to gain a deeper
understanding of the material. Although it emphasizes theory, three chapters are devoted to computational aspects. Many illustrative examples and exercises enrich the text.
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Intermediate Value Theorem!
April 4th 2010, 08:06 PM #1
Apr 2010
Intermediate Value Theorem!
Please! Can some show how Ax^3 + Bx^2 + CX + D has atleast one real root using the Intermediate Value Theorem?
I don't know about Intermediate Value Theorem, but by the fundamental theorem of polynomials, there are as many complex roots as the degree of the polynomial.
So every cubic has 3 complex roots.
But since roots that have an imaginary part always appear as complex conjugates, that means that in this case, at most there are 2 roots with imaginary part. That means there must be at least one
real root.
I highly suspect that the proof of your fundamental theorem of polynomials *uses* the fact that any odd degree polynomial has a root in R. At least the only proof I know to prove the fundamental
theorem of algebra, which uses a combination of Galois theory and Sylow theory, indeed uses this fact. I would appreciate it if you could have a check and tell me whether it uses this fact or
And there's a tiny bit problem in the statement. A should be nonzero otherwise the statement is false.
If A is nonzero, then WLOG a>0. The goal is to find two points x_1,x_2 such that f(x_1)>0 but f(x_2)<0. By IVT, there exists some zero in between. x_1 is chosen as big as possible, and x_2 is
chosen as small (negative and the abs is as big) as possible. You can set any bound you like, as long as you can prove f(x_1)>0,f(x_2)<0.
I highly suspect that the proof of your fundamental theorem of polynomials *uses* the fact that any odd degree polynomial has a root in R. At least the only proof I know to prove the fundamental
theorem of algebra, which uses a combination of Galois theory and Sylow theory, indeed uses this fact. I would appreciate it if you could have a check and tell me whether it uses this fact or
And there's a tiny bit problem in the statement. A should be nonzero otherwise the statement is false.
If A is nonzero, then WLOG a>0. The goal is to find two points x_1,x_2 such that f(x_1)>0 but f(x_2)<0. By IVT, there exists some zero in between. x_1 is chosen as big as possible, and x_2 is
chosen as small (negative and the abs is as big) as possible. You can set any bound you like, as long as you can prove f(x_1)>0,f(x_2)<0.
A quicker way to state this (it really is just masking what FancyMouse said, it's quicker if you're allowed to use it) is that $\lim_{x\to\pm\infty}P(x)=\pm\text{sgn } A\text{ }\infty$. And thus,
it is clear that you may choose $|T|$ large enough so that (WLOG the other case is analgous) $P(x)<0, x<T$ and $P(x)>0,\text{ }x>T$. Then, apply the IVT.
By the way, as stated this is NOT true. As FancyMouse said, A must be non-zero to guarentee this statement.
If A= 0, B= 1, C= 0, and D= 1, the polynomial has NO real roots.
April 4th 2010, 08:26 PM #2
April 4th 2010, 09:56 PM #3
Apr 2010
April 4th 2010, 10:01 PM #4
April 5th 2010, 04:59 AM #5
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Topic: Missouri State University Problem Corner
Replies: 5 Last Post: Dec 17, 2012 2:23 AM
Messages: [ Previous | Next ]
Re: Missouri State University Problem Corner
Posted: Dec 17, 2012 2:23 AM
Michael Press wrote: (but not necessarily in this order)
> Brian Chandler <imaginatorium@despammed.com> wrote:
> > Michael Press wrote:
> > > William Elliot <marsh@panix.com> wrote:
> > > > > Three unit spheres are mutually tangent to one another and to a
> > > > > hemisphere, both along the spherical part of the hemisphere and
> > > > > along its equatorial plane. Find the radius of the hemisphere.
Well, the question is slightly vaguely worded, but I take "tangent to
a hemisphere" to mean "touching a part of the hemisphere", so the bit
about the equatorial plane needs to be inside the "equator".
> > There's a real question: what position _is_ the centre of a
> > hemisphere...
> You labeled it C. I labeled it O. No question here.
Just a nitpick: is the "centre" of a hemisphere the centre of its
"equatorial plane"? It could be the centre of gravity of the
hemisphere, for example (wherever that is, exactly)... But OK, I'll
call it O too.
> > Three unit spheres touch; therefore their centres (S1, S2, S3) are at
> > the vertices of an equilateral triangle of side 2. They also sit on
> > the flat face of the hemisphere; so the height of each of the centres
> > over the flat face is 1, and symmetry implies that the centre (C) of
> > the hemisphere is under the centre (T) of the equilateral triangle,
> > and the centres S1, S2, S3, and C form a pyramid of height 1 on a
> > base of side 2.
> >
> > Since each sphere touches the curved surface of the hemisphere, the
> > normal to the point of contact goes through the centre of the sphere
> > and the centre of the flat face. Therefore the radius of the
> > hemisphere is the length of a sloping edge of the pyramid (e) plus the
> > radius of a sphere (1).
> >
> > (Using r() for square root...)
> > Two applications of Pythagoras' theorem give us the distance from the
> > centre of the triangle to a vertex:
> >
> > d = 2 / r(3) (30-60-90 triangle; longer right side = 1)
> >
> > And sloping edge
> >
> > e = r( d^2 + 1^2) = r( 4/3 + 1) = r(7/3)
> >
> > So radius of hemisphere is 1 + r(7/3)
> Why?
Because the distance from the centre (O) of the flat face to the
surface of the hemisphere equals the distance to the centre of a
sphere (=sloping edge of the pyramid (e)) plus the distance (1) from
this centre to the surface of the hemisphere. (Because they're in a
straight line)
> > > That looks large. I get a different answer.
> >
> > Hmm.
> >
> > > Let r denote the radius of the hemisphere.
> > > Label the center of the hemisphere O.
> >
> Looks like the difference is whether the unit spheres
> are internally or externally tangent to the hemisphere.
Right. It you phrase the question as three touching spheres resting on
a plane, then there are two hemispherical bubble on the plane which
touch the spheres, one inside, one outside, and their radii are:
e +- 1 QED
Brian Chandler
Date Subject Author
12/4/12 Missouri State University Problem Corner Les
12/9/12 Re: Missouri State University Problem Corner William Elliot
12/15/12 Re: Missouri State University Problem Corner Michael Press
12/16/12 Re: Missouri State University Problem Corner Brian Chandler
12/16/12 Re: Missouri State University Problem Corner Michael Press
12/17/12 Re: Missouri State University Problem Corner Brian Chandler | {"url":"http://mathforum.org/kb/thread.jspa?threadID=2418593&messageID=7937975","timestamp":"2014-04-17T19:38:01Z","content_type":null,"content_length":"25550","record_id":"<urn:uuid:4262251c-6ded-4517-bb51-affcc12969a3>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00174-ip-10-147-4-33.ec2.internal.warc.gz"} |
Delay/Wait/Pause in Java + AirBrushing
October 27th, 2009, 06:22 AM #1
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I have two problems I am trying to solve.
I am using the program RealJ and JDK 1.6
(I seem to have a hard time finding tutorials on the internet because of this could anyone recommend a way to search?)
First thing I am trying to do is make Java wait. I cant seem to find a workable code. And I cant seem to slow it down by simply slugging it with complex number problems) Could anyone tell me what
I can use to make Java wait?
The other thing I am trying to do is AirBrushing (like Microsoft Paint)
The idea is to draw a random dot in a random location over the area of a circle.
Its extremely easy for a square..
public void mouseMoved (MouseEvent e)
for(int i=1; i<120; i++)
{ q=(int)(Math.random()*120+1+599);
This simulates the general idea..but I'm not much for doing math. How would I do it over a circle?
I figured select a starting point(mousex,mousey) and draw a 1v1 square at a random direction,random distance over a fixed max distance...but I cannot for the life of me figure out how to do this
(Help with sin/cos would be good)
Last edited by literallyjer; October 27th, 2009 at 07:31 AM.
I've tried using that but the problem is I cannot repaint while it is being used as it halts the entire program (except for showStatus(""); I noticed)
how can I repaint like this?
if (x>200 && y>200)
for(int i=1; i<20; i++)
if (i%2 != 0)
catch(Exception ee) {}
else if (i%2 == 0)
catch(Exception ee) {}
Each time it sleeps the showStatus changes but it does not repaint.
So are you saying that you want a certain action to happen every 90ms but you want the repaint method to happen all the time?
// Json
Create a separate thread that will do the action every 90ms, but have your main thread do the re-painting.
For some reason it didn't click that you were using Swing. You have to be really careful when using multiple threads with Swing. I suggest you look into SwingUtilities.invokeLater() and
The Java API: http://java.sun.com/javase/6/docs/api/
I would avoid the multiple threads solution and just have an update method and a repaint method. This is how I write my games btw.
You call both methods all the time but in the update method you check the time since last update and make sure that the code inside the method only gets run every 90 milliseconds. That should
sort you out.
// Json
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Transformation Of Probability Density Functions
1. The problem statement, all variables and given/known data
Let X and Y be random variables. The pdfs are [itex]f_X(x)=2(1-x)[/itex] and [itex]f_Y(y) = 2(1-y)[/itex]. Both distributions are defined on [0,1].
Let Z = X + Y. Find the pdf for Z, [itex]f_Z(z)[/itex].
2. Relevant equations
I'm using ideas, not equations.
3. The attempt at a solution
I'm dying of curiosity about where I'm going wrong. I'm so sure of each step, but my answer can't be correct because [itex]\int_0^2 f_Z(z)\,dz[/itex] is zero! Here's my logic.
Consider the cdf (cumulative distribution function) for Z:
F_Z(z) = P(Z\le z) = P(X+Y \le z)
Here, [itex]F_Z(z)[/itex] is the volume above the triangle shown in the image I attached to this message (in case something happens to the attachment, it's the triangle in quadrant 1 bounded by x=0,
y=0 and x+y=z.)
The volume above the shaded region represents [itex]F_Z(z)[/itex].
F_Z(z) = \int_{x=0}^{x=z} 2(1-x)\int_{y=0}^{y=z-x} 2(1-y)\,dy\,dx
Performing the integrals gives [itex]F_Z(z) = \frac{1}{6}z^4 - \frac{4}{3}z^3 + 2z^2[/itex]. Then taking the derivative of the cdf gives the pdf:
f_Z(z) = \partial_z F_Z(z) = \frac{2}{3}z^3 - 4z^2 +4z
Unfortunately, this can't be right because the integral of this function over [0,2] gives zero.
I also would've expected that the maximum of [itex]f_Z(z)[/itex] would be at z=0 since individually, X and Y are most likely to be zero.
I checked my algebra and calculus with Mathematica; it looked fine. I think there's a conceptual problem. There must be something I don't understand or some point I'm not clear about.
What did I do wrong? | {"url":"http://www.physicsforums.com/showthread.php?t=152331","timestamp":"2014-04-20T08:33:19Z","content_type":null,"content_length":"28025","record_id":"<urn:uuid:79b4057d-19aa-49e1-a957-8d2b931454e2>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00431-ip-10-147-4-33.ec2.internal.warc.gz"} |
Algebra One
Remember all those rationals and irrationals, the ones we got confused?
I still get them wrong and I am not very amused.
Digits and decimals, too many D's
But wait until we start on those x, y, and z's.
Powers of 10 are expressed by exponents in many ways,
But, remember, there is only one and that is to have them raised.
Counting all those zeros makes me very crazy,
So use that scientific notation and don't be so lazy.
Collections of elements in sets are a breeze.
Elements follow a pattern just like 1, 2, and 3's.
A variable is a symbol like x which has 1 value, not 2.
If you put the right variable in an open sentence, the statement will be true.
Factoring numbers can be such a bore!
But, prime numbers are the best since the factors are only 1 and itself, and nothing more.
Basic axioms of algebra leave you in awe,
But not to understand them is one major flaw.
The most important axiom is the distributive one.
It's the one we used the most, but it wasn't that fun.
Reciprocals are the numbers that always do a flip upside down.
Inverses are very different and in their own special way they turn around.
Numbers and variables are jumbled disorderly,
Are called equations and are only solved algebraically
Solving inequalities can make you want to die,
But, there are only 3 choices: greater than, less than, or equal to, so there is no need to cry.
Polynomials can leave you in such a disarray,
But, just remember there are coefficients and constant terms and then you will be straightened out today.
Products of binomials can take you so long to do a few.
But, just skip the steps and use FOIL without a big to do.
Figuring out ordered pairs can leave you in such a mess.
But, if you remember the x-axis, y-axis, and origin, you'll do them with success.
Systems of linear equations can be solved in 4 different ways.
Substitution, addition method, determinants, and graphing with different rays.
The slope always equals rise over run.
If you remember this you'll always get those problems done.
Quotients of 2 polynomials are called Rational Algebraic Expression.
If you don't reduce them fully they will leave you in a great big depression.
Square roots and cubic roots leave me very puzzled, But the index and radicand leave me troubled.
Square-free integers can't be broken down anymore.
They are like 2, 3, and 5, but never integers like 16 or 4.
There are many other rules, guidelines and steps to Algebra 1
But, we still have Geometry, Algebra 2 and Analysis to continue the fun!!!
This piece has been published in Teen Ink’s monthly print magazine.
Join the Discussion
This article has 155 comments. Post your own!
gman123 said...
Jan. 19, 2011 at 9:06 am:
this poem really desribes how challenging algebra one is
I got scared at first when I saw the title of this poem, being a freshman in high school, but as I read it, I laughed.
It's been a while since somebody has made me laugh. Be proud of your work.
I just sent this to my friend Tyler.
you should make this lyrics to a song.
parody mebbe...
oh, goodness. I couldn't STAND geometry. Far too many constructions for my liking.
Ha, this was great! Please do one on Algebraic Graph Theory when you get the chance. :)
DIE ALGEBRA!!!! D8
Excellentness=) Algebra is the bane of student existence... at least we won't have to do it in the real world =P
I'm a junior in algebra 2 this year and I HATE it. I wanna go back to Geometry. =.=
memeko said...
Nov. 27, 2010 at 8:43 pm:
i need to memorize this by tuesday for our speech choir..
JOJOkester said...
Nov. 23, 2010 at 11:05 am:
this reminds me so much of my algebra 1 class only if you put in something about a funny talking phillipino teacher. AWESOME POEM!!!
Very creative! I wrote a poem similar to this a few weeks ago but I never posted it lol
clever =.=
uh, i hated geometry, and now it turns out the curriculum we used was old, isnt used anymore and is harder than the newer versions!
i hated algebra with a firey passion
have you done algebra 2 yet?
more algebra, but harder (although honestly, not by much, but if you hated algebra 1, then you'll prob hate algebra 2 as well)
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Stephen A
Stephen Arthur Cook was born on December 14, 1939 in Buffalo, NY. Cook’s father worked as a chemist for a subsidiary of Union Carbide, and was also an adjunct professor at SUNY Buffalo. His mother
worked as a homemaker and also as an occasional English teacher at Erie Community College. When he was ten, Cook moved with his family to Clarence, NY, which was the home of Wilson Greatbatch, the
inventor of the implantable pacemaker. As a teenager, Stephen developed an interest in electronics and worked forGreatcatch, who was then experimenting with transistor-based circuitry.
Cook entered the University of Michigan in 1957, majoring in science engineering. He was introduced to computer programming in a freshman course taught by Bernard Galler. With a fellow student he
wrote a program to test Goldbach’s conjecture that every even integer greater than two is the sum of two primes. Stephen eventually changed his major to mathematics, and received his Bachelor’s
degree in 1961. He subsequently entered graduate studies in the Mathematics Department at Harvard University, where he encountered influences that would shape the direction of his future research,
including Michael Rabin’s early work on computational complexity, Alan Cobham’s characterization of polynomial time computable functions, and his supervisor Hao Wang’s investigations in automated
theorem proving. Cook received his Ph.D. in 1966. His thesis, titled On the Minimum Computation time of Functions, addresses the intrinsic computational complexity of multiplication. One contribution
of the thesis was an improvement of Andrei Toom’s multiplication algorithm, which is now known as Toom-Cook multiplication. This algorithm is still a subject of study and is of practical importance
in high-precision arithmetic.
After graduating, he joined the Mathematics Department at the University of California, Berkeley, leaving there in 1970 to take the position of Associate Professor in Computer Science at the
University of Toronto. A year later, Cook presented his seminal paper, “The complexity of theorem proving procedures,” at the 3rd Annual ACM Symposium on Theory of Computing [1]. This paper marked
the introduction of the theory of NP-completeness, which henceforth occupied a central place in theoretical computer science. (Leonid Levin independently introduced NP-completeness in 1973.) It was
also an early contribution to the theory of propositional proof complexity, an area in which Cook continued to do extensive research over the next 40 years.
The theory of NP-completeness provides a way to characterize the difficulty of computational problems with respect to the time, that is, the maximum number of computational steps required to solve a
problem, as a function of input size. Cook’s paper addresses the fact that many problems which are difficult to solve in this sense have solutions which are easy to verify once they are found. In
current terminology, problems which are easy to solve comprise the class P, while those which are easy to verify comprise the class NP. Cook showed that certain problems in NP, now known as
NP-complete problems, are as hard to solve as any others in NP, in the sense that if any one of these problems is easy to solve, then all problems in NP are easy to solve. Cook’s paper also was the
source of the celebrated and still unsolved P versus NP question, which asks whether there are indeed problems in NP which are not in P, that is, problems whose solutions are easily verified but are
not easily solved. A simple example which elucidates the relationship between the classes P and NP is the popular Sudoku puzzle (see here).
Cook’s paper had an immediate impact. In 1972 Richard Karp published a paper in which he showed that twenty-one problems in combinatorics, optimization and graph theory which were believed to be
computationally difficult were indeed NP-complete. Seven years later, Michael Garey and David Johnson published the book Computers and Intractability: A Guide to the Theory of NP-Completeness, which
included a compendium of over three hundred problems which by then had been shown to be NP-compete. It would be hard to estimate how many problems have now been proved to be NP-complete, but they
certainly number in the thousands. More significantly, the use of NP-completeness as a tool to understand computational difficulty spans virtually all areas of computer science.
The 1970 paper (“The complexity of theorem proving procedures”) introduced concepts and techniques which have had a lasting impact in various fields of computer science. In the paper, Cook introduces
a canonical NP-complete problem, the satisfiability problem for Boolean formulas (SAT). The study of the SAT problem has become a field in its own right, and the development and use of specialized
programs known as SAT-solvers has become an important practical approach to problems in areas such as verification and circuit design. In order to show that other problems are NP-complete, Cook
developed the method of resource-bounded reducibility. This technique is an essential tool in computational complexity theory and is the foundation of complexity-based approaches to cryptographic
The impact of the P versus NP problem has extended beyond the field of computer science. In particular, it has been recognized as a mathematical problem of fundamental significance. In the year 2000,
Fields Medalist Steve Smale listed the P versus NP problem as the third in his list of eighteen unsolved problems in mathematics for the 21st century. In the same year, the Clay Mathematics Institute
announced the Millennium Prize Problems. In the words of the Institute, the prize was proposed “to record some of the most difficult problems with which mathematicians were grappling at the turn of
the second millennium.’’ The P versus NP was included as one of these seven fundamental problems.
Cook has also made important contributions to areas of mathematical logic related to computational complexity. The paper “The relative efficiency of propositional proof systems,”[2] co-authored with
his student Robert Reckhow, laid the foundations of propositional proof complexity. His 1975 paper, ``Feasibly constructive proofs and the propositional calculus’’ [3] introduces a logical system
which characterizes feasible reasoning, and demonstrates how such reasoning is related to efficiency in propositional proof systems. He has also done significant work in areas including automata
theory, parallel computation, program language semantics and verification, computational algebra, and computability and complexity in higher types.
Professor Cook’s work has received extensive recognition. He was awarded the ACM Turing Award in 1982. He is a fellow of the Royal Society of Canada and the Royal Society of London, and is a member
of the National Academy of Sciences (U.S.), the American Academy of Arts and Sciences, and the Gottingen Academy of Sciences. He has been a recipient of the Canada Council Izaak Walton Killam
Memorial Prize in 1997, the CRM-Fields Prize in 1999, the Royal Society of Canada John L. Synge Award in 2006, the NSERC Award of Excellence in 2007, and the Czech Academy of Sciences Bernard Bolzano
Medal in2008. He was an NSERC Steacie Fellow in 1978-79 and was awarded a Killam Research Fellowship in 1982-83.
In 1985, Stephen Cook was promoted to the position of University Professor at the University of Toronto, and now holds the position of Distinguished University Professor in the Computer Science and
Mathematics Departments. Over the course of his career he has mentored over thirty Ph.D. students, and continues to be active in teaching, research and graduate supervision. His book, Logical
Foundations of Proof Complexity, co-authored with Phuong Nguyen, appeared in 2010. He currently lives in Toronto with his wife Linda, and has two sons. He is an avid sailor and a long-time member of
the Royal Canadian Yacht Club.
Enduring Links of Interest
• Clay Foundation Millenium Problem Prize page on the P vs NP problem
• Oral history interview with Stephen A. Cook at the Charles Babbage Institute
• Stephen A. Cook at the Mathematics Genealogy Project
Author: Bruce Kapron | {"url":"http://amturing.acm.org/award_winners/cook_n991950.cfm","timestamp":"2014-04-16T07:14:10Z","content_type":null,"content_length":"21225","record_id":"<urn:uuid:58697a85-3fee-4cd6-86d0-ab4fe851df87>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00489-ip-10-147-4-33.ec2.internal.warc.gz"} |
Universal polygraphic factorization of strict ω-categories relative to a cobase
up vote 2 down vote favorite
Recall from 1 that a cofibration of strict ω-categories is a retract of relative $I$-cell complexes, where $I$ denotes the set of boundary inclusions $\partial D^n \hookrightarrow D^n$, where $D_n$
denotes the free-standing $n$-cell (the $n$-disk or the $n$-globe). The class of trivial fibrations is the class of maps with the right lifting property with respect to $I$.
It is a theorem of Métayer that the cofibrant strict ω-categories are those strict ω-categories generated by polygraphs. Indeed, we have a comonadic cofibrant replacement $Q$, called the standard
polygraphic resolution in 2, associated with the adjunction $$\mathbf{Pol} \rightleftarrows \mathbf{\omega\operatorname{-}Cat},$$ which gives a terminal factorization of every map $\emptyset \to X$
as a cofibration followed by a trivial fibration. That is, there is a terminal cofibrant strict ω-category in the category of cofibrant strict ω-categories trivially fibrant over any strict
What I'd like to know is if there is a cofibrant replacement for strict ω-categories under a fixed strict ω-category $X$.
That is, can we find a universal/terminal factorization of any strict ω-functor $X\to Y$ as a pair $X\to Q_X Y \to Y$ of a cofibration followed by a trivial fibration, where $Q_X$ is some comonad
given by an adjunction $$\mathbf{Pol}_{X/} \rightleftarrows \mathbf{\omega\operatorname{-}Cat}_{X/},$$ where $\mathbf{Pol}_{X/}$ is some category of "relative polygraphs under $X$"?
higher-category-theory word-problem presentations-of-groups homotopy-theory
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Browse other questions tagged higher-category-theory word-problem presentations-of-groups homotopy-theory or ask your own question. | {"url":"http://mathoverflow.net/questions/83507/universal-polygraphic-factorization-of-strict-categories-relative-to-a-cobase?answertab=oldest","timestamp":"2014-04-20T14:12:10Z","content_type":null,"content_length":"48422","record_id":"<urn:uuid:c933a506-ee3a-4413-9b59-f5cb99400da4>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00503-ip-10-147-4-33.ec2.internal.warc.gz"} |
Untitled Document
HEPTAGRAM CONSTRUCTION
It is supposedly impossible to construct an exact heptagram with only a ruler and a compass. Below is a virtually perfect heptagram construction based on the dimensions of the great pyramid.
Draw a circle with center at point A. Construct a vertical and horizontal axis. Mark point B at the intersection point of the vertical axis and the edge of the circle. Draw circle BA. Repeat this
process until 11 segments equal in length to segment AB are constructed. Segment AC is seven times the length of segment AB. Construct a horizontal line through point C. Segment AD is 10 times the
length of segment AB. Construct circle AD. Segment AE is 11 times the length of segment AB. Arc segment AE down to the point where it intersects the right horizontal axis (point F). Construct the
midpoint of segment AF (point G). Draw a perpendicular line from point G, intersecting horizontal line C at point H. With a baselength of 11 and a height of 7, the angular dimensions of triangle AHF
are the same as the great pyramid. Construct the midpoint of segment AH (point I). Construct the midpoint of segment AI (point J). The dark shaded triangle has the same angular dimensions as the
larger triangle and the great pyramid. Arc segment AJ to the point where it intersects the lower vertical axis (point K). Construct a horizontal segment from point K, intersecting the large circle at
points 3 and 6.
Having fixed the distance of the horizontal segment from the center of the circle through the above operation, construct a vesica pisces with points 3 and 6 as the centers of the large outer circles.
Mark the points where these circles intersect the central circle (points 2 and 7). Construct a circle with a radius from point 3 to point 2. Construct a circle with a radius from point 6 to the point
7. Mark the two lower points where the smaller circles intersect the central circle (points 4 and 5). The first point of the heptagram is point D from the diagram above, where the vertical axis
intersects the top of the central circle.
The acute angles at the points of an exact heptagram are 25° 42' 51" (180°/7). The acute angles formed by the construction above range from 25° 42' 14" to 25° 43' 05". Given a diameter of 20 for the
circle around the heptagram, the length of each arm of a perfect heptagram is 18.019. The segments formed by the construction above range from 18.018 to 18.021.
Given a height of 280 cubits and a baselength of 440 cubits (7/11), the base angles of the great pyramid are 51° 50' 34". The ratio between the height of 7 and the half base of 5.5 is 1.272727.
Beginning with an exact heptagram and performing the operation described above in reverse produces the following dimensions for the pyramid triangle. If segment AK is arced to the point it crosses
the vertical axis of point J in the diagram above, the ratio between the height and the half base of the resultant triangle is 1.272401. This produces a base angle of 51° 50' 08". Given a baselength
of 440 cubits, this base angle produces a height of 279.928 cubits. If segment AK is arced to the point it crosses the horizontal axis of point J in the diagram above, the ratio between the height
and the half base of the resultant triangle is 1.273255. This produces a base angle of 51° 51' 15". Given a height of 280 cubits, this base angle produces a baselength of 439.817 cubits. | {"url":"http://home.hiwaay.net/~jalison/hepta.html","timestamp":"2014-04-19T19:33:54Z","content_type":null,"content_length":"4818","record_id":"<urn:uuid:af29b74a-5d16-4f4c-9692-408e4ca827bf>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00145-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to calculate poker odds quickly.
Poker is one of those games which many people will have tried and enjoyed, but may not understand completely. It is a game of real skill, and true poker players will spend time studying the different
hands and learn how to quickly calculate the odds in order to assess their chance of winning a hand.
When playing, a player will have to decide whether to call or fold. One way of making this decision is to check the sum of money held in the pot, divide it by your own ‘pot odds’ and see if this is
equal to, or is more than the odds of getting the cards needed to win (hand odds). Being able to calculate the pot odds quickly can be an essential skill. This will allow the poker player to ensure
that they bet only if they have a good chance of it working out in their favor.
Check the pot for the amount of money. If there is a sum of money already in the pot, such as $4 and the bet amount is $1, then the pot odds work out to be 5:1. This is a fixed calculation, although
you can take into consideration if there are others in the game that have not yet made the same decision. These are implied odds.
In order to determine the hand odds, the number of cards that are unseen needs to be divided by the number of ‘outs’ available to the player. Then 1 needs to be subtracted. The phrase ‘outs’ applies
to the cards that are remaining which will help the player to make up a winning hand; one example is if there are two hearts in the hand dealt out. If there are two more that fall on the flop, this
leaves a total of 47 unseen cards and 9 further hearts (outs) in the deck that could give a potential flush. 47 needs to be divided by 9 which gives 5.2, then a further one from this leaves 4.2, so
in order for the player to call a bet there should be at least 4.2 already in the pot.
In order to determine the chances of getting one of the remaining ‘outs’ you need to multiply the number remaining by 4. In the example already mentioned there are 9 potential outs, so this
multiplied by 4 is 36, so there is a 36% chance of getting one of the outs. When the turn happens the number of outs available needs to be multiplied by 2. If there are still 9 remaining outs then
this gives a percentage of 18, so the bet would need to be 20% or less of the pot.
One of the best ways for a beginner to get used to this system is to practice online. Taking advantage of benefits such as
Party Poker bonus codes
can give the player plenty to work with while they are getting used to the different types of poker, the different rules and assessing each hand that they are dealt. | {"url":"http://www.mzonereport.com/poker-tournament-strategy/how-to-calculate-poker-odds-quickly.html","timestamp":"2014-04-20T05:43:57Z","content_type":null,"content_length":"21780","record_id":"<urn:uuid:b7dbb779-7fe7-42a5-ae73-99e9515f2040>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00291-ip-10-147-4-33.ec2.internal.warc.gz"} |
Algebra 1 Tutors
Palo Alto, CA 94301
Experienced, credentialed math tutor (Stanford/M.I.T. grad)
...Please do make sure that you are either within my 10 mile travel radius or are willing to work with me online. I have loved tutoring/teaching math (pre-algebra,
algebra 1
, geometry, algebra 2, precalculus, calculus) for over 17 years. I am a credentialed classroom...
Offering 10 subjects including algebra 1 | {"url":"http://www.wyzant.com/Newark_CA_algebra_1_tutors.aspx","timestamp":"2014-04-19T03:06:45Z","content_type":null,"content_length":"61306","record_id":"<urn:uuid:2cdf70f6-8bb3-4e02-9ca3-8e20decbe0e2>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00333-ip-10-147-4-33.ec2.internal.warc.gz"} |
Make Your Own Globe
Flat maps may be easier to carry around, but there is still a need to make globes so that Earth's geography can be viewed without any directional or spatial distortions. Printing the location of
continents and oceans directly onto a round surface would be difficult. Instead, this map of the Earth is printed in flat, roughly triangular sections and then attached to a ball. These sections are
called gores.
Make a globe
1. Using a tape measure, determine the circumference of the ball, making sure that the tape measure circles the ball without wandering away from the "equator."
2. On a large piece of paper draw a rectangle the same length as the circumference of the ball. The height of the rectangle should be half the circumference of the ball. Draw an equator line through
the center of the rectangle, lengthwise.
3. Cut out the rectangle.
4. Place the rectangle in front of you horizontally. Fold it in half three times. Unfold the rectangle and there are eight equal sections. Draw a line along each fold. Measure the bottom edge of one
section to find its midpoint, and mark that point "A." Mark the end of the equator in that end section "B."
5. Find the midpoint between A and B as follows: Place the compass point on A. Set the compass radius to a length just short of B and draw a semicircle. Maintaining the same radius, place the
compass point on B and draw a second semicircle. The two semicircles should intersect at two points. Draw a straight line through the points where the semicircles intersect, extending the line to
a point at which it intersects the equator line. Mark this point "C."
The length of the line from A to C is the radius of the gores.
6. Attach extra paper to both ends of the original piece (These extensions should be at least the length of the gore radius) Extend the equator line out onto the extra paper at least the distance of
the gore radius. This will allow you to move your compass point out along the equator far enough to draw all of the gores.
7. Set the compass to the gore radius (the distance between A and C). Place the compass pencil on A and the compass point on C. Draw an arc from A to the top of the rectangle.
Maintaining the same compass radius, move the compass pencil to the midpoint of the bottom edge of the next section and place the compass point on the equator. Draw another arc in the same manner.
Continue moving the compass and drawing arcs for each of the eight sections.
Turn the paper upside down and repeat the above procedure to draw the opposing arcs and form the gores.
Remove the extra paper from the rectangle.
8. Create grid lines for transposing the map onto the rectangle as follows: Fold the rectangle as you did in Step 4. Fold once more in the same direction. Unfold the rectangle, and place it in front
of you horizontally. Fold it in half, top to bottom, three times. Unfold the rectangle. The rectangle should be divided into 16 sections left to right and 8 sections top to bottom. Cut out the
spaces between the gores. Transpose the map from the given to your gores using the gridlines on the diagram and the gridlines (folds) on the gores as a reference.
9. Tape the strip of gores at one end of the equator to the ball. Wrap the strip around the ball and tape the loose ends in place, taking care to align the equator line.
10. Glue each gore down against the ball so the tips meet to form the north and south poles.
Make an Orange Globe
To demonstrate the difficulty in making a flat map of a round surface make a map out of an orange peel. Try to peel an orange with an Exacto knife so that you take off the skin in one piece. Make a
flat projection of the surface of the orange by laying the peel flat. Does your peel look like the gores from the previous activity?
• paper
• ball (about the size of a playground ball)
• glue
• tape
• paints
• ruler
• tape measure
• compass
• pencil | {"url":"http://octopus.gma.org/surfing/imaging/globe.html","timestamp":"2014-04-20T08:14:51Z","content_type":null,"content_length":"16279","record_id":"<urn:uuid:e61938ad-23b7-498f-8d73-51746c63ea02>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00412-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Purpose Of This Assignment Is Three-fold: (1) ... | Chegg.com
*******PLEASE READ THIS: I definetly need the hand calculation of the first part !
Image text transcribed for accessibility: The purpose of this assignment is three-fold: (1) to give you practice manipulating the Fourier Transform of periodic functions, (2) to give you practice
manipulating the Matlab code that calculates the Fourier Transform and (3) to teach you what happens to the Fourier Transform when you multiply two functions together. Pick two functions, x1(t) = cos
(w1t) and x2(t) = sin(w2t) where w1 and w2 are different frequencies. Let w1 be the month you were born (i.e. September would be w1 = 9 rads/sec), and let w2 be the day of the month you were born. By
hand, solve for the Fourier Transform of x(t) = x1(t) · x2(t). To verify that your answer is correct, use Matlab to "rebuild" and plot x(t) using your newly derived Fourier coefficients An, and show
that this plot is in fact identical to x1(t) · x2(t). At what frequencies does x(t) have energy and how are those frequencies related to w1 and w2? In general, how does multiplying two signals appear
to change their frequency content? In this part we will again be multiplying two functions and calculating the Fourier Transform of the product. The first function is y1(t) = cos(4pit). The second
function is the square wave y2(t) shown below. By hand, calculate the Fourier Transform of the product y(t) = y1(t) · y2(t). Create a plot of |A| to w. Using what you learned in Part 1 about
multiplying signals, relate your hand calculation to the Fourier Transform plots of y1(t) and y2(t); explain this relationship clearly.
Electrical Engineering | {"url":"http://www.chegg.com/homework-help/questions-and-answers/purpose-assignment-three-fold-1-give-practice-manipulating-fourier-transform-periodic-func-q4307568","timestamp":"2014-04-18T02:26:14Z","content_type":null,"content_length":"21770","record_id":"<urn:uuid:0432e2ed-0d7e-48e5-b013-e170048839bf>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00520-ip-10-147-4-33.ec2.internal.warc.gz"} |
limit the time for proofs
Major Section: OTHER
; Limit (mini-proveall) to about 1/4 second:
(with-prover-time-limit 1/4 (mini-proveall))
; Limit (mini-proveall) to about 1/4 second, even if surrounding call of
; with-prover-time-limit provides for a more restrictive bound:
(with-prover-time-limit '(1/4) (mini-proveall))
; Limit the indicated theorem to about 1/50 second, and if the proof does not
; complete or it fails, then put down a label instead.
(mv-let (erp val state)
(thm (equal (append (append x x) x)
(append x x x))))
(if erp
(deflabel foo :doc "Attempt failed.")
(value (list :succeeded-with val))))
General Form:
(with-prover-time-limit time form &key loosen-ok)
where time evaluates to a positive rational number or to a list containing such, and form is arbitrary. Logically, (with-prover-time-limit time form) is equivalent to form. However, if the runtime
for evaluation of form exceeds the value specified by time, and if ACL2 notices this fact during a proof, then that proof will abort, for example like this:
ACL2 Error in ( DEFTHM PERM-REFLEXIVE ...): Out of time in rewrite.
If there is already a surrounding call of with-prover-time-limit that has set up an expiration time, the present with-prover-time-limit is not allowed to push that time further into the future unless
the time is specified as a list containing a rational rather than as a rational.
If you find that the time limit appears to be implemented too loosely, you are encouraged to email an example to the ACL2 implementors with instructions on how to observe the undesirable behavior.
This information can probably be used to improve ACL2 by the insertion of more checks for expiration of the time limit.
The rest of this documentation topic explains the rather subtle logical story, and is not necessary for understanding how to use with-prover-time-limit. The ACL2 state object logically contains a
field called the acl2-oracle, which is an arbitrary true list of objects. This field can be read by a function called read-acl2-oracle, which however is untouchable (see push-untouchable), meaning
that it is cannot be called by ACL2 users. The acl2-oracle field is thus ``secret''. Our claim is that any ACL2 session makes sense for some value of acl2-oracle in the initial state for that
session. Logically, with-prover-time-limit is a no-op, just returning its second value. But under the hood, it provides a ``hint'' for the acl2-oracle, so that (logically speaking) when its first
element (car) is consulted by ACL2's prover to see if the time limit has expired, it gets the ``right'' answer (specifically, either nil if all is well or else a message to print if the time limit
has expired). Logically, the acl2-oracle is then cdr'ed -- that is, its first element is popped off -- so that future results from read-acl2-oracle are independent of the one just obtained. | {"url":"http://planet.racket-lang.org/package-source/cce/dracula.plt/2/2/language/acl2-html-docs/WITH-PROVER-TIME-LIMIT.html","timestamp":"2014-04-16T16:27:37Z","content_type":null,"content_length":"4151","record_id":"<urn:uuid:7618ab9b-3534-44f3-a67a-c6639e070e11>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00659-ip-10-147-4-33.ec2.internal.warc.gz"} |
Specification and Analysis of Embedded Systems
by Henk Nieland
In order to increase the quality of software components that are typically found in telecommunication and embedded systems, CWI studies specification techniques for the unambiguous description of
these systems, and analysis techniques to establish that their implementations exhibit the expected functionality. Viability assessment is made through experiments with fundamental distributed
algorithms and concrete industrial application. The research contains a healthy mix of theoretical study and practical application.
Ever more computers are embedded within real-world technical applications, for example in avionics, process control, robotics, telecommunications and consumer products. The quality of the
installed software is crucial for their proper functioning. Several of these applications require the software to operate in real-time and in a distributed environment. Given the ever increasing
size and complexity of such systems, high demands are made upon the correctness of the embedded software.
The road to correct software starts with a specification of the problem, which provides criteria for the program to be designed. The program is called 'correct' if these criteria are satisfied.
The criteria are formulated in a specification language which is kept preferably on a high abstraction level and, for example, need not be executable. In algebraic specifications axioms provide
rules for simplification of expressions and fix the semantics of the algebraic constructions. Such formal methods are increasingly used in the specification of complex systems. They have the
advantage that all assumptions are made explicit in an early design phase, thus avoiding errors that can be extremely costly to detect and repair at later stages of product development.
The advent of distributed computer systems and parallel computation created a need for new specification techniques in order to deal with, eg, synchronization problems. A promising approach is
the use of process algebra in which concurrent processes can be formally described. CWI made important contributions to the development of process algebra and of the specification language mCRL
(micro Common Representation Language) based on it.
Current theoretical research at CWI is directed at a real-time extension of mCRL. Its expressivity is studied in applications. In particular the effectiveness of using formal methods is
demonstrated and assessed by the validation of (critical parts of) the software for selected applications from within Philips, including the future generation of IC's. In addition, CWI
participates in a project at Philips, in which the system compliance is studied of the DVB (Digital Video Broadcast) source decoder (still under development). CWI's part consists of the
development and execution of the conformance tests by means of logical models and extended finite-state models, to be derived from the relevant standards and system documentation. Furthermore,
there is an ongoing involvement in the European COST 247 programme, which focuses on the formal specification and verification, validation and testing of software in realistic problems in
contemporary distributed communication architectures.
A second branch of activity at CWI concerns proof searching and proof checking. The first aim is to increase the efficiency of current symbolic techniques to verify requirements on processes by a
fundamental understanding of proof search in simple logical systems (in casu propositional logic). Secondly, proof checking methods are developed in order to establish the correctness of
programmed systems "beyond any reasonable doubt". A recent application is the propositional logic tool Heerhugo. Heerhugowaard is the largest train station in The Netherlands operating with a VPI
(Vital Process Interlocking) system, which is a kind of programmable controller. The system has to comply with several safety requirements, for example: "trains should not derail", or: "if a
signal is green, the next one should not be red". All these requirements can be formulated as statements in propositional logic. The check on all requirements being satisfied is an NP-complete
problem. For this particular case CWI succeeded to construct a workable prover, Heerhugo. It turned out that, contrary to several other cases, Dutch Rail safety systems are of very high quality.
This prover was also successfully applied to consistency checks on two merged databases of automobile components (NedCar and Volvo).
Future research will include the study of modal logic, the development of tools and algorithms, and a better understanding of structural properties of distributed systems.
Please contact:
Jan Friso Groote - CWI
Tel: +31 20 592 4232
E-mail: JanFriso.Groote@cwi.nl | {"url":"http://www.ercim.eu/publication/Ercim_News/enw32/groote.html","timestamp":"2014-04-19T17:31:15Z","content_type":null,"content_length":"6104","record_id":"<urn:uuid:d32be493-ee81-4fb0-91b9-07a007b80e00>","cc-path":"CC-MAIN-2014-15/segments/1397609537308.32/warc/CC-MAIN-20140416005217-00378-ip-10-147-4-33.ec2.internal.warc.gz"} |
NSA backdoor in public crytographic system?
Posted by isilanes on November 17, 2007
The following shows why crytograpy methods shoud be public. There is some common misconception, that assumes that the most secure crypto methods are “proprietary” or “secret” ones. This is a terrible
error, since only knowing the “recipe” (the algorithms) behind a given method can assure us that it is actually robust.
The question can rise: how can a publicly known crytographic method be secure? By definition, everyone will know how it works! Not quite. The operation method can be known to all, and an eavesdroper
could know what method we are using, but if the method is secure, the eavesdroper will not be able to decipher a given message. It might be tempting to think that if an eavesdroper doesn’t even know
what encryption we are using, or she knows the “name”, but the method behind is secret, then the security of the message is increased. This is called security through obscurity, and is actually a
very dangerous error, because it might lead us to be less exigent in the robustness of the encryption algorithm. A communication can only be considered secure if even knowing the encryption
algorithm, an eavesdropper could not decrypt it.
To achieve this, it is vital that the encryption algorithm be publicly known, and rigorous tests applied. This is the case of the crypto standards of the North-American NIST. All the standards
“accepted” by them have to be subject to open scrutiny, which happens to be a Good Thing(tm). You’ll see it if you read the following articles in The Register and Wired.
In summary: one of the components of cryptographic methods is random number generation. One of the ones approved this year by the NIST (called Dual_EC_DRBG), relied on a set of initial numbers to
generate the “random” result (I’ll call this set P, public). This is normal, and correct. The problem comes from the fact that this set of numbers is apparently related to another (unknown) set of
numbers (that I’ll call B, backdoor), knowledge of which could empower someone to break the resulting encryption. The way I understand it, is like having the known set of numbers P = (6,12,18,24,30),
but then realizing that they are all built from the set B = (2,3). In the Dual_EC_DRBG method, some experts have realized that the set P is related to another set, but they still haven’t found what
are the elements of B.
Now, the scary part is that (life’s full of casualities) the Dual_EC_DRBG was introduced in the standard proposed, and pushed, by the NSA of the USA, aka “the eavesdroppers of the world”. So I’ll
invent a little fiction, with no relationship with the reality: imagine that a given government agency N of nation U takes a set of numbers B, and comes up with an encryption method M that produces
the apparently innocent set P from it, and then M uses P to perform encryption. If the encryption method M becomes a standard, and people all around the world use it for anything from private e-mail
to secure government or militar communications… guess who has a the key to read all these messages? (a backdoor).
Thanks $GOD, this is science fiction, is it not? | {"url":"http://handyfloss.wordpress.com/2007/11/17/nsa-backdoor-in-public-crytographic-system/","timestamp":"2014-04-21T01:58:47Z","content_type":null,"content_length":"49244","record_id":"<urn:uuid:26778802-db3c-4a3b-ad94-dc0c5f3a98b6>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00182-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum - Ask Dr. Math Archives: College Higher-Dimensional Geometry
This page:
Dr. Math
See also the
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geometric formulas
Internet Library:
linear algebra
modern algebra
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conic sections/
coordinate plane
Logic/Set Theory
Number Theory
Browse College Higher-Dimensional Geometry
Stars indicate particularly interesting answers or good places to begin browsing.
What is the name of a rectangle with two rounded ends, like a circle with top and bottom evenly truncated? For example, an 'oval' racetrack with two straightaways on opposite sides, parallel to
each other and both of the same length. I know that's not really an oval, but I'm not sure what else to call it!
Define #(n,A) to be the number of corners of an n-dimensional cube whose distance to 0 is greater than A. The limit (as n goes to infinity) #(n,A)/(2^n) = 1. How would I verify the limit
Given points A, B, and C on the surface of a unit sphere, find the point P on the great circle defined by A and B that is nearest to C.
I have the x, y, and z components of a velocity vector of an airplane, and must use this vector to calculate the bearing of the plane.
How can I find the dimensions of the smallest tetrahedron that can serve as a container for 4 spheres packed as snugly as possible?
What is the formula for the volume of a circular torus?
If I know the center and radius of a circle, and three points on the circle, can I find the parametric form of the circle equation in 3D space?
What is the parametric form of a circle in 3 space that passes through a particular 3 points?
I am trying to calculate the midpoint between cases of legionella and their nearest neighbor. How can I calculate the distance between two points below which they can be treated as if they were
in a plane rather than on a sphere?
Can we say that n planes divide space into at most 2^n regions?
I am in need of assistance in proving the volume of a truncated spherical cap (or a segment of a sphere I think it is also called).
I have been given two equations to determine the radius of the earth for a given latitude, based on ellipsoid model WGS84. I get different answers...
How can I find a vector equation of the reflection of a line in three- dimensional space?
Is there an equation to find the resultant of pitch and yaw?
I am getting a paper rewinder that runs 6,000 ft a minute, and the roll is 50' high above the floor. How many miles and feet are there in this roll of paper and how long will it take to run?
Where is the second octant? No one seems to know how to count the next octants after the first.
I know that an equation like 2x + y + z = 3 represents a plane in three dimensions. How can I sketch that plane on the xyz axes? Also, how can I sketch a system of such equations to find the
solution geometrically?
Find the volume and the areas of each of the surfaces/faces of a small section of a sphere with "dimensions" delta r, delta theta, delta phi, in spherical coordinates.
How do you mathematically turn a sphere inside out?
How do I calculate whether two lines that lie on the surface of a sphere intersect, and if they do intersect, the point of that intersection?
Could you please explain the formula for the area of a spherical polygon, and show how to determine the values of the thetas in it?
How can I find the 'spherical rectangle' defined by a pair of corner points?
Two pipes (radius = r) cross each other normally. What is the common part volume?
For what 3D figures is the derivative of the volume formula equal to the formula for surface area? With respect to which variable would you need to differentiate?
How do you calculate the surface area of an ellipsoid?
I was wondering how to calculate the surface area of a sphere in n dimensions.
How is the surface area of a sphere calculated, and why?
Can you derive the formula for the surface area of a sphere?
Can the method for finding the surface area of a pyramid be used as well to find the surface area of a cone?
I tried to derive the formula for the surface area of a cone by taking the integral the circumference of the solid of revolution, but it didn't work. What did I do wrong? Can the formula be
derived using this method?
Why does a tesseract contain eight cubes?
How do you project a regular tetrahedron perpendicularly onto a plane to get the maximum area shadow?
A vessel in a plant where I work is the frustum of a cone on its side. A liquid is contained in this section and pours out the end of the cone section, therefore the liquid only takes up a
certain portion of the cone's volume. How can I compute the volume of the liquid?
I have two questions on the transformation between (x,y) and (longitude, latitude).
How do you find the shortest distance between two polygons in 3D space?
Can you help me figure out the equations for fourth dimension figures such as the tesseract and the hypertetrahedron?
Is there a method to pick random points on a sphere so that the points wind up uniformly distributed on the sphere? I keep getting a higher density of points near the poles.
I've been looking for the equation for finding the distance between two cities, given the latitude and longitude of both cities.
We need to know how much water is in the tank at any given time.
Explain the derivation of the formula V = |a.(b x c)| (the volume of a parallelepiped is equal to the magnitude of the scalar triple product of the vectors that determine the parallelepiped;
where a, b, and c are those vectors).
Page: [<prev] 1 2 3 4 [next>] | {"url":"http://mathforum.org/library/drmath/sets/college_3d.html?start_at=81&s_keyid=40659750&f_keyid=40659751&num_to_see=40","timestamp":"2014-04-19T17:25:38Z","content_type":null,"content_length":"21641","record_id":"<urn:uuid:46463b2b-b1f4-4150-9418-75d71a09ad9f>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00142-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Repulsion of Charged Pith Balls
Two equally charged pith balls of equal mass are separated by a distance , due to their mutual electrostatic repulsion. Each ball hangs on a string of length from a common point, making an angle
with the horizontal. The value of the angle , the charge , and the tension of the string are calculated.
By trigonometry, the angle is equal to , the value of the charge is , and the tension is , where is the acceleration of gravity and is the constant in Coulomb's law. | {"url":"http://demonstrations.wolfram.com/RepulsionOfChargedPithBalls/","timestamp":"2014-04-16T04:35:29Z","content_type":null,"content_length":"43474","record_id":"<urn:uuid:94a6b35a-5b47-4c25-87e0-e5787fc8dfec>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00521-ip-10-147-4-33.ec2.internal.warc.gz"} |
Help file for the Anaddb utility of the ABINIT package.
This file explains the use and i/o parameters needed for the "Analysis of Derivative DataBase" code of the ABINIT package.
This code is able to compute interatomic force constants (hence its name), but also, more generally, many different physical properties from databases containing derivatives of the total energy
(Derivative DataBase - DDB).
The user is not supposed to know how the Derivative DataBase (DBB) has been generated. He/she should simply know what material is described by the DDB he/she wants to use.
If he/she is interested in the generation of DDB, and wants to know more about this topic, he/she will read different help files of the ABINIT package, related to the main code, to the
response-function features of the main code, to the merging code.
