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Euclidean topology
Euclidean topology
Basic concepts
For $n \in \mathbb{N}$ a natural number, write $\mathbb{R}^n$ for the Cartesian space of dimension $n$. The Euclidean topology is the topology on $\mathbb{R}^n$ characterized by the following
equivalent statements
• it is the metric topology induced from the canonical structure of a metric space on $\mathbb{R}^n$ with distance function given by $d(x,y) = \sqrt{\sum_{i = 1}^n (x_i-y_i)^2}$;
• an open subset is precisely a subset such that contains an open ball around each of its points;
• it is the product topology induced from the standard topology on the real line.
Two Cartesian spaces $\mathbb{R}^k$ and $\mathbb{R}^l$ (with the Euclidean topology) are homeomorphic precisely if $k = l$.
A proof of this statement was an early success of algebraic topology.
Revised on January 15, 2011 04:56:57 by
Toby Bartels | {"url":"http://ncatlab.org/nlab/show/Euclidean+topology","timestamp":"2014-04-18T13:17:09Z","content_type":null,"content_length":"20569","record_id":"<urn:uuid:cbcf1fc5-fd3d-4534-90e3-2a0cd47c295e>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00603-ip-10-147-4-33.ec2.internal.warc.gz"} |
How Do You Find Angle Measures for Adjacent Angles?
Learning how to find missing angle measurements is a very useful skill. In this tutorial, get some practice finding missing angle measurements by first creating an equation. Take a look!
Angles are a fundamental building block for creating all sorts of shapes! In this tutorial, learn about how an angle is formed, how to name an angle, and how an angle is measured. Take a look!
Plugging variables into an expression is essential for solving many algebra problems. See how to plug in variable values by watching this tutorial.
You can't do algebra without working with variables, but variables can be confusing. If you've ever wondered what variables are, then this tutorial is for you!
Combining like terms together is a key part of simplifying mathematical expressions, so check out this tutorial to see how you can easily pick out like terms from an expression
If two angles are complementary, that means that they add up to 90 degrees. This is very useful knowledge if you have a figure with complementary angles and you know the measurement of one of those
angles. In this tutorial, see how to use what you know about complementary angles to find a missing angle measurement! | {"url":"http://www.virtualnerd.com/pre-algebra/geometry/adjacent-angles-example.php","timestamp":"2014-04-17T13:30:45Z","content_type":null,"content_length":"26661","record_id":"<urn:uuid:e49ef029-cc9e-4872-9b3e-e10a3b7dd3da>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Diblock Copolymer Ordering Induced by Patterned Surfaces above the
Order-Disorder Transition
Yoav Tsori and David Andelman*
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences,
Tel Aviv University, 69978 Ramat Aviv, Israel
Received July 14, 2000; Revised Manuscript Received November 27, 2000
ABSTRACT: We investigate the morphology of diblock copolymers in the vicinity of flat, chemically
patterned surfaces. We use a Ginzburg-Landau free energy to describe the spatial variations of the
order parameter in terms of a general two-dimensional surface pattern above the order-disorder
transition. The propagation of several surface patterns into the bulk is investigated. The oscillation period
and decay length of the surface Fourier modes are calculated in terms of system parameters. We show
that two parallel surfaces having simple one-dimensional patterns can induce a complex three-dimensional
copolymer structure between them. Lateral order is observed parallel to a patterned surface as a result
of order perpendicular to the surface. Surfaces which have a finite chemical pattern size (e.g., a stripe of
finite width) induce lamellar ordering extending into the bulk. Close to the surface pattern the lamellae
are strongly perturbed as they try to adjust to the surface pattern.
I. Introduction
The bulk properties of diblock copolymers (BCP) are
by now well understood.1-5 These long linear macro-
molecules composed of two incompatible subchains, or | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/920/2197959.html","timestamp":"2014-04-16T23:42:58Z","content_type":null,"content_length":"8716","record_id":"<urn:uuid:f6b1a2ed-f6af-48e1-af57-5e4a640c2152>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00424-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Summer Severity Index For Goodland, Kansas
Victor J. Nouhan
National Weather Service Office
Goodland, Kansas
Several summer/growing season (May-September) parameters were combined to produce an integrated summer severity index for Goodland, Kansas, so as to observe any trend(s) of severity over the last 75
years. The period of record was from 1920-1994. This study is based upon a similar winter severity index produced for Goodland, Kansas and Pittsburgh, Pennsylvania (Nouhan 1992).
In deriving a total index value for each winter season for the Goodland winter study, each parameter was individually normalized to produce sub-index values with each scaled on a 100-point system.