It will be easier to discover the present file with the help of the tutorial, especially the second lesson on response functions.
It is worthwhile to print this help file, for ease of reading.
Copyright (C) 1998-2012 ABINIT group (XG,DCA)
This file is distributed under the terms of the GNU General Public License, see ~abinit/COPYING or http://www.gnu.org/copyleft/gpl.txt .
For the initials of contributors, see ~abinit/doc/developers/contributors.txt .
Goto :
ABINIT home Page | Suggested acknowledgments | List of input variables | Tutorial home page | Bibliography
Help files :
New user's guide | Abinit (main) | Abinit (respfn) | Mrgddb | Anaddb | AIM (Bader) | Cut3D | Optic
Content of the help file.
1. Introduction
In short, a Derivative DataBase contains a list of derivatives of the total energy with respect to three kind of perturbations : phonons, electric field and stresses. The present code analyses the
DDB, and directly gives properties of the material under investigation, like phonon spectrum, frequency-dependent dielectric tensor, thermal properties.
Given an input file (parameters described below), the user must create a "files" file which lists names for the files the job will require, including the main input file, the main output file, the
name of the DDB, and some other file names optionally used for selected capabilities of the code.
The files file (called for example ab.files) could look like:
In this example:
- the main input file is called "anaddb.in",
- the main output will be put into the file called "anaddb.out",
- the input DDB file is called "ddb",
- information to draw phonon band structures will go to band_eps
- the input GKK file is called "gkk" (used only for electron-phonon interactions)
- the base filename for electron-phonon output "anaddb.ep" (used only for electron-phonon interactions)
- the file name for ddk reference files: these are the GKK files generated in k-point derivative runs, using the prtgkk abinit input variable (used only for electron-phonon transport calculations)
Other examples are given in the ~abinit/test/v2 directory. The latter three filename information is often not used by anaddb. The maximal length of names for the main input or output files is
presently 132 characters.
The main executable file is called anaddb. Supposing that the "files" file is called anaddb.files, and that the executable is placed in your working directory, anaddb is run interactively (in Unix)
with the command
• anaddb < anaddb.files >& log
or, in the background, with the command
• anaddb < anaddb.files >& log &
where standard out and standard error are piped to the log file called "log" (piping the standard error, thanks to the '&' sign placed after '>' is really important for the analysis of eventual
failures, when not due to ABINIT, but to other sources, like disk full problem ...). The user can specify any names he/she wishes for any of these files. Variations of the above commands could be
needed, depending on the flavor of UNIX that is used on the platform that is considered for running the code.
The syntax of the input file is strictly similar to the syntax of the main abinit input files : the file is parsed, keywords are identified, comments are also identified. However, the multidataset
mode is not available.
We now list the input variables for the anaddb input file. In order to discover them, it is easier to use the different lessons of the tutorial : start with the second lesson on response functions,
then follow the lesson on elasticity and piezoelectricity, the lesson on electron-phonon interaction, and the lesson on non-linear properties.
If you are discovering this file with the help of the tutorial, you can go back to the tutorial window.
2. The list of input variables.
Alphabetical list of input variables for ANADDB.
A. alphon asr atftol atifc a2fsmear
B. brav
C. chneut
D. dieflag dipdip dosdeltae dossmear dostol dossum
E. eivec elaflag elphflag elphsmear elph_fermie enunit ep_keepbands ep_b_max ep_b_min ep_extrael ep_nqpt ep_prt_yambo ep_qptlist ep_scalprod
F. freeze_displ frmax frmin
G. gkqwrite
I. iatfix iatprj_bs iavfrq ifcana ifcflag ifltransport ifcout instrflag istrfix
K. kptrlatt kptrlatt_fine
M. mustar
N. natfix natifc natprj_bs nchan nfreq ngqpt ng2qpt ngrids nlflag nph1l nph2l nqpath nqshft nsphere nstrfix ntemper nwchan
O. outscphon
P. piezoflag polflag prtfsurf prtsrlr prtmbm prtdos prtnest
Q. qgrid_type qpath qph1l qph2l qrefine q1shft q2shft
R. ramansr relaxat relaxstr rfmeth rifcsph
S. selectz symdynmat symgkq
T. targetpol telphint temperinc tempermin thmflag thmtol
Mnemonics: ALign PHONon mode eigendisplacements
Variable type: integer
Default: 0
In case
is set to 1, ANADDB will compute linear combinations of the eigendisplacements of modes that are degenerate (twice or three times), in order to align the mode effective charges along the cartesian
axes. This option is useful in the mode-by-mode decomposition of the electrooptic tensor, and to compute the Raman susceptibilities of individual phonon modes. In case of uniaxial crystals, the
z-axis should be chosen along the optical axis.
Go to the top | List of ANADDB input variables asr
Mnemonics: Acoustic Sum Rule
Variable type: integer
Default: 1 (was 0 before v5.3)
Govern the imposition of the Acoustic Sum Rule (ASR).
• 0 => no ASR for interatomic force constants is imposed.
• 1 or 2 => the ASR for interatomic force constants is imposed by modifying the on-site interatomic force constants, in a symmetric way (asr=2), or in the more general case, unconstrained way (asr=
More detailed explanations : the total energy should be invariant under translation of the crystal as a whole. This would garantee that the three lowest phonon modes at Gamma have zero frequency
(Acoustic Sum Rule - ASR). Unfortunately, the way the DDB is generated (presence of a discrete grid of points for the evaluation of the exchange-correlation potential and energy) slightly breaks the
translational invariance. Well, in some pathological cases, the breaking can be rather important.
Two quantities are affected : the interatomic forces (or dynamical matrices), and the effective charges. The ASR for the effective charges is called the charge neutrality sum rule, and will be dealt
with by the variable chneut. The ASR for the interatomic forces can be restored, by modifying the interatomic force of the atom on itself, (called self-IFC), as soon as the dynamical matrix at Gamma
is known. This quantity should be equal to minus the sum of all interatomic forces generated by all others atoms (action-reaction law!), which is determined by the dynamical matrix at Gamma.
So, if asr is non-zero, the correction to the self-force will be determined, and the self-force will be imposed to be consistent with the ASR. This correction will work if IFCs are computed (ifcflag/
=0), as well as if the IFCs are not computed (ifcflag==0). In both cases, the phonon frequencies will not be the same as the ones determined by the output of abinit, RF case. If you want to check
that the DDB is correct, by comparing phonon frequencies from abinit and anaddb, you should turn off both asr and chneut.
Until now, we have not explained the difference between asr=1 and asr=2. This is rather subtle. In some local low-symmetry cases (basically the effective charges should be anisotropic), when the
dipole-dipole contribution is evaluated and subtracted, the ASR cannot be imposed without breaking the symmetry of the on-site interatomic forces. That explains why two options are given : the second
case (asr=2, sym) does not entirely impose the ASR, but simply the part that keeps the on-site interatomic forces symmetric (which means that the acoustic frequencies do not go to zero exactly), the
first case (asr=1, asym) imposes the ASR, but breaks the symmetry. asr=2 is to be preferred for the analysis of the interatomic force constant in real space, while asr=1 should be used to get the
phonon band structure.
(NOTE : in order to confuse even more the situation, it seems that the acoustic phonon frequencies generated by the code for both the sym and asym options are exactly the same ... likely due to an
extra symmetrisation in the diagonalisation routine. Of course, when the matrix at Gamma has been generated from IFCs coming from dynamical matrices none of which are Gamma, the breaking of the ASR
is rather severe. In order to clear the situation, one should use a diagonalisation routine for non-hermitian matrices. So, at the present status of understanding, one should always use the asr=2
option ).
Go to the top | List of ANADDB input variables
Mnemonics: ATomic Temperature Factor TOLerance
Variable type: real
Default: 0.05
The relative tolerance on the atomic temperature factors. This number will determine when the series of channel widths with which the DOS is calculated can be stopped, i.e. the mean of the relative
change going from one grid to the next bigger is smaller than
Go to the top | List of ANADDB input variables atifc
Mnemonics: AToms for IFC analysis
Variable type: integer array
Default: 0
The actual numbers of the atoms for which the interatomic force constant have to be written and eventually analysed.
WARNING : there will be an in-place change of meaning of atifc (this is confusing, and should be taken away in one future version - sorry for this).
Go to the top | List of ANADDB input variables a2fsmear
Mnemonics: Alpha2F SMEARing factor
Characteristic: ENERGY
Variable type: real
Default: 0.00002
Smearing width for the Eliashberg alpha^2F function (similar to a phonon DOS), which is sampled on a finite q and k grid. The Dirac delta functions in energy are replaced by Gaussians of width
a2fsmear (by default in Hartree).
Go to the top | List of ANADDB input variables brav
Mnemonics: BRAVais
Variable type: integer
Default: 1
Allows to specify the Bravais lattice of the crystal, in order to help to generate a grid of special q points.
• 1 => all the lattices (including FCC, BCC and hexagonal)
• 2 => specific for Face Centered lattices
• 3 => specific for Body Centered lattices
• 4 => specific for the Hexagonal lattice
Note that in the latter case, the rprim of the unit cell have to be 1.0 0.0 0.0 -.5 sqrt(3)/2 0.0 0.0 0.0 1.0 in order for the code to work properly.
Warning : the generation of q-points in anaddb is rather old-fashioned, and should be replaced by routines used by the main abinit code.
Go to the top | List of ANADDB input variables
Mnemonics: Integer for CHarge NEUTrality treatment
Variable type: integer parameter
Default is 0.
Set the treatment of the Charge Neutrality requirement for the effective charges.
• chneut=0 => no ASR for effective charges is imposed
• chneut=1 => the ASR for effective charges is imposed by giving to each atom an equal portion of the missing charge. See Eq.(48) in Phys. Rev. B55, 10355 (1997).
• chneut=2 => the ASR for effective charges is imposed by giving to each atom a portion of the missing charge proportional to the screening charge already present. See Eq.(49) in Phys. Rev. B55,
10355 (1997).
More detailed explanation : the sum of the effective charges in the unit cell should be equal to zero. It is not the case in the DDB, and this sum rule is sometimes strongly violated. In particular,
this will make the lowest frequencies at Gamma non-zero. There is no "best" way of imposing the ASR on effective charges.
Go to the top | List of ANADDB input variables
Mnemonics: DIElectric FLAG
Variable type: integer
Default: 0
Integer. Frequency-dependent dielectric tensor flag.
• 0 => No dielectric tensor is calculated.
• 1 => The frequency-dependent dielectric tensor is calculated. The frequencies are defined by the nfreq, frmin, frmax variables. Also, the generalized Lyddane-Sachs-Teller relation will be used as
an independent check of the dielectric tensor at zero frequency (this for the directions defined in the phonon list 2. See nph2l).
• 2 => Only the electronic dielectric tensor is calculated. It corresponds to a zero-frequency homogeneous field, with quenched atomic positions. For large band gap materials, this quantity is
measurable because the highest phonon frequency is on the order of a few tenths of eV, and the band gap is larger than 5eV.
• 3 => Compute and print the relaxed-ion dielectric tensor. Requirements for preceding response-function DDB generation run: electric-field and full atomic-displacement responses. Set rfstrs = 1,
2, or 3 (preferably 3). Set rfatpol and rfdir to do a full calculation of phonons at Q=0 (needed because the inverse of force-constant tensor is required). Note that the relaxed-ion dielectric
tensor computed here can also be obtained as the zero-frequency limit of the frequency-dependent dielectric tensor using input variables dieflag=1 and frmin=0.0. (The results obtained using these
two approaches should agree to good numerical precision.) The ability to compute and print the static dielectric tensor here is provided for completeness and consistency with the other tensor
quantities that are computed in this section of the code.
• 4 => Calculate dielectric tensor of both relaxed ion and free stress. We need information of internal strain and elastic tensor (relaxed ion) in this computation. So please set: elaflag=2,3,4 or
5 and instrflag=1
Go to the top | List of ANADDB input variables dipdip
Mnemonics: DIPole-DIPole interaction
Variable type: integer
Default: 1
• 0 => the dipole-dipole interaction is not handled separately in the treatment of the interatomic forces. This option is available for testing purposes or if effective charge and/or dielectric
tensor is not available in the derivative database. It gives results much less accurate than dipdip=1.
• 1 => the dipole-dipole interaction is subtracted from the dynamical matrices before Fourier transform, so that only the short-range part is handled in real space. Of course, it is reintroduced
analytically when the phonon spectrum is interpolated, or if the interatomic force constants have to be analysed in real space.
Go to the top | List of ANADDB input variables dosdeltae
Mnemonics: DOS DELTA in Energy
Variable type: real
Default: 4.5E-06 Hartree = 1 cm
The input variable
is used to define the step of the frequency grid used to calculate the phonon density of states when
Go to the top | List of ANADDB input variables dossmear
Mnemonics: DOS SMEARing value
Characteristic: Energy
Variable type: real
Default: 4.5E-05 Hartree = 10 cm
^-1 dossmear
defines the gaussian broadening used to calculate the phonon density of states when
Go to the top | List of ANADDB input variables dostol
Mnemonics: DOS TOLerance
Variable type: real
Default: 0.25
The relative tolerance on the phonon density of state. This number will determine when the series of grids with which the DOS is calculated can be stopped, i.e. the mean of the relative change going
from one grid to the next bigger is smaller than
Go to the top | List of ANADDB input variables dossum
Mnemonics: DOS SUM
Variable type: integer
Default: 0
Set the flag to calculate the two phonon dos density of states. Sum and Difference for the Gamma point. The DOS is converged and based on that, the sum and different is reported in the output file
Go to the top | List of ANADDB input variables eivec
Mnemonics: EIgenVECtors
Variable type: integer
Default: 0
• 0 => do not write the phonon eigenvectors;
• 1 or 2 => write the phonon eigenvectors;
• 3 => write the phonon eigenvectors, in the lwf-formatted file;
• 4 => generate output files for band2eps (drawing tool for the phonon band structure);
Go to the top | List of ANADDB input variables elaflag
Mnemonics: ELAstic tensor FLAG
Variable type: integer
Default: 0
Flag for calculation of elastic and compliance tensors
• 0 => No elastic or compliance tensor will be calculated.
• 1 => Only clamped-ion elastic and compliance tensors will be calculated. Requirements for preceding response-function DDB generation run: Strain perturbation. Set rfstrs to 1, 2, or 3. Note that
rfstrs=3 is recommended so that responses to both uniaxial and shear strains will be computed.
• 2 => Both relaxed- and clamped-ion elastic and compliance tensor will be calculated, but only the relaxed-ion quantities will be printed. The input variable instrflag should also be set to 1,
because the internal-strain tensor is needed to compute the relaxed-ion corrections. Requirements for preceding response-function DDB generation run: Strain and atomic-displacement responses at Q
=0. Set rfstrs = 1, 2, or 3 (preferably 3). Set rfatpol and rfdir to do a full calculation of phonons at Q=0 (needed because the inverse of force-constant tensor is required).
• 3 => Both relaxed and clamped-ion elastic and compliance tensors will be printed out. The input variable instrflag should also be set to 1. Requirements for preceeding response-function DDB
generation run: Same as for elaflag=2.
• 4 => Calculate the elastic and compliance tensors (relaxed ion) at fixed displacement field, the relaxed-ion tensors at fixed electric field will be printed out too, for comparison. When elaflag=
4, we need the information of internal strain and relaxed-ion dielectric tensor to build the whole tensor, so we need set instrflag=1 and dieflag=3 or 4 .
• 5 => Calculate the relaxed ion elastic and compliance tensors, considering the stress left inside cell. At the same time, bare relaxed ion tensors will still be printed out for comparison. In
this calculation, stress tensor is needed to compute the correction term, so one supposed to merge the first order derivative data base (DDB file) with the second order derivative data base (DDB
file) into a new DDB file, which can contain both information. And the program will also check for the users.
Go to the top | List of ANADDB input variables elphflag
Mnemonics: ELectron-PHonon FLAG
Variable type: integer
Default: 0
is 1, anaddb performs an analysis of the electron-phonon coupling.
Go to the top | List of ANADDB input variables elphsmear
Mnemonics: ELectron-PHonon SMEARing factor
Characteristic: ENERGY
Variable type: real
Default: 0.01 Hartree
Smearing width for the Fermi surface integration (in Hartree by default).
Go to the top | List of ANADDB input variables elph_fermie
Mnemonics: ELectron-PHonon FERMI Energy
Characteristic: ENERGY
Variable type: real
Default: 0.0
If non-zero, will fix artificially the value of the Fermi energy (e.g. for semiconductors), in the electron-phonon case. Note that elph_fermie and
should not be used at the same time. (
Go to the top | List of ANADDB input variables enunit
Mnemonics: ENergy UNITs
Variable type: integer
Default: 0
Give the energy for the phonon frequency output (in the output file, not in the console log file, for which Hartree units are used).
• 0 => Hartree and cm-1;
• 1 => meV and Thz;
• 2 => Hartree, cm-1, meV, Thz, and Kelvin.
Go to the top | List of ANADDB input variables ep_alter_int_gam
Mnemonics: Electron Phonon ALTERnative INTegration of GAMma matrices
Variable type: integer
Default: 0
When set,
= 1, and
is given, an alternative integration method is used on the Fermi Surface. Does not work yet at all, and is in heavy development. Maybe for version 6.2
Go to the top | List of ANADDB input variables ep_b_max
Mnemonics: Electron Phonon integration Band MAXimum
Variable type: integer
Default: 0
When set, and
is equal to 2, this variable determines the k-point integration weights which are used in the electron-phonon part of the code. Instead of weighting according to a distance from the Fermi surface, an
equal weight is given to all k-points, for all bands between
and ep_b_max.
Go to the top | List of ANADDB input variables ep_b_min
Mnemonics: Electron Phonon integration Band MINimum
Variable type: integer
Default: 0
As for
, but ep_b_min is the lower bound on the band integration, instead of the upper bound. See also
telphint Go to the top | List of ANADDB input variables ep_extrael
Mnemonics: Electron-Phonon EXTRA ELectrons
Variable type: real
Default: 0.0
If non-zero, will fix artificially the number of extra electrons per unit cell, according to a doped case. (e.g. for semiconductors), in the electron-phonon case. Note that ep_extrael and
should not be used at the same time. (
Go to the top | List of ANADDB input variables ep_keepbands
Mnemonics: Electron-Phonon KEEP dependence on electron BANDS
Variable type: integer
Default: 0
This flag determines whether the dependency of the electron-phonon matrix elements on the electron band index is kept (
1), or whether it is summed over immediately with appropriate Fermi Surface weights. For transport calculations
must be set to 1.
Go to the top | List of ANADDB input variables ep_nqpt
Mnemonics: Electron Phonon Number of Q PoinTs
Variable type: integer
Default: 0
In case a non-uniform grid of q-points is being used, for direct calculation of the electron-phonon quantities without interpolation, this specifies the number of q-points to be found in the GKK
file, independently of the normal anaddb input (ngqpt)
Go to the top | List of ANADDB input variables ep_prt_yambo
Mnemonics: Electron Phonon PRinTout YAMBO data
Variable type: integer
Default: 0
For electron-phonon calculations, print out matrix elements for use by the yambo code.
Go to the top | List of ANADDB input variables ep_qptlist
Mnemonics: Electron Phonon Q PoinT LIST
Variable type: real array of 3*ep_nqpt elements
Default: *0
In case a non-uniform grid of q-points is being used, for direct calculation of the electron-phonon quantities without interpolation, this specifies the q-points to be found in the GKK file,
independently of the normal anaddb input (ngqpt), in reduced coordinates of the reciprocal space lattice.
Go to the top | List of ANADDB input variables ep_scalprod
Mnemonics: DO SCALar PRODuct for gkk matrix elements
Variable type: integer
Default: 0
The input variable
is a flag determining whether the scalar product of the electron-phonon matrix elements (gkk) with the phonon displacement vectors is done before or after interpolation. Doing so before (
1) makes phonon linewidths smoother but introduces an error, as the interpolated phonons and gkk are not diagonalized in the same basis. Doing so afterwards (
0) eliminates the diagonalization error, but sometimes gives small spikes in the phonon linewidths near band crossings or high symmetry points. I do not know why...
Go to the top | List of ANADDB input variables freeze_displ
Mnemonics: FREEZE DISPLacement of phonons into supercells
Variable type: real number
Default: 0.0
If different from zero,
will be used as the amplitude of a phonon displacement. For each q-point and mode in the
list, a file will be created containing a supercell of atoms with the corresponding phonon displacements frozen in. This is typically useful to freeze a soft phonon mode, then let it relax in abinit
is unitless, but has a physical meaning: it is related to the Bose distribution n_B and the frequency w_qs of the phonon mode. At a given temperature T,
will give the mean square displacement of atoms (along with the displacement vectors, which are in Bohr). In atomic units
= sqrt((0.5 + n_B(w_qs/kT) / w_qs) Typical values are 50-200 for a frequency of a few hundred cm-1 and room temperature. If all you want is to break the symmetry in the right direction, any
reasonable value (10-50) should be ok.
: this will create a
of files (3*natom*nph1l), so it should be used with a small number
of q-points for interpolation.
Go to the top | List of ANADDB input variables frmax
Mnemonics: FRequency : MAXimum
Variable type: real number
Default: 10.0
Value of the largest frequency for the frequency-dependent dielectric tensor, in Hartree.
Go to the top | List of ANADDB input variables frmin
Mnemonics: FRequency : MINimum
Variable type: real number
Default: 0.0
Value of the lowest frequency for the frequency-dependent dielectric tensor, in Hartree.
Go to the top | List of ANADDB input variables gkqwrite
Mnemonics: GKk for input Q grid to be WRITtEn to disk
Variable type: integer
Default: 0
Flag to write out the reciprocal space matrix elements to a disk file named gkqfile. This reduces strongly the memory needed for an electron-phonon run.
Mnemonics: Integer for the printing of AVerage FReQuency
Variable type: integer
Default: 0
Used only when
When this flag is set to 1, the "average frequency" is printed out (as a function of temperature, with phonon internal energy, free energy, entropy, ...). The average frequency is defined as:
Omega_average = Sum_over_q_and_i [Cv_iq Omega_iq]/Cv
- Omega_iq is the frequency of the ith mode for q-point q
- Cv is the specific heat
- Cv_iq is the contribution to the specific heat of the ith mode for q-point q
The "average frequency" can be used to have an estimation of the average Gruneisen parameter: Gamma_average=-d(log(Omega_average))/d(log(V)).
Go to the top | List of ANADDB input variables iatfix
Mnemonics: Indices of the AToms that are FIXed
Variable type: integer array (1:
Default: 0
Indices of the atoms that are fixed during a structural relaxation at constrained polarization. See
Go to the top | List of ANADDB input variables iatprj_bs
Mnemonics: Indices of the AToms for the PRoJection of the phonon Band Structure
Variable type: integer array (1:
Default: 0
Indices of the atoms that are chosen for projection of the phonon eigenvectors, giving a weighted phonon band structure file.
Go to the top | List of ANADDB input variables ifcana
Mnemonics: IFC ANAlysis
Variable type: integer
Default: 0
• 0 => no analysis of interatomic force constants;
• 1 => analysis of interatomic force constants.
If the analysis is activated, one get the trace of the matrices between pairs of atoms, if dipdip is 1, get also the trace of the short-range and electrostatic part, and calculate the ratio with the
full matrice; then define a local coordinate reference (using the next-neighbour coordinates), and express the interatomic force constant matrix between pairs of atoms in that local coordinate
reference (the first vector is along the bond; the second vector is along the perpendicular force exerted on the generic atom by a longitudinal displacement of the neighbouring atom - in case it does
not vanish; the third vector is perpendicular to the two other) also calculate ratios with respect to the longitudinal force constant ( the (1,1) element of the matrix in local coordinates).
Go to the top | List of ANADDB input variables
Mnemonics: Interatomic Force Constants FLAG
Variable type: integer
Default: 0
• 0 => do all calculations directly from the DDB, without the use of the interatomic force constant.
• 1 => calculate and use the interatomic force constants for interpolating the phonon spectrum and dynamical matrices at every q wavevector, and eventually analyse the interatomic force constants,
according to the informations given by atifc, dipdip, ifcana, ifcout, natifc, nsphere, rifcsph.
More detailed explanations : if the dynamical matrices are known on a regular set of wavevectors, they can be used to get the interatomic forces, which are simply their Fourier transform. When
non-analyticities can been removed by the use of effective charge at Gamma (option offered by putting
to 1), the interatomic forces are known to decay rather fast (in real space). The interatomic forces generated from a small set of dynamical matrices could be of sufficient range to allow the
remaining interatomic forces to be neglected. This gives a practical way to interpolate the content of a small set of dynamical matrices, because dynamical matrices can everywhere be generated
starting from this set of interatomic force constants. It is suggested to always use
=1. The
=0 option is available for checking purpose, and if there is not enough information in the DDB.
Go to the top | List of ANADDB input variables ifcout
Mnemonics: IFC OUTput
Variable type: integer
Default: 0
For each atom in the list
(generic atoms),
give the number of neighbouring atoms for which the ifc's will be output (written) and eventually analysed. The neighbouring atoms are selected by decreasing distance with respect to the generic
Go to the top | List of ANADDB input variables ifltransport
Mnemonics: IFLag for TRANSPORT
Variable type: integer
Default: 0
if ifltransport=1 anaddb calculates the transport properties: electrical and thermal resistivities from electron-phonon interactions (needs
= 1)
Go to the top | List of ANADDB input variables instrflag
Mnemonics: INternal STRain FLAG
Variable type: integer
Default: 0
Internal strain tensor flag.
• 0 => No internal-strain calculation.
• 1 => Print out both force-response and displacement-response internal-strain tensor. Requirements for preceding response-function DDB generation run: Strain and full atomic-displacement
responses. Set rfstrs = 1, 2, or 3 (preferably 3). Set rfatpol and rfdir to do a full calculation of phonons at Q=0.
Go to the top | List of ANADDB input variables istrfix
Mnemonics: Index of STRain FIXed
Variable type: integer array istrfix(1:
Default: 0
Indices of the elements of the strain tensor that are fixed during a structural relaxation at constrained polarisation :
• 0 => No elastic or compliance tensor will be calculated.
• 1 => Only clamped-ion elastic and compliance tensors will be calculated. Requirements for preceding response-function DDB generation run: Strain perturbation. Set rfstrs to 1, 2, or 3. Note that
rfstrs=3 is recommended so that responses to both uniaxial and shear strains will be computed.
• 2 => Both relaxed- and clamped-ion elastic and compliance tensor will be calculated, but only the relaxed-ion quantities will be printed. The input variable instrflag should also be set to 1,
because the internal-strain tensor is needed to compute the relaxed-ion corrections. Requirements for preceding response-function DDB generation run: Strain and atomic-displacement responses at Q
=0. Set rfstrs = 1, 2, or 3 (preferably 3). Set rfatpolrfatpol and rfdir to do a full calculation of phonons at Q=0 (needed because the inverse of force-constant tensor is required).
• 3 => Both relaxed and clamped-ion elastic and compliance tensors will be printed out. The input variable instrflag should also be set to 1. Requirements for preceeding response-function DDB
generation run: Same as for elaflag=2'.
• 4 => Calculate the elastic and compliance tensors (relaxed ion) at fixed displacement field, the relaxed-ion tensors at fixed electric field will be printed out too, for comparison. When elaflag=
4, we need the information of internal strain and relaxed-ion dielectric tensor to build the whole tensor, so we need set instrflag=1 and dieflag=3 or 4 .
• 5 => Calculate the relaxed ion elastic and compliance tensors, considering the stress left inside cell. At the same time, bare relaxed ion tensors will still be printed out for comparison. In
this calculation, stress tensor is needed to compute the correction term, so one supposed to merge the first order derivative data base (DDB file) with the second order derivative data base (DDB
file) into a new DDB file, which can contain both information. And the program will also check for the users.
Go to the top | List of ANADDB input variables kptrlatt
Mnemonics: K PoinT Reciprocal LATTice
Variable type: integer
Default: 9*0
Un normalized lattice vectors for the k-point grid in reciprocal space (see
abinit variable definition
as well). Input needed in electron-phonon calculations using nesting functions or tetrahedron integration.
Go to the top | List of ANADDB input variables kptrlatt_fine
Mnemonics: K PoinT Reciprocal LATTice for FINE grid
Variable type: integer
Default: 9*0
As kptrlatt above, but for a finer grid of k-points. Under development. Does not work yet, as of June 2010.
Go to the top | List of ANADDB input variables mustar
Mnemonics: MU STAR
Variable type: real
Default: 0.1
Average electron-electron interaction strength, for the computation of the superconducting Tc using Mc-Millan's formula.
Go to the top | List of ANADDB input variables natfix
Mnemonics: Number of AToms FIXed
Variable type: integer
Default: 0
Number of atoms that are fixed during a structural optimisation at constrained polarization. See
Go to the top | List of ANADDB input variables natifc
Mnemonics: Number of AToms for IFC analysis
Variable type: integer
Default: 0
Give the number of atoms for which ifc's are written and eventually analysed. The list of these atoms is provided by
atifc Go to the top | List of ANADDB input variables natprj_bs
Mnemonics: Number of AToms for PRoJection of the Band Structure
Variable type: integer
Default: 0
Give the number of atoms for which atomic-projected phonon band structures will be output. The list of these atoms is provided by
iatprj_bs Go to the top | List of ANADDB input variables nchan
Mnemonics: Number of CHANnels
Variable type: integer
Default: 800
The number of channels of width 1 cm-1 used in calculating the phonon density of states through the histogram method, or, equivalently, the largest frequency sampled. The first channel begins at 0.
Go to the top | List of ANADDB input variables nfreq
Mnemonics: Number of FREQuencies
Variable type: integer
Default: 1
Number of frequencies wanted for the frequency-dependent dielectric tensor. Should be positive. See
. The code will take
equidistant values from
Go to the top | List of ANADDB input variables ngqpt
Mnemonics: Number of Grids points for Q PoinTs
Variable type: integer array
Default: 3*0 (will not work)
The Monkhorst-Pack grid linear dimensions, for the DDB (coarse grid).
Go to the top | List of ANADDB input variables ng2qpt
Mnemonics: Number of Grids points for Q PoinTs (grid 2)
Variable type: integer array
Default: 3*0 (will not work)
The Monkhorst-Pack grid linear dimensions, for the finer of the series of fine grids. Used for the integration of thermodynamical functions (Bose-Einstein distribution) or for the DOS.
Go to the top | List of ANADDB input variables ngrids
Mnemonics: Number of GRIDS
Variable type: integer
Default: 4
This number define the series of grids that will be used for the estimation of the phonon DOS. The coarsest will be tried first, then the next, ... then the one described by
. The intermediate grids are defined for igrid=1...
, by the numbers ngqpt_igrid(ii)=(ng2qpt(ii)*igrid)/
ngrids Go to the top | List of ANADDB input variables nlflag
Mnemonics: Non-Linear FLAG
Variable type: integer
Default: 0
Non-linear properties flag.
• 0 => do not compute non-linear properties ;
• 1 => the electrooptic tensor, Raman susceptibilities and non-linear optical susceptibilities are calculated;
• 2 => only the non-linear optical susceptibilities and first-order changes of the dielectric tensor induced by an atomic displacement are calculated;
Go to the top | List of ANADDB input variables nph1l
Mnemonics: Number of PHonons in List 1
Variable type: integer
Default: 0
The number of wavevectors in phonon list 1, used for interpolation of the phonon frequencies. The values of these wavevectors will be specified by
The dynamical matrix for these wavevectors, obtained either directly from the DDB - if
=0 - or through the interatomic forces interpolation - if
=1 -), will be diagonalized, and the corresponding eigenfrequencies will be printed.
Go to the top | List of ANADDB input variables nph2l
Mnemonics: Number of PHonons in List 2
Variable type: integer
Default: 0
The number of wavevectors in phonon list 2, defining the directions along which the non-analytical splitting of phonon frequencies at Gamma will be calculated. The actual values of the wavevector
directions will be specified by qph2l. These are actually all wavectors at Gamma, but obtained by a limit along a different direction in the Brillouin-zone. It is important to note that
non-analyticities in the dynamical matrices are present at Gamma, due to the long-range Coulomb forces. So, going to Gamma along different directions can give different results.
The wavevectors in list 2 will be used to :
- generate and diagonalize a dynamical matrix, and print the corresponding eigenvalues.
- calculate the generalized Lyddane-Sachs-Teller relation. Note that if the three first numbers are zero, then the code will do a calculation at Gamma without non-analyticities.
Go to the top | List of ANADDB input variables
Mnemonics: Number of Q wavevectors defining a PATH
Variable type: integer
Default: 0
Number of q-points in the array
defining the path along which the phonon band structure and phonon linewidths are interpolated.
Go to the top | List of ANADDB input variables nqshft
Mnemonics: Number of Q SHiFTs
Variable type: integer
Default: 1
The number of vector shifts of the simple Monkhorst and Pack grid, needed to generate the coarse grid of q points (for the series of fine grids, the number of shifts it is always taken to be 1).
Usually, put it to 1. Use 2 if BCC sampling (Warning : not BCC lattice, BCC *sampling*), and 4 for FCC sampling (Warning : not FCC lattice, FCC *sampling*).
Go to the top | List of ANADDB input variables nsphere
Mnemonics: Number of atoms in SPHERe
Variable type: integer
Default: 0
Number of atoms included in the cut-off sphere for interatomic force constant, see also the alternative
. If
= 0 : maximum extent allowed by the grid .
This number defines the atoms for which the short range part of the interatomic force constants, after imposition of the acoustic sum rule, will not be put to zero. This option is available for
testing purposes (evaluate the range of the interatomic force constants), because the acoustic sum rule will be violated if some atoms are no more included in the inverse Fourier Transform.
Go to the top | List of ANADDB input variables
Mnemonics: Number of STRain components FIXed
Variable type: integer
Default: 0
Number of strain component that are fixed during a structural optimisation at constrained polarization. See
Go to the top | List of ANADDB input variables ntemper
Mnemonics: Number of TEMPERatures
Variable type: integer
Default: 10
Number of temperatures at which the thermodynamical quantities have to be evaluated. Now also used for the output of transport quantities in electron-phonon calculations. The full grid is specified
with the
Go to the top | List of ANADDB input variables nwchan
Mnemonics: Number of Widths of CHANnels
Variable type: integer
Default: 10
Integer. The width of the largest channel used to sample the frequencies. The code will generate different sets of channels, with decreasing widths (by step of 1 cm-1), from this channel width to 1,
eventually. It considers to have converged when the convergence criterion based on
have been fulfilled.
Mnemonics: OUTput files for Self Consistent PHONons
Variable type: integer
Default: 0
If set to 1, the phonon frequency and eigenvector files needed for a Self Consistent phonon run (as in Souvatzis PRL
095901) will be output to files appended _PHFRQ and _PHVEC. The third file needed is appended _PCINFO for Primitive Cell INFOrmation.
Go to the top | List of ANADDB input variables piezoflag
Mnemonics: PIEZOelectric tensor FLAG
Variable type: integer
Default: 0
Flag for calculation of piezoelectric tensors
• 0 => No piezoelectric tensor will be calculated.
• 1 => Only the clamped-ion piezoelectric tensor is computed and printed. Requirements for preceding response-function DDB generation run: Strain and electric-field responses. For the
electric-field part, one needs results from a prior 'ddk perturbation' run. Note that even if only a limited number of piezoelectric tensor terms are wanted (as determined by rfstrs and rfdir in
this calculation) it is necessary to set rfdir = 1 1 1 in the d/dk calculation for most structures. The only obvious exception to this requirement is cases in which the primitive lattice vectors
are all aligned with the cartesian axes. The code will omit terms in the output piezoelectric tensor for which the available d/dk set is incomplete. Thus: Set rfstrs to 1, 2, or 3i (preferably 3)
• 2 => Both relaxed- and clamped-ion elastic and compliance tensor will be calculated, but only the relaxed-ion quantities will be printed. The input variable instrflag should also be set to 1,
because the internal-strain tensor is needed to compute the relaxed-ion corrections. Requirements for preceding response-function DDB generation run: Strain, electric-field and full
atomic-displacement responses at Q=0. Set rfstrs = 1, 2, or 3 (preferably 3). Set rfelfd = 3. Set rfatpol and rfdir to do a full calculation of phonons at Q=0 (needed because the inverse of
force-constant tensor is required).
• 3 => Both relaxed and clamped-ion piezoelectric tensors will be printed out. The input variable instrflag should also be set to 1. Requirements for preceding response-function DDB generation run:
Same as for piezoflag=2.
• 4 => Calculate the piezoelectric d tensor (relaxed ion). In order to calculate the piezoelectric d tensor, we need information of internal strain and elastic tensor (relaxed ion). So we should
set elaflag= 2,3,4, or 5 and instrflag=1. The subroutine will also do a check for you, and print warning message without stopping even if flags were not correctly set.
• 5 => Calculate the piezoelectric g tensor (relaxed ion). In this computation, we need information of internal strain, elastic tensor (relaxed ion) and dielectric tensor (relaxed ion). So we
should set: instrflag=1, elaflag=2,3,4 or 5, dieflag=3 or 4. The subroutine will also do a check for you, and print warning message without stopping even if flags were not correctly set.
• 6 => Calculate the piezoelectric h tensor (relaxed ion). In this calculation, we need information of internal strain and dielectric tensor (relaxed ion). So we need set: instrflag=1 and dieflag=3
or 4. The subroutine will also do a check for you, and print warning message without stopping even if flags were not correctly set.
• 7 => calculate all the possible piezoelectric tensors, including e (clamped and relaxed ion), d, g and h tensors. The flags should be set to satisfy the above rules from 1 to 6.
Go to the top | List of ANADDB input variables piezoflag
Mnemonics: PIEZOelectric tensor FLAG
Variable type: integer
Default: 0
Flag for calculation of piezoelectric tensors
• 0 => No piezoelectric tensor will be calculated.
• 1 => Only the clamped-ion piezoelectric tensor is computed and printed. Requirements for preceding response-function DDB generation run: Strain and electric-field responses. For the
electric-field part, one needs results from a prior 'ddk perturbation' run. Note that even if only a limited number of piezoelectric tensor terms are wanted (as determined by rfstrs and rfdir in
this calculation) it is necessary to set rfdir = 1 1 1 in the d/dk calculation for most structures. The only obvious exception to this requirement is cases in which the primitive lattice vectors
are all aligned with the cartesian axes. The code will omit terms in the output piezoelectric tensor for which the available d/dk set is incomplete. Thus: Set rfstrs to 1, 2, or 3i (preferably 3)
• 2 => Both relaxed- and clamped-ion elastic and compliance tensor will be calculated, but only the relaxed-ion quantities will be printed. The input variable instrflag should also be set to 1,
because the internal-strain tensor is needed to compute the relaxed-ion corrections. Requirements for preceding response-function DDB generation run: Strain, electric-field and full
atomic-displacement responses at Q=0. Set rfstrs = 1, 2, or 3 (preferably 3). Set rfelfd = 3. Set rfatpol and rfdir to do a full calculation of phonons at Q=0 (needed because the inverse of
force-constant tensor is required).
• 3 => Both relaxed and clamped-ion piezoelectric tensors will be printed out. The input variable instrflag should also be set to 1. Requirements for preceding response-function DDB generation run:
Same as for piezoflag=2.
• 4 => Calculate the piezoelectric d tensor (relaxed ion). In order to calculate the piezoelectric d tensor, we need information of internal strain and elastic tensor (relaxed ion). So we should
set elaflag= 2,3,4, or 5 and instrflag=1. The subroutine will also do a check for you, and print warning message without stopping even if flags were not correctly set.
• 5 => Calculate the piezoelectric g tensor (relaxed ion). In this computation, we need information of internal strain, elastic tensor (relaxed ion) and dielectric tensor (relaxed ion). So we
should set: instrflag=1, elaflag=2,3,4 or 5, dieflag=3 or 4. The subroutine will also do a check for you, and print warning message without stopping even if flags were not correctly set.
• 6 => Calculate the piezoelectric h tensor (relaxed ion). In this calculation, we need information of internal strain and dielectric tensor (relaxed ion). So we need set: instrflag=1 and dieflag=3
or 4. The subroutine will also do a check for you, and print warning message without stopping even if flags were not correctly set.
• 7 => calculate all the possible piezoelectric tensors, including e (clamped and relaxed ion), d, g and h tensors. The flags should be set to satisfy the above rules from 1 to 6.
Go to the top | List of ANADDB input variables polflag
Mnemonics: POLarization FLAG
Variable type: integer
Default: 0
If activated, compute polarization in cartesian coordinates, and update lattice constants and atomic positions in order to perform a structural optimization at constrained polarization.
More detailed explanation : ANADDB can use the formalism described in Na Sai et al, PRB 66, 104108 (2002), to perform structural relaxations under the constraint that the polarization is equal to a
value specified by the input variable targetpol. The user starts from a given configurationof a crystal and performs a ground-state calculation of the Hellman-Feynman forces and stresses and the
Berry phase polarization as well as a linear response calculation of the whole matrix of second-order energy derivatives with respect to atomic displacement, strains and electric field.
In case polflag=1, ANADDB solves the linear system of equations (13) of the Na Sai paper, and computes new atomic positions (if relaxat=1) and lattice constant (if relaxstr=1). Then, the user uses
these parameters to perform a new ground-state and linear-response calculation. This must be repeated until convergence is reached. THe user can also fix some atomic positions, or strains, thanks to
the input variables natfix, nstrfix, iatfix, istrfix.
In case both relaxat and relaxstr are 0, while polflag=1, ANADDB only computes the polarization in cartesian coordinates.
As described in the Na Sai's paper, it is important to use the finite difference expression of the ddk (berryopt=2 or -2) in the linear response calculation of the effective charges and the
piezoelectric tensor.
Go to the top | List of ANADDB input variables
Mnemonics: PRinT the phonon Density Of States
Variable type: integer
Default : 0
The prtdos variable is used to calculate the phonon density of states, PHDOS, by Fourier interpolating the interatomic force constants on the (dense) q-mesh defined by ng2qpt. Note that the variable
ifcflag must be set to 1 since the interatomic force constants are supposed to be known.
The available options are:
• 0 => no output of PHDOS (default);
• 1 => calculate PHDOS using the gaussian method and the broadening defined by dossmear.
The step of the frequency grid employed to calculate the DOS can be defined through the input variable
Go to the top | List of ANADDB input variables prtfsurf
Mnemonics: PRinT the Fermi SURFace
Variable type: integer
Default: 0
Only for electron-phonon calculations. The available options are:
• 0 => do not write the Fermi Surface;
• 1 => write out the Fermi Surface in the BXSF format used by XCrySDen.
Further comments :
a) Only the eigenvalues for k-points inside the Irreducible Brillouin zone are required. As a consequence it is possible to use kptopt =1 during the GS calculation to reduce the computational effort.
b) Only unshifted k-grids that are orthogonal in reduced space are supported by XCrySDen. This implies that shiftk must be set to (0,0,0) during the GS calculation with nshiftk=1. Furthermore if
kptrlatt is used to generate the k-grid, all the off-diagonal elements of this array must be zero.
Go to the top | List of ANADDB input variables prtmbm
Mnemonics: PRinT Mode-By-Mode decomposition of the electrooptic tensor
Variable type: integer
Default: 0
• 0 => do not write the mode-by-mode decomposition of the electrooptic tensor;
• 1 => write out the contribution of the individual zone-center phonon modes to the electrooptic tensor.
Go to the top | List of ANADDB input variables prtnest
Mnemonics: PRinT the NESTing function
Variable type: integer
Default: 0
Only for electron-phonon calculations. This input variable is used to calculate the nesting function defined as: \chi_{nm}(q) = \sum_k \delta(\epsilon_{k,n}-epsilon_F) \delta(\epsilon_{k+q,m}-\
epsilon_F). The nesting factor is calculated for every point of the k-grid employed during the previous GS calculation. The values are subsequently interpolated along the trajectory in q space
defined by
, and written in the _NEST file using the X-Y format (
=1). It is also possible to analyze the behavior of the function in the reciprocal unit cell saving the values in the NEST_XSF file that can be read using
=2). Note that in the present implementation what is really printed to file is the "total nesting" defined as \sum_{nm} \chi_{nm}(q). Limitations: the k-grid defined by
must be orthogonal in reciprocal space, moreover off-diagonal elements are not allowed, i.e kptrlatt 4 0 0 0 4 0 0 0 4 is fine while kprtlatt = 1 0 0 0 1 1 0 -1 1 will not work.
• 0 => do not write the nesting function;
• 1 => write only the nesting function along the q-path in the X-Y format;
• 2 => write out the nesting function both in the X-Y and in the XSF format.
Go to the top | List of ANADDB input variables prtsrlr
Mnemonics: PRinT the Short-Range/Long-Range decomposition of phonon FREQuencies
Variable type: integer
Default: 0
Only if
=1. The available options are:
• 0 => do not write the SR/LR decomposition of phonon frequencies;
• 1 => write out the SR/LR decomposition of the square of phonon frequencies at each q-point specified in qph1l.
For details see
Europhys. Lett., 33 (9), pp. 713-718 (1996)
. See also
Go to the top | List of ANADDB input variables qgrid_type
Mnemonics: Q wavevectors defining a PATH
Variable type: integer
Default: 0
is set to 1, the electron-phonon part of anaddb will use the
variables to determine which q-points to calculate the electron-phonon coupling for. This is an alternative to a regular grid as in the rest of anaddb (using
Go to the top | List of ANADDB input variables qpath
Mnemonics: Q wavevectors defining a PATH
Variable type: real array
It is used to generate the path along which the phonon band structure and phonon linewidths are interpolated. There are
-1 segments to be defined, each of which starts from the end point of the previous one. The number of divisions in each segment is automatically calculated inside the code to respect the proportion
between the segments. The same circuit is used for the output of the nesting function if
Go to the top | List of ANADDB input variables qph1l
Mnemonics: Q for PHonon List 1
Variable type: real array
Default: 0
List of
wavevectors, at which the phonon frequencies will be interpolated. Defined by 4 numbers: the wavevector is made by the three first numbers divided by the fourth one (a normalisation factor). The
coordinates are defined with respect to the unit vectors that spans the Brillouin zone. Note that this set of axes can be non-orthogonal and not normed. The normalisation factor makes easier the
input of wavevector such as (1/3,1/3,1/3), represented by 1.0 1.0 1.0 3.0 .
The internal representation of this array is as follows : for each wavevector, the three first numbers are stored in the array qph1l(3,nph1l), while the fourth is stored in the array qnrml1(nph1l).
Go to the top | List of ANADDB input variables qph2l
Mnemonics: PHonon List 2
Variable type: real array qph2l(4,
Default: all 0
List of phonon wavevector
along which the non-analytical correction to the Gamma-point phonon frequencies will be calculated (for insulators). Four numbers, as for
, but where the last one, that correspond to the normalisation factor, is 0.0 For the anaddb code, this has the meaning that the three previous values define a direction. The direction is in
CARTESIAN COORDINATES, unlike the non-Gamma wavevectors defined in the first list of vectors...
Note that if the three first numbers are zero, then the code will do a calculation at Gamma without non-analyticities.
Also note that the code automatically set the imaginary part of the dynamical matrix to zero. This is useful to compute the phonon frequencies when half of the k-points has been used, by the virtue
of the time-reversal symmetry (which may induce parasitic imaginary parts...).
The internal representation of this array is as follows : for each wavevector, the three first numbers are stored in the array qph2l(3,nph2l), while the fourth is stored in the array qnrml2(nph2l).
Go to the top | List of ANADDB input variables
Mnemonics: Q-point REFINEment order (experimental)
Variable type: integer
Default: 0
is superior to 1, attempts to initialize a first set of dynamical matrices from the DDB file, with a q-point grid which is
divided by
(e.g. ngqpt 4 4 4 qrefine 2 starts with a 2x2x2 grid). The dynamical matrices are interpolated onto the full
grid and any additional information found in the DDB file is imposed, before proceeding to normal band structure and other interpolations. Should implement Gaal-Nagy's algorithm in PRB
Go to the top | List of ANADDB input variables q1shft
Mnemonics: Q shifts for the grid number 1
Variable type: real array
Default: all 0.0
This vector gives the shifts needed to define the coarse q-point grid.
a) Case nqshft=1 In general, 0.5 0.5 0.5 with the ngqpt's even will give very economical grids. On the other hand, is it sometimes better for phonons to have the Gamma point in the grid. In that
case, 0.0 0.0 0.0 should be OK. For the hexagonal lattice, the above mentioned quantities become 0.0 0.0 0.5 and 0.0 0.0 0.0 .
b) Case nqshft=2 The two q1shft vectors must form a BCC lattice. For example, use 0.0 0.0 0.0 and 0.5 0.5 0.5
c) Case nqshft=4 The four q1shft vectors must form a FCC lattice. For example, use 0.0 0.0 0.0 , 0.0 0.5 0.5 , 0.5 0.0 0.5 , 0.5 0.5 0.0 or 0.5 0.5 0.5 , 0.0 0.0 0.5 , 0.0 0.5 0.0 , 0.5 0.0 0.0 (the
latter is referred to as shifted)
Further comments : by using this technique, it is possible to increase smoothly the number of q-points, at least less abruptly than relying on series of grids like (for the full cubic symmetry):
1x1x1 => (0 0 0)
2x2x2 (shifted) => (.25 .25 .25)
2x2x2 => 1x1x1 + (.5 0 0) (.5 .5 0) (.5 .5 0)
4x4x4 => 2x2x2 + (.25 0 0) (.25 .25 0) (.25 .5 0) (.25 .25 .25) (.25 .25 .5) (.25 .5 .5)
with respectively 1, 1, 4 and 10 q-points, corresponding to a number of points in the full BZ of 1, 8, 8 and 64. Indeed, the following grids are made available :
1x1x1 with nqshft=2 => (0 0 0) (.5 .5 .5)
1x1x1 with nqshft=4 => (0 0 0) (.5 .5 0)
1x1x1 with nqshft=4 (shifted) => (.5 0 0) (.5 .5 .5)
2x2x2 with nqshft=2 => 2x2x2 + (.25 .25 .25)
2x2x2 with nqshft=4 => 2x2x2 + (.25 .25 0) (.25 .25 .5)
2x2x2 with nqshft=4 (shifted) => (.25 0 0) (.25 .25 .25) (.5 .5 .25) (.25 .5 0)
with respectively 2, 2, 2, 5, 6 and 4 q-points, corresponding to a number of points in the full BZ of 2, 4, 4, 16, 32 and 32.
For a FCC lattice, it is possible to sample only the Gamma point by using a 1x1x1 BCC sampling (nqshft=2).
Go to the top | List of ANADDB input variables
Mnemonics: Q points SHiFTs for the grids 2
Variable type: real array q2shft(3)
Default: all 0
Similar to
, but for the series of fine grids. Note that
for this series of grids corresponds to 1.
Go to the top | List of ANADDB input variables ramansr
Mnemonics: RAMAN Sum-Rule
Variable type: integer
Default: 0
Govern the imposition of the sum-rule on the Raman tensors.
As in the case of the Born effective charges, the first-order derivatives of the linear dielectric susceptibility with respect to an atomic displacement must vanish when they are summed over all
atoms. This sum rule is broken in most calculations. By putting
equal to 1 or 2, this sum rule is imposed by giving each atom a part of the discrepancy.
• 0 => no sum rule is imposed;
• 1 => impose the sum rule on the Raman tensors, giving each atom an equal part of the discrepancy;
• 2 => impose the sum rule on the Raman tensors, giving each atom a part of the discrepancy proportional to the magnitude of its contribution to the Raman tensor.
For the time being,
=1 is the preferred choice.
Go to the top | List of ANADDB input variables relaxat
Mnemonics: RELAXation of AToms
Variable type: integer
Default: 0
=1, relax atomic positions during a structural relaxation at constrained polarization. See
Go to the top | List of ANADDB input variables relaxstr
Mnemonics: RELAXation of STRain
Variable type: integer
Default: 0
=1, relax lattice constants (lengths/angles) during a structural relaxation at constrained polarization. See
Go to the top | List of ANADDB input variables relaxstr
Mnemonics: RELAXation of STRain
Variable type: integer
Default: 0
=1, relax lattice constants (lengths/angles) during a structural relaxation at constrained polarization. See
Go to the top | List of ANADDB input variables rfmeth
Mnemonics: Response-Function METHod
Variable type: integer
Default: 1
Select a particular set of Data Blocks in the DDB. (PRESENTLY, ONLY OPTION 1 IS AVAILABLE)
• 1 => Blocks obtained by a non-stationary formulation.
• 2 => Blocks obtained by a stationary formulation.
For more detailed explanations, see abinit_help If the information in the DDB is available, always use the option 2. If not, you can try option 1, which is less accurate.
Go to the top | List of ANADDB input variables rifcsph
Mnemonics: Radius of the Interatomic Force Constant SPHere
Variable type: real
Default: zero
Cut-off radius for the sphere for interatomic force constant, see also the alternative
. If
= 0 : maximum extent allowed by the grid .
This number defines the atoms for which the short range part of the interatomic force constants, after imposition of the acoustic sum rule, will not be put to zero.
Go to the top | List of ANADDB input variables
Mnemonics: SeLECT Z
Variable type: integer
Default: 0
Select some parts of the effective charge tensor. (This is done after the application or non-application of the ASR for effective charges). The transformed effective charges are then used for all the
subsequent calculations.
Go to the top | List of ANADDB input variables selectz
Mnemonics: SeLECT Z
Variable type: integer
Default: 0
Select some parts of the effective charge tensor. (This is done after the application or non-application of the ASR for effective charges). The transformed effective charges are then used for all the
subsequent calculations.
• 0 => The effective charge tensor is left as it is.
• 1 => For each atom, the effective charge tensor is made isotropic, by calculating the trace of the matrix, dividing it by 3, and using this number in a diagonal effective charge tensor.
• 2 => For each atom, the effective charge tensor is made symmetric, by simply averaging on symmetrical elements.
Note : this is for analysis the effect of anisotropy in the effective charge. The result with non-zero selectz are unphysical.
Go to the top | List of ANADDB input variables symdynmat
Mnemonics: SYMmetrize the DYNamical MATrix
Variable type: integer
Default: 1 (was 0 before v5.3)
is equal to 1, the dynamical matrix is symmetrized before the diagonalization.
This is especially useful when the set of primitive vectors of the unit cell and their opposite do not reflect the symmetries of the Bravais lattice (typical case : body-centered tetragonal lattices
; FCC and BCC lattices might be treated with the proper setting of the
variable), and the interpolation procedure based on interatomic force constant is used : there are some slight symmetry breaking effects. The latter can be bypassed by this additional symmetrization.
Go to the top | List of ANADDB input variables symgkq
Mnemonics: SYMmetrize the GKk matrix elements for each Q
Variable type: integer
Default: 1
is equal to 1, the electron-phonon matrix elements are symmetrized over the small group of the q-point they correspond to. This should always be used, except for debugging or test purposes.
Go to the top | List of ANADDB input variables targetpol
Mnemonics: TARGET POLarization
Variable type: real targetpol(1:3)
Default: 0.0
Target value of the polarization in cartesian coordinates and in C/m^2. See
Go to the top | List of ANADDB input variables telphint
Mnemonics: Technique for ELectron-PHonon INTegration
Variable type: integer
Default: 1
Flag controlling the Fermi surface integration technique used for electron-phonon quantities.
• 0 = tetrahedron method (no adjustable parameter)
• 1 = Gaussian smearing (see elphsmear)
• 2 = uniformly weighted band window between ep_b_min and ep_b_max, for all k-points
Go to the top | List of ANADDB input variables temperinc
Mnemonics: TEMPERature INCrease
Variable type: real
Default: 2.0
Increment of the temperature in Kelvin, for thermodynamical and el-phon transport properties. See the associated
Go to the top | List of ANADDB input variables tempermin
Mnemonics: TEMPERature MINimum
Variable type: real
Default: 1.0
Lowest temperature (Kelvin) at which the thermodynamical quantities have to be evaluated. Cannot be zero.
The highest temperature is defined using temperinc and ntemper.
Go to the top | List of ANADDB input variables
Mnemonics: THerMal FLAG
Variable type: integer
Default: 0
Flag controlling the calculation of thermal quantities.
• When thmflag == 1, the code will compute, using the histogram method :
□ the normalized phonon DOS
□ the phonon internal energy, free energy, entropy, constant volume heat capacity as a function of the temperature
□ the Debye-Waller factors (tensors) for each atom, as a function of the temperature
□ the mean-square velocity tensor for each atom, as a function of temperature
□ the "average frequency" as a function of the temperature (if iavfrq=1)
• When thmflag == 2, all the phonon frequencies for the q points in the second grid are printed.
• When thmflag == 3, 5 or 7, the thermal corrections to the electronic eigenvalues are calculated. If thmflag==3, the list of phonon wavevector from the first list is used (with equal weight for
all wavevectors in this list), while if thmflag==5 or 7, the first grid of wavevectors is used, possibly folded to the irreducible Brillouin Zone if symmetry operations are present, or if they
are recomputed (this happens for thmflag==7).
• When thmflag == 4 or 6, the temperature broadening (electron lifetime) of the electronic eigenvalues is calculated. If thmflag==4, the list of phonon wavevector from the first list is used (with
equal weight for all wavevectors in this list), while if thmflag==6, the first grid of wavevectors is used, possibly folded to the irreducible Brillouin Zone if symmetry operations are present or
if they are recomputed (this happens for thmflag==8).
WARNING : The use of symmetries for the temperature dependence of the eigenenergies is tricky ! It can only be valid for the k points that respect the symmetries (i.e. the Gamma point), provided one
also averages over the degenerate states.
Input variables that may be needed if this flag is activated : dostol, nchan, ntemper, temperinc, tempermin, as well as the wavevector grid number 2 definition, ng2qpt, ngrids, q2shft.
Go to the top | List of ANADDB input variables
Mnemonics: THerModynamic TOLerance
Variable type: real
Default: 0.05
The relative tolerance on the thermodynamical functions This number will determine when the series of channel widths with which the DOS is calculated can be stopped, i.e. the mean of the relative
change going from one grid to the next bigger is smaller than
Go to the top | List of ANADDB input variables
Goto :
ABINIT home Page | Suggested acknowledgments | List of input variables | Tutorial home page | Bibliography
Help files :
New user's guide | Abinit (main) | Abinit (respfn) | Mrgddb | Anaddb | AIM (Bader) | Cut3D | Optic | {"url":"http://www.abinit.org/documentation/helpfiles/for-v6.12/users/anaddb_help.html","timestamp":"2014-04-20T13:38:49Z","content_type":null,"content_length":"115658","record_id":"<urn:uuid:bd1aa0ff-0937-48ad-a7a6-6eb3aa7df13e>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00088-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Difference Between Regression and Correlation
Regression vs Correlation
In statistics, determining the relation between two random variables is important. It gives the ability to make predictions about one variable relative to others. Regression analysis and correlation
are applied in weather forecasts, financial market behaviour, establishment of physical relationships by experiments, and in much more real world scenarios.
What is Regression?
Regression is a statistical method used to draw the relation between two variables. Often when data are collected there might be variables which are dependent on others. The exact relation between
those variables can only be established by the regression methods. Determining this relationship helps to understand and predict the behaviour of one variable to the other.
Most common application of the regression analysis is to estimate the value of the dependent variable for a given value or range of values of the independent variables. For example, using regression
we can establish the relation between the commodity price and the consumption, based on the data collected from a random sample. Regression analysis produces the regression function of a data set,
which is a mathematical model that best fits to the data available. This can easily be represented by a scatter plot. Graphically, regression is equivalent to finding the best fitting curve for the
give data set. The function of the curve is the regression function. Using the mathematical model, the demand of a commodity can be predicted for a given price.
Therefore, the regression analysis is widely used in predicting and forecasting. It is also used to establish relationships in experimental data, in the fields of physics, chemistry, and many natural
sciences and engineering disciplines. If the relationship or the regression function is a linear function, then the process is known as a linear regression. In the scatter plot, it can be represented
as a straight line. If the function is not a linear combination of the parameters, then the regression is non-linear.
What is Correlation?
Correlation is a measure of strength of the relationship between two variables. The correlation coefficient quantifies the degree of change in one variable based on the change in the other variable.
In statistics, correlation is connected to the concept of dependence, which is the statistical relationship between two variables.
The Pearsons’s correlation coefficient or just the correlation coefficient r is a value between -1 and 1 (-1≤r≤+1) . It is the most commonly used correlation coefficient and valid only for a linear
relationship between the variables. If r=0, no relationship exist, and if r≥0, the relation is directly proportional; i.e. the value of one variable increases with the increase of the other. If r≤0,
the relationship is inversely proportional; i.e. one variable decreases as the other increases.
Because of the linearity condition, correlation coefficient r can also be used to establish the presence of a linear relationship between the variables.
What is the difference between Regression and Correlation?
Regression gives the form of the relationship between two random variables, and the correlation gives the degree of strength of the relationship.
Regression analysis produces a regression function, which helps to extrapolate and predict results while correlation may only provide information on what direction it may change.
The more accurate linear regression models are given by the analysis, if the correlation coefficient is higher. (|r|≥0.8)
Related posts: | {"url":"http://www.differencebetween.com/difference-between-regression-and-vs-correlation/","timestamp":"2014-04-20T21:41:01Z","content_type":null,"content_length":"90205","record_id":"<urn:uuid:21703333-74bf-4e93-87ea-9409a90537f3>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00663-ip-10-147-4-33.ec2.internal.warc.gz"} |
matrix theory
matrix theory
Let $A$ be a Lawvere theory with generic object $T$. The full subcategory of $A$ generated by the cartesian powers of $T^n$ is also a Lawvere theory, that we denote by $M_n(A)$. In the case of an
annular theory (the theory of modules over a ring that we also call $A$), this is the construction of $n\times n$ matrices over $A$. If we denote by $M_n$ the application of this construction to the
initial theory (the theory of sets), then we may identify $M_n(A)$ with the tensor product theory $M_n\otimes A$.
It is an amusing exercise to present $M_n$ in terms of generating operations and relations between them.
Revised on June 23, 2009 19:28:33 by
Toby Bartels | {"url":"http://www.ncatlab.org/nlab/show/matrix+theory","timestamp":"2014-04-16T13:46:35Z","content_type":null,"content_length":"14582","record_id":"<urn:uuid:6e0705a1-b0d1-42e0-80e7-a39a58b2a9a0>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00161-ip-10-147-4-33.ec2.internal.warc.gz"} |
Minimum Area of Steel required for Spread Footings
imsengr (Structural) 16 Feb 07 15:29
Hi everyone,
I'm designing isolated spread footings for an industrial plant and my research and discussions with colleagues are giving me differing answers to the minimum amount of steel area required for
Group I says the area of steel must not be less than the temperature or shrinkage steel, i.e. rho, or the reinforcemet ratio, must be greater than 0.0018, and as long as it is that, the steel area is
ok. In fact, in the footing examples provided in Notes on ACI 318-89 (Pg. 24-11) and ACI 318-95 (Pg. 371), this appears to the cases.
Group II says that the footing is a flexural member, and thus, must meet the minimum reinforcement for flexural members. The old requirement was 200/fy, but the newer ACI Codes called for not less
3*SQRT (f'c)/fy * bw * d, and not less than:
Now, which I should I follow? Advice from structural engineers who work a lot with foundations, and anyone else, is most welcome.
Thanks, y'all. | {"url":"http://www.eng-tips.com/viewthread.cfm?qid=178679","timestamp":"2014-04-20T00:52:33Z","content_type":null,"content_length":"33200","record_id":"<urn:uuid:1e33754d-d84e-47ff-8a60-8ff694764c9c>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00526-ip-10-147-4-33.ec2.internal.warc.gz"} |
Infeasible models
A linear program is infeasible if there exists no solution that satisfies all of the constraints -- in other words, if no feasible solution can be constructed. Since any real operation that you are
modelling must remain within the constraints of reality, infeasibility most often indicates an error of some kind. Simplex-based LP software like lp_solve efficiently detects when no feasible
solution is possible.
The source of infeasibility is often difficult to track down. It may stem from an error in specifying some of the constraints in your model, or from some wrong numbers in your data. It can be the
result of a combination of factors, such as the demands at some customers being too high relative to the supplies at some warehouses.
Upon detecting infeasibility, LP codes typically show you the most recent infeasible solution that they have encountered. Sometimes this solution provides a good clue as to the source of
infeasibility. If it fails to satisfy certain capacity constraints, for example, then you would do well to check whether the capacity is sufficient to meet the demand; perhaps a demand number has
been mistyped, or an incorrect expression for the capacity has been used in the capacity constraint, or the model simply lacks any provision for coping with increasing demands. More often,
unfortunately, LP codes respond to an infeasible problem by returning a meaninglessly infeasible solution, such as one that violates material balances. lp_solve is behaving also as such.
lp_solve currently doesn't provide analysis routines to detect infeasible constraints however that doesn't mean that it stops there.
A useful approach is to forestall meaningless infeasibilities by explicitly modelling those sources of infeasibility that you view as realistic. As a simple example, you could add a new "slack"
variable on each capacity constraint, having a very high penalty cost. Then infeasibilities in your capacities would be signalled by positive values for these slacks at the optimal solution, rather
than by a mysterious lack of feasibility in the linear program as a whole. Modelling approaches that use this technique are called sometimes "elastic programming" or "elastic filter".
So in practice, if a constraint is a < constraint, add a variable to the model and give it for that constraint a -1 coefficient for that variable. In the objective you give it a relative large cost.
If a constraint is a > constraint, add a variable to the model and give it for that constraint a +1 coefficient for that variable. In the objective you give it a relative large cost. If a constraint
is an equal constraint, add two variables to the model and give it for that constraint respectively a -1 and +1 coefficient for that variable. In the objective you give them a relative large cost. Or
you only add one variable and give it an -infinite lower bound.
This will result in an automatic relaxation of the constraint(s) when needed (if that constraint would make the model infeasible). To make sure that these added variables only get non-zero values
when the constraint is violating, the value in the objective must be relative large. Like that this variable gets a penalty cost and it will only become non-zero when really needed. Note that the
signs of these objective coefficients must be positive when minimizing and negative when maximizing. Don't make these costs too big also because that introduces instabilities. If none of these added
variables have a non-zero value then the model was initially feasible. When at least one is non-zero then the original model is infeasible. Note that the objective value will then not be very useful.
However you could subtract the cost * value of all these variables from the objective to obtain the objective value of the relaxed model.
Note that a model can also become infeasible because of bounds set on variables. Above approach doesn't relax these.
min: x + y;
c1: x >= 6;
c2: y >= 6;
c3: x + y <= 11;
This model is clearly infeasible. Now introduce extra variables to locate the infeasibility:
min: x + y + 1000 e1 + 1000 e2 + 1000 e3;
c1: x + e1 >= 6;
c2: y + e2 >= 6;
c3: x + y - e3 <= 11;
The result of this model is:
Value of objective function: 1011
Actual values of the variables:
x 5
y 6
e1 1
e2 0
e3 0
With this simple example model, multiple solutions were possible. Here, the first constraint was relaxed since e1 is non-zero. Only this one constraint had to be relaxed to make the model feasible.
The objective value of 1011 isn't saying very much. However if we subtract 1000 e1 + 1000 e2 + 1000 e3 from it, then it becomes 11 which is the value of the original objective function (x + y). | {"url":"http://lpsolve.sourceforge.net/5.5/Infeasible.htm","timestamp":"2014-04-23T15:07:42Z","content_type":null,"content_length":"6000","record_id":"<urn:uuid:27d10634-faa9-442e-98dc-da9005071c76>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00638-ip-10-147-4-33.ec2.internal.warc.gz"} |
First-class polymorphic function values in shapeless (2 of 3) — Natural Transformations in Scala
Posted by Miles Sabin on 10th May 2012
Last time we saw that Scala’s standard function values weren’t going to help us in our goal of mapping over an HList because they’re insufficiently polymorphic. In this article I’m going to start
exploring how we can address that problem. The techniques I’m going to explain are fairly well known and extremely useful where applicable, but ultimately they’re not quite enough to get us all the
way there — however they will set the scene for a solution which is.
As our running examples of polymorphic functions, let’s take the singleton function from last time (which given an argument of type T should return a single element Set[T] containing it), the
headOption function (which given an argument of type List[T] gives us back its head as an Option[T]), the identity function (which returns its argument unchanged), and a generic size function which
will compute an integer size appropriate to the type of its argument (eg. the size of a List or a String will be its length).
This is how we expect them to behave in the REPL,
scala> singleton("foo")
res0: Set[String] = Set(foo)
scala> identity(1.0)
res1: Double = 1.0
scala> headOption(List(1, 2, 3))
res2: Option[Int] = Some(1)
scala> size("foo")
res3: Int = 3
scala> size(List(1, 2, 3, 4))
res4: Int = 4
The function-like signatures for each of these are as follows,
singleton (∀T) T => Set[T]
identity (∀T) T => T
headOption (∀T) List[T] => Option[T]
size (∀T) T => Int
I say “function-like” here because, of course, Scala can’t directly express generic function value types of this form — that’s the problem we’re trying to solve.
Polymorphism lost, polymorphism regained
Recall from the preceeding article that the explanation for Scala’s function values being monomorphic is that the polymorphism of the FunctionN traits is fixed at the point at which they’re
instantiated rather than the point at which they’re applied. This follows immediately from the position that their type parameters occur in their definition. For example, for Function1,
trait Function1[-T, +R] {
def apply(t : T) : R
As you can see, the argument and result type parameters are declared at the trait level and hence are fixed for each invocation of the apply method.
The natural move at this point is try to shift the type parameters off Function1 and onto the apply method making it polymorphic independently of its enclosing trait — as we saw last time, the
combination of polymorphic methods and call site eta-expansion gets us something that looks very much like a polymorphic function value.
We still want to be left with a first class type, values of which can be passed as arguments to higher-order functions, so we have to keep an enclosing type of some sort. A first naive pass at this
might look something like,
trait PolyFunction1 {
def apply[T, R](t : T) : R
But this won’t do at all, as we discover as soon as we try to implement it.
The problem we immediately run up against is that the result type of the apply method of our PolyFunction1 trait is completely unconstrained. But the signatures we’re trying to implement require that
the result type be determined by the argument type (singleton, identity, headOption) or constant (size). There’s no way that we can map those signatures into the form required for the common trait.
Unconstrained result type parameters are in any case problematic when it comes to type inference, as I discussed in an earlier article, so let’s start by focussing on the singleton case where it’s
easy to view the result type as a simple function of the argument type. This leads us to a second pass at a polymorphic function trait which captures that idea directly in terms of a higher-kinded
trait-level type parameter — the higher-kinded type parameter is going act as a type-level function,
trait PolyFunction1[F[_]] {
def apply[T](t : T) : F[T]
We can now define singleton as follows,
object singleton extends PolyFunction1[Set] {
def apply[T](t : T) : Set[T] = Set(t)
and this behaves more or less as you’d expect — in particular, note the inferred result types,
scala> singleton(23)
res0: Set[Int] = Set(23)
scala> singleton("foo")
res1: Set[String] = Set(foo)
So far so good. Now, can we squeeze identity into the same mould? To do that we need to find a higher-kinded type for the F type-argument to PolyFunction1 such that F[T] = T — a type-level identity
function in fact! Scala’s type aliases make such a type extremely straightforward to define — it’s just,
type Id[T] = T
Now we can define and apply the identity function like so,
object identity extends PolyFunction1[Id] {
def apply[T](t : T) : T = t
scala> identity(23)
res0: Int = 23
scala> identity("foo")
res1: java.lang.String = foo
Next up is headOption. In this case we have a signature that has constraints on its argument type as well, not just on its result type as was the case for singleton and identity. Hopefully, though,
it should be clear that we can repeat the same trick, and view both the argument type and the result type as functions of a common underlying type. This leads us to a third pass at the polymorphic
function trait which this time has two higher-kinded trait-level type parameters — one to constrain the argument type and one to constrain the result type,
trait PolyFunction1[F[_], G[_]] {
def apply[T](f : F[T]) : G[T]
And now we can define our first three functions as follows,
object singleton extends PolyFunction1[Id, Set] {
def apply[T](t : T) : Set[T] = Set(t)
object identity extends PolyFunction1[Id, Id] {
def apply[T](t : T) : T = t
object headOption extends PolyFunction1[List, Option] {
def apply[T](l : List[T]) : Option[T] = l.headOption
scala> singleton("foo")
res0: Set[java.lang.String] = Set(foo)
scala> identity(1.0)
res1: Double = 1.0
scala> headOption(List(1, 2, 3))
res2: Option[Int] = Some(1)
That just leaves the size function. Handling that entails making the constant result type Int take the form of a higher-kinded type as well. We can do that with the help of a type lambda representing
a type-level function from an arbitrary type T to some constant type C,
type Const[C] = {
type λ[T] = C
For the particular case of type Const[Int], it is a structural type with a higher-kinded type member λ[_] which is equal to Int no matter what type argument it is applied to. So the type Const[Int]#λ
[T] will be equal to type Int whatever type we substitute for T. Here’s short REPL session demonstrating that,
scala> implicitly[Const[Int]#λ[String] =:= Int]
res0: =:=[Int,Int] = <function1>
scala> implicitly[Const[Int]#λ[Boolean] =:= Int]
res1: =:=[Int,Int] = <function1>
This is a type-level rendering of the value-level constant function that you might also know as the K combinator from the SKI calculus (or as a “Kestrel” if you’re a Ray Smullyan fan),
def const[T](t : T)(x : T) = t
scala> val const3 = const(3) _
const3: Int => Int = <function1>
scala> const3(23)
res6: Int = 3
With this in hand we can begin to define a size function that implements the same PolyFunction1 trait as singleton, identity and headOption,
object size extends PolyFunction1[Id, Const[Int]#λ] {
def apply[T](t : T) : Int = 0
scala> size(List(1, 2, 3, 4))
res0: Int = 0
scala> size("foo")
res1: Int = 0
We have the signature right, at least, but what about the implementation of the apply method? Just returning a constant 0 isn’t particularly interesting. Unfortunately we don’t have much to work with
— the use of Id on the argument side is what allows this function to be applicable to both Lists and Strings, but the direct consequence of that generality is that within the body of the method we
have no knowledge about the type of the argument, so we have no immediate way of computing an appropriate result.
We can pattern match here of course, but as we’ll see that’s not a particularly desirable solution. For now let’s just go with that, and note a distinct lingering code smell,
object size extends PolyFunction1[Id, Const[Int]#λ] {
def apply[T](t : T) : Int = t match {
case l : List[_] => l.length
case s : String => s.length
case _ => 0
scala> size(List(1, 2, 3, 4))
res0: Int = 4
scala> size("foo")
res1: Int = 3
scala> size(23)
res2: Int = 0
A spoon full of sugar
Parenthetically, I’d like to flag up a small syntactic tweak that we can make to the PolyFunction1 trait which takes advantage of symbolic names in Scala and the ability to write types with two type
arguments using an infix notation. The latter allows us to write types of the form T[X, Y] as X T Y. And if we choose the name T carefully this can give us a very syntactically elegant way of
expressing the concept we’re trying to render.
Here we’re talking about the types of function-like things, so something which puns on Scala’s function arrow symbol => is a good choice — let’s use ~>. Our trait now looks like this,
trait ~>[F[_], G[_]] {
def apply[T](f : F[T]) : G[T]
and our definitions look a lot more visibly function-like,
object singleton extends (Id ~> Set) {
def apply[T](t : T) : Set[T] = Set(t)
object identity extends (Id ~> Id) {
def apply[T](t : T) : T = t
object headOption extends (List ~> Option) {
def apply[T](l : List[T]) : Option[T] = l.headOption
object size extends (Id ~> Const[Int]#λ) {
def apply[T](t : T) : Int = t match {
case l : List[_] => l.length
case s : String => s.length
case _ => 0
I’ve been careful to describe these things as “function-like” values rather than as functions to emphasize that they don’t and can’t conform to Scala’s standard FunctionN types. The immediate upshot
of this is that they can’t be directly passed as arguments to any higher-order function which expects to receive an ordinary Scala function argument. For example,
scala> List(1, 2, 3) map singleton
:11: error: type mismatch;
found : singleton.type (with underlying type object singleton)
required: Int => ?
List(1, 2, 3) map singleton
We can fix this however — whilst ~> can’t extend Function1, we can use an implicit conversion to do a job similar to the one that eta-expansion does for polymorphic methods,
implicit def polyToMono[F[_], G[_], T](f : F ~> G) : F[T] => G[T] = f(_)
This is along the right lines, but unfortunately due to a current limitation in Scala’s type inference this won’t work for functions like singleton that are parametrized with Id or Const because
those types will never be inferred for F[_] or G[_]. We can help out the Scala compiler with a few additional implicit conversions to cover all the relevant permutations of those cases,
implicit def polyToMono2[G[_], T](f : Id ~> G) : T => G[T] = f(_)
implicit def polyToMono3[F[_], T](f : F ~> Id) : F[T] => T = f(_)
implicit def polyToMono4[T](f : Id ~> Id) : T => T = f[T](_)
implicit def polyToMono5[F[_], G, T](f : F ~> Const[G]#λ) : F[T] => G = f(_)
implicit def polyToMono6[G, T](f : Id ~> Const[G]#λ) : T => G = f(_)
With these in place we can map singleton over an ordinary Scala List,
scala> List(1, 2, 3) map singleton
res0: List[Set[Int]] = List(Set(1), Set(2), Set(3))
Natural transformations and their discontents
This encoding of polymorphic function values in Scala has been around for quite some time — in fact more or less from the point at which higher-kinded types arrived in the language. And, as a
representation of a natural transformation, it’s been put to good use in scalaz.
So we’re done, right? Well, no not really. Whilst function-like values of this form are undoubtedly useful, they have a number of shortcomings which make them less than ideal in general. Let’s have a
look at some of them now.
The first problem we saw earlier in the implementation of the size function,
object size extends (Id ~> Const[Int]#λ) {
def apply[T](t : T) : Int = t match {
case l : List[_] => l.length
case s : String => s.length
case _ => 0
Because the apply method’s type parameter T is completely unconstrained, the type of the argument t within the method body is effectively Any. In other words, the compiler knows nothing at all about
its shape, specifically it can’t know that it has a length method yielding an Int.
We can pattern match to recover some type information as we’ve done above, but this is unsatisfactory for several reasons. First, we have to be careful to handle all cases or risk being hit by a
MatchError — that forces us to include a possibly artifical default case, or take our chances at runtime. It’s also hopelessly non-modular — if we want to add cases for additional types then we have
to modify this definition rather than adding orthogonal code to handle the new cases. Not good.
Second we have to be aware of the limitations of pattern matching in the face of type erasure. For example, suppose we had wanted to define the size of a List[String] as the sum of the lengths of its
String elements. We might try something like,
object size extends (Id ~> Const[Int]#λ) {
def apply[T](t : T) : Int = t match {
case l : List[String] => l.map(_.length).sum
case l : List[_] => l.length
case s : String => s.length
case _ => 0
We get a warning from this definition,
warning: non variable type-argument String in type pattern List[String] is
unchecked since it is eliminated by erasure
case l : List[String] => l.map(_.length).sum
but let’s carry on regardless and try it out on the REPL,
scala> size(List("foo", "bar", "baz"))
res0: Int = 9
So far so good. But suppose we try with a list of non-Strings,
scala> size2(List(1, 2, 3))
java.lang.ClassCastException: java.lang.Integer cannot be cast to java.lang.String
Oops! — that unchecked warning was telling us that Scala’s pattern matching runtime infrastructure (or rather, the parts of the JVM’s runtime infrastructure which support Scala’s pattern matching) is
unable to verify the types of a List‘s elements because the element type is erased at runtime. Consequently the first case, List[String], is always selected, and this fails at runtime if handed a
list of anything other than Strings. Also not good.
The conclusion to draw from this is that implementing a polymorphic function of this form via pattern matching is unworkable if we want type safety, modularity or type-specific cases which are
distinguished by particular type arguments of a common type constructor.
If pattern matching is ruled out, then what can we do? Well, we can do anything which doesn’t depend on the shape of T. That’s trivially the case for the identity function — it simply returns it’s
argument unexamined. And it’s also the case for methods which are themselves polymorphic in the the same way that our apply method is. That’s what’s happening in the definition of singleton. Here it
is again with a little less syntactic sugar,
object singleton extends (Id ~> Set) {
def apply[T](t : T) : Set[T] = Set.apply[T](t)
The apply method (parametric in T) is implemented in terms of the Set companion object’s apply method (also parametric in T). No information is needed about the shape of T here because the Set
factory method doesn’t need to examine its arguments, it just needs to wrap them in a fresh container.
Things like headOption are also fine, this time because we do have a constraint on the type of the argument to the apply method — here we know that the outer type constructor is List[_]. That means
that we can implement the method in terms of any methods defined on List which don’t themselves need to know anything about the shape of T (this excludes methods like sum and product for example).
Clearly this gives us quite a bit to work with, and if you can manage with the constraints that parametricity imposes, then it’s definitely the way to go. But it’s a real constraint with sharp teeth
— we won’t be able to implement polymorphic functions like size in this way.
The fact that our problems are caused by lacking information about the shape of T might encourage us to explore the option of modifying the ~> trait to add bounds on T (ordinary type bounds or view
or context bounds). This is an interesting exercise to attempt, but ultimately it adds a lot of complexity without solving the problem fully generally. I encourage you to give it a try — if you do,
you’ll very quickly find yourself wanting to be able to abstract over function signatures more generally, and that’s going to take us in a different direction as we’ll see in the next part of this
4 comments
1. Great article and thank you. For the issues with matching on size, couldn’t a scalaz like type class be used ( which defaults to 0 if no implicit conversion exists, either through type magic or
creating a more generic implicit conversion from any to the typeclass) I don’t know if this is possible and you are the scala type guru.
2. You’re anticipating part three … type classes are indeed a key part of the solution.
3. Is part three going to happen? I’m just desperately clinging to the cliff-hanger. :D
4. Yes, I’m afraid this is turning into a bit of a Duke Nukem Forever … I’ll try to get to it soon. | {"url":"http://www.chuusai.com/2012/05/10/shapeless-polymorphic-function-values-2/","timestamp":"2014-04-21T14:41:45Z","content_type":null,"content_length":"42588","record_id":"<urn:uuid:0395475d-970b-416a-8079-2fa705fe87eb>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00125-ip-10-147-4-33.ec2.internal.warc.gz"} |
Geometry and the imagination
You are currently browsing the tag archive for the ‘Kahn’ tag.
Jeremy Kahn kindly sent me a more detailed overview of his argument with Vlad Markovic, that I blogged earlier about here (also see Jesse Johnson’s blog for other commentary). With his permission,
this is reproduced below in its entirety.
Editorial note: I have latexified Jeremy’s email; hence “dhat-mu” becomes $\hat{d}\mu$, “boundary-hat” becomes $\hat{d}$, and “boundary-tilde” becomes $\tilde{d}$. I also linkified the link to
Caroline Series’ paper.
Hi Danny,
I was busy with the conference on Thursday and Friday, and taking a break on Saturday, and now I’ve finally had a chance to read your blog, and reply to your message. I decided (especially as Jesse
had requested it) to write out a complete outline of the theorem. I’m sending a copy of this message to you, Jesse Johnson, Ian Agol, and Francois Labourie: you are all welcome to reproduce it, as
long as it is reproduced in its entirety, and states clearly that this is joint work with Vladimir Markovic. Of course, time and energy permitting, I’ll be happy to answer any questions.
Here is an outline of the argument, working backwards to make it clearer:
1. We want to construct a surface made out of skew pants, each of which has complex half-length close to $R$, and which are joined together so that the complex twist-bends are within $o(1/R)$ of $1$.
Using a paper of Caroline
Series (published in the Pacific J. of Mathematics) we show that these surfaces are quasi-isometrically embedded in the universal cover of the three-manifold.
2. Consider the following two conditions on two Borel measures $\mu$ and $u$ on a metric space $X$ with the same (finite) total measure:
A. For every Borel subset $A$ of $X$, $\mu(A)$ is less than or equal to the $u$-measure of an $\epsilon$ neighborhood of $A$.
B. There is a measure space $(Y, \eta)$ and functions $f: Y \to X$ and $g: Y \to X$ such that $\mu$ and $u$ are the push-forwards by $f$ and $g$ respectively of the measure $\eta$, and the distance
in $X$ between $f(y)$ and $g(y)$ is less than $\epsilon$ for almost every $y \in Y$.
It is easy to show that B implies A (also that A is symmetric in $\mu$ and $u$!). In the case where $\mu$ and $u$ are discrete and integral measures (the measure of every point is a non-negative
integer), we can show that A implies B (and $Y$ will be a finite set with the counting measure) using Hall’s marriage theorem. In fact, the statement that A implies B for discrete and integral
measures is easily shown to be equivalent to Hall’s marriage theorem. I don’t know if A implies B in general because I don’t know how to replace the inductive algorithm for Hall’s marriage theorem
with a method that works for a relation between two general measure spaces.
We call $\mu$ and $u$$\epsilon$-equivalent if they satisfy condition A, and note that the condition is additively transitive: if $\mu$ is $\epsilon$-equivalent to $u$, and $u$ is $\delta$-equivalent
to $\rho$, then $\mu$ and $\rho$ are $(\epsilon+\delta)$-equivalent.
3. Suppose that $\gamma$ is one boundary component of a pair of skew pants $P$. We can form the common orthogonals in $P$ from $\gamma$ to each of other other two cuffs. For each common orthogonal,
at the point where it meets $\gamma$, we can find a unit normal vector to $\gamma$ that points along this common orthogonal. The two resulting normal vectors are related by a translation along the
half-length of $\gamma$ (the suitable square root of the loxodromic element for $\gamma$), so we will call them a pair of opposite unit normal vectors (or pounv for short) and they live in the live
in the bundle of pounv’s which is conformally equivalent to the complex plane mod the lattice generated by the half-length of $\gamma$ and $2\pi i$. We give the bundle of pounv’s the Euclidean metric
inherited from the complex plane, and also the Lebesgue measure.
4. Given a measure on pants we can produce a measure on the union pounv bundles of the boundary geodesics as follows: if the measure is a unit atom on one pair of skew pants, the resulting measure on
pounv bundles is a unit atom on the pounv bundle of each the cuffs, at the pounv described in step 3. We extend to a general measure by linearity. This produces a linear operator we will call the $\
hat{d}$ operator.
If we are given a positive integral formal sum of pants (or a multi-set of pants) we can think of it as an integral measure on the space of pants.
5. On the pounv bundle for each closed geodesic we can apply a translation of $1 + i \pi$; we will call this translation $\tau$. We can think of $\tau$ as a map from the union of the pounv bundles to
6. Let $\mu$ be an integral measure on pants with cuff half-lengths close to $R$. We can apply the $\hat{d}$ operator described in step 4 to obtain a measure on the union of pounv bundles of all the
boundary geodesics; we will call the measure $\hat{d}\mu$. If $\hat{d}\mu$ and the translation of $\hat{d}\mu$ by $\tau$ are $\epsilon/R$ equivalent, then we can take two oriented pants for each pair
of pants in our multi-set (taking each of the two possible orientations) and then fit all of these oriented pants into an oriented surface of the type described in step 1. We use Hall’s marriage
theorem as described in step 2, and a very small amount of combinatorics.
If the measure $\hat{d}\mu$, restricted to a given pounv bundle, is $\epsilon/R$ equivalent to a rescaling of Lebesgue measure on that torus, then $\hat{d}\mu$ and $\tau$ of $\hat{d}\mu$ are $2\
epsilon/R$-equivalent, which is what we wanted.
This is as far as I got in the first talk at Utah, so it would be best to stop and take a breath for a moment. We haven’t really done anything, but we’ve reformulated the problem: the type of surface
we want has been well-defined, and the problem of finding this surface has been reformulated as finding a measure on pairs of pants that satisfies a given criterion.
7. A two-frame for $M$ will comprise a tangent vector and a normal vector both at the same point, unit length and orthogonal. Given a two-frame we can rotate the tangent vector 120 degrees around the
normal vector, using the right-hand rule; the orbit of this action is an ordered triple of two-frames, which will call a tripod. We can also rotate 120 degrees in the opposite direction, and obtain
an anti-tripod.
8. A connected pair of two-frames is a pair of two frames along with a geodesic segment connecting them. Given $\epsilon$ and $r$, with $r$ large in terms of $\epsilon$, we can find a weighting
function on connected two-frames such that the following properties hold whenever the weight is non-zero:
A. The length of the connecting segment is within $\epsilon$ of $r$.
B. If the normal vector of one two-frame is parallel translated along the connecting segment, then it forms an angle of less then $\epsilon$ with the normal vector of the other two-frame.
C. The angle between the the tangent vector of the two frame and (the tangent vector to) the connecting geodesic segment is exponentially small in $r$.
D. Given a pair of two-frames, the sum of the weights of the connecting geodesic segments is exponentially close (in $r$) to 1.
E. The weighting is geometrically natural, in that it depends only the length of the connecting segment, the angle between the parallel translated normal vectors, and the angles between the
connecting segment and the tangent vectors.
We will describe the (relatively simple) weighting function in the end; we will use the exponential mixing of geodesic flow to obtain property D.
9. Given a tripod and an anti-tripod, we can form three pairs of two-frames by pairing the frames in order, and then we can measures (or weightings) on the connected pairs of two-frames, and then
form the product measure (or weighting) by multiplying the weights of the three connections. This gives us a weighting on “connected pairs of tripods” (really a tripod and an anti-tripod) that is
supported on connections that satisfy properties A, B, and C.
10. We call a perfect connection between two two-frames a geodesic segment that has a length of $r$, and angle of zero between the segment and the tangent vectors, and translates one normal vector to
the other. If a tripod and an anti-tripod were connected by three perfect connection, then they would be a 1-dimensional retract of a flat pair of pants with three cuffs of equal length $R$, where
$R$ is approximately $r + \log \cos \pi/6$ when $r$ is large. If the tripod and anti-tripod are connected by arcs that satisfy properties A and B, then the connected pair of tripods is still a
retract of a skew pair of pants, whose cuffs have half-length within $\epsilon$ (or $10\epsilon$) of $R$. Thus there is a map from good connected pairs of tripods to good pairs of pants, which we
will denote by $\pi$.
11. We can let $\tilde{\mu}$ be the measure on connected pairs of tripods, given by integrating the weighting of steps 8 and 9 with respect to the Liouville measure on pairs of tripods (or pairs of
two-frames). We then push this measure forward by $\pi$ to obtain a measure $\mu$ on pairs of pants; after finding a rational approximation and clearing denominators, it will be the $\mu$ that was
asked for in step 6. We will show that $\hat{d}\mu$ (taking the original irrational $\mu$) is $\epsilon/R$-equivalent to a rescaling of Lebesgue measure on each pounv bundle and thereby complete the
12. A partially connected pair of tripods $T$ is a pair of tripods where we have connected two out of the three pairs of two-frames. To a partially connected pair of tripods we can assign a single
closed geodesic $\gamma$ that is homotopic to the concatenation (at both ends) of the two connecting segments. If we connect the third pair of two-frames and apply $\pi$ we obtain a pair of pants $P$
, and we can then find a pair of opposite unit normal vectors for gamma pointing to the two cuffs of $P$ (as described in step 3). We will describe a method for predicting the pounv for $\gamma$ and
$P$ knowing only the partially connected tripod $T$: First, lift $T$ to the solid torus cover of $M$ determined by $\gamma$, and then follow geodesic segments from the tangent vectors of the two
unconnected two frames of (the lift of) $T$ to the ideal boundary of this $\gamma$-cover. We can connect these two points in the boundary by two geodesics, each of which goes about half-way around
this solid torus cover. We can then find the common orthogonals from each of these geodesics to (the lift of) $\gamma$, and then obtain two normal vectors to $\gamma$ pointing along these common
orthogonals; it is easy to verify that these are half-way along $\gamma$ from each other (in the complex sense) and hence form a pounv. Property C of the connections between two-frames (and hence
tripods) implies that this predicted pounv will be exponentially close (in $r$) to the actually pounv of any pair of pants $P$.
To summarize: given a good connected pair of tripods, we get a good pair of pants $P$, and taking one cuff gamma of $P$, we get a pounv for $\gamma$ as described in step 3. But we only need two out
of the three connecting segments to get $\gamma$, and using the third pair of two frames, without even knowing the third connecting segment, we can predict the pounv for $\gamma$ and $P$ to very high
13. We can then define the $\tilde{d}$ operator from measures on partially connected pairs of tripods to measures on the pounv bundles for the associated geodesics; this operator is just the linear
extension of the operation in step 12. Given a connected pair of tripods, we can get three partially connected pairs of tripods in the obvious way; we can thereby extend $\tilde{d}$ to map measures
on connected pairs of tripods to measures on the bundles of pounv’s; because the predicted pounv described in step 12 is exponentially close to the actual pounv described in step 3, the two measures
$\tilde{d} \tilde{\mu}$ and $\hat{d}\mu$ are $\exp(-\alpha r)$-equivalent, by the B => A of step 2.
14. For each closed geodesic $\gamma$, we can lift all the partially connected tripods that give $\gamma$ to the $\gamma$ cover of $M$ described in step 12. There is a natural torus action on the
normal bundle of $\gamma$, and this extends to an action on all of the solid torus cover associated to $\gamma$. Moreover, it acts on the (lifts of) partially connected tripods, and it does not
change the weightings of the two established connecting segments, because of property E of the weighting function.
This is the crucial point: the effective weighting on a partially connected pair of tripods is not just the product of the weights of the two established connections, but that product times the sum
of the weights of all possible third connections. By property D of the weighting function, this sum, while not constant, is exponentially close to being constant, so the effective weighting is
exponentially close to being invariant under the torus action. Because the predicted pounv for a partially connected pair of tripods is equivariant for the torus action, the measure $\tilde{d} \tilde
{\mu}$ is exponentially close to a torus invariant measure on the pounv bundle (which is necessary a rescaling of Lebesgue measure), in the sense that the Radon-Nikodym derivative is exponentially
close to 1. It is then an easy lemma that the two measures are exponentially close in the sense of step 2. And then we’re finished: $\hat{d}\mu$ is exponentially close to $\tilde{d} \tilde{\mu}$,
which is exponentially close to a rescaling of Lebesgue measure, which is what we wanted (with
overkill) in step 6.
15. It remains only to define the weighting function described in step 8, which is surprisingly simple: We take some left-invariant metric on $\text{PSL}_2(\bf{C})$, and hence on the two-frame bundle
for $M$ and its universal cover. Given a connected pair of two-frames in $M$, we lift to the universal cover, to obtain two two-frames $v$ and $w$. We then flow $v$ and $w$ forward by the frame flow
for time $r/4$ to obtain $v'$ and $w'$. We let $V$ be the $\epsilon$ neighborhood of $v'$, and $W$ be the $\epsilon$ neighborhood of $w'$, with the tangent vector of $w'$ replaced by its negation.
Then the weighting of the connection is the volume of the intersection of $W$ with the image of $V$ under the frame flow for time $r/2$.
Properties A, B, and C are not difficult to verify. Property D follows immediately from exponential mixing: If we have $v$ and $w$ downstairs without any connection, and similarly define $v'$, $w'$,
$V$ and $W$, then the sum of the weights of the possible connections will just be the volume of the intersection of the downstairs $W$ with the frame flow of $V$. By exponential mixing, this
converges at the rate $\exp(-\alpha r)$ to the square of the volume of an $\epsilon$ neighborhood, divided by the volume of $M$.
We can normalize the weights by dividing by this constant.
I will try to add comments as they occur to me.
One obvious comment to make is that the argument is remarkably short, and does not depend on any very delicate or complicated analytic estimates (maybe the argument that the glued up surfaces are
quasi-geodesic is the most delicate part). It is fair to say that it defies the conventional wisdom in that respect — I was personally very surprised that the general method could be made to work,
especially in light of the failure of Bowen’s program. Kudos to Jeremy and Vlad for their boldness and ingenuity.
Another comment to make is that the matching argument is surprisingly robust and general, and I expect it to have many broader applications. One thing I was confused about in my last post seems to be
resolved by Jeremy’s sketch above — if I understand it correctly, one first (almost) pairs continuous measures, and only then approximates them by discrete integral measures (with a little bit of
combinatorics at the end). And one really does need exponential mixing rather than just mixing.
Incidentally, apropos the matching argument, there are some interesting and well-known variations where things go haywire. For example, papers by Burago-Kleiner and (Curt) McMullen show that there
are examples of separated nets in Euclidean space which are not bilipschitz to a lattice (though, interestingly, Curt shows that they are Holder equivalent). No such examples exist in hyperbolic
space, because of — nonamenability and Hall’s marriage theorem! Roughly, when trying to match up points in two nets in hyperbolic space, one doesn’t need to look very far because the number of
options grows exponentially. This is one reason why Kahn-Markovic need to control the matchings of their measures carefully, because it must be done on a very small scale (where the exponential
growth does not kick in).
I thought I would also mention that in case my previous comments lead one to believe otherwise, exponential mixing of the geodesic flow on a hyperbolic manifold is somewhat delicate. Exponential
mixing under a flow $g_t$ on a space $X$ preserving a probability measure $\mu$ means that for all (sufficiently nice) functions $f$ and $h$ on $X$, the correlations $\rho(h,f,t):= \int_X h(x)f(g_tx)
d\mu - \int_X h(x) d\mu \int_X f(x) d\mu$ are bounded in absolute value by an expression of the form $C_1e^{-tC_2}$ for suitable constants $C_1,C_2$ (which might depend on the analytic quality of $f$
and $h$). For example, one takes $X$ to be the unit tangent bundle of a hyperbolic manifold, and $g_t$ the geodesic flow (i.e. the flow which pushes vectors along the geodesics they are tangent to,
at constant speed). Exponential mixing should be contrasted with the much slower mixing of the horocycle flow on a hyperbolic surface, for which the correlation is bounded by an expression like $C_1
(\log t)^{C_2}t^{-1}$. The geodesic flow on a hyperbolic manifold is an example of what is called an Anosov flow; i.e. the tangent bundle $TM$ splits equivariantly under the flow into three
subbundles $E^0, E^s, E^u$ where $E^0$ is $1$-dimensional and tangent to the flow, $E^s$ is contracted uniformly exponentially by the flow, and $E^u$ is expanded uniformly exponentially by the flow.
The best one knows for (certain) Anosov flows (by Chernov) is that the flow is stretched exponentially mixing, i.e. with an estimate of the form $C_1e^{-\sqrt{t}C_2}$. One knows exponential mixing
for the geodesic flow on variable negative curvature surfaces by Dolgopyat, and on certain locally symmetric spaces, using representation theory. See Pollicott’s lecture notes here for more details.
I don’t know if exponential mixing for geodesic flows is known on manifolds of variable negative curvature in high dimensions. Also I’d appreciate it if any reader who knows some ergodic theory can
confirm/deny/clarify this paragraph . . .
(Update 8/12): Jeremy tells me that he and Vladimir only need “sufficiently high degree polynomial” mixing, so perhaps there is a decent chance the methods can be extended to variable negative
(Update 10/29): The paper is now available from the arXiv.
I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that every complete hyperbolic $3$-manifold $M$ contains a
closed $\pi_1$-injective surface. Equivalently, $\pi_1(M)$ contains a closed surface subgroup. Apparently, Jeremy made the announcement at an FRG conference in Utah. This answers a long-standing
question in $3$-manifold topology, which is a variation on some problems originally posed by Waldhausen. If one further knew that hyperbolic $3$-manifold groups were LERF, one would be able to deduce
that all hyperbolic $3$-manifolds are virtually Haken, and (by a recent theorem of Agol), virtually fibered. Dani Wise (and others) have programs to show that hyperbolic $3$-manifold groups are LERF;
if successful, this would therefore resolve some of the most important outstanding problems in $3$-manifold topology (in fact, I would say: the most important outstanding problems, by a substantial
In fact, the argument appears to work for hyperbolic manifolds of every dimension $\ge 3$, and possibly more generally still. Details on the argument of Markovic-Kahn are scarce (Vlad informs me that
they expect to have a preprint in a few weeks) but the sketch of the argument presented by Kahn is compelling. Roughly speaking, the argument (as summarized by Ian Agol in a comment at Jesse’s blog)
takes the following form:
1. Given $M$, for a sufficiently big constant $R$, one can find “many” immersed, almost totally-geodesic pairs of pants (i.e. thrice-punctured spheres) with geodesic boundary components (i.e.