The mildest occurrence of a parameter was given a near zero value and the most severe a near 100 value. All scaled sub-index values were then combined with their respective weighting factors to
produce a total index value for a given winter season (Nouhan 1992).
A similar approach was used in this project, except parameters related to summer and the growing season, were selected. The chosen parameters with their weighting factors are 1) Summer (S) mean
(June, July, and August temperatures combined) 40 percent, 2) Averaged May/September (MS) mean temperatures for each season 10 percent, 3) Extremes (high temperature) index (E) 25 percent, and 4)
Growing season (May-Sept) rainfall (R) 25 percent. The following points were considered in the selection and weighting of the above parameters: A) Data availability, B) Public impact, and C) Impact
on the agricultural community (Goodland's economy is mainly agriculturally based).
Departures from the normal summer mean temperature directly relate to energy demands due to air-conditioning use. Crop growth and stress factors are also directly related to this parameter.
Averaged May/Sept mean temperatures were included mainly as a longevity factor to the growing season (defined as May 1-Sept 30 for the purpose of this study).
The Extremes index in this study simply incorporates the number of days with maximum temperature > 90 deg F and > 100 deg F for each season. This index is a good proxy to measure human and crop
stress as well as crop water consumption (i.e., irrigation use). Days over 100 deg F were weighted two and a half more times than days over 90 deg F because of much higher heat stress 100+ deg F heat
places on outdoor human activities. Also, since 100+ deg F heat at Goodland normally occurs with low relative humidities, much higher evapotranspiration rates can be expected from crops. For the
Extremes index, 90+ deg F days were assigned 2 points while 100+ deg F days were assigned 5 points each. The total number of points was then tallied for each season. A graduated point scheme based on
interim values between 90 and 110 deg F would have been more desirable, but much more labor intensive, since Local Climate Data from the National Climate Data Center does not list the number of
occurrences for any interim values (i.e., number of days > 95 deg F and etc.).
Dew point and overnight low temperatures were other possible parameters for this study. Goodland, however, rarely experiences dew points and overnight low temperatures >65 deg F due to its high
plains location at 3760 ft above MSL and its close proximity to the Rocky Mountain rain shadow. In fact, high temperatures >95 deg F are normally accompanied by a reduction in dew point to below 60
deg F and occasionally below 50 deg F. Consequently, dew point and overnight low temperatures were not chosen for this location. Growing season rainfall, the last parameter selected for this study,
was exclusively chosen as a summer crop (mainly corn at Goodland) stress proxy.
2. PROCEDURE
First, the mean for each parameter was calculated. Second, the standard deviation (s) for each parameter was computed to show relative variability and skewness.
Since no parameter in this study was exactly normally distributed, a splitsplit standard deviation procedure was employed to include unique aspects of each individual parameter distribution. This
method is a proxy to, and much less rigorous than employing an incomplete gamma distribution (Wilks 1995) on each parameter. In using the split standard deviation technique, the extreme highest and
lowest values of each parameter were assumed to be +3s and -3s respectively (despite actual calculations based on true s values derived in step 2). The difference between the mean and extreme
endpoint values of each parameter (Table 1a and 1b) were then divided by 3, resulting in two approximate s values per parameter, one for above (s+) and the other for below (s-) the mean (end of Table
1b). Endpoints were then assigned Z values (similar to those found in a normal distribution table) as close to +3s and -3s as possible based on the precision of rounding for each respective parameter
unit. The result of this step was to help later calibrate the most extreme endpoints of every parameter as close as possible to sub-index values of zero or 100. This maximizes the range of the
overall summer severity index between opposing extreme seasons.
An important exception to this procedure was the E parameter. The distribution of this particular parameter was skewed far enough to the right (with the left tail somewhat bounded by zero), that if
one were to use the actual s value calculated from step 2, the lowest value of E would only be around 2s below the mean. Consequently, using the s- and s+ values for E, the 1923, 1934 and 1936
seasons were assigned Z values (see step 4) greater than 3.0, but less than 4.0 above (or below) the mean respectively. The assigned Z values for these seasons were approximately based on the
numerical ranking of each respective seasonal E value divided by the total range of E from the entire record. Since the median (Ott 1977) of the E distribution was significantly less than the mean,
values of s for above the mean were assigned increasingly larger increments to reflect each seasons relative position within the distribution.
Next, a statistical "Z" score (Ott 1977) for all data values was derived using the appropriate mean and s for each parameter. The equation used was:
where x represents a value from any of the four parameters and s represents the corresponding s- or s+ for the respective parameter (including incremental s+ values for E). Note, that "Z" values
derived in this step from the split standard deviation approach used in step 3 are close but not the same as theoretical normal curve Z values. Lastly, this pseudo-normal distribution was used in
this study because there was 75 years of data for each parameter. Locations with less than 30 years of data should calculate t values (from Student's t distribution) and then proceed with the rest of
the procedure using t values in place of Z (Ott 1977).