“cuffs”) of length very close to $2R$. In fact, one can further insist that the complex length of the boundary geodesic is very close to $2R$ (i.e. holonomy transport around this geodesic does
not rotate the normal bundle very much).
2. Conversely, given any geodesic of complex length very close to $2R$, one can find many such pairs of pants that it bounds, and moreover one can find them so that the normal to the geodesic
pointing in to the surface is prescribed.
3. If one takes a sufficiently big collection of such geodesic pairs of pants, one has enough of them in oppositely-aligned pairs along each boundary component, that they can be matched up (by some
version of Hall’s marriage theorem), and furthermore, matched up with a definite prescribed “twist” along the boundary components
4. One checks that the resulting (closed) surface is sufficiently close to totally geodesic that the ambient negative curvature certifies it is $\pi_1$-injective
Many aspects of this argument have a lot in common with some previous attempts on the surface subgroup conjecture, including one recent approach by Bowen (note: Bowen’s approach is known to have some
fatal difficulties; the “twist” in 3. above specifically addresses some of them). All of these points deserve some comments.
First, where do the pairs of pants come from? If $P$ is a totally geodesic pair of pants with boundary components of length close to $2R$, the pants $P$ retract onto a geodesic spine, i.e. an
immersed totally geodesic theta graph, whose edges all have length close to $2R$, and which meet at angles very close to $120$ degrees. One can cut this spine up into two pieces, which are obtained
by exponentiating the edges of an infinitesimal (almost)-planar tripod for length $R$.
Given a tripod $T$ in some plane in the tangent space at some point of $M$, one can exponentiate the edges for length $R$ to construct such a half-spine; if $T$ and $T'$ are a pair of tripods for
which the exponentiated endpoints nearly match up, with almost opposite tangent vectors, then the resulting half-spines can be glued up to make a spine, and thickened to make a pair of pants. One key
idea is to use the exponential mixing property of the geodesic flow on a hyperbolic manifold, e.g. as proved by Pollicott. Given some tolerance $\epsilon$, once $R$ is sufficiently large, the mixing
result shows that the set of such pairs of tripods for which such a matching occurs have a definite density in the space of all pairs (and in fact, are more and more equidistributed in this space, in
probability). In fact, one may even insist that two of the pairs of prongs join up to make some specific closed geodesic of length almost $2R$, and vary the pair of third prongs a very small amount
so that they glue up. This takes care of the first two points; this seems quite uncontroversial (exponential mixing comes in, I suspect, to know that one doesn’t need to wiggle the pair of third
prongs much, having paired the first two pairs).
The matching (i.e. the gluing up of opposite pant cuffs) apparently is done by some variant of Hall’s marriage theorem. One needs to know (I think) that for any finite set of cuffs to be glued, the
set of other cuffs that they could potentially be glued to is at least as big in cardinality. This probably needs some thought, but it is plausibly true: given a cuff, it can be glued to any cuff
which is almost oppositely aligned to it, and since there is some tolerance in the angle of gluing — this is where dimension at least $3$ is necessary — and moreover, since oriented cuffs are almost
equidistributed, one can always find “more” cuffs that are opposite, up to a bit of tolerance, to any given subset of cuffs (of course, more details are necessary here). There is an extra wrinkle to
the argument, which is that the gluing must be done with a “twist” of a definite amount, so that cuffs are not glued up in such a way that the perpendicular geodesic arcs joining pairs of cuffs match
(Update 8/8: I think there must necessarily be more details to the matching argument, as very loosely described above. There are at least two additional issues that must be dealt with in order to
perform a matching: a parity issue (since each pants has an odd number of cuffs) and a homology issue (if the argument relativizes, so that one fixes some collection of cuffs in advance and glues up
everything else, one concludes a posteriori that the union of the unglued cuffs is homologically inessential). Probably the parity issue (and more subtle divisibility issues) can be solved by gluing
with real-valued weights, then approximating a real solution by a rational solution, and multiplying through to clear denominators. Maybe the homology issue does not arise, if in fact the argument
doesn’t relativize.) Both these issues suggest that one does not specify in advance a collection of pants to be glued up, but rather wants to glue up a definite number of pants from some subset.)
This issue of a twist is important for the 4th point, which is perhaps the most delicate. In order to know that the resulting surface is $\pi_1$-injective, one must use geometry. A closed (immersed)
surface in a hyperbolic manifold which is (locally) very close to being totally geodesic is $\pi_1$-injective. One way to see this is to observe that a geodesic loop in the surface is almost geodesic
in the manifold; the ambient negative curvature means that the geodesic can be shrunk (by the negative of the gradient of length in the space of loops) to become geodesic in the ambient manifold; if
it is close to being geodesic at the start, it very quickly becomes totally geodesic, without getting much shorter. Any closed geodesic in a hyperbolic manifold is essential.
If one builds a surface by gluing up almost totally geodesic pieces in such a way that there is almost no angle along the gluing, the resulting surface is almost geodesic, and therefore injective.
However, one must be very careful to control the geometry of the pieces that are glued, and this is hard to do if the injectivity radius is very small. A geodesic pair of pants has area $2\pi$ no
matter how long its boundary components are. So if the boundary components have length $2R$, then at the points where they are thinnest, they are only $e^{-R}$ across. If cuffs are glued where the
pants are thinnest, even if the gluing angle is very small, the surfaces themselves might twist through a big angle in a very short time. So one needs to make sure that the thinnest part of one pants
are glued up to a thicker part of the next, which is glued to a thicker part of the next . . . and so on. This is the point of introducing the twist before gluing: the twists accumulate, and before
one has glued $R$ pieces together, one has entered the thick part of some pants, where the injectivity radius is bounded below by some universal constant.
Anyway, this seems like a really spectacular development, with an excellent chance of working out. Some of the ingredients — e.g. the exponential mixing of the geodesic flow — work just as well in
variable negative curvature. In fact, some version of it should work for arbitrary hyperbolic groups (using Mineyev’s flow space). Without knowing more details of the argument, one can’t say how
delicate the last part of the argument is, and how far it generalizes (but readers are invited to speculate . . .)
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Alessandro Di Bucchianico
In order to facilitate student learning of statistical concepts, I have been developing web-based software with the help of Marko Boon (IT specialist and responsible for the Java implementations) and
Emiel van Berkum. We currently have the following software:
Quality Control in Production (1F200)
This was an introductory course on quality control for first-year management science students. The emphasis was on quality management, in particular process design, rather than statistical issues.
The textbook for this course was the excellent text Creating Quality. Process Design for Results by W.J. Kolarik.
Production Quality of Series Production(1R360)
This was a course for third-year management science students in the special programme Series Production. This course became superfluous with the introduction of the first-year compulsory course
Statistics for Chemical Engineering (2S070)
This was an introductory probability and statistics course, with heavy emphasis on regression analysis. This course will not be taught anymore, since statistics teaching for chemical engineering
students has been integrated in their lab work.
Kansrekening (2DI25)
This is an introductory probability for computer science students. Information is available through Studyweb. Together with the new course 2DI35 (to be taught for the first time in 2008-2009), this
is a replacement for the course 2S970.
Statistics 1 for Chemical Engineering (2DS00)
This is an introductory, hands-on statistics course, with heavy emphasis on regression analysis, including nonlinear regresssion.
Statistics 2 for Chemical Engineering (2DS01)
This was a short advanced course in statistical methods, with emphasis on analysis of variance and design of experiments (fractional factorial designs, response surface methods, mixture designs). We
will try to link these subjects as much as possible with recent developments in combinatorial chemistry and high-throughput analysis. We use the web teaching tool Statlab for interactively teaching
DOE . The course has been integrated into the course 6BV04.
Mathematical Statistics 2 (2S220)
This was an advanced mathematical statistics course for mathematics students on asymptotics, empirical process theory and kernel estimators.
Wiskunde 5 voor Informatica (2S970)
This was an introductory probability and statistics course for computer science students. Information is available through Studyweb. The course 2S970 is superseded by the new course 2DI25 Probability
Theory and 2DI35 Statistics.
Mathematical Statistics (2S990)
This introductory statistics course for mathematics students has been superseded by 2WS05.
Industrial Statistics (2WS02)
This was a course in the SPOR Master's Programme, which I teach jointly with Emiel van Berkum. The topics are design of experiments (DOE; fractional factorial designs, response surface methods,
optimal designs) and statistical process control (SPC; historical development of quality control, basic acceptance sampling schemes, measurement analysis (Gauge R & R), capability analysis:
capability indices, testing for capability, Six Sigma metrics, control charts: Shewhart control charts for grouped and individual data, CUSUM and EWMA control charts, control charts for attribute
data, control charts for specific situations (correlated data (in particular time series models), tool-wear charts).
Mathematical Statistics (2WS05; former code 2S990)
This course is being taught as part of the Bachelor's programme Applied Mathematics, and is also included in the Mathematics minor for. The contents include estimation theory (sufficiency,
Rao-Blackwell, ML estimation of functions of parameters, robust estimation, delta method, confidence intervals), elementary testing theory (Neyman-Pearson lemma, likelihood ratio tests), and one- and
two-sample problems.
Robust Design (2WS08)
This was a course for our Master's Program Industrial Mathematics. The course provided an introduction to robust design by covering the first 7 chapters of the well-known book by Phadke.
Applied Statistics (2WS10)
This was an elective course for the joint Master's programme of the three Dutch technical universities. The contents of this course follow a two-year scheme. In 2005-2006 the course was an
introduction to SPC (Statistical Process Control). The main topics were historical development of quality control, basic acceptance sampling schemes, measurement analysis (Gauge R & R), capability
analysis, Six Sigma metrics, Shewhart control charts for grouped and individual data, CUSUM and EWMA control charts, control charts for attribute data, control charts for specific situations
(correlated data (in particular time series models), tool-wear charts). I taught this part with assistance from Professor Albers and Dr. Kallenberg from Twente University and did so again in
2007-2008. In 2006-2007 Dr. Lopuhaä of Delft University taught the course with survival analysis as content.
Reliability (2WS16; formerly 2WS00)
This course was taught as part of the Bachelor's programme Applied Mathematics. In this course, basic topics in reliability theory and survival analysis like redundancy, life time distributions,
censoring, Cox proportional hazards model etc. were covered.
Vaardigheidsblok Anorganische Chemie (6BV04)
This is not a regular course, but a practical part of the chemical engineering curriculum. The statistical theory of design of experiments is an integrated part of this course. Students will directly
apply basic principles of experimental design to experiments that they must perform themselves. I taught this course together with Koo Rijpkema. The responsible teachers from the Chemical Engineering
Department are Christian Müller and Pieter Magusin. | {"url":"http://www.win.tue.nl/~adibucch/teaching.html","timestamp":"2014-04-19T10:47:26Z","content_type":null,"content_length":"14426","record_id":"<urn:uuid:fc27d87d-671c-47ef-a8ce-61459aa7acb7>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00333-ip-10-147-4-33.ec2.internal.warc.gz"} |
For all his accomplishments, Einstein never lived to see his fondest dream come true. Our century’s best-known physicist spent most of his life searching for a comprehensive set of laws that would
explain the behavior of nature on all levels, from quasar to quark. He had, in his twenties, already shown that space and time were intertwined. He then succeeded in showing how gravity is intimately
related to the geometry of this curved space-time. But he failed when he tried to weave all aspects of nature--all its forces and fundamental rules--into one seamless cloth. The new science of
quantum mechanics simply wouldn’t fit, no matter how hard he, or anyone else, tried.
Today physicists are still stuck in the same quagmire. Nature seems to play by two sets of rules, and they are incompatible. It’s as if physicists were being asked to go bowling with tiddlywinks or
to jump-start a car with an eggbeater. The tools that work so well in one realm are totally inappropriate in another. Not only can’t they win at this game, they can’t even begin to play.
Einstein’s theory of gravity--also known as general relativity-- still describes the universe on its grandest scale with a power that continues to astound physicists. The structure and dynamics of
stars, galaxies, black holes, the very shape and evolution of the universe--all are explored using the tools Einstein developed. Gravity, according to this theory, is not the result of invisible
tendrils of attraction emanating from a mass, keeping planet to sun or boulder to Earth. Rather, gravity is the result of warps in space-time. Massive objects indent the flexible backdrop of
space-time like boulders sitting on a rubber mat. The wells they create naturally attract and frequently capture nearby objects, just as potholes attract cars. The language of general relativity
speaks of a gently curving space-time, a landscape of hills and basins, a continuous flow of smooth, connected forms. The alphabet of this language is geometry; its vocabulary consists of lines,
angles, surfaces, curves.
Zoom in on matters subatomic, however, and the landscape suddenly changes. Einstein’s rules no longer apply. Atoms and nuclear particles buzz around like angry bees. Their energy and motion are
served up in discrete bits, jumpy and blurred, their exact behavior and position forever uncertain. The words always and never, used so readily in describing the physics of our everyday world, are
replaced with the terms usually and seldom. The language that describes this lumpy landscape is quantum mechanics. Keeping track of such a mad gambol of particles requires a vocabulary that deals
with statistical relationships, the probabilities of events. Its alphabet is algebraic symbols and quantum numbers: 1, 1/2, 2.
Trying to do general relativity with the rules of quantum mechanics (or vice versa) would be like using the formula for the area of a circle to compute your chances of winning the lottery, or
employing probability theory to measure the area of a house. Yet physicists find themselves in just such a position. They can’t proceed until they find a common vocabulary that will enable the
quantum theorist to talk freely with the relativist, allowing the lumpy microcosm to join with the smooth macrocosm in an all-embracing theory of quantum gravity. In fact, given such strikingly
different pictures of reality, it’s somewhat surprising that physics has been able to progress at all.
Certainly a theory of quantum gravity is not needed to help us understand events in our everyday world, such as the flight of a rocket or the path of a bowling ball rolling down an alley. Current
laws of physics are quite sufficient to handle those types of problems. Applying laws any more precise would be wasteful, as if you were to use an atomic clock to get you to the airport on time. But
quantum gravity is required in any situation where extreme subtleties are involved, or where gravity is concentrated and the effects of errors are vastly multiplied. Such situations include some of
science’s most vexing mysteries.
It’s well established, for example, that gravity controls the motions of stars and galaxies, but what does gravity do when all the matter in a star is squeezed tighter and tighter, until the size of
the star becomes atomic rather than celestial? That squeezing may be what happens when a particularly massive star explodes as a supernova and, within a wink, its remnant core collapses into a black
hole, that gravitational abyss from which no light or matter can escape. What lies at the heart of this black hole? Einstein’s theory of general relativity blows up when it attempts to describe its
inner recesses. The calculations go awry. The only thing that theorists get for their trouble is a basketful of infinities.
And what if we could turn back the cosmic clock some 15 billion years, to the time of the Big Bang, when all the matter and energy in the visible universe was tucked away in a space no bigger than a
subatomic speck. How did gravity act under those hellishly confined conditions? And how did such behavior produce the universe we presently see around us? No one can yet say. A complete understanding
of gravity’s behavior on subatomic scales will not arrive until physicists can merge general relativity with quantum mechanics and thus fashion a successful theory of quantum gravity.
That is why the work of Syracuse University physicists Abhay Ashtekar and Lee Smolin, and their colleague Carlo Rovelli of the University of Pittsburgh and the University of Trento in Italy, is
creating a bit of a stir within the physics community. Over the last few years these three men have been carrying out a series of calculations that could be moving physics many steps closer to its
cherished goal, finding a path through the mathematical roadblocks that have frustrated theorists for decades in their pursuit of quantum gravity. And what is emerging from their initial explorations
is a tantalizing picture of what space might look like on the tiniest levels. Instead of a space-time that’s immeasurably smooth, their calculations hint that it might have a fine- grained structure,
a texture that resembles a carpet woven out of an endless series of ultrasmall loops, interlinked in every direction. For years physicists and science writers alike have spoken of the fabric of
space-time; incredibly enough, they might literally be right.
The need for a theory of quantum gravity is so compelling that some of the most imaginative, stubborn, and celebrated physicists in twentieth-century science have worked on the problem at one time or
another. Serious work began in the late 1940s, right after the war. And the most popular tactic in the attempts to merge gravity with quantum mechanics was to view gravity much like the other forces
that already fit nicely into the quantum fold--namely, electromagnetism and the strong and weak nuclear forces.
Everything in the quantum mechanical universe--energy, motion, spin, and so forth--comes in indivisible bits. Forces fit naturally into this framework. Instead of viewing magnetism, say, as the
result of invisible lines of force emanating from a magnet, the quantum world transforms the notion of force into an exchange of force particles--a subatomic tennis game. In electromagnetism this
diminutive tennis ball is the photon, a particle that constantly bounces between charged particles, generating a force of either attraction or repulsion. In the same way, gravity is conveyed among
masses by the continual transmission and absorption of gravitons, particles that exist, for now, only hypothetically; they have not been detected.
Mathematically, physicists treated these particles as tiny excitations, or perturbations--small waves moving about the large, calm ocean of space. But when it came to quantum gravity, a major problem
arose: theories that treat forces as particles assume that every event in the subatomic world takes place on a fixed, unchanging background of space and time. Space-time is the stage upon which the
actors, particles such as photons and gravitons, flit to and fro. Take light, for example, says Ashtekar. We imagine that space and time are just sitting here. Turn on a switch, and the light comes.
Turn off the switch, and the light disappears. Space-time is not a participant.
But in general relativity the distinction between stage and actor doesn’t exist. Physicists were saying that the force of gravity arises whenever particles are exchanged. But according to Einstein,
gravity was the very geometry of space-time. Thus the graviton became both actor and stage simultaneously. A graviton could enter onto the stage of space-time, but by doing so it ended up bending and
warping the stage as if it were so much Jell-O. This dual role made gravity nearly impossible to handle with the mathematical techniques that the physicists were using for other forces. When they
tried, their results made no sense whatsoever; the odds of a certain event’s occurring, for example, could turn out to be greater than 100 percent.
General relativists recognized this problem early on and argued with particle theorists that the job had to be done in a different way altogether, one that allowed for the geometry of space-time to
become an active player instead of merely a passive stage. A proper theory of quantum gravity, they said, should allow space to evolve and change in response to forces or the presence of mass. Oxford
mathematician Roger Penrose chastised the particle physicists for attempting to steamroll general relativity flat and then wave the magic wand of quantum theory over the resulting corpse. Gravity
could simply not be handled like the other forces; it was different. Rallying behind Penrose, relativists returned to the classical theory of general relativity itself and worked on putting the
equations into a form that could be treated by quantum mechanics directly, but without necessarily assuming that gravity has to boil down ultimately to an exchange of particles. Unfortunately, upon
setting up their own equations, they found them absolutely impossible to solve, as if they had erected a beautiful house without any doors through which to enter.
Mathematicians, like other tinkerers, need tools to pry open the meaning of their equations. Let’s say you have an equation--for example, x2 = 4. To find out what x is, you take the square root of 4.
The same approach works for any value of x, but if you didn’t know about square roots--if you didn’t have the tool--you wouldn’t be able to solve the equation. The relativists had set up equations
that amounted to elegant statements about how gravity would behave under quantum conditions. They were internally consistent. The grammar was right. They made sense. The only problem was that the
physicists didn’t have the mathematical tools to generate solutions.
Relativists might have been stymied for decades had it not been for a breakthrough in 1985 that changed the way in which they thought about quantum gravity. That year Ashtekar, a relativist, erected
the first crossable bridge between general relativity and quantum mechanics. It is a bridge that he had wanted to build ever since college.
Ashtekar was born in 1949 in the small town of Shirpur near India’s west coast; he was drawn to physics through the popular books of cosmologist George Gamow. That he had a flair for physics was
apparent soon after he entered the University of Bombay. Finding a mistake in a classic text written by Nobel laureate Richard Feynman, he boldly wrote the great physicist to inform him of the error.
Feynman actually replied and agreed the book was wrong. It was so uplifting that I still have the letter, says Ashtekar.
Ashtekar’s interest in cosmology naturally led to his study of general relativity, because it is through Einstein’s equations that cosmologists can understand how the universe expands and why it
looks the way it does. By the time he arrived in the United States in 1969 to pursue his graduate degree, he already knew that the field of relativity, far removed from the public spotlight, best
suited his reflective personality and mathematical inclinations. Relativity has the reputation of being a ‘gentlemanly’ pursuit, he says with a smile. You can freely talk with your colleagues and
never worry about someone stealing your results, a situation in stark contrast to the more rough-and-tumble atmosphere of high-energy particle physics.
Quantum gravity was a particular attraction. There’s a sort of innocent arrogance when you’re young, Ashtekar says, encouraging you to tackle the most difficult problems. He struggled with it
throughout the 1970s, as he graduated from the University of Chicago and moved on to a series of professional appointments. But quantum gravity eluded him, just as it had his fellow relativists.
What was missing, he suspected, was one key idea, perhaps something on the same level as the insights that led to the development of quantum mechanics. Before 1900, physicists were perplexed by the
confusing experimental data on the way light was absorbed and emitted. Then German physicist Max Planck proposed that energy did not flow continuously in an unbroken stream but came instead in
discrete packets, or quanta (from Latin, meaning how much). Indeed, when light was thought of as a barrage of particles, called photons, the experiments suddenly made sense. Planck derived a
quantity--known as Planck’s constant--to describe the minimum amount of energy possible in the quantum universe, the finest possible grain.
Ashtekar’s insight came in the form of a mathematical breakthrough rather than a novel physical idea. His new approach arrived by way of a University of Chicago graduate student named Amitabha Sen,
now a physicist with Motorola in Washington, D.C. What Sen developed was a way of dealing with geometric curvatures that allowed him to describe better the motion of an electron caught within a
gravitational field. I had an intuition about Sen’s approach, that it would be extremely valuable in general relativity, recalls Ashtekar.
He was right. Inspired by Sen’s work, Ashtekar was able to introduce two new mathematical functions, or relationships--in effect, a novel geometric language in which to rewrite Einstein’s theory of
general relativity. As Ashtekar well knew, physical insights often depend on the proper choice of mathematics. Newton’s laws dealing with the motions of the planets depended critically on a new kind
of mathematics--calculus--that could describe forces and objects in a state of constant change. Einstein, in turn, might never have connected gravity to curved space-time if he hadn’t come across
Riemannian geometry, the geometry of curved surfaces.
To see how the proper mathematics can make a complex problem simpler, imagine an everyday problem: Take an airplane circling an airport from three miles away. If you want to describe its motion using
the geometry of a flat grid, the result is very messy. Every time the plane changes position, its longitude and latitude change, too. If you designate its east-west position x, and its north-south
position y, then the equation that describes its route is x2 + y2 = 32. The coordinates are constantly changing. But let’s say you shift to a different geometry: a graph with radial, or circular,
coordinates. In that case you don’t have to worry about x’s and y’s at all. The plane is simply three miles from the center of a circle, and the equation that describes its flight path is no more
complicated than r = 3 (radius = 3).
In a sense, Ashtekar found a way of rewriting Einstein’s equations using new mathematical variables. It was a task that required several years of contemplation and blind alleys, followed by weeks of
filling up his office blackboard with new equations. However, it was worth the wait. Transformed by Ashtekar, Einstein’s equations came to strongly resemble equations already easily handled in
quantum mechanics. Indeed, the quartet of equations that Ashtekar derived were similar in many respects to equations introduced by James Clerk Maxwell more than a century ago that showed electricity
and magnetism to be just two different aspects of the same force. Electromagnetism had been the first force that physicists successfully merged with the quantum world; with general relativity now
looking more like electromagnetism, the union with quantum mechanics appeared more promising than ever.
In the abstract and frequently arcane world of quantum gravity, Ashtekar’s name is now regularly invoked. Papers in the Journal of Classical and Quantum Gravity, a bible in the field, regularly refer
to Ashtekar’s theory of gravity, the Ashtekar formulation of general relativity, and Ashtekar’s variables.
The mathematics itself is not a new invention; similar kinds of tools have already been used in other areas of physics. Technically, mathematicians refer to the two tools that Ashtekar introduced as
a connection and a frame field. A connection (the more important of the two) is a way of defining the geometry of an object--how the surface of a sphere or saddle curves, for instance--a valuable
commodity when dealing with curving warps in space-time. Just as the equation x2 + y2 = 32 described our circle, so more-complicated equations describe more-complex kinds of curves. A connection is a
clever mathematical device that allows you more easily to map and measure curvatures, including the curvatures of space-time. Even Einstein stumbled when he tried to rewrite general relativity in
terms of connections. Ashtekar’s great accomplishment was finding a unique pair of mathematical forms that got the job done.
Oddly enough, there were no shouts of Eureka! when Ashtekar published his results in 1986. More attention was then being paid to the new (and far more popular) kid on the block, superstrings.
Superstrings is more than a theory of quantum gravity, the straightforward union of general relativity with quantum mechanics. It is, at the same time, an all- encompassing theory of everything. In
other words, it attempts to show how gravity and all the other forces are just different manifestations of one ancestral force--a unified force--that briefly existed at the dawn of time. Over the
last few years, however, superstring theory has fallen on hard times. Not only is its mathematics intractable, but there doesn’t seem to be a unique superstring solution that applies to our universe
alone. There are zillions of string theories! exclaims Smolin.
Consequently, Ashtekar’s approach began to look more attractive-- and more doable. It isn’t a theory of everything, describing all the forces by one law. It’s simply a means of examining how gravity
might act as you examine smaller and smaller slices of space, until you enter the lilliputian territory ruled by quantum mechanics.
In early 1986, before his new form of general relativity was officially published, Ashtekar presented a series of lectures on the idea at a quantum gravity workshop held at the Institute for
Theoretical Physics, located at the University of California at Santa Barbara. Lee Smolin, a young and enthusiastic investigator in the field, was in the audience.
Ten years earlier, when Smolin had arrived at Harvard to work on his graduate degree, he’d gone against the advice of all his professors to pursue quantum gravity, a subject then considered far from
the paths of glory in physics. As Smolin puts it, You didn’t know if you were 5 years, 50 years, or 100 years from an answer. The Santa Barbara meeting was a turning point for him. After Ashtekar
described his reformation of general relativity, Smolin and another young relativist at the workshop, Ted Jacobson, now with the University of Maryland, immediately teamed up to clear a path to
possible solutions. They didn’t think they could actually solve Einstein’s equations using Ashtekar’s new framework, but, almost accidentally, they did. Jacobson remembers sitting in his kitchen with
Smolin, reams of paper spread out over the table, finding solution after solution for equations once deemed impossible to solve. They were carrying out the first, tentative translations in the new
quantum language.
Interest in the method spread, swiftly generating converts, the most important of whom was Carlo Rovelli of Verona. Rovelli had been attracted to science relatively late, not until the age of 20,
after participating in Italy’s student rebellions in the early 1970s. We lost the revolution, so I decided to try physics, he says. While working as a postdoc in 1986, he wangled an Italian
fellowship (and funds from his father) for travel to the United States to work specifically with Ashtekar and Smolin. Affable, creative, and easygoing, Rovelli quickly settled into the role of
go-between, helping mesh the analytic powers of the quiet, contemplative Ashtekar with the creativity of the brash, impetuous Smolin.
Imagine Johann Sebastian Bach joining forces with Thelonious Monk. As physicists, Ashtekar and Smolin present a similar contrast. Ashtekar’s attention to detail and form is reflected in his Syracuse
office. The room is a scientific monastery. There are no stray papers in sight; tape dispenser, stapler, and pencil holder line up in regimental order on the desk. Only a single poster graces the far
wall, a portrait of Wolfgang Amadeus Mozart. A man ahead of his time, remarks Ashtekar.
Just three doors down from Ashtekar’s office, another room appears as if it had been caught in the calamitous path of Hurricane Andrew. Books, clothes, and journals litter the floor and every
available surface. I don’t really use this room much as an office, more like a closet, says Smolin sheepishly, running a hand through his unruly hair. Like the subatomic particles that he studies,
Smolin is never at rest. You catch him on the run.
With the arrival of Rovelli, the disparate twosome turned into a more balanced triumvirate. If Ashtekar is the baroque composer and Smolin the jazz musician, more impulsive and experimental, then
Rovelli is someone like trumpeter Wynton Marsalis, who is equally at home playing either jazz or the classics. The way each of us organizes our thoughts is incredibly different, which can be
frustrating, says Rovelli. Yet we understand together what we couldn’t understand separately.
Like advance scouts exploring a new territory, Rovelli and Smolin began to plumb Ashtekar’s equations ever more deeply, figuring out what they might be able to say about space and time. Smolin had
earlier noticed that the solutions he was finding shared an uncanny resemblance to solutions to classic mathematical problems involving knots. Smolin explored this relationship in vain for a year,
and he explained it to Rovelli when he arrived. Within a day Rovelli was able to respond: I know how to do it.
Rovelli proposed a new technique that used loops--which are closely related to knots--as a basis for quantum theory. Combining the Ashtekar equations with the loop technique spawned a new set of
equations in which each seemed to represent a possible configuration of space-time. And if the mathematics looked similar, that was a broad hint that the physical reality might be similar, too. The
solutions that worked best described simple open loops, linked together. Realizing this, Rovelli and Smolin came to confront what other quantum theorists had suspected for decades: that our everyday
notions about space may have to be altered. What we found exceeded our wildest expectations, says Smolin.
It’s natural to think of space as a continuous and uniform medium. Swing your arm through the air and the motion proceeds freely and fluidly from one point in space to the next. But that sense of
space as a smooth continuum could be merely an illusion. Rovelli and Smolin believe that space, at the very tiniest of sub-submicroscopic levels, is actually constructed out of loops, separate and
discrete units.
That space might have a texture is not an entirely new idea. In the 1950s Princeton theorist John Archibald Wheeler, now the doyen of relativity in the United States (he coined the term black hole),
suggested that space might consist of a sort of space-time foam, a froth of space- time bubbles. But the space-time foam was based on a simple estimate, explains Wheeler. What Ashtekar, Smolin, and
Rovelli have done is spell out the mathematics of that foam. By applying the loop formulation of quantum theory to the problem, they were the first to derive discrete units of space directly from the
equations of general relativity.
Once you get adjusted to the notion of spatial building blocks, it seems quite natural; it’s what quantum mechanics is all about. A slab of iron, for example, looks quite solid and uniform to our
eyes, but when examined down to a billionth of a centimeter, it is nothing more than empty space peppered with distinct particles, such as protons and neutrons. These, in turn, can be further
subdivided into quarks. Now space joins the quantum party, but only at an amazingly small scale; the diameter of a quantum loop is a minuscule 10-33 centimeter (a million-billion-billion- billionths
of a centimeter). And that number, in turn, is a measure of the Planck length--the minimum grain size conceivable in our universe, derived from the minimum unit of energy.
If an atom were blown up to the size of our galaxy, which spans some 100,000 light-years, one of these quantum loops would still be no bigger than a human cell. So it’s not surprising that space
looks so smooth, just as a T-shirt seen from a distance looks smooth, says Rovelli. If matter were squeezed to such a tiny dimension, gravity--usually the weakest of nature’s forces--would overwhelm
all the other forces. Yet nothing will ever be known about that fateful transition until a theory of quantum gravity is successfully forged.
And what is a quantum loop? In many ways it resembles the lines of magnetic force surrounding a bar magnet, the halo of lines that is so apparent when you sprinkle iron filings around the bar. Each
loop, in fact, can be thought of as the gravitational equivalent of a magnetic line of force--a gravitational excitation. Nothing exists inside or outside a loop line, not even empty space; the loop
itself defines space.
According to Smolin, it is difficult to talk about the properties of one loop of space, just as you can’t talk about the temperature or density of a single atom. Temperature and density become
meaningful only when you’re dealing with trillions and trillions of atoms. Similarly, the space so familiar to us emerges only when considering countless numbers of loops, all interconnecting for
inches, miles, and light-years on end. Einstein had described space-time as a smooth mat, but the concept of quantum loops suggests that it’s more like a net--a net with the finest of meshes.
That’s exactly what Ashtekar, Rovelli, and Smolin described in a paper entitled Weaving a Classical Geometry with Quantum Threads. If there were a microscope powerful enough to examine quantum space,
they informed us, we would begin to perceive it as a never-ending carpet, spreading outward in every direction. At first the loop-space team thought this carpet might be constructed like a textile,
with infinitely long threads interwoven to form the fabric of space-time. A quantum loop would then be the smallest cell in this weave. Ashtekar even took a weaving lesson to gain more insight into
this imagery. But the three researchers eventually concluded that the carpet’s construction would more resemble chain mail, the flexible armor worn by medieval soldiers. Each loop of the carpet would
be separate and distinct, yet linked to its neighbors. To get a better idea of how this works, Rovelli built a three-dimensional model, a stunning mesh of metal circles, using hundreds of key
rings--every available key ring in Verona, he jokes.
Given this fabric, it becomes possible to think how the weave can be used. Gravity, for instance, might be the result of a bit of embroidery on the weave; you might imagine a graviton as a single
loop of embroidery stitched into the net. A large collection of gravitons would distort the weave, just as mass distorts space-time. More intricate knots or distortions in the quantum threads might
represent other types of physical effects, although that is extremely speculative at the moment. And the long-held suspicion in physics that nothing can be smaller than the Planck length starts to
make sense when picturing the quantum loops; if a particle were smaller than a loop, there would be no scaffolding on which to hang it. Space-time simply doesn’t exist where loop lines are absent,
any more than a blanket exists between the weave of its threads.
Since Ashtekar first published his groundbreaking paper seven years ago, dozens of theorists have written more than 200 papers dissecting, amending, and extending the topic. Researchers from around
the globe--from Sweden, England, India, Japan, Germany, South America--arrive monthly at Syracuse and Pittsburgh to learn from the loop-space gurus. Once I read Ashtekar’s paper, I couldn’t think of
gravity in any other way. I’m surprised it wasn’t done earlier, says Jerzy Lewandowski, a Fulbright scholar now at the University of Florida.
That’s not to say that everyone is greeting the new development with open arms. Both general relativists and quantum theorists alike have some serious concerns about quantum loops. Ted Jacobson, who
had so eagerly embraced Ashtekar’s approach at first, now suspects that the solutions he worked on with Smolin may not be physically significant, more a mathematical trick than a peek at reality.
Just because equations don’t lead to nonsensical results doesn’t mean they lead to physically correct results, either. For the moment, the Syracuse and Pittsburgh researchers seem to be driven more
by intuition and hope, he cautions. I don’t believe their mathematics yet supports the conclusion that the loops correspond to discrete space.
Ashtekar agrees that the status of this new field is far from settled. And yet he argues: If a new variable appreciably simplifies a problem in physics, it’s often telling us something very deep,
that nature is really built out of those variables.
Others are more wary of the science. They acknowledge that the mathematics of loop space is beautiful but wonder when some full-fledged physics is going to get done. They have to tie their method to
something that could, at least in theory with some sort of thought experiment, be observed in the real world, says Bryce DeWitt, a quantum theorist with the University of Texas at Austin and one of
the founding fathers of the field of quantum gravity. Only then will we know whether this approach is useful to pursue.
Ashtekar, Rovelli, and Smolin believe such criticism is fair but stress that they are far from formulating a complete theory of quantum gravity. It’s uncharted territory, points out Ashtekar.
Conceptual revolutions don’t happen quickly. In fact, to simplify their initial calculations, Ashtekar and his colleagues have been working in a timeless space, a space without a clock. Before they
can start making predictions about how the space-time fabric might behave at the quantum level (predictions being the engine that drives science forward), they must figure out a way to bring time
back into their equations. They need a quantum clock. And that may require some new mathematics, one of the reasons Ashtekar and Smolin are moving to Penn State next fall. The Penn State mathematics
department has experts in knot theory, complex analysis, and operator algebras, all areas important to our work, says Ashtekar. The university lured him with the offer to establish a research center
in general relativity and quantum gravity.
The desire to crack the problem of quantum gravity is certainly seductive, although the theory can never be tested directly; to reach the temperatures and pressures at which the law of quantum
gravity kicks in, physicists would have to duplicate the conditions of the Big Bang, a technological feat not expected anytime soon. The best we can hope to do are indirect tests, says Ashtekar,
figuring out how the quantum- mechanical weave state would manifest itself in our everyday physics. It’s a tall order, but possible in the coming years.
Still, there are already hints that a viable theory of quantum gravity could lead to some interesting insights. Nearly 20 years ago Stephen Hawking startled the astronomical community by announcing
that black holes ain’t so black. According to the Cambridge physicist, black holes--those bottomless gravity wells from which nothing can supposedly ever escape--slowly emit radiation and actually
evaporate away. No one ever expected black holes to behave in this crazy way, but that seems to be the conclusion when quantum rules are applied to the strongest gravitational field that nature can
offer. It tells us something deep about how the world is put together, notes Smolin. Black hole evaporation is a hint of the sort of surprises in store for physicists when a full-blown theory of
quantum gravity is at last achieved. Might it drastically change our view of the universe? Absolutely, answers Smolin. Our current theory of the Big Bang may look as quaint as Ptolemy’s
Earth-centered model of the solar system.
The loop-space investigators are generating a lot of press these days, but other schemes for quantum gravity are being actively pursued as well. Roger Penrose has offered an idea whereby the
continuum of space-time is somehow built up from more fundamental processes that involve particles with spin. He calls it his twistor theory. Others, such as Hawking, are looking for answers by
applying the laws of quantum mechanics to the universe at large, in hopes of re-creating the time in our cosmic history, many eons ago, when quantum gravity reigned supreme. And superstring theory is
still the richest, if most complicated, candidate around.
Of course, the possibility remains that none of these approaches will pan out. Maybe physicists will again have to experience a change in their basic understanding of the physical world as
revolutionary and startling as the shift from classical to quantum mechanics.
Smolin himself confesses that he leans toward this view. I’m surprised that the loop-space theory has gone this far, because I’ve always strongly believed that almost anything we now invent, educated
as we are in a mostly classical framework, is unlikely to be radical enough.
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Do I have to be good in math to be good at science?
The short answer: No. Before the longer answer, here's the full comment/question from the teacher:
Question: Do I have to be good in math to be good at science?
The reason I ask this is that many scientifically inclined students I know are not going to pursue science as they get older because they perceive themselves to be less than stellar in math. This is
a shame. And a waste. Please stress to the young, hormonally-infused people who read your blog (not the adults-one hopes they've figured it out already) that science is a process, just like running,
and eating and IMing on the phone. And math is a tool -- a really great, useful tool -- that's part of the process. Splitting hairs? I don't think so. I've got a few 5th graders who I've shared your
blog with, and they were enjoying themselves until they hit the math and freaked out. Nooooooo, I say, fear not the many zeros and exponents. It all makes sense as you practice it (even I can say
So ... address my question. My 5th graders will thank you. And, of course, so will I!
I whole-heartedly second everything the teacher said. The 'enjoying themselves until they hit the math and freaked out' is also not limited to the 5th graders. I've heard from some distinctly older
folks about this too. It's why I'll be making some changes to my posting practices.
For the students, there are two different questions to think about. One is, what does it mean to be 'good at math', and the other is 'what am I doing when I'm doing science?'. I'll share some of my
thoughts and invite students and teachers to share theirs as well. Questions, as always, welcome.
First, 'what am I doing when I'm doing science?' I can speak first hand, as can your 5th graders. They may not realize it though. Science is about trying to understand the world around you in a
sharable way. Do you see the word 'math' there? When you watch clouds and notice that some of them get taller and puffier over time, and the ones that get tallest, puffiest, and darkest are the ones
that give you thunderstorms -- you're doing science. Or you watch bees and notice how they behave. One thing you notice is that bees normally fly around, or collect pollen, or build their nest. But
then you notice one particular kind of bee occasionally digs through sand. This is how a friend discovered a new species of bee. That's science. Then, of course, she collected some of those bees and
examined them to see how it was they could dig through sand (which really is a bizarre thing for a bee to do). More science, plus a pretty electron microscope picture we have on our wall at home.
When math does show up, it is as a servant of either understanding the world, or for doing the sharing. So my latest scary post looked at just how much cooling you might get from clouds. Without the
math, we could tell that there could be some cooling, but not whether it would be enough to erase the greenhouse gas warming. With the math, we understand that it probably can't erase the warming,
and just how big the changes would have to be in order to do the erasing. It's a tool for our understanding here. You'll be doing this, probably already are, in daily life as well. As soon as you
want to decide whether you have enough money to buy something that's 10% off, you're doing math this way. Same for deciding whether you have enough to by 12 of something that costs $9.95 each.
It can also be a tool for doing the sharing. In saying that a cloud is tall, how do you decide what 'tall' means? You're doing math here. Either measurement or geometry. Again, may not be the sort of
thing you're encountering yet in daily life, but you certainly will. Any time you try to decide whether one thing is bigger than another, you're doing this kind of math.
The second question is what it means to be 'good at math'. The real answer for science is that no, you don't have to be 'good at math'. But I expect your 5th graders are also not thinking about the
right kind of good at math in the first place. A friend recently observed that when he was in elementary school, he thought he was bad at math. He learned later, when he was earning a PhD in computer
science, that he was good at math -- he just wasn't good at arithmetic. Still isn't good with arithmetic, but he can do his mathematics just fine. There is a huge world of mathematics outside of what
you're encountering in 5th grade, and many people discover that the new world contains things that are both interesting and fun for them to work on.
Then there's my baseball career, which is similar to what other folks experienced for mathematics. In 5th grade, I played baseball. I was without any question one of the worst players in the league,
and possibly
worst. I liked the game, and understood pretty much what I was supposed to do. But my body just wouldn't do it. Time passes and I'm playing softball on a team at work. And I'm not horrible. In fact,
one year I'm our team's representative to the league All Star game. (Which got rained out :-) Many scientists I know have similar experiences with their mathematics history -- not being good at math
in elementary or even high school, but later on, high school or college the old math started making sense. This doesn't help your grade on today's math test. But when you're doing your science and
talking about your interesting clouds, or bees, nobody cares what grade you got on today's math test.
Then there's a final version, which fits some good scientists I know. Namely, they never did get 'good at math'. Some find their way into kinds of science that don't involve much math. Such areas do
exist. And some just grit their teeth and do the math they need to in order to do the way more fun science. The thing is, you can do it. It's like how in sports you have practices and practice skills
for the game. The fun thing is playing the game. But you have to do your skills practices to play the game the best you can. (Just think about the thousands of hours of practice time the Olympic
athletes have put in!) Musicians play scales, which almost never show up in real music. But it's a skill that is important, so you practice it and it helps you do the fun part better. At worst, math
is like this for you as you go about doing the much more fun science. (Plus you can usually find a friend who
good at the math and work together on your fun scientific idea.)
Final example of doing science, as Beth (who discovered the bees I mentioned above) called while I was writing this note and passed along this story. Her son John, who is now an artist, discovered a
species of wasp. He was 13 at the time. The discovery shows, Beth was pointing out, that young people can be better observers than adults. 13 year old John was in the house on a field expedition to
India, along with Beth (who is a professional hymenopterist -- studies bees, wasps, ants and such; bees being her favorite) and a very experienced, very famous hymenopterist who specialized in wasps.
But the new species was discovered by the 13 year old. The wasps were flying around the house, getting caught in netting, and the like. The adults figured these were just some very common Indian
wasp, rather like flies for being so very common -- not some new species. The 13 year old watched them carefully, and thought that the color pattern was new and interesting. Certainly he hadn't seen
the pattern before (and, thanks to mom's library, he'd seen more than a few types of wasp). Eventually he caught one and passed it on to the adults to identify. It worked out that nobody had ever
identified this wasp to science before -- and the professional scientists in the house hadn't realized this. It was the 13 year old with good eyes and persistence (and using no math :-) who made the
For 5th graders' math concerns, the future holds three possibilities:
a) even if you're currently not good with math and find it unpleasant, in a few years you might be good at it (think how much you've learned in the last 10 years!)
b) in a few years you encounter new kinds of math, and those could be fun and interesting, or you might turn out to be good at them, or both
c) math never does become fun or interesting, and you never get very good at it, so you grit your teeth some to get through what you need to on the way to doing the way more fun and interesting
Regardless of which one turns out to be true for you, at worst it means you have to do some exercises you don't like. You
do it. Just might be more work, or might mean that you work in a particular area of science (one that doesn't use as much math), or with somebody else who does the math better or more easily than
you. And this last is true for us all. I have a project myself where I might be turning to a math person for help -- and I was always very good with math and always enjoyed it. Still, there are
people who know more than me in an area of math that I might need in order to get my science done. This is normal, too.
For the science, though, there is no reason ever to stop. The universe is a very interesting place! That means there are many different kinds of things to study. I'm not a big fan of going to a beach
and staring at bees, though Beth would consider that a vacation. On the other hand, I do like running a whole bunch of numbers from satellites through programs I make up to try to figure out what sea
ice is doing; there might be a few people who wouldn't consider that much fun. And there are many different ways to study the universe. So keep learning about the universe, and sharing what you
learn. Do it for as many parts of the universe as you like, and in as many different ways as you like.
A book that might be a help to students thinking they're not good at math, or they can't do it, etc.. is Sheila Tobias'
Overcoming Math Anxiety
More specific to pre-teen and early teen age girls is
Math Doesn't Suck
7 comments:
No. No you don't have to be good at maths to do science. You just have to know someone who is good at maths.
I managed to sail through my science degrees while believing that double differentiation was the ability to tell twins apart.
Just avoid the dirtyfithlystinkingrotten math-heavy subjects and you'll be fine. Just have a tame mathematician on call.
I'm a physicist - an experimentalist. I work with some heavy duty Russian mathematicians who generally do the mathematical/theoretical heavy lifting in our group.
I've noticed that among physicists there are some who approach the subject very much using maths as their 'medium' for thinking about physics and others (like me) who have a more 'intuitive'
sense of the physical processes and start from the 'intuition' to work out the maths (when we have to!). Both types have their strengths and weaknesses and we work best in combination IMHO.
Incidentally I was almost bottom in my class at doing times tables at 9 years of age. When it came to rote learning I sucked! I was pretty good at the more abstract types of maths I learned later
There are many areas of science which are not at all maths intensive. There are certainly mathematical types working in some subsections of those fields but there are many other folk doing great
work who would rarely need to venture beyond ordinary arithmetic. The working scientist these days has access to loads of great tools and software that take a lot of the mathematical sweat out of
our day!
Thanx for bringing this up. I was all right at math in school, but I didn't really like anything but geometry. Then I discovered I also really liked physics. I learned more math reading physics
than reading math, because I wanted to understand physics. Today, after a loooong time in the physics world, I have discovered that I quite like math.
Thank you for the comments folks, more welcome!
I'll add that Chris Nedin is a paleontologist who blogs at Ediacaran.
If there were any field that has a reputation for (demanding!) mathematics, it would be physics, but see here that two physicists have written in, with agreement on their own paths in the
We are at the HP calculator moment for math. Symbolic algebra programs are better than 99.99% of scientists (and faster than 100%) at computation. What a student has to learn is using mathematics
to formulate a problem.
That really depends on what is meant by "good".
Math really is the 'language' of science.
Most areas of science require (at a minimum) basic (working) knowledge of algebra, geometry and (very basic) statistics.
Whether one needs to be really "good" at math depends on the field and even sub-field.
In fields like cosmology, for example, it is really hard to see how anyone could make a significant contribution these days without understanding the tensor math of general relativity.
But, certainly, there are fields that are much less math intensive.
When Horatio was teaching secondary (physical) science, he used to tell his students that the more math they took, the better prepared they would be for a career in science.
While telling people that they need to know math to do science may discourage some from pursuing science as a career, it can also be counterproductive to tell young adults that they don't need to
know a lot of math because it can essentially "cut them off" from certain scientific career paths.
I agree that the more math students learn, the better.
Certainly there are areas, general relativistic cosmology is probably one, where you need a lot. Then again, I've been surprised at times. A coworker was doing some cosmology of just that sort
with his college age daughter on a recent break. She doesn't know tensor calculus. But, it turned out, some interesting and important things can be approached by way of straight geometric
So I can't be as prescriptive as I used to be about even what math you need for any particular field. | {"url":"http://moregrumbinescience.blogspot.com/2010/02/do-i-have-to-be-good-in-math-to-be-good.html","timestamp":"2014-04-17T09:38:07Z","content_type":null,"content_length":"123635","record_id":"<urn:uuid:0cf690ea-9397-4969-b750-853f2c88a15f>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00296-ip-10-147-4-33.ec2.internal.warc.gz"} |
General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
October 10th 2011, 03:44 AM
General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Can anyone help with the following problem from Artin - Algebra Ch9 on Linear Groups.
Is $GL_n$( $\mathbb{C}$) isomorphic to a subgroup of $GL_ {2n}$( $\mathbb{R}$)?
How do I approach proving this one way or the other?
October 10th 2011, 04:08 AM
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Start with $n=1$, and see where that takes you. Can you find a 2x2 real matrix which squares to -I, where I is the 2x2 real identity matrix.
One you have found this matrix, you want to "expand" $GL_n(\mathbb{R})$ by this matrix. Try and work out what I mean by "expand"...
October 10th 2011, 04:56 AM
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Taking your advice, it looks like something like the following may work for n = 1
$\phi$: (a + ib) $\rightarrow$$\left(\begin{array}{cc}a&b\\-b&a\end{array}\right)$:
should be OK for n = 1
But how to 'expand' $GL_{2n}$( $\mathbb{R}$)???
Can you help?
October 10th 2011, 05:06 AM
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Let A and B be nxn matrices. Then let $\phi: (A+iB) \mapsto \left( \begin{array}{cc} A & B\\-B & A\end{array} \right)$...
October 10th 2011, 05:19 AM
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Will check this out!
Thanks so much for the help. Appreciate your assistance!
Will now try to go further with Artin's chapter on the Linear Groups!
October 10th 2011, 01:29 PM
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Perhaps a more conceptual way of looking at it, if that kind of thing makes you happy, is that if $V,W$ are isomorphic vector spaces then $\text{GL}(V),\text{GL}(W)$ are isomorphic groups. Now,
evidently $\dim_\mathbb{R}\mathbb{C}^n=2n$ so that $\mathbb{C}^n\cong\mathbb{R}^{2n}$ as real vector spaces, and so $\text{GL}_n(\mathbb{C})\cong \text{GL}(\mathbb{C}^n)\cong\text{GL}(\mathbb{R}^
{ 2n})\cong\text{GL}_{2n}(\mathbb{R})$.