Signs of Z for each parameter were determined by how each contributes toward higher final index values. Positive departures of S, E, and MS are associated with severe summers, so, the sign of Z for
these parameters correctly reflect their contribution. Conversely, negative departures of R contribute to greater summer severity at Goodland. Therefore, the opposite sign of Z for this parameter
must be taken to properly account for a positive contribution toward the final index.
Next, the fraction (Zp) of the area under the normal curve right (+Z) or left (-Z) of the mean for each data point were found. This was done by taking the fractional value listed for each Z from a
normal curve statistics table (Ott 1977). Zp values derived from a negative Z must be preceded by a negative sign to reflect the proper contribution toward the final index.
Next, the individual sub-indices (ix) were computed for each parameter using the following equation, where x represents any of the four parameters:
ix = 2(50+16.5Z)+(50+100Zp) (2)
If Z exceeds (is less than) 3s (-3s) but is less (greater) than or equal to 4s (-4s), then equation 3 must be used:
ix = 2[xx.x+0.5 (Z- or + 3] + (50 + 100Zp)] (3)
> 3
In equation (3) xx.x denotes a sub-index of 99.6 (0.4) and the constant 0.5 is multiplied by Z-3 (Z+3). These equations were formulated too essentially to adjust Zp values closer to the mean (flatten
the normal curve somewhat).
Equation (2) was used for all except for 5 parameter values. The remaining values exceeded +3s, requiring equation 3. Note that both equations linearly smooth the total area under the normal curve
right of each individual Z value. Individual indices could have been computed by multiplying the result of .50 + or - Zp by 100. However, the resulting index values obtained would have varied too
greatly between -1s and +1s, because index values near 1s would be too high and those near -1s values too low (Table 2). In order to expand the rapidly changing portion of the index scale over a
greater range of Z, a linear smoothing term was included in the first term of the numerator in both equations (2) and (3). This smoothing term provided a linear value simply based on an input Z
value. The 16.5 Z constant in equation (2) kept the smoothing term from rising (falling) above 99.5 (0.5) for inputs of < +3s (> -3s) and also kept the smoothing term always less (more) than the
Normal Curve (NC) term (second term of numerator in both equations (2) and (3)) for +Z (-Z) values. The smoothing term in equation (3) simply increments the max (min) sub-index value of 99.6 (0.4)
from equation 2 by 0.1 (-0.1) for each 0.25s above (below) +3.0s (-3.0s).
Inclusion of the smoothing term in equation (2) expanded the range where ix undergoes the most rapid change from 0.0s to +1.0s (-1.0s) to 0.0s to +2.0s (-2.0s) and further linearized ix values up
(down) to +2.5s (-2.5s) compared to straight NC values. A close examination of Tables 3a & b (Table 3a) shows that the smoothed ix values differ the most from NC values between +1.0s and +1.5s and
between -1.0s and -1.5s (i.e., greatest smoothing intervals). Both values are the same at Z=0 and begin to approach each other again beyond +2.5s and -2.5s.
Table 3b
Physical Implications of Qualitative Terms
Very Dismal
Many outdoor activities cancelled or postponed due to cool, rainy weather. Potentially, equal to or greater than 50 percent of all crops affected or not planted with near zero irrigation needs.
Some outdoor activities cancelled or postponed due to cool, rainy weather. Potentially, 25 to 49 percent of all crops affected or not planted with well below normal irrigation needs.
Somewhat cooler and/or rainier than normal, but with little disruption on outdoor activities. Normally, less than 25 percent of crops affected or not planted with below normal irrigation needs.
Overall, what would be considered normal summer/growing season conditions with near normal irrigation needs.
Less than 25 percent of dryland crops affected by heat stress. Little additional water needs for irrigated crops.
Many daytime outdoor activities curtailed due to heat. At least 25 to 49 percent of dryland crops affected by heat stress. Perhaps some additional water needs for irrigated crops.
Very Oppressive
Most daytime outdoor activities curtailed due to heat. At least 50 to 74 percent of dryland crops affected by heat stress. Irrigation systems may not be able to keep up with water demands of
irrigated crops.
Extremely Oppressive
Similar to "Very Oppressive" except generally equal to or greater than 75 percent of dryland and even some irrigated crops likely affected by heat stress.