October 10th 2011, 02:34 PM
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups
Perhaps a more conceptual way of looking at it, if that kind of thing makes you happy, is that if $V,W$ are isomorphic vector spaces then $\text{GL}(V),\text{GL}(W)$ are isomorphic groups. Now,
evidently $\dim_\mathbb{R}\mathbb{C}^n=2n$ so that $\mathbb{C}^n\cong\mathbb{R}^{2n}$ as real vector spaces, and so $\text{GL}_n(\mathbb{C})\cong \text{GL}(\mathbb{C}^n)\cong\text{GL}(\mathbb{R}^
{ 2n})\cong\text{GL}_{2n}(\mathbb{R})$.
do you mean GL:Vect-->Grp is a functor?
October 10th 2011, 03:06 PM
Re: General Linear Group GLn _ Problem from Artin's Chapter on Linear Groups | {"url":"http://mathhelpforum.com/advanced-algebra/189974-general-linear-group-gln-_-problem-artins-chapter-linear-groups-print.html","timestamp":"2014-04-16T08:43:03Z","content_type":null,"content_length":"17534","record_id":"<urn:uuid:9e99efa4-1600-43c0-aafc-71780ce0455c>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00642-ip-10-147-4-33.ec2.internal.warc.gz"} |
ASTM D7759 - 12a
Standard Guide for Nuclear Surface Moisture and Density Gauge Calibration
Active Standard ASTM D7759 | Developed by Subcommittee: D18.08
Book of Standards Volume: 04.09
ASTM D7759
Significance and Use
5.1 Gauge calibration is performed for the following purposes:
5.1.1 To formulate a mathematical equation, or density calibration equation, that relates the gauge density system response (the “density count”) to the soil-equivalent density of the standard on
which this response is elicited.
5.1.2 To formulate a mathematical equation, or water content calibration equation, that relates the gauge water content system response (the “water content count”) to the water mass per unit volume
value of the standard on which this response is elicited.
5.2 Gauge verification is performed for the following purposes:
5.2.1 To indicate to the party or agency performing the verification when the mathematical relationship between the in-place density reading indicated by the gauge and the corresponding gauge density
test count needs to be adjusted so that the gauge calibration meets the required level of measurement uncertainty.
5.2.2 To indicate to the party or agency performing the verification when the mathematical relationship between the water mass per unit volume indicated by the gauge and the corresponding gauge water
content test count needs to be adjusted so that the gauge calibration meets the required level of measurement uncertainty.
5.2.3 Gauge verification and calibration require specialized training and equipment. Gauge calibration and verification should only be conducted by those trained in the proper operation of the gauge,
the calibration or verification standards, and any tables, charts, graphs, or computer programs required for the proper execution of these operations.
1. Scope
1.1 This guide describes the process and objective of calibrating the density system of a nuclear surface moisture and density gauge, or formulating the mathematical relationship between the density
system response (the “density count”) of a nuclear surface moisture and density gauge and the corresponding density value of the density standard upon which the density system response was observed.
1.2 This guide describes the process and objective of calibrating the water content system of a nuclear surface moisture and density gauge, or formulating the mathematical relationship between the
water content system response (the “water content count”) of a nuclear surface moisture and density gauge and the corresponding water mass per unit volume value of the water content standard upon
which the water content system response was observed.
1.3 This guide describes the process and objective of verifying the density system of a nuclear surface moisture and density gauge.
1.4 This guide describes the process and objective of verifying the water content system of a nuclear surface moisture and density gauge.
1.5 This guide describes two mathematical processes by which the gauge measurement precision may be computed or measured.
1.6 This guide offers guidance for developing and reporting estimates of uncertainties in measurements made with gauges that have undergone calibration or verification.
1.7 All observed and calculated values shall confirm to the guide for significant digits and rounding established in Practice D6026.
1.8 This guide does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this guide to establish appropriate safety and health
practices and determine the applicability of regulatory limitations prior to use.
1.9 This guide offers an organized collection of information or a series of options and does not recommend specific course of action. This document cannot replace education or experience and should
be used in conjunction with professional judgment. Not all aspects of this guide may be applicable in all circumstances. This ASTM standard is not intended to represent or replace the standard of
care by which the adequacy of a given professional service must be judged, nor should this document be applied without consideration of a project’s many unique aspects. The word “Standard” in the
title of this document means only that the document has been approved through the ASTM consensus process.
ICS Code
ICS Number Code 17.040.30 (Measuring instruments)
UNSPSC Code
UNSPSC Code
DOI: 10.1520/D7759-12A
ASTM International is a member of CrossRef.
ASTM D7759 | {"url":"http://www.astm.org/Standards/D7759.htm","timestamp":"2014-04-16T22:35:00Z","content_type":null,"content_length":"32584","record_id":"<urn:uuid:773b4878-aa46-4ab6-9192-37d179ad5c06>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00380-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Simpsons Has Been Tricking You Into Learning Math for Decades | Underwire | WIRED
It’s no secret that the longest-running American sitcom is also one of the smartest. Academics have pored over The Simpsons for its insights into philosophy and psychology, but it took physicist
Simon Singh, the author of previous books about cryptography, the Big Bang, and Fermat’s Enigma, to tap a vein of knowledge that runs even deeper in the animated world of Springfield: math.
In the engaging (and educational) The Simpsons and Their Mathematical Secrets, Singh delves into the academic backgrounds of some of the most poindextrous Simpsons writers -- Al Jean, J. Stewart
Burns, Jeff Westbrook, and David X. Cohen among them -- who are equipped with advanced degrees in math and science. Naturally, they’ve been using their platform to advance what Cohen calls "a
decades-long conspiracy to secretly educate cartoon viewers."
Hoyvin-glavin! Click through the gallery above for some key Simpsons moments of witty math infusion.
Above: "Treehouse of Horror VI: Homer^3" (1995)
Simon Singh calls this segment "the most intense and elegant integration of mathematics into The Simpsons since the series began" – no small praise from someone who combed through the entire run
in search of math content. In the very meta storyline, Homer leaves behind 2-D Springfield when he enters a portal to the abstract and mysterious third dimension, which is rendered with
then-cutting-edge computer animation.
Homer’s stunned family can hear but not see him as he wanders along a z-axis, and eccentric genius Professor Frink is the only one who can explain the concept of depth: "This forms a
three-dimensional object known as a cube, or a Frinkahedron in honor of its discoverer."
"The Wizard of Evergreen Terrace" (1998)
Some of the most elaborate mathematical concepts on The Simpsons are distilled into freeze-frame gags, and this one is particularly dense.
Among Homer’s scribbles during an inventing fervor, elite math nerds (or those reading Singh’s explanations) will recognize playful equations for very complex mathematical problems: the
long-elusive mass of the Higgs boson (which 14 years later is no longer speculative), Fermat’s last theorem, the density of the universe, and how to transform a doughnut into a sphere according
to the rules of topology.
Image via The Simpsons and Their Mathematical Secrets, Nicole Gastonguay and Na Kim
"They Saved Lisa’s Brain" (1999)
Dr. Stephen Hawking makes a guest appearance in this episode, intervening to save Lisa from an angry mob after Springfield’s power-hungry Mensa chapter runs amok.
In the last scene, Hawking offers Homer some seemingly ridiculous praise: “Your theory of a doughnut-shaped universe is intriguing ... I may have to steal it." In fact, that structure has serious
cosmological support, though mathematicians prefer to call the three-dimensional shape a torus. Mmm... all-encompassing torus of unfathomable size...
"Bye Bye Nerdie" (2001)
Bumbling Professor Frink struggles to bring order to a raucous crowd of scientists at Springfield’s 12th Annual Big Science Thing, before he shocks the room to silence by yelling, "Pi is exactly
The obvious joke is that only such a preposterous inaccuracy could quiet a group of math geeks, but there’s another layer. Writer Al Jean based the line on an actual attempt to legislate an
official value for pi, known as the Indiana Pi Bill of 1897, in which an amateur mathematician suggested "squaring the circle" by rounding π to 3.2. The baffled state House of Representatives
allowed the measure to pass, but luckily a Purdue University math professor intervened before the Senate ratified the absurdity into law."
"Marge in Chains" (1993)
When Marge goes on trial for shoplifting, sketchy lawyer Lionel Hutz tries to discredit the memory of convenience store owner Apu Nahasapeemapetilon in court. Unfortunately, the Kwik-E-Mart
proprietor has a savant’s memory: "In fact, I can recite pi to 40,000 places. The last digit is 1."
Apu’s claim is a reference to the pi-reciting record holder at the time, mnemonist Hideaki Tomoyori (the current record for reciting the infinite decimal from memory is 67,890 places, set by Chao
Lu in 2005). However, punctuating the punch line with the actual 40,000th digit of pi was a challenge for the writers, since in 1993, they couldn’t simply Google the known digits. NASA
mathematician David Bailey saved the day by mailing a printed copy of the first 40,000 decimal places of pi to confirm that the last digit is, in fact, 1.
"Girls Just Want to Have Sums" (2006)
In a nod to the public firestorm provoked by Harvard President Lawrence Summers’ 2005 comments about why women are underrepresented in academia, Principal Skinner triggers a hate campaign when he
suggests that girls are inherently inferior to boys in “the real subjects,” math and science.
In a questionable effort to fight sexism at Springfield Elementary, an education reformer separates girls to learn a supposedly more feminine form of math. Lisa chafes at the lack of actual
problem-solving, and sneaks into the male half of the school to excel in math under cover.
The writers found no easy way to resolve the controversy in the episode – just as Lisa is about to offer her opinion about why women continue to be underrepresented in STEM fields in a climactic
speech, Martin Prince interrupts with a flute solo.
"$pringfield (Or, How I Learned to Stop Worrying and Love Legalized Gambling)" (1993)
Wearing Henry Kissinger’s glasses (the former secretary of state dropped them in the toilet while touring Springfield’s nuclear power plant) seems to endow Homer with his intellect. So he spouts
an impressive-sounding geometric formula – “the sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side” – that turns out to be totally
false. Back to proofs, Simpson.
The scene references The Wizard of Oz, when the Scarecrow, emboldened by the brains supposedly endowed by his new diploma, spouts the same bastardized version of the Pythagorean theorem.
“Bart the Genius” (1990)
In the first regular Simpsons episode, math-infused sequences underscore a plot about Bart’s academic frustration. When an aptitude test is foisted on the class, Bart’s visualization of a classic
trains-traveling-at-different-speeds problem devolves into a numerical nightmare. Fed up, he swaps his answer sheet with that of the brainy Martin – and finds himself fast-tracked to a school for
gifted students.
• "Bart the Genius” (1990)
At his new school, Bart is out of his depth among brainy students who negotiate lunch exchanges using rarefied units of measurement like gills and picoliters. In this scene, calculus-deficient
Bart fails to understand his classmates’ glee when the teacher derives the equation y = (r^3)/3. The solution (r dr r) is also a punch line (har-de-har-har), but it doesn’t take advanced calculus
to understand that a joke’s lulz decelerate the more it’s explained.
□ It’s no secret that the longest-running American sitcom is also one of the smartest. Academics have pored over The Simpsons for its insights into philosophy and psychology, but it took
physicist Simon Singh, the author of previous books about cryptography, the Big Bang, and Fermat’s Enigma, to tap a vein of knowledge that runs even deeper in the animated world of
Springfield: math.
In the engaging (and educational) The Simpsons and Their Mathematical Secrets, Singh delves into the academic backgrounds of some of the most poindextrous Simpsons writers -- Al Jean, J.
Stewart Burns, Jeff Westbrook, and David X. Cohen among them -- who are equipped with advanced degrees in math and science. Naturally, they’ve been using their platform to advance what Cohen
calls "a decades-long conspiracy to secretly educate cartoon viewers."
Hoyvin-glavin! Click through the gallery above for some key Simpsons moments of witty math infusion.
Above: "Treehouse of Horror VI: Homer^3" (1995)
Simon Singh calls this segment "the most intense and elegant integration of mathematics into The Simpsons since the series began" – no small praise from someone who combed through the entire
run in search of math content. In the very meta storyline, Homer leaves behind 2-D Springfield when he enters a portal to the abstract and mysterious third dimension, which is rendered with
then-cutting-edge computer animation.
Homer’s stunned family can hear but not see him as he wanders along a z-axis, and eccentric genius Professor Frink is the only one who can explain the concept of depth: "This forms a
three-dimensional object known as a cube, or a Frinkahedron in honor of its discoverer."
"The Wizard of Evergreen Terrace" (1998)
Some of the most elaborate mathematical concepts on The Simpsons are distilled into freeze-frame gags, and this one is particularly dense.
Among Homer’s scribbles during an inventing fervor, elite math nerds (or those reading Singh’s explanations) will recognize playful equations for very complex mathematical problems: the
long-elusive mass of the Higgs boson (which 14 years later is no longer speculative), Fermat’s last theorem, the density of the universe, and how to transform a doughnut into a sphere
according to the rules of topology.
Image via The Simpsons and Their Mathematical Secrets, Nicole Gastonguay and Na Kim
"They Saved Lisa’s Brain" (1999)
Dr. Stephen Hawking makes a guest appearance in this episode, intervening to save Lisa from an angry mob after Springfield’s power-hungry Mensa chapter runs amok.
In the last scene, Hawking offers Homer some seemingly ridiculous praise: “Your theory of a doughnut-shaped universe is intriguing ... I may have to steal it." In fact, that structure has
serious cosmological support, though mathematicians prefer to call the three-dimensional shape a torus. Mmm... all-encompassing torus of unfathomable size...
"Bye Bye Nerdie" (2001)
Bumbling Professor Frink struggles to bring order to a raucous crowd of scientists at Springfield’s 12th Annual Big Science Thing, before he shocks the room to silence by yelling, "Pi is
exactly three!"
The obvious joke is that only such a preposterous inaccuracy could quiet a group of math geeks, but there’s another layer. Writer Al Jean based the line on an actual attempt to legislate an
official value for pi, known as the Indiana Pi Bill of 1897, in which an amateur mathematician suggested "squaring the circle" by rounding π to 3.2. The baffled state House of Representatives
allowed the measure to pass, but luckily a Purdue University math professor intervened before the Senate ratified the absurdity into law."
"Marge in Chains" (1993)
When Marge goes on trial for shoplifting, sketchy lawyer Lionel Hutz tries to discredit the memory of convenience store owner Apu Nahasapeemapetilon in court. Unfortunately, the Kwik-E-Mart
proprietor has a savant’s memory: "In fact, I can recite pi to 40,000 places. The last digit is 1."
Apu’s claim is a reference to the pi-reciting record holder at the time, mnemonist Hideaki Tomoyori (the current record for reciting the infinite decimal from memory is 67,890 places, set by
Chao Lu in 2005). However, punctuating the punch line with the actual 40,000th digit of pi was a challenge for the writers, since in 1993, they couldn’t simply Google the known digits. NASA
mathematician David Bailey saved the day by mailing a printed copy of the first 40,000 decimal places of pi to confirm that the last digit is, in fact, 1.
"Girls Just Want to Have Sums" (2006)
In a nod to the public firestorm provoked by Harvard President Lawrence Summers’ 2005 comments about why women are underrepresented in academia, Principal Skinner triggers a hate campaign
when he suggests that girls are inherently inferior to boys in “the real subjects,” math and science.
In a questionable effort to fight sexism at Springfield Elementary, an education reformer separates girls to learn a supposedly more feminine form of math. Lisa chafes at the lack of actual
problem-solving, and sneaks into the male half of the school to excel in math under cover.
The writers found no easy way to resolve the controversy in the episode – just as Lisa is about to offer her opinion about why women continue to be underrepresented in STEM fields in a
climactic speech, Martin Prince interrupts with a flute solo.
"$pringfield (Or, How I Learned to Stop Worrying and Love Legalized Gambling)" (1993)
Wearing Henry Kissinger’s glasses (the former secretary of state dropped them in the toilet while touring Springfield’s nuclear power plant) seems to endow Homer with his intellect. So he
spouts an impressive-sounding geometric formula – “the sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side” – that turns out to
be totally false. Back to proofs, Simpson.
The scene references The Wizard of Oz, when the Scarecrow, emboldened by the brains supposedly endowed by his new diploma, spouts the same bastardized version of the Pythagorean theorem.
“Bart the Genius” (1990)
In the first regular Simpsons episode, math-infused sequences underscore a plot about Bart’s academic frustration. When an aptitude test is foisted on the class, Bart’s visualization of a
classic trains-traveling-at-different-speeds problem devolves into a numerical nightmare. Fed up, he swaps his answer sheet with that of the brainy Martin – and finds himself fast-tracked to
a school for gifted students.
"Bart the Genius” (1990)
At his new school, Bart is out of his depth among brainy students who negotiate lunch exchanges using rarefied units of measurement like gills and picoliters. In this scene,
calculus-deficient Bart fails to understand his classmates’ glee when the teacher derives the equation y = (r^3)/3. The solution (r dr r) is also a punch line (har-de-har-har), but it doesn’t
take advanced calculus to understand that a joke’s lulz decelerate the more it’s explained. | {"url":"http://www.wired.com/2013/11/simpsons-math/","timestamp":"2014-04-20T06:31:24Z","content_type":null,"content_length":"120204","record_id":"<urn:uuid:49c4b0ee-2f72-41f8-84b5-642a151fbdd1>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00609-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mastering Matrices
April 7, 2013
By Slawa Rokicki
R has many ways to store information. Most of the time, our data comes in the form of a dataset, which we bring into R as a data.frame object. However, there are times when we want to use matrices as
well. This post will show you
how matrices can be useful and how to manipulate them easily
First of all, the big difference between matrices and dataframes is that all of the rows and columns of a matrix must have the same class (numeric, character, etc). In a dataframe, you can have some
of each. See my initial post about objects,
You can
convert from one to the other using as.data.frame() or as.matrix()
. Be careful though, that if you convert a dataframe with different classes of columns, then your matrix will just be all character:
In order to have a numeric matrix, I'm going to just take the first 6 columns of the
dataframe. I can
delete columns of a matrix or dataframe
in two ways:
These two lines are doing the exact same thing. In the first, I am subsetting the dataframe
by taking all rows and the first 6 columns of the dataframe, then I'm converting that subset to a matrix. In the second, I'm taking all rows and all columns
the 7th column. Note that if I wanted to drop even more columns, I would just use the
function like this:
Note now that since I have taken out the one character column in my dataframe before I convert it to a matrix, I will get a numeric matrix instead of a character matrix:
This kind of operation for deleting columns works the same way in both matrices and dataframes. However,
to add a column to a dataframe or matrix is different
. In a dataframe, you can use the $ operator to identify columns, like
is the vector corresponding to the Married column. However, you can't use the $ operator on matrices. You will get the following error that the "$ operator is invalid for atomic vectors", which I see
all the time when I'm converting back and forth from dataframes to matrices and make a mistake:
All this message means is that the object you're using is a matrix (
) and you can't use the $ operator on a matrix. If you get this message, you can either use
to convert your matrix to a dataframe, or you can adjust what you are doing to accomodate the rules of matrices.
For adding columns to a matrix, you use cbind()
, and likewise for rows,
So let's say I want to add an age squared column. In the dataframe, I do:
which instantly names the new column "agesq". Now for a matrix, there are two ways to do this, via indexing by number or by name of the original column:
mydata.mat<-cbind(mydata.mat, mydata.mat[,2]^2)
mydata.mat<-cbind(mydata.mat, mydata.mat[,"Age"]^2)
In the first line, I'm taking all rows and the second column of the
matrix and squaring it, then I'm column binding it to the original matrix. In the second line, I'm doing the exact same thing, except that instead of indexing with a number, I can use the name of the
column "Age". I get the following after running both statements:
Notice that the last two columns of this matrix do not have names, which can be rectified, by using the
colnames(mydata.mat)[7:8]<-c("AgeSq", "AgeSqAgain")
I don't want to rename everything, so I take the 7th and 8th columns and name those appropriately.
Finally, what can matrices do for us? One important aspect of matrices is of course matrix multiplication, which is how we do any multivariable regression analysis. I'll do a post soon on regression
analysis by hand in R. But another reason is that
matrices are great way to store values that you return during the course of running a loop
For example, say I want to show how great the central limit theorem is. I'll generate deviates from some other distribution, say the Poisson, and take the mean of the draws each time. I'll do this
1000 times and then show what the histogram looks like. In a problem like this, I'll use a loop. I'll also use a matrix to store the mean each time.
Ok, we start out by initializing a matrix. We'll create a matrix of all NAs with 1000 rows and 3 columns using the
mat1<-matrix(NA, nrow=1000, ncol=3)
Next, we'll set up the
loop. Let's look at it first and then go through the logic:
for(i in 1:nrow(mat1)){
mat1[i,]<-c(mean(vec1), mean(vec2), mean(vec3))
So in the first line, we're saying for each value of i going from
, do the stuff in the loop. We could have written 1:1000, but it's nice to leave it as
since we may want to change the size of
later and this way the loop will still be fine.
Next, we draw from a Poisson distribution three times, each time a larger number of draws (first 1 draw, then 10, then 100), and each time with a lambda of 1.
Finally, and this is where the matrix comes in, we'll take the mean of each one of those vectors and store it. We will store the three values in the ith row of the mat1 matrix, filling in all three
columns. In a longer way, I could have done:
and it would have come out the same, but the first way is nicer since it's more compact.
Remember that matrices are just columns and rows of vectors, so you can always assign a vector to a row
as long as it's the same length
. When you concatenate numbers (using the
function), you make a vector, which is why it works.
Now, let's see how the old CLT is working by plotting some histograms:
hist(mat1[,1], main="n=1")
hist(mat1[,2], main="n=10")
hist(mat1[,3], main="n=100")
Again, with the histograms, I can plot each column at a time by subsetting the
Pretty nice! Other very helpful places to better understand matrices:
for the author, please follow the link and comment on his blog:
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A conjecture on intersection of some intervals.
up vote 9 down vote favorite
It was proved here that if $a\in \mathbb{N}_{\geq3}$ then
$$\bigcap_{i = 1}^{a} \bigcup_{j = 0}^{i-1} \left[\frac{1+aj}{i},\frac{a(j+1)-1}{i}\right] = \varnothing \tag{1}$$
It may be conjectured that forcing $i\ne b$, where $1\leq b< a$, renders $(1)$ untrue, that is, the result is not an empty interval.
We look at the diagram here from the link above for $a=5$ to get a better picture:
Red is the interval, Yellow are the gaps between the intervals that cause the intersection to be a null set, white gaps do not effect the intersection. The conjecture here says that if we were to
remove any one of the top $4$ strips, then there will form a region of intersection.
I tried analyzing the gaps but everything seems to meet up at a dead end.
What tools may one employ to handle such problems?
co.combinatorics lo.logic conjectures
It looks a bit Cantor-setish.. – Felix Goldberg May 26 '13 at 10:46
3 Another way to state the original is that for every number $x\in [0,1]$ there is some $1 \le i \le a$ so that $i x$ is within $1/a$ of an integer $j$. Your conjecture is that for any $1 \le b \lt
a$, there is some $x \in [0,1]$ so that the only one of $x, 2x, ..., ax$ which is within $1/a$ of an integer is $bx$. – Douglas Zare May 26 '13 at 12:10
Which is just $x=\frac ca+\frac 1a^2$ when $b$ and $a$ are relatively prime with $c$ given by $bc\equiv -1\mod a$. So, when $a$ is prime, it is certainly true. – fedja May 26 '13 at 16:12
Sorry, $\frac ca+\frac 1{a^2}$, of course... – fedja May 26 '13 at 16:13
It's interesting that fedja's answer for $(a,b)=1$ and mine correspond to different endpoints of the interval of solutions for $b=2, a=5$. – Douglas Zare May 27 '13 at 3:03
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1 Answer
active oldest votes
As I commented, the original result can be restated as that for every $x \in [0,1]$ (or $x\in \mathbb R$) there is some $1\le i\le a$ so that $ix$ is within $1/a$ of an integer (and the
proof is the pigeonhole principle -- two of the fractional parts of $0,x,2x,...,ax$ must be within $1/a$ of each other, so their difference is within $1/a$ of an integer). Your
conjecture is that for any $1\le b \lt a$ there is some $x$ so that the only one of $x,2x,...,ax$ within $1/a$ of an integer is $bx$. For example, for $a=5$ and $b=2$, then if we take $x
\in [\frac{44}{100},\frac{45}{100}] \cup [\frac{55}{100},\frac{56}{100}]$ then the only one of the first $5$ multiples within $1/5$ of an integer is the second.
up vote 6 If $2b \gt a$ then $x=1/b$ works. Since $b\lt a$, $i/b$ is within $1/a$ of an integer only when $i$ is a multiple of $b$.
down vote
accepted For $2b \le a$ we can modify this to $x=1/b + 1/(2ab) = \frac{2a+1}{2ab}$. This is designed so that $bx = \frac{2ab + b}{2ab} = 1 + \frac{1}{2a}$ is within $1/a$ of $1$, but $2bx = 2 + \
frac{1}{a}$ just barely misses being within $1/a$ of $2$. Larger multiples of $bx$ are also too large, $bix - i = \frac{i}{2a} \ge \frac{1}{a},$ while $(bi-1)x$ is too small to be within
$1/a$ of $i$ when $bi-1 \le a$. $i - \frac{(bi-1)(2a+1)}{2ab} = \frac{2a - (bi-1)}{2ab} \ge \frac{a}{2ab} = \frac{1}{2b} \ge \frac{1}{a}.$
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Not the answer you're looking for? Browse other questions tagged co.combinatorics lo.logic conjectures or ask your own question. | {"url":"http://mathoverflow.net/questions/131892/a-conjecture-on-intersection-of-some-intervals?answertab=oldest","timestamp":"2014-04-16T14:16:26Z","content_type":null,"content_length":"58273","record_id":"<urn:uuid:dcf23e2d-a8f8-42ef-b8ca-784b381066cb>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00120-ip-10-147-4-33.ec2.internal.warc.gz"} |
Polygon area calculator (Coordinate Geometry)
The calculator on the right will find the area of any polygon if you know the coordinates of each vertex. This will work for triangles, regular and irregular polygons, convex or concave polygons.
It uses the same method as in Area of a polygon but does the arithmetic for you.
1. Enter the vertices in order, either clockwise or counter-clockwise starting at any vertex.
2. Enter the x,y coordinates of each vertex into the table. Empty rows will be ignored.
3. Click on "Calculate".
Unlike the manual method, you do not need to enter the first vertex again at the end, and you can go in either direction around the polygon. The internal programming of the calculator takes care of
it all for you.
There are other, often easier ways to calculate the area of triangles and regular polygons. See
The calculator will produce the wrong answer for crossed polygons, where one side crosses over another, as shown below.
(C) 2009 Copyright Math Open Reference. All rights reserved
Math Open Reference now has a Common Core alignment.
See which resources are available on this site for each element of the Common Core standards.
Check it out | {"url":"http://www.mathopenref.com/coordpolygonareacalc.html","timestamp":"2014-04-16T07:13:13Z","content_type":null,"content_length":"8588","record_id":"<urn:uuid:dbeaeaac-8633-48f0-9fed-0ac8039844d7>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00391-ip-10-147-4-33.ec2.internal.warc.gz"} |
Building Blocks - Classifying Angles - First Glance
Two parallel lines intersected by a
corresponding pairs
of angles that are
Adjacent angles
share a common
and a common
Two intersecting lines form pairs of adjacent angles that are
. Also, two intersecting lines form pairs of congruent angles, called
vertical angles
Click each term below to see an example. | {"url":"http://www.math.com/school/subject3/lessons/S3U1L5GL.html","timestamp":"2014-04-18T15:38:49Z","content_type":null,"content_length":"25161","record_id":"<urn:uuid:e266dd89-cd76-4675-b42f-b41691e22573>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00610-ip-10-147-4-33.ec2.internal.warc.gz"} |
Command line calc program and PI constant
Ask Ubuntu is a question and answer site for Ubuntu users and developers. It's 100% free, no registration required.
I am using calc often (from the terminal) and I was wondering if there is a PI constant in there, predefined, somewhere?
up vote 5 down vote favorite command-line calculator
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I am using calc often (from the terminal) and I was wondering if there is a PI constant in there, predefined, somewhere?
If you are talking about calc from the apcalc package the constant Pi is defined like this:
up vote 4 down vote accepted note, the parentheses are used even though the function does not take any arguments.
not too clear for the GUI desktop calculator,gcalctool, either, which is: Ctrl+P
add comment
If you are talking about calc from the apcalc package the constant Pi is defined like this:
note, the parentheses are used even though the function does not take any arguments.
not too clear for the GUI desktop calculator,gcalctool, either, which is: Ctrl+P
you also can choose kcalc there is a scientific - mode in kcalc there again you can choose mathematical constants e.g. PI, Euler, golden ratio ...
if you want to tickle more out of PI in own written program you can use then the library of http://gmplib.org/ this library you can use when compiling with gcc resp. g++
up vote 0 down vote
have fun !
add comment
you also can choose kcalc there is a scientific - mode in kcalc there again you can choose mathematical constants e.g. PI, Euler, golden ratio ...
if you want to tickle more out of PI in own written program you can use then the library of http://gmplib.org/ this library you can use when compiling with gcc resp. g++ | {"url":"http://askubuntu.com/questions/101764/command-line-calc-program-and-pi-constant?answertab=active","timestamp":"2014-04-16T07:58:03Z","content_type":null,"content_length":"69460","record_id":"<urn:uuid:56b8f006-e051-4fdd-b3ae-bc82bb615ef4>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00080-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mata matrices
In Stata, can I use Mata matrices that are larger than Stata’s matsize?
How can I perform clustermat on a Mata matrix?
How can I send clustermat a matrix larger than the currently set matsize?
Title Mata matrices larger than matsize in Stata
Author Jean Marie Linhart and Kenneth Higbee, StataCorp
Date July 2006
You can save a Mata matrix exceeding Stata’s matsize to Stata.
. mata: st_matrix("MyMatrix", J(12000,2,7.5))
. mat dir
Matrix operators will not work on such large matrices. However, matrix functions and a few other commands work just fine:
. local r = rowsof(MyMatrix)
. local c = colsof(MyMatrix)
. display "rows = `r' cols = `c'"
rows = 12000 cols = 2
. display issymmetric(MyMatrix)
You can use Mata to define and then perform clustermat on a distance matrix. The size of this matrix can exceed Stata’s matsize. The following example shows how to do this.
What if Stata’s cluster command had no option for Euclidean (L2) distance? You could overcome this obstacle by creating an L2 distance matrix based on your data and then using clustermat to obtain
the desired cluster analysis. However, what if the number of observations exceeds Stata’s matsize? Using regular Stata matrix commands, you would be unable to create a matrix large enough to
accommodate your data.
Using Mata to create the distance matrix allows you to exceed Stata’s matsize and yet still use clustermat. Below you will see a Mata function that accepts as arguments a variable list and a matrix
name for the L2 distance matrix it creates and stores back in Stata.
Here we create 810 observations on four variables (x1, x2, x3, and x4) and set Stata’s matsize to 200.
. clear
. set seed 12345
. set obs 810
obs was 0, now 810
. forvalues i = 1/4 {
2. gen x`i' = invnormal(uniform())
3. }
. set matsize 200
We have 810 observations and our matsize is only 200. In Intercooled Stata, the maximum matsize value is 800; with 810 observations we appear to be stuck. How would we proceed? The answer is to use
Mata. First we define the Mata function that will take our data and compute the L2 distance matrix.
. mata:
------------------------------------- mata (type end to exit) ------
: void function l2dist(string varlist, string Distmat)
> {
> /* creates a matrix with name given by Distmat that is the
> * L2 distance for the observations of the variables specified
> * in varlist. All observations are used.
> */
> real matrix Dist
> real matrix Data
> V = st_varindex(tokens(varlist))
> Data = J(1,1,0)
> st_view(Data,.,V)
> Dist = J(rows(Data), rows(Data),0)
> for(i=1; i<=rows(Data); i++) {
> for(j=1; j<=i; j++){
> Dist[i,j] = sqrt(rowsum((Data[i,.]
> - Data[j,.]):^2))
> Dist[j,i] = Dist[i,j]
> }
> }
> st_matrix(Distmat, Dist)
> }
: end
Now we use this Mata function to compute the distance matrix and to store that as a Stata matrix.
. mata: l2dist("x1 x2 x3 x4", "Dist")
Even though Stata’s matsize is currently set at 200, we were able to create an 810 × 810 matrix.
. mat dir
And clustermat can successfully use the matrix.
. clustermat single Dist, add
cluster name: _cl_1 | {"url":"http://www.stata.com/support/faqs/mata/matrices-larger-than-matsize/","timestamp":"2014-04-16T10:16:48Z","content_type":null,"content_length":"28204","record_id":"<urn:uuid:a7bd086e-f34b-4553-81e7-6eefecbdb8eb>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00260-ip-10-147-4-33.ec2.internal.warc.gz"} |
Really want to do I.T but....
04-06-2008 #1
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Join Date
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Really want to do I.T but....
Hi how are you?I need some advice Sorry if it of topic.
The thing is that I am doing a major in Hospitality and Tourism management at college here in Jamaica. I recently started to wonder why I was so interested in Information Technology and
programming when i was doing a tourism major. People started to ask me why i am not doing I.T.
I am not really good at math so thats why i decided to choose a different career path but i am still drawn back to programming and I.T. I was wondering how much math, physics, etc. would i need
to know to even consider I.T. I was told by colleagues that i need to be good in math to do it.
Ive always wanted to do programming but been afraid because of how bad i am at math, and also what skills would i need since i have no prior formal education in I.T. What would u recommend i do ?
How much math and physics? I really want to do I.T.
I appreciate your response. Thanks in advance
You don't need a major in programming to develop a successful programming career. It certainly helps, but it's not mandatory. If you are happy in your current major, any programming knowledge you
acquire will only diversify your resume and open new doors in the area of Tourism.
To do general programming, your math skills need only be at the level of elementary algebra. The stuff you learned on your 5th grade maybe. Everything else will be determined by what projects you
are involved. Naturally, developing a Weather Simulator will push your math skills to the limit. 3D games are also another type of math intensive programming, I reckon (never tried it). But most
programs will not require this level of math skills.
Unfortunately, a major will indeed demand a lot from you in terms of maths even if you don't apply these skills when you graduate.
The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
The programmer comes home with 12 loaves of bread.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
3d graphics require you to understand vector and matrix calculation, so yes, there's math.
Majoring in CS will require a lot of math. Vectors, matrices, the formal foundation of algebra, non-trivial calculus, that stuff.
All the buzzt!
"There is not now, nor has there ever been, nor will there ever be, any programming language in which it is the least bit difficult to write bad code."
- Flon's Law
If you are looking for formal education in CSE be prepared for a lot of math. I have had mathematics for 5 semesters and we have covered all sorts of topics from complex numbers, vectors,
matrices to calculus and probability and many others.
>+++++++++[<++++++++>-]<.>+++++++[<++++>-]<+.+++++++..+++.[-]>++++++++[<++++>-] <.>+++++++++++[<++++++++>-]<-.--------.+++.------.--------.[-]>++++++++[<++++>- ]<+.[-]++++++++++.
Actually the math's vectors and matrixes are a lot different that the 3d graphics ones. My brother has to learn them in the university in maths. The vectors and matrixes are way overcomplicated
there, I don't see how those things could be of any use. Like a math's vector consists of speed and direction, but in 3d vectors are usually a simple XYZ, which either determine a position,
speed, direction, whatever. You can determine EVERYTHING simply with XYZ and that's what I like about 3d programming.
"The Internet treats censorship as damage and routes around it." - John Gilmore
Over here in Portugal, we call each discipline a 'chair'.
Any major in CS has about 3 math related chairs. One of them is named 'Analysis'. We don't call this one a chair. The students adopted slang is 'bench'. You have to flunk everything else to pass
this one
Naturally an exaggeration. But serves to described how hard it can be for someone not math inclined... which is a serious problme in this country btw.
The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
The programmer comes home with 12 loaves of bread.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
Actually the math's vectors and matrixes are a lot different that the 3d graphics ones. My brother has to learn them in the university in maths. The vectors and matrixes are way overcomplicated
there, I don't see how those things could be of any use. Like a math's vector consists of speed and direction, but in 3d vectors are usually a simple XYZ, which either determine a position,
speed, direction, whatever. You can determine EVERYTHING simply with XYZ and that's what I like about 3d programming.
Well, it's the same mathematical principles, but the matrices and vectors used for 3D math are OFTEN quite simple scenarios of vector/matrix usage - it can get complicated, but for most intents
and purposes it's sufficient to understand basic vector/matrix math.
And of course for a lot of programming, all you need is basic "middle-school" level math skills - and a mindset that allows you to break down a large problem into several smaller ones.
Compilers can produce warnings - make the compiler programmers happy: Use them!
Please don't PM me for help - and no, I don't do help over instant messengers.
I'm doing IT and we don't do any maths. TBH I wish we did as it would be more useful to me than other stuff, like writing business reports telling some fictional manager not to leave his company
servers in a rotting shed.
Yes. IT is a different animal. However I seem to recall you still needing some math skills, over here.
The programmer’s wife tells him: “Run to the store and pick up a loaf of bread. If they have eggs, get a dozen.”
The programmer comes home with 12 loaves of bread.
Originally Posted by brewbuck:
Reimplementing a large system in another language to get a 25% performance boost is nonsense. It would be cheaper to just get a computer which is 25% faster.
Yeah, its true you do. Like being able to understand how to create a sum in excel. The most complex maths we covered on my course so far was learning how to subtract binary integers. The
networking guys get all the maths fun, but then none of them seem to like it for some reason. I think they must be mad.
I'm nearing the end of CS and the hardest of my maths were my electives (Graph theory [dropped], Numerical Analysis [struggling] ) But I personally like 3D math as I'm learning from a book on the
subject, I like more specific and applicable maths that also explain some of the theory behind it (Quaternions, I love you).
But I forgot most of Calculus. But taking it made later maths less intensive (try taking Numerical analysis without knowing what an integral or derivative is)
The vectors and matrixes are way overcomplicated there, I don't see how those things could be of any use. Like a math's vector consists of speed and direction, but in 3d vectors are usually a
simple XYZ, which either determine a position, speed, direction, whatever.
Actually, 3d graphics vectors are typically 4-dimensional, with the first three components being XYZ, and the fourth being W, identifying the vector as specifying a position or a distance.
(Distances aren't translated by translation matrices.)
The vectors we learned in maths are simply made up of coordinate components, like the graphics programming ones. The vectors you talk about I only know from physics.
All the buzzt!
"There is not now, nor has there ever been, nor will there ever be, any programming language in which it is the least bit difficult to write bad code."
- Flon's Law
Unless you want to do everything in Computers (which is mostly impossible) you can always avoid math related stuff (unless you kinda love the same stuff).
I have been doing it. I hate math. I can still do so many things not involving maths.
I really have to thank you guys, you really know your stuff.Since you say its not necessarily math intensive, im really goin to consider it. Regardless of what i decide im still gonna do some
programming in my spare time. Its giving me alot to think about. Thanks again. Any other opinions is always welcome. ><
Actually the math's vectors and matrixes are a lot different that the 3d graphics ones. My brother has to learn them in the university in maths. The vectors and matrixes are way overcomplicated
there, I don't see how those things could be of any use. Like a math's vector consists of speed and direction, but in 3d vectors are usually a simple XYZ, which either determine a position,
speed, direction, whatever. You can determine EVERYTHING simply with XYZ and that's what I like about 3d programming.
A vector has a direction and magnitude. The x, y, and z represent the direction, the length represents the magnitude. Indeed, the maths and graphics vectors are the same thing.
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pooja888 @ PaGaLGuY
Hey guys please tell how to solve this questions..options given are very close to each other..
7 Comments
just keep the first 2 digits and check 43/38 = 1.13 , so ....
can anybody tell how much of trigonometry you need to know for cat and other exams ?????
3 Comments
the basic class 10 is good enough.
Good read for Venn Daigrams:
Venn diagram is a diagram using closed curves like circle,rectangle,square,etc. to represent all possible logical relations between finite set of objects. John Venn is the person who introduced this
concept. Venn diagrams have been widely used in many fields like probability, set theory,...
Good read on Cubes:
Question on Cubes like finding the number of coloured sides, etc. are very common in MBA examinations. Let us try to address some key areas of this topic. A cube is a three-dimensional shape with six
square or rectangular sides.
replied to CMAT Preparation Group
The larger radius is A/root(2) and smaller radius is A/2
Why 42 months when said for 3 years?13% SI per annum half yearly does not make sense
replied to [OFFICIAL] XAT 2013 Registrations & Discussions
replied to NMAT 2013 FINAL consolidated scores thread(11t... | {"url":"http://www.pagalguy.com/u/pooja888","timestamp":"2014-04-19T17:04:47Z","content_type":null,"content_length":"125050","record_id":"<urn:uuid:4b4e16c5-8bb8-4d86-b1c7-4824cbe34c4f>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00445-ip-10-147-4-33.ec2.internal.warc.gz"} |
Negative Index & Binomial Theorem
August 4th 2008, 03:39 PM
Negative Index & Binomial Theorem
hey everyone, I'm teaching myself calculus, because I felt it would be useful, even if I'm not going to use it in accounting. I am learning about the binomial theorem and differentiating negative
Here is the current problem I'm working on
1.) y=x^-2
2.) y+dy=(x+dx)^-2
3.) =x^-2(1+ dx/x)^-2
4.) =x^-2 [1-2dx/x + 2(2-1)/1*2 * (dx/x)^2 - etc. ]
5.) =x^-2 -2x^-3*dx + 3x^-4(dx)^2 -4x^-5(dx)^3 +etc...
Ignoring small units
6.) y+dy=x^-2 -2x^-3*dx
Subtracting original y=x^-2
7.) dy= -2x^-3*dx
8.) dy/dx= -2x^-3
the binomial theorem as it is explained in the book follows,
(a+b)^n=a^n + n(a^n-1b/1) + n(n-1)(a^n-2b^2/2) + n(n-1)(n-2)(a^n-3b/3) + etc....
I understand how one moves from Step 1 to 2, that being said how does (x+dx)^-2 become x^-2(1+dx/x)^-2? Secondly, could someone how one goes about applying the binomial theorem to a negative
any help would be greatly appreciated
August 4th 2008, 10:47 PM
$(x+dx)^{-2} = \left(x \cdot \left(1+\frac{dx}x \right) \right)^{-2} = x^{-2} \cdot \left(1+\frac{dx}x \right)^{-2}$
For example:
$(a+b)^{-2} = \frac1{(a+b)^2} = \frac1{a^2+2ab+b^2}$
August 5th 2008, 08:19 AM
Following the rule wouldn't it be,
1.) (x+dx)'2
2.) 1/(x'2) + 1/(2x*dx) + (dx/1)'2
minus insignificant value
3.) 1/x'2 + 2x*dx
In your equation (x*(1+dx/x))'-2 shouldn't it be 1 over x? The rule you used doesn't seem to work for the problem.
August 5th 2008, 11:20 AM
You can continue the first example using the second example:
$(x+dx)^{-2} = \left(x \cdot \left(1+\frac{dx}x \right) \right)^{-2} = x^{-2} \cdot \left(1+\frac{dx}x \right)^{-2}$$~ = ~$
$\frac1{x^2} \cdot \dfrac1{\left(1+\dfrac{dx}x \right)^{2}} = \frac1{x^2} \cdot \dfrac1{1+\dfrac{2dx}x + \dfrac{(dx)^2}{x^2}}$
But you can't split the last fraction into three different fractions with a single summand as denominator. Remember: The line of a fraction sets automatically parantheses - even though you can't
see them. | {"url":"http://mathhelpforum.com/calculus/45296-negative-index-binomial-theorem-print.html","timestamp":"2014-04-16T14:19:24Z","content_type":null,"content_length":"8473","record_id":"<urn:uuid:10b38a5a-ea75-4710-9709-32d81f5c64fc>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00519-ip-10-147-4-33.ec2.internal.warc.gz"} |
InfoLogic's MathMagic adds support for Zoho Writer equations
InfoLogic’s MathMagic is a desktop based equation editor software. Its latest version 6.8 for Mac users has introduced a special menu item for Zoho Writer. MathMagic can now be used as a front-end
equation editor by Zoho Writer users (Zoho Writer has allowed writing of LaTeX equations since May 2008). Whether or not you are fluent in LaTeX, you can now write mathematical equations and symbols
using MathMagic
easily and fast, then simply copy-paste it into Zoho Writer’s Equation
Editor window. LaTeX expressions from Zoho Writer’s Equation Editor can also be copy-pasted into MathMagic window to edit the equation on a WYSIWYG interface.
InfoLogic says MathMagic’s Windows version will also support “Copy as Zoho equation” in the near future. We believe this new menu will help both MathMagic & Zoho Writer users.
4 thoughts on “InfoLogic's MathMagic adds support for Zoho Writer equations”
1. This is great! Wished I had this back in college.
2. This is great! Wished I had this back in college.
3. Your Equation Editor is one of your biggest strength in Writer.
4. Your Equation Editor is one of your biggest strength in Writer. | {"url":"https://www.zoho.com/general/blog/infologic-s-mathmagic-adds-support-for-zoho-writer-equations.html","timestamp":"2014-04-20T21:21:35Z","content_type":null,"content_length":"30710","record_id":"<urn:uuid:2fad1db7-2bb4-4ae5-b331-51a9bc4342dc>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00532-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Department of Mathematics & Statistics
Speaker: Yongjun Xing
Title: On the Spread of Real Symmetric Matrices with Entries in an Interval
Date: Thursday, November 29, 2007
Time: 1.30 pm
Location: College West 307.20
Abstract: The spread of a matrix has extensive and practical applications in combina-
torial optimization problems and cybernetics problems. There are many papers on the
spread of a symmetric matrix, but restricting the entries of such n n symmetric matrices
to each lie in OE a; b seems to be a new view of this problem. As a first step, we show
that the entries must equal a or b in the case when the spread is maximum. Next, when
the spread attains the upper bound of Mirsky's seminal result, we describe the structure
of those matrices. Then we focus our study on the maximal value of the spread and the
corresponding structure of the matrix that achieves the maximum spread over all real sym-
metric, n n matrices, whose entries lie in a given interval. Matlab is used as a tool to
aid the verification of some cases.
Supervisor: S Fallat
Coffee & cookies will be served in the Lounge prior to the lecture | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/896/1261039.html","timestamp":"2014-04-20T13:59:15Z","content_type":null,"content_length":"8241","record_id":"<urn:uuid:f1f1321f-4133-4e25-a2d3-d3ddce3d4bb7>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00184-ip-10-147-4-33.ec2.internal.warc.gz"} |
category-extras-0.53.6: Various modules and constructs inspired by category theory Contents Index
Portability portable
Control.Category.Braided Stability experimental
Maintainer Edward Kmett <ekmett@gmail.com>
class Braided k p where
braid :: k (p a b) (p b a)
class Braided k p => Symmetric k p
swap :: Symmetric k p => k (p a b) (p b a)
class Braided k p where
A braided (co)(monoidal or associative) category can commute the arguments of its bi-endofunctor. Obeys the laws:
idr . braid = idl
idl . braid = idr
braid . coidr = coidl
braid . coidl = coidr
associate . braid . associate = second braid . associate . first braid
coassociate . braid . coassociate = first braid . coassociate . second braid
braid :: k (p a b) (p b a)
class Braided k p => Symmetric k p
If we have a symmetric (co)Monoidal category, you get the additional law:
swap . swap = id
swap :: Symmetric k p => k (p a b) (p b a)
Produced by Haddock version 2.1.0 | {"url":"http://comonad.com/haskell/category-extras/dist/doc/html/category-extras/Control-Category-Braided.html","timestamp":"2014-04-21T04:32:40Z","content_type":null,"content_length":"9606","record_id":"<urn:uuid:e5bc9411-2530-4fd9-a836-c0b5ed274ff8>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00186-ip-10-147-4-33.ec2.internal.warc.gz"} |
root of the equation
(p^2)(x^2)-12x+p+7=0 has the root 3/2..find the value of p. please help,show step by step the answer that u get..thanks
Last edited by mastermin346; February 26th 2010 at 01:16 AM. Reason: wrong write
To explain HallsOfIvy's solution, remember that the roots of an equation are the values of $x$ when $y = 0$. Here, you are given an equation in the form $y = f(x)$ with $y = 0$ and $f(x) = (p^2)(x^2)
-12x+p+7$ for some $p$. Since you are given that one root is $\frac{3}{2}$, you know that when $y = 0$ (which is our case), $x = \frac{3}{2}$. So you can just substitute $x = \frac{3}{2}$ into $f(x)$
and solve for $p$, to find which values of $p$ (there may be more than one) satisfy the equation $f(x) = 0$ when $x = \frac{3}{2}$. | {"url":"http://mathhelpforum.com/algebra/130847-root-equation.html","timestamp":"2014-04-21T10:32:32Z","content_type":null,"content_length":"38144","record_id":"<urn:uuid:8e012cd1-ee31-4388-80e4-ad863e81d81e>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00134-ip-10-147-4-33.ec2.internal.warc.gz"} |
GD' 94 Graph Drawing Competition
rfc@mocha.newcastle.edu.au (Bob Cohen)
Fri, 29 Jul 1994 02:56:42 GMT
From comp.compilers
| List of all articles for this month |
Newsgroups: comp.compilers
From: rfc@mocha.newcastle.edu.au (Bob Cohen)
Keywords: tools, graphics
Organization: Uni of Newcastle, Australia
Date: Fri, 29 Jul 1994 02:56:42 GMT
GD' 94 Graph Drawing Competition
Organized by Peter Eades and Joe Marks
A graph drawing competition will take place in conjunction with the
Graph Drawing '94 DIMACS Workshop. Entries in four different
categories (trees, planar graphs, general directed graphs, general
undirected graphs) will be judged by a panel of experts. All drawings
should be submitted electronically to marks@merl.com as PostScript
files before midnight, October 2, 1994. (Entries may also be faxed to
+1 617-621-7550.) At the workshop, prizes will be awarded for the best
drawing in each category. The winning entries will also be included in
the workshop proceedings.
How to Enter the Competition:
The files digraph, planar, tree, and undirected, available by anonymous
ftp from ftp.cs.brown.edu:/pub/gd94/competition contain representations
of four graphs. In each graph, each node is represented by an integer.
For each node v, there is a record of the form
v u1 u2 ... uk
which represents
* (for a directed graph) arcs v->u1, v->u2, ..., v->uk,
* (for an undirected graph) edges v-u1, v-u2, ..., v-uk.
Note that for an undirected graph, each edge v-u is represented twice:
once in the record for v, once in the record for u.
Submit drawings of any or all of these graphs to marks@merl.com as
PostScript files before midnight, October 2. (Entries may also be faxed
to Joe Marks at +1 617-621-7550.)
Each node in your drawing should be labelled with its integer name, so
that the judges can check that you have drawn the right graph without
solving the Isomorphism Problem.
Your drawings *must* each include identification information:
(1) your name and email address, and
(2) some information as to how the drawing was obtained
(e.g., "manually computed layout using MacDraw", or "Sugiyama
layout using GraphEd, followed by some manual editing").
This information should be included in the PostScript file, or on the
Corporate sponsors: IBM Canada and Mitsubishi Electric Research Labs.
Post a followup to this message
Return to the comp.compilers page.
Search the comp.compilers archives again. | {"url":"http://compilers.iecc.com/comparch/article/94-07-079","timestamp":"2014-04-18T05:36:50Z","content_type":null,"content_length":"6259","record_id":"<urn:uuid:3bcd7b04-7bd1-4c45-9a02-7f6718105ea5>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00336-ip-10-147-4-33.ec2.internal.warc.gz"} |
Binomial summation proof
Show that: $\sum_{k=m}^n \binom{k}{m}\binom{n}{k}p^k(1-p)^{n-k}=\binom{n}{m}p^m$ You may be able to use this result: $\binom{k}{m}\binom{n}{k}=\binom{n}{m}\binom{n-m}{k-m}$
Start with the Binomial Theorem: $(x+y)^n = \sum_k \binom{n}{k}x^k y^{n-k}$ Differentiate m times with respect to x: $(n-m+1) \cdots (n-1)(n) (x+y)^{n-m} = \sum_k \binom{n}{k} (k-m+1) \cdots (k-1)(k)
x^{k-m}y^{n-k}$ Divide by m!: $\binom{n}{m} (x+y)^{n-m} = \sum_k \binom{n}{k} \binom{k}{m} x^{k-m}y^{n-k}$ Now let $x = p$ and $y = 1-p$: $\binom{n}{m} (1)^{n-m} = \sum_k \binom{n}{k} \binom{k}{m} p^
{k-m}(1-p)^{n-k}$ Multiply by $p^m$: $\binom{n}{m} p^{m} = \sum_k \binom{n}{k} \binom{k}{m} p^k (1-p)^{n-k}$ | {"url":"http://mathhelpforum.com/statistics/147836-binomial-summation-proof.html","timestamp":"2014-04-19T15:35:42Z","content_type":null,"content_length":"35146","record_id":"<urn:uuid:2f7efdaa-b180-4cad-83bd-2de164e2fc14>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00286-ip-10-147-4-33.ec2.internal.warc.gz"} |
Isomorphism ring :P
December 23rd 2009, 09:38 AM #1
Junior Member
Oct 2009
Suppose $r$ and $s$ are positive integer numbers with $gcd(r,s)=1$, then show that the mapping $\varphi : Z_{rs}\rightarrow Z_r \times Z_s$ with $\varphi (n) = n (1,1)$ is isomorphism ring!
Last edited by GTK X Hunter; December 23rd 2009 at 09:49 AM.
What you mean is that it is a ring isomorphism.
I'll let you show that it's a ring homomorphism. To see that it's an isomorphism, note that $\ker \varphi = \{n \in \mathbb{Z}_{rs} : n(1,1) = (1,1)\} = \{n \in \mathbb{Z}_{rs} : n \equiv 1 \mod
r \mbox{ and } n \equiv 1 \mod s\}$$= \{n \in \mathbb{Z}_{rs} : n \equiv 1 \mod r\} \cap \{ n \equiv 1 \mod s\} = \{1\}$. (Chinese remainder theorem!) So the map is injective. Since we have $|\
mathbb{Z}_{rs}| = |\mathbb{Z}_{r} \times \mathbb{Z}_{s}| = rs$ we are done.
December 23rd 2009, 10:39 AM #2 | {"url":"http://mathhelpforum.com/advanced-algebra/121486-isomorphism-ring-p.html","timestamp":"2014-04-18T23:44:25Z","content_type":null,"content_length":"34131","record_id":"<urn:uuid:c3c371e1-aaba-4554-af2b-e7bdbe13a088>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00259-ip-10-147-4-33.ec2.internal.warc.gz"} |
Neural Network Training
I'm trying to learn about neural networks and AIs. So far, I think I'm understanding most everything but I am having trouble figuring out one thing. How do I teach a neural network(feedforward
network)? For example, let's say I have a three layer network with 2 perceptrons in the input layer, 4 in the hidden layer, and 1 in the output layer, and I know the desired output of the output
layer only, how do I adjust the weights of all of the other layers? Do I need to know the desired output for each individual neuron (which I hope I don't because that would get annoying quick with
larger nets) or is it possible to do it off of only knowing the final desired output(s)?
Thanks for the replies. I'm going to check out those methods and see what I can do with them.
Topic archived. No new replies allowed. | {"url":"http://www.cplusplus.com/forum/general/104691/","timestamp":"2014-04-17T18:51:45Z","content_type":null,"content_length":"8821","record_id":"<urn:uuid:41458722-09c3-4bbe-92a5-08191d5837a7>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00500-ip-10-147-4-33.ec2.internal.warc.gz"} |
Squares, Rectangles, Parallelograms and Other Polygons
From EscherMath
K-12: Materials at high school level.
We will be talking about all kinds of polygons. Below we will talk about and show some examples of the most common ones including triangles, quadrilaterals (4-sided shapes), etc.
Triangles are just shapes with 3 (straight) sides. They can be big or small and can look somewhat different. Depending on the angles and the sides we can sort the triangles into different types.
The simplest way to sort triangles is by their angle size:
• Acute triangle: An acute triangle is one in which all the angles are acute (less than 90°).
• Obtuse triangle: An obtuse triangle is one in which one of the angles is obtuse (more than 90°).
• Right triangle: A right triangle is one in which one of the angles is a right angle (exactly 90°).
It is not hard to see that every triangle falls into exactly one of these three groups. Every triangle is either an acute triangle, an obtuse triangle, or a right triangle.
Another way to group triangles is by looking at the lengths of their sides:
• Equilateral triangle: An equilateral triangle is one in which all three sides have the same length.
• Isoceles triangle: An isoceles triangle is one in which two sides have the same length.
• Scalene triangle: A scalene triangle is one in which all three sides have different lengths.
Note that every equilateral triangle is automatically also isosceles.
4-sided polygons (Quadrilaterals)
After triangles, the type of shape we will encounter the most is the quadrilateral:
• Quadrilateral : A polygon with four sides.
Note that in general the 4-gons can look pretty strange and irregular. There are a handful of special cases we care about. (See below)
• Square: A quadrilateral with four equal-length sides and four right angles.
Note that the square can be of differnt sizes and you are allowed to rotate it. Above are several examples of squares.
• Rectangle: A quadrilateral having four right angles.
Again, the rectangles come in different sizes and we can draw them at an angle if we want. As long as all the angles are 90 degrees, it will be a reactangle.
• Parallelogram: A quadrilateral with two pairs of parallel sides.
We haven't technically defined what parallel means, but it means that the two segments never meet, even if you were to continue tem infinitely far. Known examples are for instance the left and the
right side of a ladder, or the double lines you see on some freeways.
A quadrilateral having all four sides of equal length.
If you have played cards this is the shape of diamonds :)
Polygons in General
Polygon: A polygon is a closed planar figure made by joining line segments. The segments may not cross, and each segment must connect to exactly two others at its endpoints.
Below are some examples of things that are polygons and things that are not polygons.
The left figure is not closed, and the figures in the middle are not made of line segments. The figure on the right is not a polygon, since its sides intersect each other.
• Vertex: A vertex of a polygon is a point where two sides come together.
The word "vertex" is more precise than the common term "corner", because "corner" has many other uses in English. The plural of "vertex" is "vertices" - a triangle has three vertices.
The way we identify a polygon is usually by the number of sides it possesses, which is the same as its number of angles.
Classifying polygons by number of sides is important enough that there are special words for polygons with small numbers of sides:
│# of sides │ Name │
│3 │Triangle │
│4 │Quadrilateral │
│5 │Pentagon │
│6 │Hexagon │
│7 │Heptagon │
│8 │Octagon │
│9 │Enneagon │
│10 │Decagon │
There are (ridiculous) names for polygons with many more sides (see wikipedia:Polygon), but generally for larger numbers of sides, one uses the number of sides followed by "-gon". We talk about
7-gons and 8-gons for instance (instead of the harder to remember names heptagon or octagon). | {"url":"http://euler.slu.edu/escher/index.php/Squares,_Rectangles,_Parallelograms_and_Other_Polygons","timestamp":"2014-04-19T14:29:14Z","content_type":null,"content_length":"25065","record_id":"<urn:uuid:707da950-7e33-4779-8b6d-c118b90133fa>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00540-ip-10-147-4-33.ec2.internal.warc.gz"} |
Beverly, MA Statistics Tutor
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Analemma and equation of time
I have no idea what you are talking about. What ellipse are you referring to? The analemma makes a figure 8, so I don't see any ellipses other than the two ellipses that form the figure 8.
Sorry for the misunderstanding, but english is not my mother tongue. I agree analemma makes a figuer 8. On this figure 8, you have on the floor 365 marks for the 365 days of the year. You have also
other indications like months, equinoxes, solstices, etc.
But, on the figure 8, the position of the solstice is not marked by a single dot but with a little ellipse (in dotted line). Why ?
My first ideas were :
- Solstice doesn't strictly occur at the same day (20, 21 or 22 june / december) for a given year.
- in addition, near solstices the sun declination speed is very low, so the positions of the sunspot on the ground are too close day by day to be represented individually, as singles positions. So we
draw an ellipse which the "envelope" of the different possible positions for solstice.
I hope that my words are clearer now... | {"url":"http://www.physicsforums.com/showthread.php?p=3806182","timestamp":"2014-04-20T21:21:15Z","content_type":null,"content_length":"44041","record_id":"<urn:uuid:f2a4f05e-e61d-41ff-b4a4-f0d060e2ebcc>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00464-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sparse solutions to linear inverse problems with multiple measurement vectors
Results 1 - 10 of 115
, 2006
"... This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of
elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that ..."
Cited by 298 (1 self)
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This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of
elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary
signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently
solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the
optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact
of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and
numerical analysis.
, 2004
"... Abstract. A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the
same time, the problem balances the error in approximation against the total number of elementary signals th ..."
Cited by 213 (4 self)
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Abstract. A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the same
time, the problem balances the error in approximation against the total number of elementary signals that participate. These elementary signals typically model coherent structures in the input
signals, and they are chosen from a large, linearly dependent collection. The first part of this paper proposes a greedy pursuit algorithm, called Simultaneous Orthogonal Matching Pursuit, for
simultaneous sparse approximation. Then it presents some numerical experiments that demonstrate how a sparse model for the input signals can be identified more reliably given several input signals.
Afterward, the paper proves that the S-OMP algorithm can compute provably good solutions to several simultaneous sparse approximation problems. The second part of the paper develops another
algorithmic approach called convex relaxation, and it provides theoretical results on the performance of convex relaxation for simultaneous sparse approximation. Date: Typeset on March 17, 2005. Key
words and phrases. Greedy algorithms, Orthogonal Matching Pursuit, multiple measurement vectors, simultaneous
, 2007
"... The data of interest are assumed to be represented as N-dimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed
accurately using only a small number M ≪ N of basis-function coefficients associated with B. Compressive sensing ..."
Cited by 132 (15 self)
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The data of interest are assumed to be represented as N-dimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed accurately
using only a small number M ≪ N of basis-function coefficients associated with B. Compressive sensing is a framework whereby one does not measure one of the aforementioned N-dimensional signals
directly, but rather a set of related measurements, with the new measurements a linear combination of the original underlying N-dimensional signal. The number of required compressive-sensing
measurements is typically much smaller than N, offering the potential to simplify the sensing system. Let f denote the unknown underlying N-dimensional signal, and g a vector of compressive-sensing
measurements, then one may approximate f accurately by utilizing knowledge of the (under-determined) linear relationship between f and g, in addition to knowledge of the fact that f is compressible
in B. In this paper we employ a Bayesian formalism for estimating the underlying signal f based on compressive-sensing measurements g. The proposed framework has the following properties: (i) in
addition to estimating the underlying signal f, “error bars ” are also estimated, these giving a measure of confidence in the inverted signal; (ii) using knowledge of the error bars, a principled
means is provided for determining when a sufficient
, 2008
"... Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given
measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structu ..."
Cited by 112 (43 self)
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Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements
is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper we develop a general
framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m
possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated
as that of recovering a block-sparse vector whose non-zero elements appear in fixed blocks. We then propose a mixed ℓ2/ℓ1 program for block sparse recovery. Our main result is an equivalence
condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a
generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modeling errors. A special
case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as
equivalence conditions under which the entire set can be determined efficiently.
"... Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper
focuses on the large-scale matrix factorization problem that consists of learning the basis set, adapting it t ..."
Cited by 97 (18 self)
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Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses
on the large-scale matrix factorization problem that consists of learning the basis set, adapting it to specific data. Variations of this problem include dictionary learning in signal processing,
non-negative matrix factorization and sparse principal component analysis. In this paper, we propose to address these tasks with a new online optimization algorithm, based on stochastic
approximations, which scales up gracefully to large datasets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range
of learning problems. A proof of convergence is presented, along with experiments with natural images and genomic data demonstrating that it leads to state-of-the-art performance in terms of speed
and optimization for both small and large datasets.
, 2005
"... Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we
introduce a new theory for distributed compressed sensing (DCS) that enables new distributed coding algori ..."
Cited by 84 (21 self)
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Compressed sensing is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. In this paper we
introduce a new theory for distributed compressed sensing (DCS) that enables new distributed coding algorithms for multi-signal ensembles that exploit both intra- and inter-signal correlation
structures. The DCS theory rests on a new concept that we term the joint sparsity of a signal ensemble. We study in detail three simple models for jointly sparse signals, propose algorithms for joint
recovery of multiple signals from incoherent projections, and characterize theoretically and empirically the number of measurements per sensor required for accurate reconstruction. We establish a
parallel with the Slepian-Wolf theorem from information theory and establish upper and lower bounds on the measurement rates required for encoding jointly sparse signals. In two of our three models,
the results are asymptotically best-possible, meaning that both the upper and lower bounds match the performance of our practical algorithms. Moreover, simulations indicate that the asymptotics take
effect with just a moderate number of signals. In some sense DCS is a framework for distributed compression of sources with memory, which has remained a challenging problem for some time. DCS is
immediately applicable to a range of problems in sensor networks and arrays.
- IEEE Trans. Signal Process , 2006
"... Abstract — Multiple measurement vector (MMV) is a relatively new problem in sparse representations. Efficient methods have been proposed. Considering many theoretical results that are available
in a simple case – single measure vector (SMV) – the theoretical analysis regarding MMV is lacking. In th ..."
Cited by 69 (2 self)
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Abstract — Multiple measurement vector (MMV) is a relatively new problem in sparse representations. Efficient methods have been proposed. Considering many theoretical results that are available in a
simple case – single measure vector (SMV) – the theoretical analysis regarding MMV is lacking. In this paper, some known results of SMV are generalized to MMV. Some of these new results take
advantages of additional information in the formulation of MMV. We consider the uniqueness under both an ℓ0-norm like criterion and an ℓ1-norm like criterion. The consequent equivalence between the
ℓ0-norm approach and the ℓ1-norm approach indicates a computationally efficient way of finding the sparsest representation in an over-complete dictionary. For greedy algorithms, it is proven that
under certain conditions, orthogonal matching pursuit (OMP) can find the sparsest representation of an MMV with computational efficiency, just like in SMV. Simulations show that the predictions made
by the proved theorems tend to be very conservative; this is consistent with some recent theoretical advances in probability. The connections will be discussed.
, 2008
"... The rapid developing area of compressed sensing suggests that a sparse vector lying in a high dimensional space can be accurately and efficiently recovered from only a small set of non-adaptive
linear measurements, under appropriate conditions on the measurement matrix. The vector model has been ext ..."
Cited by 63 (38 self)
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The rapid developing area of compressed sensing suggests that a sparse vector lying in a high dimensional space can be accurately and efficiently recovered from only a small set of non-adaptive
linear measurements, under appropriate conditions on the measurement matrix. The vector model has been extended both theoretically and practically to a finite set of sparse vectors sharing a common
sparsity pattern. In this paper, we treat a broader framework in which the goal is to recover a possibly infinite set of jointly sparse vectors. Extending existing algorithms to this model is
difficult due to the infinite structure of the sparse vector set. Instead, we prove that the entire infinite set of sparse vectors can be recovered by solving a single, reduced-size
finite-dimensional problem, corresponding to recovery of a finite set of sparse vectors. We then show that the problem can be further reduced to the basic model of a single sparse vector by randomly
combining the measurements. Our approach is exact for both countable and uncountable sets as it does not rely on discretization or heuristic techniques. To efficiently find the single sparse vector
produced by the last reduction step, we suggest an empirical boosting strategy that improves the recovery ability of any given sub-optimal method for recovering a sparse vector. Numerical experiments
on random data demonstrate that when applied to infinite sets our strategy outperforms discretization techniques in terms of both run time and empirical recovery rate. In the finite model, our
boosting algorithm has fast run time and much higher recovery rate than known popular methods.
"... We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling.
Prior recovery methods for this sampling strategy either require knowledge of band locations or impose stric ..."
Cited by 61 (51 self)
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We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling. Prior
recovery methods for this sampling strategy either require knowledge of band locations or impose strict limitations on the possible spectral supports. In this paper, only the number of bands and
their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multi-coset sampling so that a unique multiband signal matches the given
samples. To recover the signal, the continuous reconstruction is replaced by a single finitedimensional problem without the need for discretization. The resulting problem is studied within the
framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate
required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal
rate. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.
- IEEE Trans. on Inform. Theory , 2009
"... Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal
basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampli ..."
Cited by 56 (9 self)
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Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal
basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper we consider a more general signal model
and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the Nyquist-Shannon sampling
theorem in that they connect the number of required samples to certain model parameters. Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition for the
number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the
existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least
of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness
properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically | {"url":"http://citeseerx.ist.psu.edu/showciting?doi=10.1.1.154.4555","timestamp":"2014-04-18T08:40:47Z","content_type":null,"content_length":"43962","record_id":"<urn:uuid:939a0b14-2f0d-4e23-a92a-08032c43abfa>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00049-ip-10-147-4-33.ec2.internal.warc.gz"} |
Module 8: Exploring Algorithms
Module Objectives
In this module you will:
a) Solve a task involving scientific notation and Order of Operations
b) Explore algorithms related to subtraction, multiplication, and division.
Task 1: Distances in Space
The table below gives you the average distance of 3 planets from the Sun.
Mercury Venus Earth
5.7x10^7 km 1.08 x10^8 km 1.5x10^8 km
How far are each of those planets from the Sun in million kilometers?
How far are each of those planets from the Sun in thousand kilometers?
Susan said, "Venus is more than twice as far from the Sun than Mercury is."
Tyrone said, "Earth is more than twice as from the Sun than Mercury is?
Are they correct? Why or why not?
Scientific notation is used to express large numbers in terms of powers of ten. The Grade 5 Common Core Standards ask students to:
5.NBT.1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.NBT.2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or
divided by a power of 10. Use whole-number exponents to denote powers of 10.
How does the work that you did above with the distances between the planets relate to each of these Standards?
Submit your work in a Word document.
Task 2:
Algorithms and Strategies
Here is an article on how place value understanding influences students' use of algorithms. Read the article. Here are the figures that go with it.
Here is work on multiplication strategies. Complete the attached task.
Here is some information on division strategies. Here are more strategies. Use three of the division strategies described to solve the task:
444 divided by 16.
As you were solving these tasks using various strategies, how did your understanding of place value and number sense help you?
Submit your work in the same Word document as Task 1.
Task 3:
Final Project
Go back to the previous module for directions. This is due August 6.
Work due at the end of this module:
Task 1-
Final project- submit through Moodle2 or e-mail | {"url":"http://coedpages.uncc.edu/abpolly/6311/module-8-algorithms.htm","timestamp":"2014-04-20T04:03:39Z","content_type":null,"content_length":"7415","record_id":"<urn:uuid:a01abefe-6384-43d7-8310-99e2f0a735e8>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00497-ip-10-147-4-33.ec2.internal.warc.gz"} |
Polynomials and polynomial functions
Next: Polynomials in one variable Up: Pre-requisites Previous: Conic sections
We now revise the definition and elementary properties of polynomials and polynomial functions. The fundamental ideas of calculus consist of extending these notions to a larger class of functions.
There are two ways of approaching the concept of function. The ancient way is through formulae, while the modern approach is through the study of functions of sets of points. Most functions that we
study arise naturally and can be defined formally (i. e. by formulae or expressions). On the other hand, many of the properties demanded of functions are best defined by thinking of them as set
functions. Moreover, most of the formulae have a ``life of their own''; the formal expressions have a more general validity than as functions alone. Thus the study of formulae becomes algebra while
the study of functions becomes analysis. Calculus is thus seen differently by algebraists and analysts. The fundamental example in both cases is that of a polynomial which we study below.
Kapil H. Paranjape 2001-01-20 | {"url":"http://www.imsc.res.in/~kapil/geometry/prereq1/node4.html","timestamp":"2014-04-17T19:59:35Z","content_type":null,"content_length":"3569","record_id":"<urn:uuid:7d65d0b2-48c9-4e73-a7e6-c389ffcd1693>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00191-ip-10-147-4-33.ec2.internal.warc.gz"} |
CGTalk - trying to use the API in python.. again
Once more I'm trying to use the Maya API via python .. and again I can't get it to work properly. What I want to do is using the api Matrix class to create the inverse of the one I retrieved from the
xform command.
#get the inverse matrix using the maya api
def inverseMatrix(m):
newM = []
for i in range(0,4):
for y in range(0,4):
ma = om.MMatrix(newM).inverse()
for i in range(0,4):
for y in range(0,4):
return newM
I'm converting the matrix from float[16] to float[4][4] but the function seems to want 'float const [4][4]' and I have no idea how to create it. I know there is the MScriptUtility class for making
pointers and references but it doesnt seem to help there.
Does anybody know how to do this? I guess I could use some math library for python but I wanted to get more familiar using the API .. :/
Thanks for any hints! | {"url":"http://forums.cgsociety.org/archive/index.php/t-592173.html","timestamp":"2014-04-17T16:03:59Z","content_type":null,"content_length":"6027","record_id":"<urn:uuid:636679b0-490e-4c58-ad91-173ae6884172>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00537-ip-10-147-4-33.ec2.internal.warc.gz"} |
Find Parametric Equation for Moving Particle
October 11th 2010, 08:04 AM
Find Parametric Equation for Moving Particle
Hello. I am familiar with parametric equations but the way this one is being asked is throwing me off.
Find parametric equations for the path of a particle that moves along the circle
(x-1)^2 + (y+2)^2 = 4 three times clockwise, starting at point (1,-4).
October 11th 2010, 08:05 AM
i assume you know the (counter-clockwise) way to parameterize a circle, just do it the other way. you then want to choose the angle so that you get 3 revolutions out of it, beginning at the
indicated point. How's that?
October 11th 2010, 08:37 AM
Hello, lindsmitch!
$\text{Find parametric equations for the path of a particle}$
$\text{that moves along the circle: }\:(x-1)^2 + (y+2)^2 \:=\: 4$
$\text{ three times clockwise, starting at point (1, -4)}$
The path is a circle, center (1,-2) and radius 2.
The curve starts at "6 o'clock" and moves clockwise for 3 revolutions.
There is a variety of ways to write the parametric equations.
. . I'll use the easiest way (for me).
. . $\begin{Bmatrix}{x &=& 1 + 2\cos\theta \\ y &=& \text{-}2 + 2\sin\theta \end{Bmatrix}\quad \text{ for }\,\theta = \frac{3\pi}{2}\,\text{ to }\,\theta = \text{-}\frac{9\pi}{2}$
October 11th 2010, 08:59 AM
Thank you both very much. That was very helpful.
In the future, how would I have arrived at those parametric equations Soroban? Did you just use the conversion x = r * cos(theta) and y = r*sin(theta)? | {"url":"http://mathhelpforum.com/calculus/159164-find-parametric-equation-moving-particle-print.html","timestamp":"2014-04-18T13:23:40Z","content_type":null,"content_length":"7192","record_id":"<urn:uuid:0caabbb0-689b-4fdb-8786-772c14a3ff49>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00433-ip-10-147-4-33.ec2.internal.warc.gz"} |
Integral Equation
Copyright © University of Cambridge. All rights reserved.
'Integral Equation' printed from http://nrich.maths.org/
A differential equation involves functions and derivatives. Integral equations involve functions and integrals.
In this example you have an integral equation. First you have to differentiate both sides. Then you have to solve a differential equation where you should be able to spot the solution immediately.
Then you have to check that you have found a solution by substituting the function you have found into the integral equation. | {"url":"http://nrich.maths.org/4933/note?nomenu=1","timestamp":"2014-04-20T16:11:10Z","content_type":null,"content_length":"3229","record_id":"<urn:uuid:3684d9b0-4feb-480b-8ee7-46009545b5a9>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00026-ip-10-147-4-33.ec2.internal.warc.gz"} |
Taylor Polynomial and Remainder
December 9th 2010, 04:47 PM #1
Junior Member
Oct 2009
Taylor Polynomial and Remainder
I'm not sure how to present an answer to this kind of question. It states:
Present the 3rd-Degree Taylor Polynomial $P_{a,2}(h)$ as well as the Lagrange remainder for the function $f(x,y) = x^3y + sin(xy)$.
Then let $a = (1, \pi)$ and write this polynomial in matrix form.
Thanks a lot!
Well, I don't know how to apply the definitions to find an answer and I haven't read anything on the matrix forms for the taylor polynomial.
The third degree Taylor polynomial for f(x, y), at $(1, \pi)$ is
$f(1, \pi)+ \frac{\partial f(1, \pi)}{\partial x}(x- 1)+ \frac{\partial f(1, \pi)}{\partial y}(y- \pi)+ \frac{1}{2}\frac{\partial^2 f(1, \pi)}{\partial x^2}(x- a)^2+$$\frac{1}{2}\frac{\partial^2
f(1, \pi)}{\partial x\partial y}(x- a)(y-\pi)$$+ \frac{1}{2}\frac{\partial^2 f(1,\pi)}{\partial y^2}(y- \pi)^2+$$\frac{1}{6}\frac{\partial^3 f(1,\pi)}{\partial x^3}(x- 1)^3+ \frac{1}{6}\frac{\
partial^3 f(1, \pi)}{\partial x^2\partial y}(x-1)^2)(y- \pi)+$$\frac{1}{6}\frac{\partial^3 f(1,\pi)}{\partial x\partial y^2}(x-1)(y-\pi)^2+ \frac{1}{6}\frac{\partial f(1,\pi)}{\partial y^3}(y- \
December 9th 2010, 05:17 PM #2
December 9th 2010, 05:23 PM #3
Junior Member
Oct 2009
December 10th 2010, 03:05 AM #4
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/calculus/165847-taylor-polynomial-remainder.html","timestamp":"2014-04-16T20:14:21Z","content_type":null,"content_length":"40876","record_id":"<urn:uuid:f0f3f908-5528-433f-a39d-cb7ffbed32a3>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00053-ip-10-147-4-33.ec2.internal.warc.gz"} |
predicate logic
“Exponentiation on reals has no left identity.” What does this mean exactly?
Hello thehollow89Exponentiation on reals takes a real number, $x$, and raises a base, $a$, say, to the power of $x$. So, using $\circ$ notation, exponentiation can be defined as the binary operation:
$a\circ x = a^x$. A left identity is an element $i$ in the domain of $\circ$, such that $i \circ x = x, \forall x$ in the domain. So to say that exponentiation on reals has no left identity is to say
that there is no real number $i$ for which $i^x = x, \forall x \in \mathbb{R}$. Grandad | {"url":"http://mathhelpforum.com/discrete-math/76981-predicate-logic.html","timestamp":"2014-04-18T00:36:54Z","content_type":null,"content_length":"34897","record_id":"<urn:uuid:ca921fa8-aee1-436e-a31c-495edbc57184>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00347-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Topic: Day 3 Math Forum Advanced Summer Institute
Replies: 0
Day 3 Math Forum Advanced Summer Institute
Posted: Jul 9, 1998 12:42 PM
The participants started a rainy day 3 of the Math Forum Advanced
Summer Institute in connections, and then on individual projects.
Dave then spoke on some uses of Javascript and Dynamic HTML
(http://forum.swarthmore.edu/workshops/sum98/js.explain.html). He
began by explaining that Javascript and Java differ: Javascript was
created by Netscape with the purpose of augmenting HTML, whereas Java
was created by Sun MicroSystems. Javascript is useful for the
valuation of data in input forms. Dave encouraged the participants to
adapt any of the examples available
(http://forum.swarthmore.edu/workshops/sum98/script.html), copying and
pasting source code. He noted, in the checklist function,
checkChecker(), that the first line reads
thevalue = document.forum.list1.selectedIndex;
"list1" refers to the name of the given to the pulldown list code, so
it should be changed accordingly. Dave also mentioned a few other
sites for examples of Javascript code
(http://forum.swarthmore.edu/workshops/sum98/js.examples.html). To see
an example of one participant's work with forms, view Rob Rumppe's
online quiz
Following Dave's session, Bob Panoff demonstrated some the Shodor
Foundation's work. Project Interactivate
(http://www.shodor.org/interactivate/) explores fractals by iterating
line deformations, providing a tool for teaching pattern recognition
and self-similarity (http://www.shodor.org/interactivate/applets/).
The Project Interactivate lessons are organized into three segments:
"What?" asks questions to be investigated; "How?" explains the how to
use the applet; and "Why?" develops other applications. In addition,
there are lessons linked to the table of contents of several math
texts (http://www.shodor.org/interactivate/toc.html). He began with a
paper and pencil exercise, where participants drew a line with a
simple "kink" in it and then successively replaced each segment with
the same kink to the scale of that segment. Bob noted that this
exercise becomes tedious quickly and thus defined the point where
using applets becomes beneficial. In one example, Bob demonstrated
three fractals, each of which differed only slightly from the other at
the outset; when iterated, however, they resulted in very different
images -- the perimeter of the first approaching infinite length but
enclosing finite area, the second copying over itself, the last one
increasingly space-filling. He drew qualitative parallels from these
three illustrations of "sensitivity to initial conditions" to the
regeneration of healthy skin, the growth of a benign tumor, and the
spread of a malignant tumor, respectively. Bob requested feedback on
the pages, particularly on how to expand and clarify the "What?" and
"Why?" segments. The participants suggested adding information that
links the explorations to real world uses (such as the three examples
above), developing exercises that require more participation from the
students (similar to Sketchpad), and organizing feedback with
annotations so that others may view some participant results.
After lunch, Judy discussed developing effective online units. She
finds that, in particular, listing a table of contents and background
information (such as target age groups) speed navigation. When
inserting applets, it helps students to see instructions alongside the
applet. Here, Bob noted that applets can be more easily available to
several links when they stand alone. Ron Knott and Suzanne Alejandre
then demonstrated some of their online work on Fibonacci numbers
and "traffic jam"
In the next session, Nicole and Tushar introduced displaying
Mathematica, Maple, and Mathview notebooks on the Web
(http://forum.swarthmore.edu/spimsow/). Users can rotate graphs and
change equations to experiment with graphs already created. Of the
three, Mathview, which requires a plug-in
(http://www.maplesoft.com/www/mathview.html), is the least powerful,
but also the least expensive and most interactive. To put a Maple
notebook on web pages, generate the notebook in Maple and save it as
an HTML file. Maple will then create three Maple HTML files and a
graphics file, which must be uploaded through FTP. To see Mathematica
on the web, download the application helper Math Reader
(http://www.wolfram.com). Nicole and Tushar noted that with Maple and
Mathematica notebooks, it is not possible to create an interactive
environment. For more on implementing math software on the web, view
their example notebooks
(http://forum.swarthmore.edu/spimsow/notebooks.html), read their
synopsis of comparisons
(http://forum.swarthmore.edu/spimsow/synopsis.html), or e-mail them
directly at <nicole@forum.swarthmore.edu> and
<parlikar@forum.swarthmore.edu>, respectively.
Steve then began the conversation about building online community
After looking at several online discussion communities, participants
moved upstairs to have a face-to-face discussion, a running record of
which appears in a <geometry-institutes> thread and has seeded some
Betsy Teeple and Richard Tchen | {"url":"http://mathforum.org/kb/thread.jspa?threadID=350605","timestamp":"2014-04-16T14:22:34Z","content_type":null,"content_length":"20511","record_id":"<urn:uuid:849fcfa9-c365-4390-93a8-28dcf37fe92d>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00381-ip-10-147-4-33.ec2.internal.warc.gz"} |
Definition of right triangle in English:
right triangle
Syllabification: right tri·an·gle
North American
• A triangle with a right angle.
More example sentences
□ To fit the National Gallery's East Building on a trapezoid-shaped site, architect I.M. Pei based his design on a division of a trapezoid into an isosceles triangle and a smaller right
□ The base of the wedge is an isosceles right triangle in a vertical plane.
□ Indeed, if there is one thing that someone might remember from grade school mathematics, it's the fact that ‘the square of the hypotenuse of a right triangle is equal to the sum of the
squares of the two adjacent sides.’
More definitions of right triangle
Definition of right triangle in:
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Word of the day astrogation
Pronunciation: ˌastrə(ʊ)ˈgeɪʃ(ə)n
(in science fiction) navigation in outer space | {"url":"http://www.oxforddictionaries.com/definition/american_english/right-triangle","timestamp":"2014-04-20T11:24:01Z","content_type":null,"content_length":"114229","record_id":"<urn:uuid:3c3048e6-7885-4673-8855-997426ebd13f>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00407-ip-10-147-4-33.ec2.internal.warc.gz"} |
Why is Mathematics Possible: Tim Gowers's Take on the Matter
Why is Mathematics Possible: Tim Gowers’s Take on the Matter
In a previous post I mentioned the question of why is mathematics possible. Among the interesting comments to the post, here is a comment by Tim Gowers:
“Maybe the following would be a way of rephrasing your question. We know that undecidability results don’t show that mathematics is impossible, since we are interested in a tiny fraction of
mathematical statements, and in practice only in a tiny fraction of possible proofs (roughly speaking, the comprehensible ones). But why is it that these two classes match up so well? Why is it that
nice mathematical statements so often have proofs that are of the kind that we are able to discover?
I think part of the answer is that the evolution of mathematics follows what we can prove: it is quite easy to come up with simple statements that look hopeless (is zeta(3) a normal number? etc.
etc.), so the match-up just mentioned is far from perfect. I think we develop an instinct for what kinds of statements are likely to be amenable to our techniques, and from time to time we are
surprised by a statement like the Poincare conjecture that turns out to be very very hard.
I wonder whether the true answer might be something like this: there is a ‘random variable’ associated with ‘natural’ mathematical statements, which takes as its value the length of the shortest
‘humanly discoverable’ proof. We know that this variable can take very large values, or even be infinite, but on average it is fairly small (if only because a lot of statements you can write down
have simple counterexamples). We get interested in the statements for which the random variable takes a large but, it seems to us, not too large, value. And empirically such statements exist.
Of course, all this is wild speculation. I don’t know how to formulate a precise question, but it would be fascinating if one could somehow do rigorous justice to the intuition that most natural
statements have short proofs but a few are more difficult and a few are much more difficult. It would be something like a ‘quantitative statistical version of Godel’s theorem’.”
23 Responses to Why is Mathematics Possible: Tim Gowers’s Take on the Matter
1. The first thing you would have to do with your program is define “natural” statement. I don’t believe this is possible, as what a “natural” statement is depends heavily on the context of known
mathematics. For instance, you consider the question: “classify all finite simple groups”. It would be completely unmotivated without knowing the previous mathematical facts that (a) finite
groups are interesting structures, (b) finite groups can be constructed from finite simple groups, (c) there are a few general classes which most finite groups belong to, and a few sporadic ones,
(d) there is some chance that there are only a finite number of sporadic groups, so this problem has a hope of being solvable.
□ In (c), I meant finite simple groups, of course.
☆ “I don’t believe this is possible, as what a “natural” statement is depends heavily on the context of known mathematics.”
I may be misunderstanding you, but that seems like a non-sequitur. Or rather, it seems like a (correct) argument that any satisfactory definition of “natural” would have to be relative to
a given corpus of knowledge. Is there any reason to suppose that it is impossible to come up with such a definition?
☆ I should add that if what you think is impossible is coming up with a definition that is precise enough to prove rigorously some facts about how the lengths of nice proofs of natural
statements are distributed, then I agree that it is very unlikely to be possible to do that except perhaps in some toy models of mathematics. That is, while I think it may be possible to
define “natural” in an appropriate relative way, there is still a big difficulty about which corpus of knowledge one chooses when formulating a theorem. (In principle it could be a
statement about arbitrary corpora satisfying certain conditions, but that seems even harder.) Maybe one could define a “natural building process” that builds up a corpus of mathematics
from a few basic facts, and then argue that at each stage one expects a certain distribution of difficulty levels for the current open problems.
☆ You’re right. I hadn’t thought of it, but “naturalness” should be defined relative to a corpus of knowledge. And ease of proof should also be defined relative to a corpus of knowledge.
This lets us discuss a situation that I find really interesting, where a statement is natural with respect to one corpus of knowledge, but whose ease of proof depends on a completely
different corpus of knowledge.
2. To the extent that mathematics has to do with reasoning about possible existence, or inference from pure hypothesis, a line of thinking going back to Aristotle and developed greatly by C.S.
Peirce may have some bearing on the question of How and Why Mathematics is Possible. In that line of thought, hypothesis formation is treated as a case of “abductive” inference, whose job in
science generally is to supply suitable raw materials for deduction and induction to develop and test. In this light, a large part of our original question becomes, as Peirce once expressed it —
Is it reasonable to believe that “we can trust to the human mind’s having such a power of guessing right that before very many hypotheses shall have been tried, intelligent guessing may be
expected to lead us to the one which will support all tests, leaving the vast majority of possible hypotheses unexamined”? (Peirce, Collected Papers, CP 6.530).
The question may fit the situation in mathematics slightly better if we modify the word hypothesis to say proof.
□ I copied out a more substantial excerpt from that paper here:
C.S. Peirce • The Proper Treatment of Hypotheses
The question of naturalness arises in many areas, from AI and cognitive science to logic and the philosophy of science, most often under the heading of “Natural Kinds”. Given a universe of
discourse X, the lattice of All Kinds would be its power set, and we seek the portion of that ordering which constitutes the Natural Kinds, the extensions of concepts or hypotheses that are
worth considering in practice.
To the same purpose, Peirce uses the criterion of “admissible hypotheses that seem the simplest to the human mind”.
□ The following project report outlines the three types of inference — Abductive, Deductive, and Inductive — as treated by Aristotle and Peirce, at least insofar as these patterns of reasoning
can be analyzed in syllogistic forms. I did this work by way of exploring how a propositional logic engine might be used to assist in scientific inquiry.
• Functional Logic : Inquiry and Analogy
It looks a bit cobbled together to my eyes today and probably could use a rewrite, but I did put a lot of work into the diagrams and remain rather pleased with those.
□ It looks like that site is down at present. There’s another copy of the paper here:
3. Are you sure that equation is correct in the top-left corner of the picture?
□ Of course it is, it is the standard formula for the solutions of the quadratic equation 4a^2x + 4abx + b^2-b+4ac = 0. :)
☆ Damn it, I meant 4a^2x^2 + 4abx + b^2-b+4ac = 0
4. yes think these are key questions on the boundaries/frontiers of math. the terra incognita. there is some strong connection here to kolmogorov theory & chaitins constant. suggest some answer to
the question lies in a new category called quasi algorithms. automated theorem proving seems to impinge on these questions also.
5. Pingback: The Kadison-Singer Conjecture has beed Proved by Adam Marcus, Dan Spielman, and Nikhil Srivastava | Combinatorics and more
6. There is a remark by Dawkins about “improving a design by evolution” and “improving a design by an engineer”, namely, in the first case the design can be changed only through small steps with the
restriction that all of these “immediate” steps must themselves be functioning well. In spirit it is similar to the “natural building process” metaphor used before by Tim, but I think it still
worth adding this remark as it emphasizes the constrained nature of new steps. For example in his logic textbook Manin remarks that most definition used in mathematics has very short (less than
4) maximal length of nested alternating quantifiers. If it is true it could be because of some cognitive constraint which maybe always present when people generate new mathematics. Of course it
is not clear if this sort of thinking can answer the original question but it would suggest to look for the nature of limitations of the small steps by which human mathematics advances.
7. Maybe it’s because even pure mathematics doesn’t just study arbitrary concepts, but concepts that are grounded in experience (e.g. groups <- symmetry, manifolds <- phase spaces, natural numbers
<- counting, coordinates <- the space in which we live). This seems to be a good starting point for "naturalness". That it coincides with provability maybe suggests that either the universe is
somehow logically or rationally constructed, and/or that mathematics simplifies these concepts to the point of tractability.
8. I think asking why math is possible but not easy is like asking why reading novels is possible but not easy. Novels will be written to be as complex as people can understand, but not more so. And
we will solve the hardest math problems we can but no more.
If we were much less good at math, then we would find arithmetic to be interesting, and logarithms to be at the frontier, like Fermat’s last theorem.
If we were much better at math, then Fermat’s last theorem would be boring like arithmetic, and we would be working on vastly harder problems, like NP vs coNP or something.
9. somehow related, but other question is: are some mathematical “facts” true only by accident, without any relation to “other parts of mathematics”? I do not mean “something we may choose as we
wish”, as in for example one may choose to work without axiom of choice framework” but something which is visible in mathematics ( as a theorem, interesting fact etc) and has no explanation nor
proof at all. Naive example – why a map may be colored by “4″ colors? Why 4? Why even number here? etc.
There are two examples that comes in mind:
1. independence of some theorems ( sometimes the axioms) from given set of sentences. AFAIK all of such examples have in common that they are independent from artificially chosen set of “axioms”
or “theory” constructed by the humans. But are they “truly independent”? Is there any form of “true independence” in a place of “relative independence” fro what we choose as a base system? I
presume not. So if we start from other set of axioms or with another theory, we obtain other “independence relations”.
2. some strange facts ( cryptomorphism by G.C.Rota in matroid theory comes in mind here) which at the end of the day gives us insight that they are parts of the more general view in which they
has his own explanations.
So the only possibility here, in mathematics, is the relative independence and curious coincidence which has his own explanations, OR, there exists truly random mathematical facts which are
“independent” in “global, objective” meaning – such as there exists some computational functions? It looks like there is no such thing – truly random mathematical relation or theorem…
10. Consider a system of axioms A. We can choose different axioms A’, and obtain a theory which is equivalent to that derived from A. A bigger change would be to replace one of the axioms of A with a
statement that is undecidable from A, and it is possible that the replaced axiom becomes now undecidable, from the new axioms. We can change the axioms, somehow similar to changing a basis in a
vector space. In the case of a vector space, for a particular problem it may be more convenient to work with a particular basis. Similarly, we choose the axioms which are of most interest to the
questions we want to investigate. The questions tend to be interconnected, and cluster together in the space of statements of a particular field. By choosing the axioms within the cluster, it is
more likely to ask questions that are “closer”, in terms of length of a proof, to our axioms, than questions that require lengthy proofs, or that are undecidable, although the latter ones may be
much more than the former. Of course, it is possible to have exceptions: other clusters of interesting questions may exist in the space of statements of the field. Then, we may have a derived
field or a subfield of interest, and if a theorem connects the second cluster to the first one, then we would call it “the fundamental theorem of …” whatever the second cluster is called. If we
don’t know such a theorem, maybe we can conjecture a connection.
My main conjecture is that the statements of interest cluster together, and we tend to pick the axioms within the cluster.
11. Sellisele “rängale” küsimusele puudub ühene (kuitahes keerukas) vastus.
Oletame, et:
Maailm on loodud teleoloogilises otstarbekuses – siis tuleb neist ka järjepidevalt kinni hoida.
“Arengule” – on antud SUUND (spinn – paremakäeline jne.) – ja ka “hea ja kuri” – on sellega suunalt määratud.
VALIME Galilei teisenduse (x´= x – vt) arenduse (eelnevas suunises):
Lorentz-teisendusi SAAB esitada funktsionaalsel kujul: f(l(ct)) = L(ct)(1 – (v/c)cosa));
(Mina) esitan: f(ct) = ct(1 – v/c)cosa)); milline kuju ON “arenduslik” sihil v/c.
ERINEVUSED tekivad “arengu pööritamisel: 1) matemaatilises keerukustamises; 2) lihtsuses.
Cartesiuse tasandil:
{ x = ctcosa; y = ctsina;}
Lorentz: { x´= L(x – vt); y´= y;}
Mina: { x´= x – vt; y´= k y;}
On kerge näha, et eelnevas k = 1/L;
KUI matemaatiline VORM ei ole otstarbekas (muudab 2 etteantud mõõdet!?) – tuleb ülevaadata eeldused/postulaadid: (muutes “ainult” ristsuunaklist mõõdet – ruumis nii y kui ka z).
Täidetud saab (!) teleoloogiline otstarbekus: säilitades LOODUD matemaatilise struktuuri – arendades seda edasi: tõenäosusliku trajektoori suunas: k = (+,-)(1 – (v/c)^2)^(-1/2);
Lorentz-teisendused – EI OLE ALTERNATIIV!
Alternatiiv ei ole ka “tõenäosuslainete levimine tühjuses – RUMALUSENA.
12. Tonuke “unustas” – VÕIMALIKKUSE!?
Tõepoolest: kaasaegset matemaatikat ei saa ettegi kujutada ilma Hulgateooriata jne.
Samas: me ei tohi (mistahes!?) “matemaatika arengus” – minna vastuollu meile antud Orientatsiooniga ja Teleoloogilisde printsiibiga. (Keegi muidugi ei saa keelata meil arutlemast mistahes
mõttelisis spekulatsioones – kuid selle “mõttetus” peab olema eelnevalt deklareeritud!)
Väga sageli minnakse “loogiliste välistamiste teed”, selleks et vältida partadokse, kuid see on väär. Vaatlemegi kõige suuremat “matemaatilist välistamist”, mis (mulle) on teada:
KUI eksisteerib “lähimõju printsiip” – välistab see kaugmõju olemasolu!? Miks?
Nii nagu elusorganite areng ei eita nende algset loomist, nõnda võime ju otsida küll “monaadseid osiseid”, mis “kannaksid gravitatsiooni” – kuid see ON MÕTTETU! Mingil monaadil lihtsalt ON SEE –
ja seda “TEAB” – kogu Ilmaruum. Selleks pole vaja “teabekandjat”!
This entry was posted in Open discussion, Philosophy, What is Mathematics and tagged Foundations of Mathematics, Open discussion, Philosophy, Tim Gowers. Bookmark the permalink. | {"url":"https://gilkalai.wordpress.com/2013/06/19/why-is-mathematics-possible-tim-gowerss-take-on-the-matter/","timestamp":"2014-04-21T14:47:01Z","content_type":null,"content_length":"142736","record_id":"<urn:uuid:ce1cc941-2c5d-4837-823f-8cb8efcaa4af>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00437-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ring Test
Exercise 2. Let Z[n][i] = { a+bi | a, b belong to Z[n], i^2=-1 } (the Gaussian integers modulo n ). This software finds the group of units of this ring and the order of each element of the group. Run
the program for n = 3, 7, 11, and 23. Is the group of units cyclic for these cases? Try to guess a formula for the order of the group of units of Z[n][i] as a function of n when n is a prime and n
mod 4 = 3. Run the program for n = 9 and 27. Are the groups cyclic? Try to guess a formula for the order when n = 3^k. Run the program for n = 5, 13, 17, and 29. Is the group cyclic for these cases?
What is the largest order of any element in the group? Try to guess a formula for the order of the group of units of Z[n][i] as a function of n when n is a prime and n mod 4 = 1. Try to guess a
formula for the largest order of any element in the group of units of Z[n][i] as a function of n when n is a prime and n mod 4 = 1. On the basis of the orders of the elements of the group of units,
try to guess the isomorphism class of the group. Run the program for n = 25. Is this group cyclic? Based on the number of elements in this group and the orders of the elements, try to guess the
isomorphism class of the group. | {"url":"http://www.d.umn.edu/~jgallian/project.kai/ring.html","timestamp":"2014-04-20T13:29:42Z","content_type":null,"content_length":"3001","record_id":"<urn:uuid:425eae9c-24d5-4070-a9f0-0f70163649da>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00574-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Complex Multiplication
complex number
is a two-dimensional number and as such needs two coordinates to describe it. We usually use its , coordinates, where represents its real component, and y represents its imaginary component. When
expressed this way a complex number looks like this: .
There is another method that is more natural for understanding how complex numbers multiply. You can represent a complex number by its
—its distance from the origin—and its
its angle as measured counterclockwise from the positive real number line. These two numbers taken together uniquely determine every complex number, just as readily as .
So, now when we multiply two complex number
together we get a third complex number whose argument is just the sum of the two original arguments. Drag the green or blue complex number
around and notice how their product, represented by the red dot, has an argument equal to the sum of the green dot's angle and the blue dot's angle.
Snapshot 1: It's not so obvious how green times blue equals red.
Snapshot 2: But green and blue have angles associated with them…
Snapshot 3: … and red's angle is just the sum of green's angle and blue's angle…
Snapshot 4: … and furthermore, this is true regardless of the order. | {"url":"http://demonstrations.wolfram.com/ComplexMultiplication/","timestamp":"2014-04-17T09:41:21Z","content_type":null,"content_length":"44248","record_id":"<urn:uuid:06ecad45-34e9-4ae6-9788-e6e34b1518e4>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00338-ip-10-147-4-33.ec2.internal.warc.gz"} |
st: Fitting a two-equation zero-inflated model in GLLAMM
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st: Fitting a two-equation zero-inflated model in GLLAMM
From Serguei Kaniovski <Serguei.Kaniovski@wifo.ac.at>
To statalist@hsphsun2.harvard.edu
Subject st: Fitting a two-equation zero-inflated model in GLLAMM
Date Fri, 8 May 2009 08:37:58 +0200
Dear Stata / GLLAMM Experts!
I am trying to fit a latent variable GLS using GLLAMM. My dependent
variable "dist" is bound to the interval [0, sqrt(2)], about 11 percent of
observations are zeroes.
The zeroes denote concordant unanimous votes between a pair of parties in a
parliament. If all members of party A and all members of party B voted yes,
then dist=0. (sqrt(2) appears due to a special distance function I use).
I view zeroes is not qualitatively different from non-zeroes, but the
inflated-zero property has to be accounted for. I thus plan to use a
two-equation model, with the same set of explanatory variables but possibly
a different set for each model later.
The independent variables are "left_right" and "pro_eu", while "vote_id"
identifies the voting occasion.
I fit a level-1 GLLAMM, i.e. with random effects for each voting. Ignoring
the fact that dist is bound from above by sqrt(2), I do:
gen v = cond(dist==0,1,2)
foreach x of varlist left_right pro_eu {
gen v1_`x' = `x'*(v == 1)
gen v2_`x' = `x'*(v == 2)
gllamm dist v1_* v2_*, i(vote_id) link(logit reciprocal) family(binary
gamma) lv(v) fv(v)
Unfortunately the above specification does not converge. Is the
specification correct given what I am trying to do?
That you for your help,
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2009-05/msg00305.html","timestamp":"2014-04-18T15:43:02Z","content_type":null,"content_length":"6349","record_id":"<urn:uuid:e5a9ba81-e3dd-4c56-b378-24a5262dbc4c>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00014-ip-10-147-4-33.ec2.internal.warc.gz"} |
Test Average Word Problem
Date: 02/20/2004 at 10:22:51
From: Rich
Subject: Difficult Word Problem
In a certain class there are more than 20 and fewer than 40 students.
On a recent test the average passing mark was 75. The average failing
mark was 48 and the class average was 66. The teacher then raised
every grade 5 points. As a result the average passing mark became
77.5 and the average failing mark became 45. If 65 is the established
minimum for passing, how many students had their grades changed from
failing to passing?
Date: 02/20/2004 at 11:22:59
From: Doctor Greenie
Subject: Re: Difficult Word Problem
Hi, Rich --
Thanks for sending the interesting problem. It was unlike any I had
seen before; and in solving it I got to use one of my favorite
mathematical "tricks".
Here's what we have....
Before changing the grades:
average passing grade: 75
class average: 66
average failing grade: 48
After changing the grades:
average passing grade: 77.5
class average: 71
average failing grade: 45
All of these numbers were given specifically in the problem except
for the class average after the grades were raised; because every
grade was raised by 5 points, the class average was raised by 5
What we do with these numbers is determine the ratio of passing
grades to failing grades before and after. To do this, we use my pet
technique for solving "mixture" problems.
For the case before the grades were raised, the average passing grade
is 9 points above the class average, and the average failing grade is
18 points below the class average. So what we have is a "mixture"
problem, where a certain number of students with average grades 18
points below average and another number of students with average
grades 9 points above average combine to produce that average.
With the average for one group twice as far from the overall average
as the average for the other group, the ratio of students in the two
groups must be 2:1. And since the overall average is closer to the
average of the students who passed, the larger group of students is
the group who passed. So now we know that we have "2x" students
who passed and "x" students who failed. This means the total number
of students in the class is 3x, so the total number of students is
divisible by 3.
For the case after the grades have been raised, we perform the same
analysis. The passing average and failing average are, respectively,
6.5 points above and 26 points below the overall average. These two
numbers are in the ratio 1:4; this means that now the ratio of
students who passed to students who failed is 4:1; and this means the
number of students in the class is divisible by 5.
So now we know that the number of students in the class is divisible
by both 3 and 5, and has to be between 20 and 40. Can you determine
how many students there are from that information?
Once you know the total number of students, can you use the pass:fail
ratios to determine how many students passed and failed with each of
the two scorings? That will let you answer the actual question posed
in the problem.
I hope this helps. Please write back if you have any further
questions about any of this.
- Doctor Greenie, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/65416.html","timestamp":"2014-04-16T22:28:34Z","content_type":null,"content_length":"8560","record_id":"<urn:uuid:0c950112-1d2c-405e-b50e-09f115d2395a>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00061-ip-10-147-4-33.ec2.internal.warc.gz"} |
Platonic Solids
This article explains how to calculate the coordinates of the corners of the Platonic solids: tetrahedron, cube (hexahedron), octahedron, dodecahedron, and icosahedron.
Several of my writings involve the Platonic solids. For instance, some of the programs included with my book Visual Basic Graphics Programming draw these objects, though the book does not explain how
to find the coordinates of the solids' vertices, it merely uses them. Much to my surprise, I received several requests for these derivations.
This article explains how to calculate the positions of the corners of the Platonic solids. Deriving these values requires only relatively simple algebra and a little trigonometry. Even so, they can
be a bit involved so mistakes are certainly possible. After using the formulas shown here to find the points, I verified the results by writing test programs to verify that all of the edges on a side
were the same length.
For programs that draw three-dimensional objects, see my book Visual Basic Graphics Programming.
A regular polygon is a two-dimensional shape where each edge has the same length and the edges all make the same angles with respect to each other. Figure 1.1 shows two quadrilaterals. The square on
the left is a regular polygon because all its sides are the same length and they all meet at 90 degree angles. The parallelogram on the right is not regular. While its sides have the same length,
they do not all meet at the same angles.
Figure 1.1. A regular polygon (left) and a polygon that is not regular (right).
The Platonic solids were defined by the Greek mathematician and philosopher Plato (427-347 BC). They are all of the three-dimensional solids that you can define using faces that are identical regular
polygons. These solids are also known as the three-dimensional regular polytopes.
The Platonic solids include the tetrahedron (4 triangular faces), cube or hexahedron (6 square faces), octahedron (8 triangular faces), dodecahedron (12 pentagonal faces), and the icosahedron (20
triangular faces).
There are no other regular polytopes in three dimensions. You may have seen solids made up of more faces, each of which is identical. For example, some game stores sell 30-sided and even 100-sided
dice. The faces of these solids are parallelograms not regular polygons, so they are not regular polytopes. Similarly you can make geodesic domes using identical triangles. The triangles are not
equilateral (they have different side lengths), however, so the dome is not a regular polytope.
There are other regular polytopes in higher dimensions. For example, "cubes" are defined for all higher dimensional spaces. There are four-dimensional cubes, five-dimensional cubes, etc. There are
even some regular polytopes that do not correspond to any two- or three-dimensional polytopes.
Two-dimensional space has an infinite number of regular polytopes because you can make a regular polygon with any number of sides: triangle, square, pentagon, hexagon, and so forth. Four-dimensional
space has the next most regular polytopes, although I can't remember how many it has or how many faces they have. I think I remember it having four-dimensional counterparts to tetrahedrons, cubes,
and octahedrons (I think all dimensions have those) but I don't remember what else. If someone knows, email me.
The Platonic solids have some rather interesting dual relationships. To make the dual of a solid, place a vertex in the center of each of the solid's faces. Then connect each vertex to the vertices
on the adjacent faces. For the Platonic solids, the result is another Platonic solid.
Figure 1.2 shows a cube and its dual: an octahedron.
Figure 1.2. A cube and its dual, an octahedron.
Table 1.1 lists the Platonic solids and their duals.
│ Solid │ Dual │
│ Tetrahedron │ Tetrahedron │
│ Cube │ Octahedron │
│ Octahedron │ Cube │
│ Dodecahedron │ Icosahedron │
│ Icosahedron │ Dodecahedron │
Table 1.1 The duals of the Platonic solids.
The number of vertices, faces, and edges in duals have a reciprocal relationship. For example, a cube has 6 faces and 8 vertices while an octahedron has 8 faces and 6 vertices. Both have 12 edges.
Table 1.2 lists the number of faces, vertices, and edges in the Platonic solids.
│ Solid │ Faces │ Vertices │ Edges │
│ Tetrahedron │ 4 │ 4 │ 6 │
│ Cube │ 6 │ 8 │ 12 │
│ Octahedron │ 8 │ 6 │ 12 │
│ Dodecahedron │ 12 │ 20 │ 30 │
│ Icosahedron │ 20 │ 12 │ 30 │
Table 1.2 The number of faces, vertices, and edges in the Platonic solids.
The number of faces, edges, and vertices are related to the shape and arrangement of the faces. Define the following values:
F = Total faces in the solid
E = Total edges in the solid
V = Total vertices in the solid
EF = Number of edges on each face
VF = Number of vertices on each face
SE = Number of faces that share each edge (always 2)
SV = Number of faces that share each vertex
E = F * EF / SE
V = F * VF / SV
For example, an icosahedron has 20 triangular faces with each vertex shared by 5 faces so F = 20, EF = 3, VF = 3, SE = 2, and SV = 5. Plugging these numbers into the previous equations gives:
E = 20 * 3 / 2 = 30
V = 20 * 3 / 5 = 12
A tetrahedron has four faces that are equilateral triangles. In an equilateral triangle, the sides meet at 60 degree angles. Figure 2.1 shows a tetrahedron.
Figure 2.1. A tetrahedron.
It is a trigonometric fact that in a triangle with angles of 30, 60, and 90 degrees, the sides have relative lengths of 1, 2, and Sqr(3) as shown in Figure 2.2. These lengths are not absolute--they
are relative to each other. For example, if the short side has length 4, the other sides have lengths 4 * 2 and 4 * Sqr(3).
Figure 2.2. The 30-60-90 triangle has sides with relative lengths of 1, 2, Sqr(3).
Imagine looking at the tetrahedon from the top as shown in Figure 2.3. By symmetry, the projections of the sides of the tetrahedron bisect the angles in the solid's base. Those angles are each 60
degrees, so the bisected angles are 30 degrees. Since one of the other angles in this smaller triangle is 90 degrees, the remaining angle must be 180 - 90 - 30 = 60 degrees.
Now suppose the sides of the tetrahedron have length 2. Then the longer side next to the right angle in the smaller triangle has length 1 because it is half of the larger triangle's side. This
corresponds to the side with length Sqr(3) in the 30-60-90 triangle with side lengths 1, 2, Sqr(3). For the sides of this smaller triangle to have the correct relative lengths, the other two sides
must have lengths 1/Sqr(3) and 2/Sqr(3). To verify that these lengths are correct, multiply the lengths 1/Sqr(3), 2/Sqr(3), and 1 by Sqr(3) and you will see that the sides have relative lengths 1, 2,
Sqr(3). This situation is shown in Figure 2.3.
Figure 2.3. Top view of a tetrahedron.
Now suppose you want the top point of the tetrahedron to lie in the Z axis, and the bottom face to lie in the X-Y plane. Then the values shown in Figure 2.3 give the coordinates of all four points
with the exception of the Z coordinate of the top point as shown in Figure 2.4.
Figure 2.4. Most of the coordinates for a tetrahedron's vertices.
Now imagine looking at the tetrahedron from the side, looking parallel to the X axis. Figure 2.5 shows the tetrahedron from almost this angle. The view is tilted just a little so you can still see
all four sides.
Figure 2.5. Calculating the Z coordinate of the tetrahedron's top point.
We have assumed that the tetrahedron's edge lengths are 2 so the distance A in Figure 2.5 is 2. We know from Figures 2.3 and 2.4 that the distance B is Sqr(3) - 1/Sqr(3). Because sides A, B, and z
form a right triangle, z * z + B * B = A * A. Solving for z gives z = Sqr(A * A - B * B). Plugging in the values of A and B gives:
z = Sqr(2 * 2 - (Sqr(3) - 1/Sqr(3)) * (Sqr(3) - 1/Sqr(3)))
= Sqr(4 - (3 - 2 + 1/3))
= Sqr(3 - 1/3)
= Sqr(8/3)
= 2 * Sqr(2/3)
Thus the coordinates of the tetrahedron's vertices are:
( 0, Sqr(3) - 1/Sqr(3), 0)
(-1, - 1/Sqr(3), 0)
( 1, - 1/Sqr(3), 0)
( 0, 0, 2 * Sqr(2/3))
To verify that these values are correct, you can write a program that calculates the distances between each pair of these points. All of the distances should be 2. Click here to download a program
that performs this verification.
The cube is an easy one. If you want the cube's sides to be 2 units long and you want the cube centered at the origin, the cube has vertices at:
(-1, -1, -1)
(-1, 1, -1)
( 1, -1, -1)
( 1, 1, -1)
(-1, -1, 1)
(-1, 1, 1)
( 1, -1, 1)
( 1, 1, 1)
Click here to download a program that verifies this.
Figure 3.1. A cube.
4. Octahedron
The octahedron is also straightforward. If you place the solid's vertices on the coordinate axes as shown in Figure 4.1, the points are located at:
( 1, 0, 0)
( 0, 1, 0)
( 0, 0, 1)
(-1, 0, 0)
( 0, -1, 0)
( 0, 0, -1)
Figure 4.1. An octahedron.
Click here to download a program that verifies the points. Note that these coordinates define the dual of the previously described cube.
You can find the vertices of a dodecahedron using calculations similar to those used for the icosahedron shown in the next section but I'm too lazy to reproduce them here. If you need the vertices'
coordinates, you can take the dual of the icosahedron. Or see my book Visual Basic Graphics Programming which explains the derivation in detail.
6. Icosahedron
Figure 6.1 shows an icosahedron with its hidden surfaces removed.
Figure 6.1. An icosahedron with hidden surfaces removed.
Figure 6.2 shows the same icosahedron with hidden surfaces drawn in dashed lines and its nodes labeled a through l.
Figure 6.2. An icosahedron with nodes labeled.
The icosahedron is centered at the origin in these figures. Points a and l lie on the Z-axis so they have X and Y coordinates 0. Points b and g lie in the Y-Z plane so they have X coordinate 0.
Let Z1 be the Z coordinate of point a and let Z2 be the Z coordinate of points b through f. By symmetry, the Z coordinate of point l is -Z1 and the Z coordinate of points g through k are -Z2.
To calculate the points' X and Y coordinates, look at the icosahderon from the top. Consider only the positions of points b through f and g through k as shown in Figure 6.3.
Figure 6.3. An icosahedron viewed from the top.
The two pentagons that contain the points b through f and g through k allow you to calculate the points' X and Y coordinates. Let S be the length of one of the icosahedron's edges and consider just
the right half of the upper pentagon shown in Figure 6.4. You can use the angles t1, t2, t3, and t4 to calculate the points' coordinates.
Figure 6.4. Calculating the X and Y coordinates for points b, c, and d.
Because the nodes are arranged in a pentagon, t1 = 2 * Pi / 5 and
t2 = Pi / 2 - t1
= Pi / 2 - 2 * Pi / 5
= Pi / 10
Similarly t4 = 2 * Pi / 10 = Pi / 5 and:
t3 = -(Pi / 2 - t4) = -(Pi / 2 - Pi / 5)
= -3 * Pi / 10
Using trigonometry in the t4 triangle, R = (S/2) / Sin(t4). With this value, you can calculate H = Cos(t4) * R. Using these values, we have the coordinates for points b and d:
b = ( 0, R, Z2)
d = (S/2, -H, Z2)
Similarly the X and Y coordinates of point c are R * Cos(t2) and R * Sin(t2). For notational convenience, let these values be Cx and Cy.
Using the symmetry shown in Figure 6.3, you can calculate the coordinates for all of the rest of the points.
a = ( 0, 0, Z1)
b = ( 0, R, Z2)
c = ( Cx, Cy, Z2)
d = ( S/2, -H, Z2)
e = (-S/2, -H, Z2)
f = ( -Cx, Cy, Z2)
g = ( 0, -R, -Z2)
h = ( -Cx, -Cy, -Z2)
i = (-S/2, H, -Z2)
j = ( S/2, H, -Z2)
k = ( Cx, -Cy, -Z2)
l = ( 0, 0, -Z1)
S = the side length (given)
t1 = 2 * Pi / 5
t2 = Pi / 10
t4 = Pi / 5
t3 = -3 * Pi / 10
R = (S/2) / Sin(t4)
H = Cos(t4) * R
Cx = R * Cos(t2)
Cy = R * Sin(t2)
The only values left to calculate are Z1 and Z2. Consider the top of the icosahedron shown in Figure 6.5. Look again at Figure 6.4 to see how the value R fits into this picture.
Figure 6.5. The top of an icosahedron.
We know S and R, so we can calculate H1:
H1 = Sqr(S * S - R * R)
Now consider one of the pentagons on the side that contains the points a, b, and i as shown in Figure 6.6. Let H2 be the vertical distance between points a and i. We have already seen that the Y
coordinate of point i is H so the horizontal distance along the Y-axis between these points is H as shown in the figure.
Figure 6.6. The side of an icosahedron.
The hypotenuse of the triangle with sides H and H2 is the height of the pentagon containing points a, b, and i. Figure 6.4 shows that distance is H + R. You can now use these values to solve for H2.
H2 = Sqr((H + R) * (H + R) - H * H)
Finally, you can use the values of H1 and H2 to calculate Z1 and Z2.
Z2 = (H2 - H1) / 2
Z1 = Z2 + H1
To verify that these values are correct, you can write a program that calculates the distances between each pair of these points. All of the distances should be S. Click here to download a program
that performs this verification. | {"url":"http://www.vb-helper.com/tutorial_platonic_solids.html","timestamp":"2014-04-17T19:04:42Z","content_type":null,"content_length":"27223","record_id":"<urn:uuid:4589b5ff-97e6-4f47-8c67-b58266b5d8c1>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00038-ip-10-147-4-33.ec2.internal.warc.gz"} |
A few integration problems
April 4th 2013, 09:28 PM #1
May 2009
A few integration problems
I am not supposed to "integrate by substitution" or anything fancy yet in this section ......
"In each part, confirm that the formula is correct and state a corresponding integration formula."
a) (d/dx)[sqrt(1+ X^2)] = (X)(1+X^2)^-1/2
Integral (X)(1+X^2)^-1/2 dx
I understand how to take the integral of (1+X^2)^-1/2 , but where does the x go? is it x(dx) and becomes 1? and then whats leftover is the integral of (X)(1+X^2)^-1/2 ??? That would make sense, I
just figured that out while I was typing it but I wanted to make sure ...
b) (d/dx)[1/3 sin(1+x^3)] = x^2cos(1+x^3)
Integral x^2cos(1+x^3)dx
I got (x^3 /3)(-sin(1+x^3)(3x^2) ... I have no idea how that turns out to be what it is....
c) "Find the derivative and state a corresponding integration formula."
(d/dx)[sqrt(x^3+5)] = (3x^2)/(2)[sqrt(x^3+5)]
Integral (3x^2)/(2)[sqrt(x^3+5)] dx
Once again... I understand how to take the integral of [sqrt(x^3+5)][^-1/2] .... but I don't understand what happens to (3/2)x^2 ...?
Thank you!
Re: A few integration problems
The integration process is the reverse of differentiation.
since (X)(1+X2)-1/2 is the derivateive of your function f(x) = sqr(1+x^2) then as it is in the integration formula it will produce the antiderivative ..i.e the function from which (X)(1+X2)-1/2
came out as derivative.....this is what we call indefinite integral or antiderivative.......the dx which accompanies the integral form is the so called differential of the variable x and it is
compalsory to put it there....why? it is difficult to explain...
have a look here to understand a little bit about the differentials....
Differential of a function - Wikipedia, the free encyclopedia
Actually we integrate differentials........
Re: A few integration problems
This is what you haven't done. Instead of seeing why this...
... might be true, if it is true, you have decided to go right ahead and do this...
Now, you say...
... but are you sure? Do you mean this...
... ? If you know that, somehow, then, good. But it's true because of this...
... and explaining that is harder than what's on the menu here. (But see the bottom spoiler, below.)
So do as suggested and confirm the given derivatives. Just in case a picture helps...
... where (key in spoiler) ...
Similarly, you are invited to see that...
... is basically the same whether you read it from top down (differentiating) or bottom up (integrating). And you should use the chain rule to confirm that it's true...
Try doing the same for c):
Good luck.
That derivative of arsinh:
Hope that helps. More orthodox advice might be forthcoming too, no doubt. But the exercise is designed to make you confront the chain rule, one way or another.
Btw, it's best to read dx as just 'with respect to x', e.g. $\dfrac{d}{dx}$ means 'the derivative with respect to x of...' and $\int\ y\ dx$ means 'the integral with respect to x of...' (... of y
or whatever).
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
April 4th 2013, 11:11 PM #2
Senior Member
Feb 2013
Saudi Arabia
April 5th 2013, 04:56 AM #3
MHF Contributor
Oct 2008 | {"url":"http://mathhelpforum.com/calculus/216691-few-integration-problems.html","timestamp":"2014-04-17T21:40:30Z","content_type":null,"content_length":"43751","record_id":"<urn:uuid:185ab86f-7b92-4f0f-a604-0f89c4e536c4>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00153-ip-10-147-4-33.ec2.internal.warc.gz"} |
The world's 23 toughest math questions
It sounds like a math phobic's worst nightmare or perhaps Good Will Hunting for the ages.
Those wacky folks at he the Defense Advanced Research Projects Agency have put out a research request it calls Mathematical Challenges, that has the mighty goal of "dramatically revolutionizing
mathematics and thereby strengthening DoD's scientific and technological capabilities."
The challenges are in fact 23 questions that if answered, would offer a high potential for major mathematical breakthroughs, DARPA said. So if you have ever wanted to settle the Riemann Hypothesis,
which I won't begin to describe but it is one of the great unanswered questions in math history, experts say. Or perhaps you've always had a theory about Dark Energy, which in a nutshell holds that
the universe is ever-expanding, this may be your calling.
DARPA perhaps obviously states research grants will be awarded individually but doesn't say how much they'd be worth. The agency does say you'd need to submit your research plan by Sept. 29, 2009.
So if you're game, take your pick of the following questions and have at it.
• The Mathematics of the Brain: Develop a mathematical theory to build a functional model of the brain that is mathematically consistent and predictive rather than merely biologically inspired.
• The Dynamics of Networks: Develop the high-dimensional mathematics needed to accurately model and predict behavior in large-scale distributed networks that evolve over time occurring in
communication, biology and the social sciences.
• Capture and Harness Stochasticity in Nature: Address Mumford's call for new mathematics for the 21st century. Develop methods that capture persistence in stochastic environments.
• 21st Century Fluids: Classical fluid dynamics and the Navier-Stokes Equation were extraordinarily successful in obtaining quantitative understanding of shock waves, turbulence and solitons, but
new methods are needed to tackle complex fluids such as foams, suspensions, gels and liquid crystals.
• Biological Quantum Field Theory: Quantum and statistical methods have had great success modeling virus evolution. Can such techniques be used to model more complex systems such as bacteria? Can
these techniques be used to control pathogen evolution?
• Computational Duality: Duality in mathematics has been a profound tool for theoretical understanding. Can it be extended to develop principled computational techniques where duality and geometry
are the basis for novel algorithms?
• Occam's Razor in Many Dimensions: As data collection increases can we "do more with less" by finding lower bounds for sensing complexity in systems? This is related to questions about entropy
maximization algorithms.
• Beyond Convex Optimization: Can linear algebra be replaced by algebraic geometry in a systematic way?
• What are the Physical Consequences of Perelman's Proof of Thurston's Geometrization Theorem?: Can profound theoretical advances in understanding three dimensions be applied to construct and
manipulate structures across scales to fabricate novel materials?
• Algorithmic Origami and Biology: Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.
• Optimal Nanostructures: Develop new mathematics for constructing optimal globally symmetric structures by following simple local rules via the process of nanoscale self-assembly.
• The Mathematics of Quantum Computing, Algorithms, and Entanglement: In the last century we learned how quantum phenomena shape our world. In the coming century we need to develop the mathematics
required to control the quantum world.
• Creating a Game Theory that Scales: What new scalable mathematics is needed to replace the traditional Partial Differential Equations (PDE) approach to differential games?
• An Information Theory for Virus Evolution: Can Shannon's theory shed light on this fundamental area of biology?
• The Geometry of Genome Space: What notion of distance is needed to incorporate biological utility?
• What are the Symmetries and Action Principles for Biology?: Extend our understanding of symmetries and action principles in biology along the lines of classical thermodynamics, to include
important biological concepts such as robustness, modularity, evolvability and variability.
• Geometric Langlands and Quantum Physics: How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice
• Arithmetic Langlands, Topology, and Geometry: What is the role of homotopy theory in the classical, geometric, and quantum Langlands programs?
• Settle the Riemann Hypothesis: The Holy Grail of number theory.
• Computation at Scale: How can we develop asymptotics for a world with massively many degrees of freedom?
• Settle the Hodge Conjecture: This conjecture in algebraic geometry is a metaphor for transforming transcendental computations into algebraic ones.
• Settle the Smooth Poincare Conjecture in Dimension 4: What are the implications for space-time and cosmology? And might the answer unlock the secret of "dark energy"?
• What are the Fundamental Laws of Biology?: This question will remain front and center for the next 100 years. DARPA places this challenge last as finding these laws will undoubtedly require the
mathematics developed in answering several of the questions listed above.
Layer 8 in a box
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NASA banging, freezing next generation space telescope into shape
GAO report torches US for dumping electric waste in foreign countries | {"url":"http://www.networkworld.com/community/blog/worlds-23-toughest-math-questions","timestamp":"2014-04-17T04:08:48Z","content_type":null,"content_length":"97695","record_id":"<urn:uuid:8ac16adb-94b6-4d3d-a857-4f3a9994d51c>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00291-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability Question- could be simple/could be hard??
May 7th 2010, 12:33 AM
Probability Question- could be simple/could be hard??
Suppose you have an initial balance of £50 and play £10 stakes on a coin- heads you win ( and gain £10) / tails you lose. What is the probability of doubling your initial balance assuming you
play as long as you need to? Or indeed the probability of losing the balance?
How does the stake size affect the probabilities? What if you played at £5 will it make any difference? And what if i went for a target of £150
Also generalising, balance of B
Stake size - s
target - T ( 2xB or 3xB etc)
p= probability of a win on one trial?
Anyone got any ideas? Its either really simple or really complicated??
Many Thanks
May 7th 2010, 04:56 AM
Suppose you have an initial balance of £50 and play £10 stakes on a coin- heads you win ( and gain £10) / tails you lose. What is the probability of doubling your initial balance assuming you
play as long as you need to? Or indeed the probability of losing the balance?
How does the stake size affect the probabilities? What if you played at £5 will it make any difference? And what if i went for a target of £150
Also generalising, balance of B
Stake size - s
target - T ( 2xB or 3xB etc)
p= probability of a win on one trial?
Anyone got any ideas? Its either really simple or really complicated??
Many Thanks
You might want to read up on Gambler Ruin problem - such questions are covered there | {"url":"http://mathhelpforum.com/advanced-statistics/143502-probability-question-could-simple-could-hard-print.html","timestamp":"2014-04-21T13:55:07Z","content_type":null,"content_length":"5228","record_id":"<urn:uuid:18b472dc-821c-41b9-a7d5-bf4368425377>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00386-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Weighted Skeletons and FixedShare Decomposition #
Institute for Theoretical Computer Science
University of Technology, Graz, Austria
We introduce the concept of weighted skeleton of a polygon and present various decomposition and
optimality results for this skeletal structure when the underlying polygon is convex.
1 Introduction
Polygon decomposition is a major issue in computational geometry. Its relevance stems from breaking com
plex shapes (modeled by polygons) into subpolygons that are easier to manipulate, and from subdividing
areas of interest into parts that satisfy certain containment requirements and/or optimality properties. We
refer to [13] for a nice survey on this topic. In particular, a rich literature exists on decomposition into
convex polygons. Convex decompositions are most natural in some sense. They have many applications
and can be computed efficiently; see e.g. [7, 14, 16].
In this paper, we focus on the problem of decomposing a convex polygon such that predefined constraints
are met. More specifically, the goal is to partition a given convex ngon P into n convex parts, each part
being based on a single side of P and containing a specified 'share' of P . The share may relate, for
example, to the spanned area, to the number of contained points from a given point set, or to the total edge
length covered from a given set of curves. Possible applications of such fixedshare decompositions include | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/005/3892621.html","timestamp":"2014-04-17T17:05:46Z","content_type":null,"content_length":"8622","record_id":"<urn:uuid:a85a24f4-4535-445e-9155-736a418f9765>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00059-ip-10-147-4-33.ec2.internal.warc.gz"} |
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5 Subjects: including algebra 1, algebra 2, geometry, prealgebra | {"url":"http://www.purplemath.com/Dresher_Math_tutors.php","timestamp":"2014-04-18T01:08:30Z","content_type":null,"content_length":"23544","record_id":"<urn:uuid:a2418e35-b682-4206-a7f1-3e3b5a93429c>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00528-ip-10-147-4-33.ec2.internal.warc.gz"} |
Incorporating the DWN into the MDWD Framework
Next: Multidimensional Unit Elements Up: Digital Waveguide Networks Previous: Note
Incorporating the DWN into the MDWD Framework
At this point, the reader may have noticed more than a few similarities between the treatment of transmission lines in this chapter and the last. It should be recalled from Chapter 3 that the
multidimensional wave digital networks that numerically integrate the parallel-plate system are derived from multidimensional Kirchoff circuits under a set of coordinate transformations and spectral
mappings from the continuous to discrete domain. A DWN, on the other hand, is constructed entirely in the discrete time and space domain, and is then identified with a finite difference method
consistent with the problem.
It will be shown in this section that certain DWNs may in fact be derived from an MDKC under an alternative spectral mapping (also passivity-preserving), provided new wave digital circuit elements (
multidimensional unit elements) are also defined. In this way, the DWN may be considered to be a multidimensional wave digital network in its own right, depending on how attached one feels to the
trapezoid rule as an integration method. The behavior of TLM and wave-digital numerical integration methods have been previously compared in [131] and [71], and the material in this section also
appears in [18].
It is instructive to first reconsider the (1+1)D transmission line system in the lossless source-free case. In Figure 4.44 are presented both the type III DWN, and the MDWD network for the same
system using offset sampling, with spatial dependence expanded out. For the MDWD network, we have chosen a grid spacing of , and a time step of so as to align it with the DWN. (In other words, we
have used -- see §3.7 for details.)
Figure 4.44: Signal flow diagrams of the DWN and MDWD networks for the (1+1)D lossless source-free transmission line.
First, notice that for the DWN, we have one three-port scattering junction at each grid point, and that parallel junctions are interleaved with series junctions. Approximations to and are calculated
at alternating grid points, and at alternating multiples of the time step, . For the MDWDF, we have two two-port series adaptors at each grid point; we are approximating both and together at the same
locations, though due to offset sampling, these locations alternate from one time step to the next. Both variables are treated as currents. For both networks, all the material variation of the
transmission line is expressed in the immittances of self-loops at every junction or adaptor, the delay in which is twice that of the linking delay between adjacent grid points. It is also useful to
compare the waveguide immittances to the port resistances of the MDWDF. We have
where we recall, from the discussion of the type III waveguide network in §4.3.6, that the connecting impedances were chosen to be some constant , in this case .
Enforcing the positivity of and or and leads to identical stability conditions, and the self-loop immittances and inductances are simply related to one another by
at locations for which both quantities coexist in their respective discrete networks.
Most important, though, is the observation that, whereas the DWN can be considered to be made up of an array of lumped two-port bidirectional delay lines, the signal flow diagram of the MDWD network
in Figure 4.44 does not have such an interpretation--the port in this setting is defined only as a multidimensional object, and instances of this port in the discrete domain are not connected
port-wise. It is crucial to recognize that passivity of such a discrete network is reflected by the power conservation of the scattering operation, and not by where wave variables go in the network
after they have been scattered; in particular, they need not be paired as they are for waveguide networks, as long as the shifting operation which they undergo subsequently does not increase energy
in the network. On the other hand, as we shall see in Chapter 5, boundary conditions are much easier to implement in a lumped network.
Next: Multidimensional Unit Elements Up: Digital Waveguide Networks Previous: Note Stefan Bilbao 2002-01-22 | {"url":"https://ccrma.stanford.edu/~bilbao/master/node142.html","timestamp":"2014-04-19T14:57:54Z","content_type":null,"content_length":"12170","record_id":"<urn:uuid:eefaa06e-9790-4f21-ac68-2696ff8b4ec1>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00229-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts about Thomas Bayes on In the Dark
My earlier post on Bayesian probability seems to have generated quite a lot of readers, so this lunchtime I thought I’d add a little bit of background. The previous discussion started from the result
$P(B|AC) = K^{-1}P(B|C)P(A|BC) = K^{-1} P(AB|C)$
Although this is called Bayes’ theorem, the general form of it as stated here was actually first written down, not by Bayes but by Laplace. What Bayes’ did was derive the special case of this formula
for “inverting” the binomial distribution. This distribution gives the probability of x successes in n independent “trials” each having the same probability of success, p; each “trial” has only two
possible outcomes (“success” or “failure”). Trials like this are usually called Bernoulli trials, after Daniel Bernoulli. If we ask the question “what is the probability of exactly x successes from
the possible n?”, the answer is given by the binomial distribution:
$P_n(x|n,p)= C(n,x) p^x (1-p)^{n-x}$
$C(n,x)= n!/x!(n-x)!$
is the number of distinct combinations of x objects that can be drawn from a pool of n.
You can probably see immediately how this arises. The probability of x consecutive successes is p multiplied by itself x times, or p^x. The probability of (n-x) successive failures is similarly (1-p)
^n-x. The last two terms basically therefore tell us the probability that we have exactly x successes (since there must be n-x failures). The combinatorial factor in front takes account of the fact
that the ordering of successes and failures doesn’t matter.
The binomial distribution applies, for example, to repeated tosses of a coin, in which case p is taken to be 0.5 for a fair coin. A biased coin might have a different value of p, but as long as the
tosses are independent the formula still applies. The binomial distribution also applies to problems involving drawing balls from urns: it works exactly if the balls are replaced in the urn after
each draw, but it also applies approximately without replacement, as long as the number of draws is much smaller than the number of balls in the urn. I leave it as an exercise to calculate the
expectation value of the binomial distribution, but the result is not surprising: E(X)=np. If you toss a fair coin ten times the expectation value for the number of heads is 10 times 0.5, which is
five. No surprise there. After another bit of maths, the variance of the distribution can also be found. It is np(1-p).
So this gives us the probability of x given a fixed value of p. Bayes was interested in the inverse of this result, the probability of p given x. In other words, Bayes was interested in the answer to
the question “If I perform n independent trials and get x successes, what is the probability distribution of p?”. This is a classic example of inverse reasoning. He got the correct answer,
eventually, but by very convoluted reasoning. In my opinion it is quite difficult to justify the name Bayes’ theorem based on what he actually did, although Laplace did specifically acknowledge this
contribution when he derived the general result later, which is no doubt why the theorem is always named in Bayes’ honour.
This is not the only example in science where the wrong person’s name is attached to a result or discovery. In fact, it is almost a law of Nature that any theorem that has a name has the wrong name.
I propose that this observation should henceforth be known as Coles’ Law.
So who was the mysterious mathematician behind this result? Thomas Bayes was born in 1702, son of Joshua Bayes, who was a Fellow of the Royal Society (FRS) and one of the very first nonconformist
ministers to be ordained in England. Thomas was himself ordained and for a while worked with his father in the Presbyterian Meeting House in Leather Lane, near Holborn in London. In 1720 he was a
minister in Tunbridge Wells, in Kent. He retired from the church in 1752 and died in 1761. Thomas Bayes didn’t publish a single paper on mathematics in his own name during his lifetime but despite
this was elected a Fellow of the Royal Society (FRS) in 1742. Presumably he had Friends of the Right Sort. He did however write a paper on fluxions in 1736, which was published anonymously. This was
probably the grounds on which he was elected an FRS.
The paper containing the theorem that now bears his name was published posthumously in the Philosophical Transactions of the Royal Society of London in 1764.
P.S. I understand that the authenticity of the picture is open to question. Whoever it actually is, he looks to me a bit like Laurence Olivier… | {"url":"https://telescoper.wordpress.com/tag/thomas-bayes/","timestamp":"2014-04-18T06:05:18Z","content_type":null,"content_length":"52941","record_id":"<urn:uuid:59d49dfd-d02c-4c62-9217-e8774f03db1c>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00186-ip-10-147-4-33.ec2.internal.warc.gz"} |
XOR Encryption
Exclusive-OR encryption, while not a public-key system such as RSA, is almost unbreakable through brute force methods. It is susceptible to patterns, but this weakness can be avoided through first
compressing the file (so as to remove patterns). Exclusive-or encryption requires that both encryptor and decryptor have access to the encryption key, but the encryption algorithm, while extremely
simple, is nearly unbreakable.
Exclusive-OR encryption works by using the boolean algebra function exclusive-OR (XOR). XOR is a binary operator (meaning that it takes two arguments - similar to the addition sign, for example). By
its name, exclusive-OR, it is easy to infer (correctly, no less) that it will return true if one, and only one, of the two operators is true. The truth table is as follows:
A B A XOR B
T T F
T F T
F T T
F F F
(A truth table works like a multiplication or addition table: the top row is one list of possible inputs, the side column is one list of possible inputs. The intersection of the rows and columns
contains the result of the operation when done performed with the inputs from each row and column)
The idea behind exclusive-OR encryption is that it is impossible to reverse the operation without knowing the initial value of one of the two arguments. For example, if you XOR two variables of
unknown values, you cannot tell from the output what the values of those variables are. For example, if you take the operation A XOR B, and it returns TRUE, you cannot know whether A is FALSE and B
is TRUE, or whether B is FALSE and A is TRUE. Furthermore, even if it returns FALSE, you cannot be certain if both were TRUE or if both were FALSE.
If, however, you know either A or B it is entirely reversible, unlike logical-AND and logical-OR. For exclusive-OR, if you perform the operation A XOR TRUE and it returns a value of TRUE you know A
is FALSE, and if it returns FALSE, you know A is true. Exclusive-OR encryption works on the principle that if you have the encrypted string and the encryption key you can always decrypt correctly. If
you don't have the key, it is impossible to decrypt it without making entirely random keys and attempting each one of them until the decryption program's output is something akin to readable text.
The longer you make the encryption key, the more difficult it becomes to break it.
The actual way exclusive-OR encryption is used is to take the key and encrypt a file by repeatedly applying the key to successive segments of the file and storing the output. The output will be the
equivalent of an entirely random program, as the key is generated randomly. Once a second person has access to the key, that person is able to decrypt the files, but without it, decryption is almost
impossible. For every bit added to the length of the key, you double the number of tries it will take to break the encryption through brute force.
C++ uses ^ for bit-level exclusive-OR. To encrypt a single character you can use char x=x^key; if you have a key of one byte. To encrypt a string of characters with a longer key, you can use
something akin to the following code:
#include <iostream.h>
int main()
char string[11]="A nice cat";
char key[11]="ABCDEFGHIJ";
for(int x=0; x<10; x++)
return 0;
The program encrypts each character in the string using the ^ bit operator to exclusive-OR the string value with the key value for each character. | {"url":"http://www.cprogramming.com/tutorial/xor.html","timestamp":"2014-04-19T14:57:31Z","content_type":null,"content_length":"24139","record_id":"<urn:uuid:36360ae5-d9d3-4d3c-91c6-52a1775d75be>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00617-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: October 2010 [00108]
[Date Index] [Thread Index] [Author Index]
Re: Plotting a function dynamically in a loop
• To: mathgroup at smc.vnet.net
• Subject: [mg112905] Re: Plotting a function dynamically in a loop
• From: "Nasser M. Abbasi" <nma at 12000.org>
• Date: Tue, 5 Oct 2010 05:36:06 -0400 (EDT)
• References: <i8c8ub$g9p$1@smc.vnet.net>
• Reply-to: nma at 12000.org
On 10/4/2010 3:06 AM, ABHIJIT BHATTACHARYYA wrote:
> Hi!
> Let us consider that I have one 3D array ex[[i,j,k]] which is computed
> in time loop as given here.
> alfa=0.3; beta=0.4; gamma== 0.5;=0A For[time=0, time<= maxtime, time++,
> Do[ex[[i,j,k]] = alfa*i+beta*j+gamma*k, {i,1,imax}, {j,2,jmax+1},{k,1,kmax}]
> (* Plot ex[[]]= dynamically *)
> ]
> What I want is that I like to plot ex[[i,j,k]] at every time step in a single plot so that plot is revealed as a movie. Is it possible in
> mathematica?
> Regs
> Abhijit Bhattacharyya
This is the main difference between Mathematica and other handler
graphics based systems.
There is no "handler" to some canvas to use to send all output to. Doing
Do[Print@Plot[Sin[a x], {x, -Pi, Pi}], {a, 1, 10}]
Will just show all the plots.
But you can use many other options in Mathematics to do the same.
Such as
Animate[Plot[Sin[a x], {x, -Pi, Pi}], {a, 1, 10}]
ListAnimate[Table[Plot[Sin[a x], {x, -Pi, Pi}], {a, 1, 5}]]
or make a Dynamic expression that depends on parameter, and then change
the parameter later, this will cause the Dynamic expression to be
re-evaluated automatically each time the parameter changed. Use Pause to
slow the loop:
Dynamic[Plot[Sin[a x], {x, -Pi, Pi}]]
Do[a = i; Pause[1], {i, 0, 10}]
And many other ways to do this I am sure. | {"url":"http://forums.wolfram.com/mathgroup/archive/2010/Oct/msg00108.html","timestamp":"2014-04-17T18:28:54Z","content_type":null,"content_length":"26573","record_id":"<urn:uuid:f499e0bb-7c7c-469a-aa45-feee16e5b73a>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00320-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Acceptance/Rejection Sampling
With this Demonstration, you can visualize the rejection sampling technique, which is also known as the acceptance-rejection algorithm. Select a target distribution (the distribution from which you
would like to generate random samples) and then choose a "threshold value" that influences the likelihood that a candidate sample from a nontarget distribution will be "accepted" as if it were, in
fact, from the target distribution. You can change the number of random variates generated and view histograms of both the accepted and the rejected samples.
The exponential distribution and the Cauchy distribution for target distributions are supported on the positive half or whole real line, respectively.
Suppose that you want to generate pseudorandom numbers from a random variable with probability distribution , but that other methods (like the method of inverse transforms) do not work well (perhaps
because the cumulative distribution function associated with does not have an explicit inverse). Instead, choose a companion random variable with probability density function , making sure that and
have the same support (i.e. both vanish and fail to vanish over the same sets of real numbers).
Ideally, the random variable has been chosen in such a way that the cumulative distribution function associated with has an explicit inverse and is easy to simulate using a technique like the method
of inverse transforms.
The acceptance-rejection algorithm is then as follows: (1) independently simulate a random number with a uniform distribution over the unit interval and a realization * of the random variable ; and
then (2) using a fixed, strictly positive number , accept * as a realization of if , where and are the probability densities of the random variables and , respectively. If is not accepted, it is
rejected. This process is continued repeatedly until a target number of realizations of is generated.
Heuristically, the acceptance-rejection method works particularly efficiently when certain conditions are satisfied. First, the target random variable and the companion random variable should have
density functions that "look" as similar as possible. In this Demonstration, we use the exponential random variable as a companion random variable for the probability distributions with support on
the positive part of the real line (the gamma distribution, chi-square distribution, half-normal distribution, and log-normal distribution). We use the Cauchy distribution as the companion
distribution for the probability distributions with support over the whole real line (the normal distribution, the Student
distribution, the Gumbel distribution, and the Laplace distribution).
Second, the strictly positive constant influences the tendency of the algorithm to "accept" instead of "reject". If is particularly large, will be less than with small probability, and the algorithm
will throw away many fine proxies for . On the other hand, if is particularly close to zero and is not substantially smaller than , then will probably be accepted. It can be shown that the
acceptance-rejection algorithm produces an empirical distribution of pseudorandom numbers that converges the most rapidly to the target distribution if is chosen to be the maximum possible value of
over the (common) support of and .
The acceptance-rejection method can be generalized to the Metropolis–Hastings algorithm and is a type of Markov chain Monte Carlo simulation. For further information about the acceptance-rejection
algorithm, see [1] or [2].
[1] S. M. Ross,
, 4th ed., Boston: Elsevier, 2006.
[2] S. Ghahramani,
Fundamentals of Probability, with Stochastic Processe
s, 3rd ed., New York: Prentice Hall, 2004. | {"url":"http://demonstrations.wolfram.com/AcceptanceRejectionSampling/","timestamp":"2014-04-16T16:16:39Z","content_type":null,"content_length":"52073","record_id":"<urn:uuid:13abda76-9fda-4270-a2ac-d649d4a7ba72>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00079-ip-10-147-4-33.ec2.internal.warc.gz"} |
radical expressions and equations: simplifying radicals
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Try this one
Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
You are not logged in.
#1 2006-02-23 16:56:28
Try this one
prove that ,
u (to the power)n × v(to the power)(1-n) ≤nu + (1-n)v
n∈(o,1) and u,v > 0
by the way how do I bring this superscripts ? As word files are not working.
#2 2006-02-23 19:07:09
Re: Try this one
To do superscripts:
and you get:
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
#3 2006-02-23 19:32:17
Re: Try this one
Hence, the question would be.....
Prove that u^n x v^1-n≤nu+(1-n)v
n∈(o,1) and u,v > 0
Character is who you are when no one is looking.
#4 2006-02-24 03:51:50
Re: Try this one
If n = 0, then the inequality simplifies down to v ≤ v.
Similarly, if n = 1, then the inequality simplifies down to u ≤ u.
So u^n x v^1-n ≤ nu+(1-n)v all the time because they are actually always equal to each other.
Why did the vector cross the road?
It wanted to be normal.
#5 2006-02-25 16:28:07
Re: Try this one
No mathsy thats not quite a proof...but I can give you hints if you want...this is simple...indeed
#6 2006-03-01 16:35:07
Re: Try this one
I guess I can give you some hints about the solution...you can consider the fact that log is a concave function.
#7 2006-03-01 18:00:37
Real Member
Re: Try this one
We must prove that
Is there some unequation for
log(a+b) and log(a)+log(b)?
IPBLE: Increasing Performance By Lowering Expectations.
#8 2006-03-02 16:35:24
Real Member
Re: Try this one
Rimiq please, help us more.
IPBLE: Increasing Performance By Lowering Expectations.
Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
prove that ,u (to the power)n × v(to the power)(1-n) ≤nu + (1-n)vn∈(o,1) and u,v > 0 by the way how do I bring this superscripts ? As word files are not working.
Hence, the question would be.....Prove that un x v1-n≤nu+(1-n)v wheren∈(o,1) and u,v > 0
If n = 0, then the inequality simplifies down to v ≤ v.Similarly, if n = 1, then the inequality simplifies down to u ≤ u.So un x v1-n ≤ nu+(1-n)v all the time because they are actually always equal
to each other.
No mathsy thats not quite a proof...but I can give you hints if you want...this is simple...indeed
I guess I can give you some hints about the solution...you can consider the fact that log is a concave function.
We must prove that log(nu+(1-n)v)>=nlog(u)+(1-n)log(v)Is there some unequation forlog(a+b) and log(a)+log(b)? | {"url":"http://www.mathisfunforum.com/viewtopic.php?id=2904","timestamp":"2014-04-21T09:43:03Z","content_type":null,"content_length":"14939","record_id":"<urn:uuid:d8e8c559-9645-4106-98a7-fdbd86513a7e>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00258-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> Standardised output in multilevel SEM
Xu, Man posted on Wednesday, November 05, 2008 - 9:07 am
I was wondering how the standardised path coefficients in the multilevel SEM are derived.
1.Are level 1 standardised coefficients obstained by unstandardised beta being multiplied with ratio between sd of predictor and sd of level 1 dependent latent variable? And are level 2 standardised
coefficients obstained by unstandardised beta being multiplied with ratio between sd of predictor and sd of level 2 dependent latent variable?
2.also, if I define an extra parameter, say a variable that is the net effect of a level 2 contextual variable (level 2 beta minus level 1 beta of the same variables), should i calculate the
standardised coefficient of this variable according to which method above??
3. how do i do if I have a latent cross level interaction term and I am interested in deriving its standardised coefficient?
Sorry for asking so many questions. I appreciate your advice very much!
Bengt O. Muthen posted on Thursday, November 06, 2008 - 8:34 am
Multilevel modeling often does not use standardized coefficients. So one question is if you really need them.
1. Mplus standardizes to the variance on within for within relationships and to the variance on between for between relationships. Another choice is to standardize to the total variance,
within+between. As always, yes, you multiply the raw coefficient by the ratio of the SD's for the IV divided by the DV.
2. If you have to standardize, you should do it with respect to the SD's of the variables in the relationship, which in this case are the between-level variables, I believe.
3. Cross-level interaction means a random slope so here you have a within-level DV variance that is a function of the IV values - so it is not clear which DV SD one should use. This is probably an
argument against standardizing with respect to the DV - you can still standardize wrt the IV (multipling by the IV SD).
Kätlin Peets posted on Monday, March 21, 2011 - 1:16 pm
I have a question concerning standardized vs unstandardized between-level estimates. I am not sure which estimates (stand. or unstand) I should trust. It seems that the significance of the parameters
varies sometimes quite a bit depending on whether I look at the stand. vs unstand. output.
My second question concerns the situation where I regress the intercept on several (four) between-level covariates. It seems that some of the x and y associations become stronger (and significant)
when other covariates are in the model. However, I don't think the multicollinearity is an issue here as the highest correlation between the covariates is .27. What does it indicate (does it matter
that the number of clusters is only 33?)?
Bengt O. Muthen posted on Monday, March 21, 2011 - 5:04 pm
Standardized and unstandardized significance agree except in unusual situations. Standardized may have a more well-behaved sampling distribution and therefore be preferred. If you want to know more,
send relevant info to support.
Remember that with several covariates the slopes are partial regression coefficients, so representing the effect when other covariates are held constant. So the meaning is different.
joon hyung park posted on Tuesday, March 20, 2012 - 10:42 pm
Hi. I have same question concerning standardized vs unstandardized between-level estimates.
If I look at the unstandardized output, some parameters are not significant but if I look at STDYX (or STDY) standardization, these are significant.
However STD standardization shows same patterns as the unstandardized one.
Which estimates are preferred?
joon hyung park posted on Wednesday, March 21, 2012 - 5:18 am
Dear Dr.Muthen
You can see the output at
I found a posting saying,
"The raw and standardized coefficients have different sampling distributions. This is why their significances may differ (Linda)."
What do you suggest to use?
Linda K. Muthen posted on Wednesday, March 21, 2012 - 9:14 am
I would use raw coefficients. It depends on what is commonly reported in your discipline. Check journals where you would normally publish and see what is reported.
Melvin C Y posted on Thursday, May 10, 2012 - 4:45 am
I would like to compare the multilevel contextual effects of prior ability (1-7 scale) and SES (regression score) on achievement (irt scaled score). Generally, unstandardized estimates are preferred.
But as prior ability and SES are usually highly related, I think my best chance is to run separate analysis. My question is: how do I compare the estimates if they are measured on different metrics?
It appears that the stdYX estimates would be the best option here.
My second question is: if I add more L2 variables (all on 1-7 scale), is it still advisable to use Stdyx estimates? Or should I standardize the contextual variables first before running them in
Mplus? In this way, I can interpret the contextual variables in standardized units but keep the metrics of my L2 variables.
Thank you.
Linda K. Muthen posted on Thursday, May 10, 2012 - 1:19 pm
This question is better suited to a general discussion forum like SEMNET.
Rod Bond posted on Monday, May 13, 2013 - 8:03 am
I am running the following 2-level model:
WITHIN are height testost meanf0 df ;
mascul ON testost height df meanf0;
meanf0 on testost height;
df on testost height;
df with meanf0;
height with testost;
Clusters are participants who provide a number of judgments of mascul; The WITHIN variables are different task characteristics that do not vary between clusters (each participant completes the same
set of tasks).
I get the message: A NON-POSITIVE DEFINITE FIRST-ORDER DERIVATIVE PRODUCT MATRIX and all standard errors for the within model equal zero except those involving mascul. Am I doing something wrong
Linda K. Muthen posted on Monday, May 13, 2013 - 8:47 am
Please send the output and your license number to support@statmodel.com.
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A139250 - OEIS
A139250 Toothpick sequence (see Comments lines for definition). 378
0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67, 79, 95, 123, 155, 171, 175, 183, 195, 207, 223, 251, 283, 303, 319, 347, 383, 423, 483, 571, 651, 683, 687, 695, 707, 719, 735, 763, 795, 815, 831, 859,
895, 935, 995, 1083, 1163, 1199, 1215, 1243, 1279, 1319, 1379 (list; graph; refs; listen; history; text; internal format)
OFFSET 0,3
COMMENTS A toothpick is a copy of the closed interval [-1,1]. (In the paper, we take it to be a copy of the unit interval [-1/2, 1/2].
We start at stage 0 with no toothpicks.
At stage 1 we place a toothpick in the vertical direction, anywhere in the plane.
In general, given a configuration of toothpicks in the plane, at the next stage we add as many toothpicks as possible, subject to certain conditions.
- Each new toothpick must lie in the horizontal or vertical directions.
- Two toothpicks may never cross.
- Each new toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.
The sequence gives the number of toothpicks after n stages. A139251 (the first differences) gives the number added at the n-th stage.
Call the endpoint of a toothpick "exposed" if it does not touch any other toothpick. The growth rule may be expressed as follows: at each stage, new toothpicks is placed so their
midpoints touch every exposed endpoint.
This is equivalent to a two-dimensional cellular automaton. The animations show the fractal-like behavior.
After 2^k - 1 steps, there are 2^k exposed endpoints, all located on two lines perpendicular to the initial toothpick. At the next step, 2^k toothpicks are placed on these lines, leaving
only 4 exposed endpoints, located at the corners of a square with side length 2^(k-1) times the length of a toothpick. - M. F. Hasler, Apr 14 2009 and others. For proof, see the
Applegate-Pol-Sloane paper.
If the third condition in the definition is changed to "- Each new toothpick must have at exactly one of its endpoints touching the midpoint of an existing toothpick" then the same
sequence is obtained. The configurations of toothpicks are of course different from those in the present sequence. But if we start with the configurations of the present sequence, rotate
each toothpick a quarter-turn, and then rotate the whole configuration a quarter turn, we obtain the other configuration.
If the third condition in the definition is changed to "- Each new toothpick must have at least one of its endpoints touching the midpoint of an existing toothpick" then the sequence n^2
- n + 1 is obtained, because there are no holes left in the grid.
A "toothpick" of length 2 can be regarded as a polyedge with 2 components, both on the same line. At stage n, the toothpick structure is a polyedge with 2*a(n) components.
Conjecture: Consider the rectangles in the sieve (including the squares). The area of each rectangle (A = b*c) and the edges (b and c) are powers of 2, but at least one of the edges (b
or c) is <= 2.
In the toothpick structure, if n >> 1, we can see some patterns which looks like "canals" and "diffraction patterns". For example, see the Applegate link "A139250: the movie version",
then enter n=1008 and click "Update". See also "T-square (fractal)" in the link section. ([From Omar E. Pol, May 19 2009, Oct 01 2011]
Comment from Benoit Jubin, May 20 2009: The web page "Gallery" of Chris Moore (see link) has some nice pictures which are somewhat similar to the pictures of the present sequence. What
sequences do they correspond to?
For a connection to Sierpinski triangle and Gould's sequence A001316, see the leftist toothpick triangle A151566.
Eric Rowland comments on Mar 15 2010 that this toothpick structure can be represented as a 5-state CA on the square grid. On Mar 18 2010 David Applegate showed that three states are
enough. See links.
Equals row sums of triangle A160570 starting with offset 1; equivalent to convolving A160552: (1, 1, 3, 1, 3, 5, 7,...) with (1, 2, 2, 2,...). Equals A160762: (1, 0, 2, -2, 2, 2, 2,
-6,...) convolved with 2*n - 1: (1, 3, 5, 7,...). Starting with offset 1 equals A151548: [1, 3, 5, 7, 5, 11, 17, 15,...] convolved with A078008 signed (A151575): [1, 0, 2, -2, 6, -10,
22, -42, 86, -170, 342,...]. [from Gary W. Adamson, May 19 2009 - May 25 2009]
For a three-dimensional version of the toothpick structure, see A160160. [From Omar E. Pol, Dec 06 2009]
Contribution from Omar E. Pol, May 20 2010: (Start)
Observation about the arrangement of rectangles:
It appears there is a nice pattern formed by distinct modular substructures: a central cross surrounded by asymmetrical crosses (or "hidden crosses") of distinct sizes and also by
"nuclei" of crosses.
Conjectures: after 2^k stages, for k >= 2, and for m = 1 to k - 1, there are 4^(m-1) substructures of size s = k - m, where every substructure has 4*s rectangles. The total number of
substructures is equal to (4^(k-1)-1)/3 = A002450(k-1). For example: If k = 5 (after 32 stages) we can see that:
a) There is a central cross, of size 4, with 16 rectangles.
b) There are four hidden crosses, of size 3, where every cross has 12 rectangles.
c) There are 16 hidden crosses, of size 2, where every cross has 8 rectangles.
d) There are 64 nuclei of crosses, of size 1, where every nucleus has 4 rectangles.
Hence the total number of substructures after 32 stages is equal to 85. Note that in every arm of every substructure, in the potential growth direction, the length of the rectangles are
the powers of 2. (see illustrations in the links. See also A160124). (End)
It appears that the number of grid points that are covered after n-th stage of the toothpick structure, assuming the toothpicks have length 2*k, is equal to (2*k-2)*a(n) + A147614(n), k
> 0. See the formulas of A160420 and A160422. [Omar E. Pol, Nov 13 2010]
Version "Gullwing": on the semi-infinite square grid, at stage 1, we place a horizontal "gull" with its vertices at [(-1, 2), (0, 1), (1, 2)]. At stage 2, we place two vertical gulls. At
stage 3, we place four horizontal gulls. a(n) is also the number of gulls after n-th stage. For more information about the growth of gulls see A187220. - Omar E. Pol, Mar 10 2011.
Contribution from Omar E. Pol, Mar 12 2011 (Start):
Version "I-toothpick": we define an "I-toothpick" to consist of two connected toothpicks, as a bar of length 2. An I-toothpick with length 2 is formed by two toothpicks with length 1.
The midpoint of an I-toothpick is touched by its two toothpicks. a(n) is also the number of I-toothpicks after n-th stage in the I-toothpick structure. The I-toothpick structure is
essentially the original toothpick structure in which every toothpick is replaced by an I-toothpick. Note that in the physical model of the original toothpick structure the midpoint of a
wooden toothpick of the new generation is superimposed on the endpoint of a wooden toothpick of the old generation. However, in the physical model of the I-toothpick structure the wooden
toothpicks are not overlapping because all wooden toothpicks are connected by their endpoints. For the number of toothpicks in the I-toothpick structure see A160164 which also gives the
number of gullwing in a gullwing structure because the gullwing structure of A160164 is equivalent to the I-toothpick structure. It also appears that the gullwing sequence A187220 is a
supersequence of the original toothpick sequence A139250 (this sequence).
For the connection with the Ulam-Warburton cellular automaton see the Applegate-Pol-Sloane paper and see also A160164 and A187220.
A version in which the toothpicks are connected by their endpoints: on the semi-infinite square grid, at stage 1, we place a vertical toothpick of length 1 from (0, 0). At stage 2, we
place two horizontal toothpicks from (0,1), and so on. The arrangement looks like half I-toothpick structure. a(n) is also the number of toothpicks after n-th. - Omar E. Pol, Mar 13 2011
Version "Quarter-circle" (or Q-toothpick): a(n) is also the number Q-toothpicks after n-th stage in a Q-toothpick structure in the first quadrant. We start from (0,1) with the first
Q-toothpick centered at (1, 1). The structure is asymmetric. For a similar structure but starting from (0, 0) see A187212. See A187210 and A187220 for more information. - Omar E. Pol,
Mar 22 2011
Version "Tree": It appears that a(n) is also the number of toothpicks after n-th stage in a toothpick structure constructed following a special rule: the toothpicks of the new generation
have length 4 when are placed on the infinite square grid (note that every toothpick has four components of length 1), but after every stage, one (or two) of the four components of every
toothpick of the new generation is removed, if such component contains an endpoint of the toothpick and if such endpoint is touching the midpoint or the endpoint of another toothpick.
The truncated endpoints of the toothpicks remain exposed for ever. Note that there are three sizes of toothpicks in the structure: toothpicks of length 4, 3 and 2. A159795 gives the
total number of components in the structure after n-th stage. A153006 (the corner sequence of the original version) gives 1/4 of the total of components in the structure after n-th
stage. - Omar E. Pol, Oct 24 2011
Contribution from Omar E. Pol, Sep 16 2012 (Start):
It appears that a(n)/A147614(n) converge to 3/4.
It appears that a(n)/A160124(n) converge to 3/2.
It appears that a(n)/A139252(n) converge to 3.
It appears that A147614(n)/A160124(n) converge to 2.
It appears that A160124(n)/A139252(n) converge to 2.
It appears that A147614(n)/A139252(n) converge to 4.
REFERENCES D. Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191
L. D. Pryor, The Inheritance of Inflorescence Characters in Eucalyptus, Proceedings of the Linnean Society of New South Wales, V. 79, (1954), p. 81, 83.
Richard P. Stanley, Enumerative Combinatorics, volume 1, second edition, chapter 1, exercise 95, figure 1.28, Cambridge University Press (2012), p. 120, 166.
LINKS N. J. A. Sloane, Table of n, a(n) for n = 0..65535
David Applegate, The movie version
David Applegate, Animation of first 32 stages
David Applegate, Animation of first 64 stages
David Applegate, Animation of first 128 stages
David Applegate, Animation of first 256 stages
David Applegate, C++ program to generate these animations - creates postscript for a specific n
David Applegate, Generates many postscripts, converts them to gifs, and glues the gifs together into an animation
David Applegate, Generates bfiles for A139250, A139251, A147614
David Applegate, The b-files for A139250, A139251, A147614 side-by-side
David Applegate, A three-state CA for the toothpick structure
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, which is also available at arXiv:1004.3036v2
Barry Cipra, What comes next?, Science (AAAS) 327: 943.
Mats Granvik, Additional illustration: Number blocks where each number tells how many times a point on the square grid is crossed or connected to by a toothpick, Jun 21 2009.
Gordon Hamilton, Three integer sequences from recreational mathematics, Video (2013?).
M. F. Hasler, Illustration of initial terms
M. F. Hasler, Illustrations (Three slides)
Brian Hayes, Joshua Trees and Toothpicks
Brian Hayes, The Toothpick Sequence - Bit-Player
Benoit Jubin, Illustration of initial terms
Chris Moore, Gallery, see the section on David Griffeath's Cellular Automata.
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the toothpick structure (after 23 steps)
Omar E. Pol, Illustration of patterns in the toothpick structure (after 32 steps)
Omar E. Pol, Illustration of initial terms of A139250, A160120, A147562 (Overlapping figures)
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (triangular grid and toothpick structure)
Omar E. Pol, Illustration of the substructures in the first quadrant (As pieces of a puzzle), after 32 stages
Omar E. Pol, Illustration of the potential growth direction of the arms of the substructures, after 32 stages
L. D. Pryor, Illustration of initial terms (Fig. 2a)
L. D. Pryor, The Inheritance of Inflorescence Characters in Eucalyptus, Proceedings of the Linnean Society of New South Wales, V. 79, (1954), p. 79-89.
E. Rowland, Toothpick sequence from cellular automaton on square grid
E. Rowland, Initial stages of toothpick sequence from cellular automaton on square grid (includes Mathematica code)
K. Ryde, ToothpickTree
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Wikipedia, H tree
Wikipedia, Toothpick sequence
Wikipedia, T-square (fractal)
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata
FORMULA a(2^k) = A007583(k), if k >= 0.
a(2^k-1) = A006095(k+1), if k >= 1.
a(A000225(k)) - a((A000225(k)-1)/2) = A006516(k), if k >= 1.
a(A000668(k)) - a((A000668(k)-1)/2) = A000396(k), if k >= 1.
G.f.: (x/((1-x)*(1+2*x))) * (1 + 2*x*Product(1+x^(2^k-1)+2*x^(2^k),k=0..oo)). - N. J. A. Sloane, May 20 2009, Jun 05 2009
One can show that lim sup a(n)/n^2 = 2/3, and it appears that lim inf a(n)/n^2 is 0.451... - Benoit Jubin, Apr 15 2009 and Jan 29 2010, N. J. A. Sloane, Jan 29 2010.
Observation: a(n) mod 4 == 3 for n>=2. [From Jaume Oliver Lafont, Feb 05 2009]
a(2^k-1) = A000969(2^k-2), if k >= 1. [From Omar E. Pol, Feb 13 2010]
It appears that a(n) = (A187220(n+1) - 1)/2. - Omar E. Pol, Mar 08 2011
a(n) = 4*A153000(n-2) + 3, if n >= 2. - Omar E. Pol, Oct 01 2011
MAPLE G := (x/((1-x)*(1+2*x))) * (1 + 2*x*mul(1+x^(2^k-1)+2*x^(2^k), k=0..20)); # N. J. A. Sloane, May 20 2009, Jun 05 2009
# From N. J. A. Sloane, Dec 25, 2009: A139250 is T, A139251 is a.
a:=[0, 1, 2, 4]; T:=[0, 1, 3, 7]; M:=10;
for k from 1 to M do
a:=[op(a), 2^(k+1)];
T:=[op(T), T[nops(T)]+a[nops(a)]];
for j from 1 to 2^(k+1)-1 do
a:=[op(a), 2*a[j+1]+a[j+2]];
T:=[op(T), T[nops(T)]+a[nops(a)]];
od: od: a; T;
MATHEMATICA CoefficientList[ Series[ (x/((1 - x)*(1 + 2x))) (1 + 2x*Product[1 + x^(2^k - 1) + 2*x^(2^k), {k, 0, 20}]), {x, 0, 53}], x] (* Robert G. Wilson v, Dec 06 2010 *)
PROG (PARI) A139250(n, print_all=0)={my(p=[] /* set of "used" points. Points are written as complex numbers, c=x+iy. Toothpicks are of length 2 */,
ee=[[0, 1]] /* list of (exposed) endpoints. Exposed endpoints are listed as [c, d] where c=x+iy is the position of the endpoint, and d (unimodular) is the direction */
c, d, ne, cnt=1); print_all & print1("0, 1"); n<2 & return(n);
for(i=2, n, p=setunion(p, Set(Mat(ee~)[, 1])); /* add endpoints (discard directions) from last move to "used" points */
ne=[]; /* new (exposed) endpoints */
for( k=1, #ee, /* add endpoints of new toothpicks if not among the used points */
setsearch(p, c=ee[k][1]+d=ee[k][2]*I) | ne=setunion(ne, Set([[c, d]])); \\
setsearch(p, c-2*d) | ne=setunion(ne, Set([[c-2*d, -d]]));
); /* useing Set() we have the points sorted, so it's easy to remove those which finally are not exposed because they touch a new toothpick */
forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] & k-- & ee=vecextract(ee, Str("^"k"..", k+1)))\
cnt+=#ee; /* each exposed endpoint will give a new toothpick */ print_all & print1(", "cnt)); cnt} /* M. F. Hasler, Apr 14 2009 */
CROSSREFS Cf. A000079, A139251, A139252, A139253, A147614.
Cf. A139560, A152968, A152978, A152980, A152998, A153000, A153001, A153003, A153004, A153006, A153007.
Cf. A000217, A007583, A007683, A000396, A000225, A000668, A006516, A006095, A019988, A160570, A160552.
Cf. A000969, A001316, A151566, A160406, A160408, A160702, A078008, A151548, A001045, A147562, A160120.
Cf. A160160, A160170, A160172, A161206, A161328, A161330.
Cf. A002450, A160124. [From Omar E. Pol, May 20 2010]
Sequence in context: A169626 A160808 A151567 * A182634 A173530 A192114
Adjacent sequences: A139247 A139248 A139249 * A139251 A139252 A139253
KEYWORD nonn,look
AUTHOR Omar E. Pol, Apr 24 2008, May 08 2008, Dec 20 2008
EXTENSIONS Verified and extended, a(49)-a(53), using the given PARI code by M. F. Hasler, Apr 14 2009
Edited by N. J. A. Sloane, Apr 29 2009, incorporating comments from Omar E. Pol, M. F. Hasler, Rob Pratt, Jaume Oliver Lafont, Franklin T. Adams-Watters, R. J. Mathar, David W. Wilson,
David Applegate, Benoit Jubin and others.
Further edited by N. J. A. Sloane, Jan 28 2010
STATUS approved | {"url":"http://oeis.org/A139250","timestamp":"2014-04-17T00:49:28Z","content_type":null,"content_length":"62619","record_id":"<urn:uuid:08917d5f-cafd-455d-b686-d626e2b6ce5d>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00327-ip-10-147-4-33.ec2.internal.warc.gz"} |
Comments on The Geomblog: Top 125 questions for the next 25 yearsThis is a tricky problem. on the one hand, I feel ...As a computer scietist I am thrilled that these tw...This problem "How Far Can We Push Chemical Self...
tag:blogger.com,1999:blog-6555947.post112020042696463560..comments2014-01-12T10:46:48.153-07:00Suresh Venkatasubramanianhttps://plus.google.com/
112165457714968997350noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6555947.post-1120244110325147022005-07-01T12:55:00.000-06:002005-07-01T12:55:00.000-06:00This is a tricky problem. on the
one hand, I feel that students need more, not less, training in mathematics before they get a degree in computer science, so aligning outselves with mathematics is not a bad thing. On the other hand,
you are right in that our raison d'etre is not mathematics but computation, and there is a big difference, even if we use mathematical techhniques.<BR/><BR/>Sort of like mathematical physics I
guess...<BR/><BR/>However, the article does a decent job of explaining at an intuitive level what the P vs NP problem is. and I think this is the key point. Problems in string theory and what have
you are really mathematical problems that lie at the core of modern physics; however there is a way of expressing them 'physically' that makes these problems 'physics problems' rather than 'math
problems'. That is the distinction we need to make, that although the problems we study can be framed mathemtically, the underlying intuition is computational and (almost) physical <BR/><BR/><A>
</A><A></A>Posted by<A><B> </B></A>SureshAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6555947.post-1120241722826483282005-07-01T12:15:00.000-06:002005-07-01T12:15:00.000-06:00As a computer
scietist I am thrilled that these two problems have made the list. However, the blurb raises the following question: as theoretical computer scientists, is it in our interest that these problems are
attributed to "Mathematicians"? I've seen this happen very often, in particular when talking about the PRIMES is in P result, Shor's polynomial time quantum algorithm for factoring and other big TCS
results. <BR/><BR/>Obviously, computer scientists and mathematicians know that TCS belongs somewhere in the intersection of math and applied CS, but most people do not know that. Normally I wouldn't
really care about these semantics, but considering the funding crisis for TCS and our documented lack of public outreach, I think that attributing some of our community's greatest work to
Mathematicians (which are considered by the public at large as different than computer scientists) might not be in our interest. Furthermore these kinds of lists have a real impact on young people
who are developing a taste for science and are considering various disciplines (particularly between Math, CS and Physics).  <BR/><BR/><A></A><A></A>Posted by<A><B> </B></A>
AnonymousAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6555947.post-1120235604239310892005-07-01T10:33:00.000-06:002005-07-01T10:33:00.000-06:00This problem <BR/> "How Far Can We Push
Chemical Self-Assembly?"<BR/>also needs our help,<BR/>See webpage of Len Adleman's group . <BR/><BR/><A></A><A></A>Posted by<A><B> </B></A>AnonymousAnonymousnoreply@blogger.com | {"url":"http://geomblog.blogspot.com/feeds/112020042696463560/comments/default","timestamp":"2014-04-16T07:14:07Z","content_type":null,"content_length":"8977","record_id":"<urn:uuid:ec3d6ebc-8dd7-4617-807f-99caa0dd5427>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00371-ip-10-147-4-33.ec2.internal.warc.gz"} |
complex connection
complex connection
Let $\xi =\left(p:E\to B\right)$ be a complex-analytic $G$-principal bundle where $G$ is a complex Lie group. The complex-analytic Ehresmann connection is the analytic field of horizontal subspaces,
which is $G$-equivariant. One can consider the underlying real principal bundle ${\xi }_{R}$. The operator of complex structure becomes an automorphism $I:{\xi }_{R}\to {\xi }_{R}$ of the smooth real
${G}_{R}$-bundle ${\xi }_{R}$. One can consider the differential
$\left(dI{\right)}_{p}{T}_{p}E\to {T}_{p}E$(d I)_p T_p E\to T_p E
which on each vertical subspace $\left({T}_{p}^{V}E{\right)}_{R}$ is an operator of the complex structure on the fiber.
For a field $H$ of horizontal subspaces on ${\xi }_{R}$ the following are equivalent:
(i) $H$ is a connection on $\xi$
(ii) ${I}^{*}H=H$
(iii) ${H}_{p}=\left(dI{\right)}_{p}{H}_{p}$
One can characterize complex connections also by conditions on a covariant derivative on ${\xi }_{R}$.
Created on January 30, 2012 17:31:15 by
Zoran Škoda | {"url":"http://ncatlab.org/nlab/show/complex+connection","timestamp":"2014-04-20T05:47:41Z","content_type":null,"content_length":"14578","record_id":"<urn:uuid:399a8906-a708-480b-a480-0be321fcd9b0>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00576-ip-10-147-4-33.ec2.internal.warc.gz"} |
Repeating Decimals
5.21: Repeating Decimals
Difficulty Level:
Created by: CK-12
Practice Repeating Decimals
Have you ever had a math problem that you couldn't figure out?
Well, Josie is having a difficult time trying to figure one out. Josie is in charge of organizing the sixth and seventh grade into six teams for field day. This would seem like an easy task, except
that there are 49 students to split up.
Josie wrote the following problem.
She figured it would be easier to think of the problem in terms of an improper fraction. But that is where the trouble began. She divided to convert the improper fraction to a mixed number but the
answer came out funny.
This is Josie's answer.
The sixes kept going and going.
Josie is puzzled and isn't sure what this means at all. Do you know?
This Concept is about repeating decimals. At the end of it, you will know how to help Josie.
When we can convert a fraction by dividing the numerator by the denominator evenly to form a decimal, we call this a terminating decimal. The word “terminate” means to end. All of the fractions we
have been working with are terminating decimals.
Here the 7 is our whole number and so it is placed to the left of the decimal point. We divide 1 by 4 to get the decimal part.
$& \overset{ \quad \ .25}{4 \overline{ ) {1.00 \;}}}\\& \ \ \underline{-8}\\& \quad \ \ 20\\& \quad \underline{-20}\\& \qquad \ 0$
This is a terminating decimal. It is called that because once you added the decimal point and the zero placeholders, you were able to divide the dividend by the divisor evenly.
What do we call a decimal that is NOT a terminating decimal?
A decimal that does not end and repeats the same number over and over again is called a repeating decimal. You know that you have a repeating decimal if when you divide the numerator by the
denominator, if you keep ending up with the same number.
Convert $\frac{2}{3}$
First, this does not have a base ten denominator so we will divide the numerator by the denominator.
$& \overset{ \quad \ .666}{3 \overline{ ) {2.000 \;}}}\\& \ \underline{-18}\\& \quad \ \ 20\\& \quad \underline{-18}\\& \qquad \ 20 \\& \quad \ \ \underline{-18}\\& \qquad \quad 2$
Look at what happens as we divide!!! The same remainder keeps showing up and our quotient becomes a series of 6’s. It doesn’t matter if we keep adding zeros forever, our decimal will always repeat.
When you have a decimal that is a repeating decimal, we can add a line over the last digit in the quotient. This is a clue that the decimal repeats.
Our answer is $.66\bar{6}$
Divide these fractions and see if you end up with any repeating decimals.
Example A
Solution: $.333333$
Example B
Example C
Now back to Josie and the teams. Here is the original problem once again.
Josie is having a difficult time trying to figure one out. Josie is in charge of organizing the sixth and seventh grade into six teams for field day. This would seem like an easy task, except that
there are 49 students to split up.
Josie wrote the following problem.
She figured it would be easier to think of the problem in terms of an improper fraction. But that is where the trouble began. She divided to convert the improper fraction to a mixed number but the
answer came out funny.
This is Josie's answer.
The sixes kept going and going.
Josie is puzzled and isn't sure what this means at all. Do you know?
Josie's improper fraction converted to a decimal called a repeating decimal. This means that the values would go on an on indefinitely.
Josie can't evenly divide 49 students onto 6 teams. One team will have an extra player.
If she divides 48 students into 6 teams, there are 8 on each team.
Notice that this is the whole number in the decimal.
She can add the extra student to one of the teams and everything will be fine.
Here are the vocabulary words in this Concept.
Terminating Decimal
decimal that can be found dividing a numerator and denominator and by adding a decimal point and zero placeholders.
Repeating Decimal
a decimal where the digits in the quotient repeat themselves, can be indicated by putting a small line over the second repeating digit.
Guided Practice
Here is one for you to try on your own.
Is $4 \frac{4}{7}$
To figure this out, let's first convert it to a decimal.
$4 \frac{4}{7} = \frac{32}{7} = 4.5714285$
While this is a long decimal, it is a terminating and not a repeating decimal.
Video Review
Here are videos for review.
James Sousa Example of Fractions to a Terminating Decimal
James Sousa Another Example of Fractions to a Terminating Decimal
James Sousa Example of Fractions to a Repeating Decimal
Directions:Determine whether each fraction or mixed number is a terminating or repeating decimal.
1. $\frac{14}{3}$
2. $\frac{34}{9}$
3. $\frac{23}{3}$
4. $\frac{17}{4}$
5. $\frac{19}{6}$
6. $\frac{12}{5}$
7. $3\frac{1}{3}$
8. $8\frac{1}{2}$
9. $9\frac{2}{3}$
10. $11\frac{4}{5}$
11. $16\frac{1}{4}$
12. $\frac{44}{3}$
13. $\frac{66}{7}$
14. $\frac{18}{4}$
15. $\frac{74}{7}$
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decimal to binary conversion
decimal to binary conversion
hi i want to make a program that spits out the binary equivalent of numbers from 0-10000. Any ideas how to get started.???
Sure, lots of ideas ;-)
Look into the bit-shift operators << and >> to move individual bits into a usable position (position 0 specifically) and the boolean operators & and | to isolate one bit.
This question gets posted easily every week. Try searching for it.
If you're not up to speed on bit shifting, you can use repeated integral division of the number divided by the Radix. For example, 150: 150 / 2 = 75 with a remainder of 0. The binary
representation will come out in reverse order using the remainder values.
Hey WaltP.
Originally posted by ronin If you're not up to speed on bit shifting, you can use repeated integral division of the number divided by the Radix. For example, 150: 150 / 2 = 75 with a remainder of
0. The binary representation will come out in reverse order using the remainder values.
One wonders which is a more confusing response :D
Hey WaltP.
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Sherman Oaks Trigonometry Tutor
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Department of Physics
Professor Lynn L. Hatfield, Chairperson.
Horn Professors Estreicher and Menzel; Bucy Professor Wigmans; Professors Borst, Cheng, Gangopadhyay, Lichti, Lodhi, and Myles; Associate Professors Gibson, Glab, Holtz, Huang, Lamp, and
Papadimitriou; Assistant Professor Thacker; Joint Professors Kristiansen, Krompholz, Portnoy, Quitevis, and Temkin.
This department offers study in the following graduate degree programs: PHYSICS, Master of Science and Doctor of Philosophy. Options in applied physics leading to the M.S. and Ph.D. degrees are also
offered. These interdisciplinary options afford flexibility in course work and area of research concentration. Specializations in chemical physics (in cooperation with the Department of Chemistry and
Biochemistry) and biophysics (in cooperation with the Health Sciences Center and the University Medical Center) are also available. An M.S. degree involving industry internships is available to
selected graduate students.
All graduate students must enroll in PHYS 5101 (for the first three semesters) and PHYS 5104 (whenever on a teaching assistantship). PHYS 5312, 5322, and 5307 are tools courses that develop necessary
skills for use in other courses and in research. They are most useful when taken early.
A core curriculum consisting of PHYS 5301, 5303, 5305, and 5306 forms the nucleus of the master's and Ph.D. programs and is the basis for the comprehensive master's final examination and the Ph.D.
qualifying examination. A student selecting any of the degree options may designate a minor consisting of a minimum of 6 hours of course credit in a related area and satisfy any additional
requirements of the minor department. (These 6 hours may be taken in the Physics Department.) Full-time study towards the master's degree typically should be completed in about two years.
M.S. Degree in Physics, Thesis Option. A minimum of 24 hours of course credit plus 6 hours of thesis research with a minimum of 18 hours in the department, plus a master's thesis. The thesis is
defended in a final oral examination.
M.S. Degree in Applied Physics, Thesis Option. A minimum of 24 hours of course credit plus 6 hours of thesis research with a minimum of 9 hours in a specified applied area. This may be in a subfield
of physics or in a related discipline, with the master's thesis from that area. The thesis is defended in a final oral examination.
M.S. Degree in Applied Physics, Internship Option. 24 hours of course credit ( a separate course sequence from that above) plus two semesters of internship in a regional industry or research
laboratory arranged through the department. A departmental report is written following each internship period, and defended in an oral examination. Twelve hours of internship or report credit is
required beyond the course work.
M.S. Degree in Physics, Nonthesis Option. 36 hours of course credit with a minimum of 24 hours in the department, plus passing a comprehensive master's final examination. This option is normally
reserved for students in the Ph.D. program.
Ph.D. Degree in Physics and Ph.D. Degree in Applied Physics. 45 hours of course work in the major beyond the B.S. degree and 15 hours outside the major, plus dissertation research. The 15 hours may
be taken partially or entirely in the Physics Department. They also may be counted toward a minor. The student should consult with the graduate advisor and the research advisor about this.
The core courses for the Ph.D. degree are the same as those for the M.S. degree plus PHYS 5302 and 6306. Further selections should be made from PHYS 5304, 5307, 5311, 5322, 7304, and 5300 (which may
be repeated in different topics).
Ph.D. degree students taking the applied physics option normally take the same core courses as above. Other courses in the degree plan are worked out between the student and the graduate advisor in
consultation with the research advisor.
All students should get involved in research early by taking PHYS 7000, which may count toward the degree. Thesis hours in PHYS 6000 (6 hours required for the M.S., thesis option) and 8000 (12 hours
required for the Ph.D.) should be taken as early as possible and as part of the research.
Students seeking the Ph.D. degree must pass preliminary and qualifying examinations as described in the departmental Graduate Booklet and in accordance with Graduate School requirements. The
examination topics are from general undergraduate physics and graduate core courses. After completing the research, the candidate prepares the dissertation and makes an oral defense of it before his
or her committee and other interested persons.
Courses in Physics. (PHYS)
5001. Master's Internship (V1-12). Internship in an industrial or research laboratory setting. Arranged through the department and directly related to degree program with approval of Internship
5101. Seminar (1:1:0). Must be taken by every graduate student for at least the first four semesters. Must be taken pass-fail.
5104. Instructional Laboratory Techniques in Physics (1:1:0). Laboratory organization and explanation of instructional techniques. Does not count toward the minimum requirement of a graduate degree.
Must be taken pass-fail by all teaching assistants each semester.
5231. Solid State Device Seminar (2:2:1). The structures of simple semiconductor devices and physical description of their electrical function; includes basic device related measurement.
5300. Special Topics (3:3:0). Prerequisite: Approval of graduate advisor. Topics in semiconductor, plasma, surface, particle physics, spectroscopy, and others. May be repeated in different areas.
5301. Quantum Mechanics I (3:3:0). Experimental basis and history, wave equation, Schrödinger equation, harmonic oscillator, piece wise constant potentials, WKB approximation, central forces and
angular momentum, hydrogen atom, spin, two-level systems, and scattering. M.S. and Ph.D. core course.
5302. Quantum Mechanics II (3:3:0). Prerequisite: PHYS 5301 or equivalent. Quantum dynamics, rotations, bound-state and time-dependent perturbation theory, identical particles, atomic and molecular
structure, electromagnetic interactions, and formal scattering theory. Ph.D. core course.
5303. Electromagnetic Theory (3:3:0). Electrostatics and magnetostatics, time varying fields, Maxwell's equations and conservation laws, electromagnetic waves in materials, and waveguides. M.S. and
Ph.D. core course.
5304. Solid State Physics (3:3:0). Prerequisite: PHYS 5301 or equivalent. A survey of the microscopic properties of crystalline solids. Major topics include lattice structures, vibrational
properties, electronic band structure, and electronic transport.
5305. Statistical Physics (3:3:0). Elements of probability theory and statistics; foundations of kinetic theory. Gibb's statistical mechanics, the method of Darwin and Fowler, derivation of the laws
of macroscopic thermodynamics from statistical considerations; other selected applications in both classical and quantum physics. M.S. and Ph.D. core course.
5306. Classical Dynamics (3:3:0). Lagrangian dynamics and variational principles. Kinematics and dynamics of two-body scattering. Rigid body dynamics. Hamiltonian dynamics, canonical transformations,
and Hamilton-Jacobi theory of discrete and continuous systems. M.S. and Ph.D. core course.
5307. Methods in Physics I (3:3:0). Provides first-year graduate students the necessary skill in mathematical methods for graduate courses in physical sciences; applications such as coordinate
systems, vector and tensor analysis, matrices, group theory, functions of a complex variable, variational methods, Fourier series, integral transforms, Sturm-Liouville theory, eigenvalues and
functions, Green functions, special functions and boundary value problems. Tools course.
5309. Atomic and Molecular Physics (3:3:0). Prerequisite: PHYS 5301 or equivalent. A survey of atomic and molecular physics. Major topics include group theory, molecular orbital theory, and energy
transfer processes.
5311. Nuclear Physics (3:3:0). Prerequisite: PHYS 5301 or equivalent. Symmetries in nuclear physics, nuclear interactions, nuclear models, nuclear reactions, scattering, resonance, nuclear energy,
5312. Techniques of Graduate Research (3:1:4). Use of test equipment and shop tools; design and performance of experiments; reports; literature search; writing of scientific papers; and other
fundamental aspects of graduate research. Tools course.
5315. Electromagnetism I (3:3:0). Prerequisite: PHYS 2301, MATH 3350, 3351, or equivalent. Survey of the fundamental laws and applications of electromagnetism. For graduate students in departments
other than physics.
5316. Electromagnetism II (3:3:0). Prerequisite: PHYS 5315. Electromagnetic fields and special relativity. For graduate students in departments other than physics.
5322. Computational Physics (3:2:2). Numerical modeling of physical systems. Data acquisition and analysis. Graphics for displaying complex results. Quadrature schemes and solution of equations. Use
of minicomputers and microcomputers. Tools course.
5324. Classical Mechanics I (3:3:0). Prerequisite: PHYS 1308, MATH 3350, 3351, or equivalent. Introduction to Newtonian Mechanics, Euler-Lagrange Equations, and Hamilton's Principle. For graduate
students in departments other than physics.
5330. Semiconductor Materials and Processing (3:3:0). Survey of semiconductor materials deposition, characterization, and processing techniques with emphasis on the fundamental physical interactions
underlying device processing steps.
5332. Semiconductor Characterization and Processing Laboratory (3:1:4). A hands-on introduction to semiconductor processing technology and materials characterization techniques. Intended to accompany
PHYS 5330.
5335. Physics of Semiconductors (3:3:0). Theoretical description of the physical and electrical properties of semiconductors; Band structures, vibrational properties and phonons, defects, transport
and carrier statistics, optical properties, and quantum confinement.
5336. Device Physics (3:3:2). Principles of semiconductor devices; description of modeling of p/n junctions, transistors, and other basic units in integrated circuits; relationship between physical
structures and electrical parameters.
5371. Conceptual Physics for Teachers (3:3:0). Inquiry-based course in elementary physical principles of mechanics, heat, electricity, and magnetism.
5372. Astronomy for Teachers (3:3:0). Inquiry-based course in solar system, stellar, and galactic astronomy. Discusses history of human understanding of the universe.
6000. Master's Thesis (V1-6).
6002. Master's Report (V1-6).
6306. Advanced Electromagnetic Theory (3:3:0). Prerequisite: PHYS 5303. Classical theory of fields, special theory of relativity and electrodynamics, radiation, antennas, scattering and diffraction,
special topics. Ph.D. core course.
7000. Research (V1-12).
7304. Condensed Matter Physics (3:3:0). Prerequisite: PHYS 5304. Problems of current interest in condensed matter physics. Topics include transport properties in solids, superconductivity, magnetism,
semiconductors, and related topics.
8000. Doctor's Dissertation (V1-12).
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Differentiation using product rule
April 13th 2010, 12:59 AM #1
Sep 2008
Differentiation using product rule
Use the product rule to differentiate the given function with respect to x:
I do not know how to get to the answer from here but the answer is
Your dv/dx should read $\frac{dv}{dx} = 4(x+3)^3$
April 13th 2010, 01:08 AM #2
Senior Member
Oct 2009
April 13th 2010, 02:52 AM #3 | {"url":"http://mathhelpforum.com/calculus/138867-differentiation-using-product-rule.html","timestamp":"2014-04-18T10:47:42Z","content_type":null,"content_length":"35799","record_id":"<urn:uuid:7da33828-8c34-4e79-b510-77627f5e71d5>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00364-ip-10-147-4-33.ec2.internal.warc.gz"} |
pythagorean theorem
February 8th 2011, 04:53 PM #1
Junior Member
Nov 2010
pythagorean theorem
My question :
Give another proof of the pythagorean theorem by using the equations : a/c = e/a and b/c = d/b obtained from similar trigangles in some figure. (the figure is a triangle with sides abc. there is
a line down the middle making it into 2 right triangles. B and A are the hypotenuse's. c is split into 2 sides D and E making C = D+E)
Honestly I dont know where to start.
given data shows us that triangle with sides a,b,c is a right triangle.(because you've stated "similar triangles")
alter your given equations
use these equations to show $a^2+b^2=c^2$
which finally provides us to conclude for any right triangle, the proved equation is true.
February 8th 2011, 08:42 PM #2 | {"url":"http://mathhelpforum.com/geometry/170599-pythagorean-theorem.html","timestamp":"2014-04-19T21:15:54Z","content_type":null,"content_length":"32049","record_id":"<urn:uuid:b9560a6c-f673-4f4a-92cb-8dc43fc5b711>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00032-ip-10-147-4-33.ec2.internal.warc.gz"} |
Example 8.21: latent class analysis
Latent class analysis
is a technique used to classify observations based on patterns of categorical responses. Collins and Lanza's book,"
Latent Class and Latent Transition Analysis
," provides a readable introduction, while the UCLA ATS center has an online
statistical computing seminar
on the topic.
We consider an example analysis from the HELP dataset, where we wish to classify subjects based on their observed (manifest) status on the following variables: 1) on street or in shelter in past 180
days [homeless], 2) CESD score above 20, 3) received substance abuse treatment [satreat], or 4) linked to primary care [linkstatus]. We arbitrarily specify a three class solution.
Support for this method in SAS is available through the
proc lca
proc lta add-on routines
created and distributed by the Methodology Center at Penn State University. While it's customary in R to use researcher-written routines, it's less so for SAS; the machinery which allows
independently written
s thus has the potential to mislead users. It bears explicitly stating that third-party
s probably don't have the same level of robustness or support as those distributed by SAS Institute.
proc lca
code assumes that the data exist in the dataset
. The current coding of 0's and 1's needs to be changed to 1's and 2's.
data ds_0; set "c:\book\help.sas7bdat"; run;
data ds; set ds_0;
homeless = homeless+1;
cesdcut = (cesd > 20) + 1;
satreat = satreat+1;
linkstatus = linkstatus+1;
The call to the LCA procedure specifies the number of classes, the variables to include, the number of categories per variable, and information about the starting values and random starts. It's
highly recommended to run a "large" number of random starts to ensure that the true maximum likelihood estimate is reached (the 20 we used is likely too few for more complex models).
proc lca data=ds;
title '3 class model';
nclass 3;
items homeless cesdcut satreat linkstatus;
categories 2 2 2 2;
seed 42;
nstarts 20;
The output begins with diagnostic information, and indicates that 40% of the seeds were associated with the best fitting model.
Data Summary, Model Information, and Fit Statistics (EM
Number of subjects in dataset: 431
Number of subjects in analysis: 431
Number of measurement items: 4
Response categories per item: 2 2 2 2
Number of groups in the data: 1
Number of latent classes: 3
Rho starting values were randomly generated (seed = 42).
No parameter restrictions were specified (freely estimated).
Seed selected for best fitted model: 1486228051
Percentage of seeds associated with best fitted model: 40.00%
The model converged in 3241 iterations.
Maximum number of iterations: 5000
Convergence method: maximum absolute deviation (MAD)
Convergence criterion: 0.000001000
A number of fit statistics are provided to help with model comparison (e.g. number of classes, constraints in more complex models).
Fit statistics:
Log-likelihood: -1032.48
G-squared: 1.22
AIC: 29.22
BIC: 86.15
CAIC: 100.15
Adjusted BIC: 41.72
Entropy R-sqd.: 0.94
Degrees of freedom: 1
The results indicate that 22% of subjects are in class 1, just 8% in class 2, and 70% in class 3.
Parameter Estimates
Gamma estimates (class membership probabilities):
Class: 1 2 3
0.2163 0.0785 0.7052
The next set of output describes the classes. The prevalence for each level of each variable is described for each class. The last response category is redundant (equal to 1 minus the sum of the
other probabilities).
Rho estimates (item response probabilities):
Response category 1:
Class: 1 2 3
homeless : 0.2703 1.0000 0.5625
cesdcut : 0.1154 0.4214 0.1678
satreat : 0.0004 0.0000 1.0000
linkstatus : 0.6029 1.0000 0.5855
Response category 2:
Class: 1 2 3
homeless : 0.7297 0.0000 0.4375
cesdcut : 0.8846 0.5786 0.8322
satreat : 0.9996 1.0000 0.0000
linkstatus : 0.3971 0.0000 0.4145
Members of class 1 were primarily homeless subjects with a larger proportion of high scores on the CESD, with substance abuse treatment history, and 40% of whom linked to primary care. Class 2 (the
smallest group) was comprised of non-homeless subjects with lower CESD scores, substance abuse treatment, but no linkage. Class 3 was 44% homeless, had high levels of CESD, did not report substance
abuse treatment, and 41% linked to primary care.
We begin by reading in the data, Then we use the
function (section 1.3.1) to generate a dataframe with the variables of interest.
ds = read.csv("http://www.math.smith.edu/r/data/help.csv")
ds = within(ds, (cesdcut = ifelse(cesd>20, 1, 0)))
package supports estimation of latent class models in R. The
function, like
proc lca
, can incorporate polytomous categorical variables, but also like
proc lca
requires the variables to be coded starting with positive integers. We specify 10 repetitions (with random starting values).
res2 = poLCA(cbind(homeless=homeless+1,
cesdcut=cesdcut+1, satreat=satreat+1,
linkstatus=linkstatus+1) ~ 1,
maxiter=50000, nclass=3,
nrep=10, data=ds)
This generates the following output:
Model 1: llik = -1032.889 ... best llik = -1032.889
Model 2: llik = -1032.889 ... best llik = -1032.889
Model 3: llik = -1032.484 ... best llik = -1032.484
Model 4: llik = -1032.889 ... best llik = -1032.484
Model 5: llik = -1032.889 ... best llik = -1032.484
Model 6: llik = -1032.484 ... best llik = -1032.484
Model 7: llik = -1032.484 ... best llik = -1032.484
Model 8: llik = -1032.889 ... best llik = -1032.484
Model 9: llik = -1032.889 ... best llik = -1032.484
Model 10: llik = -1032.889 ... best llik = -1032.484
Conditional item response (column) probabilities,
by outcome variable, for each class (row)
Pr(1) Pr(2)
class 1: 0.2703 0.7297
class 2: 1.0000 0.0000
class 3: 0.5625 0.4375
Pr(1) Pr(2)
class 1: 0.1154 0.8846
class 2: 0.4213 0.5787
class 3: 0.1678 0.8322
Pr(1) Pr(2)
class 1: 0 1
class 2: 0 1
class 3: 1 0
Pr(1) Pr(2)
class 1: 0.6029 0.3971
class 2: 1.0000 0.0000
class 3: 0.5855 0.4145
Estimated class population shares
0.2162 0.0785 0.7053
Predicted class memberships (by modal posterior prob.)
0.181 0.1137 0.7053
Fit for 3 latent classes:
number of observations: 431
number of estimated parameters: 14
residual degrees of freedom: 1
maximum log-likelihood: -1032.484
AIC(3): 2092.967
BIC(3): 2149.893
G^2(3): 1.221830 (Likelihood ratio/deviance statistic)
X^2(3): 1.233247 (Chi-square goodness of fit)
The results are consistent with those found in
proc lca
. We note that, also similar to
proc lca
the global maximum likelihood estimates were reached 3 times out of 10-- this can be discerned by examination of the 10 model results. It's always a good idea to fit a large number of iterations to
ensure that the global maximum likelihood estimates have been reached.
7 comments:
Is there a way to save class membership for each subject in poLCA ?
The "poLCA()" function returns an object of class "poLCA": one of the entries is called "posterior". This consists of a matrix of posterior class membership probabilities, and should do the trick
as a way of merging back class membership for each subject.
How I should perform the trick? for merging back?
is there a way to save class membership using Proc LCA for each subject?
According to the FAQ for Proc LCA (http://methodology.psu.edu/ra/lcalta/faq):
"Can I obtain the predicted probability of membership in each latent class/status for each individual?
Yes. Posterior probabilities of class or status membership are available by using the OUTPOST option in PROC LCA and PROC LTA. See the user's guide for details on the syntax. If the goal is to
assign individuals to a latent class based on their predicted probabilities and link class membership to outcomes, note that this approach does not incorporate the uncertainty of class membership
into the analysis, thus biasing inference."
This is in response to April's "Anonymous" comment, and references the poLCA manual: http://userwww.service.emory.edu/~dlinzer/poLCA/poLCA-manual-1-3-1.pdf.
As Nick mentioned, the "poLCA()" function returns an object of class "poLCA", a list of 28 elements. One of these elements is "posterior", "an N by R matrix containing each observation’s
posterior class membership probabilities". N is defined as the “number of cases used in the model,” and R is the number of classes specified in the call to poLCA(). It is worth noting that poLCA
has a logical argument, na.rm, that specifies how to deal with missing values – if TRUE (the default), then these cases are listwise deleted before estimating the model.
Another interesting element of a “poLCA” object is “predclass,” a vector with integer values, of length N, “of predicted class memberships.”
I fully recommend reading the poLCA manual for more details; there is some latent class model theory but everything after chapter 5 should be of practical interest.
I have a few questions about adding complication to a baseline model:
1) Adding covariates:
a. I do not see G2 anymore in the output. The only fit stat I get it Log-likelihoo. Any way to get all the fit stats?
b. When running a model with covariates (even with the OUTPAR option) one can obtain a different solution compared to the baseline model without covariates. It seems that in this case arguing
about the effects of covariates cannot be done with reference to classes identified in a baseline model, but the meaning of classes should be discussed from the model with covariate, right?
c. Other programs (e.g Latent Gold) allow to use covariates as ‘inactive’ so that they do not affect the solution, but they are only used to describe class membership probabilities. Is this
possible in PROC LCA? And how can one judge absolute model fir when df is high (btw, what is the threshold after which we can say that G2 doesn’t follow a chi-square distribution?)
2) Multi-group LCA
a. I thought that adding a variable to define multiple-groups ,imposing measurement invariance, would be mathematically equivalent to adding that variable as covariate. However, from my empirical
tries, I obtain different results.
b. when running a model with multiple groups and imposing measurement invariance, class membership probabilities are given for each group separately. Is it possible to get in the output the class
membership probabilities for the whole sample? | {"url":"http://sas-and-r.blogspot.com/2011/01/example-821-latent-class-analysis.html","timestamp":"2014-04-19T01:49:32Z","content_type":null,"content_length":"179190","record_id":"<urn:uuid:e7cf5585-6f1f-485f-ab1c-0828e00d2396>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00594-ip-10-147-4-33.ec2.internal.warc.gz"} |
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