The term "affected" in this description is defined as causing lower than expected crop yields.^1
^1 Note, percentages were derived after consulting with an HMT on staff at Goodland who also is a local farmer. These percentages are only approximations.
Next, the individual indices were combined with the appropriate weighting factors to compute the total index (T) for each summer season. The equation used for this step was:
T = .40iS + .10iMS + .25iE + .25iR (4)
Lastly, the final index (I) was computed by applying a linear correction, if necessary, to center the mean close to 50.0 and to center the total range of T values equidistant from the 0 and 100 scale
endpoints. This step would likely be required to correct any errors resulting from using the split standard deviation technique. For this particular study, however, errors from each parameter
cancelled each other out, resulting in the mean of all T values very nearly at 50, and the extreme seasons nearly equidistant to the 0 and 100 scale endpoints respectively. Subsequently, no
correction was necessary at this time.
A correction factor may need to be employed in the future at Goodland if this study is extended to include additional seasons which significantly alter the means of parameters or increases the skew
of their distributions. Also, the use of this procedure on data sets from other locations will likely generate a correction factor (Nouhan 1992, 1995).
Given a large sample size (> 30 years) and the subsequent conservative nature of the means, these steps should only need to be re-accomplished every 10 years (i.e., when NCDC publishes new decade
means). Interim years can be calculated using original means and s+/s- values.
3. RESULTS
Individual parameter values (including overall means and s values), final index values, and five year running mean index values (5RM) are given in Table 1. Table 2 gives straight normal curve (NC)
values along with smoothed sub-index values (ix) for 0.5 increment values of Z.
The qualitative summer rating threshold scheme is shown in Table 3a, including the distribution of all seasons within this scheme. Each rating was determined by successive thresholds of standard
deviations (s) above and below the combined parameter mean (I). For instance, if I was within + 0.4s of the mean (0s, I=50), an individual summer would be considered "average". Physical implications
of the qualitative terms are given in Table 3b.
Decadal means are listed in Table 4. Tables 5a, 5b, 5c, and 5d gives the 10 most oppressive and dismal summers by final index, the 10 coolest and warmest summers, the 10 wettest and driest growing
seasons (May- September), and the ten wettest and driest summers (June-August). Lastly, Figure 1 shows the graphs of the final index vs. the five year running mean.
The five year running mean (5RM) data (centered on the current year) in Figure 1 indicates that summer severity at Goodland was in the "average" category during the 1920s reaching a low of 39.5
during 1928. Subsequently, the 5RM index rose dramatically through the "oppressive" category during the early 1930s and into the "very oppressive" category during the mid 1930s as indicated by an
amazing record maximum value of 83.2 in the summer of 1936. After remaining in the "very oppressive" category through 1938, the 5RM decreased sharply through 1939 and the early 1940s reaching 38.0
(moderate category) in 1943. The 5RM then remained in the normal category through 1948 only to reach another minimum of 34.4 in 1949. Afterwards, the index suddenly spiked back into the "oppressive"
category during the early 1950s as indicated by a maximum value of 77.1 in 1954. Subsequently, the 5RM has remained in the "average" or "moderate" categories reaching low values of 33.3 in 1973 and a
record low value of 28.8 (nearly into the dismal category) in 1991.
Figure 1. Final index vs five year running mean.
4. DISCUSSION
Two periods of oppressive summer severity are highlighted by this study. The first period, extending from 1931 through 1939; often referred to as the "dust bowl" by the local populace, is one of the
best examples of U.S. regional climate fluctuation this century. The second period extended from 1952 through 1956 and was shorter and not quite as severe. Outside of these two periods, with the
exception of the 1963 and 1964 seasons, no consecutive oppressive summers could be documented from the Goodland record.
When anomalous mid-tropospheric ridging occurs through summer over the Rockies and high Plains, greater sunshine, lower relative humidities, and above normal surface temperatures result over the
Goodland area (Namias 1982). This is especially true when below normal antecedent ground moisture is carried over from the previous spring (Namias 1982, Erikson 1983). Conversely, anomalous troughing
over the Rockies and high plains is associated with cooler than normal temperatures and above normal rainfall at Goodland.
The occurrence of many oppressive severe summers in the 1930s and to a lesser extent in the 1950s may be linked with the re-occurrence of certain patterns of Pacific mid-latitude and tropical sea
surface temperature anomalies. Investigation in this area is beyond the scope of his study, but is worthy of further research (assuming reliable sea surface temperature data exists for these periods)
by those who can employ more powerful statistical techniques.
5. SUMMARY
Several summer/growing season parameters were normalized (approximately) and transformed into sub-index values for each summer season for a 75-year climate series at Goodland, Kansas. The sub-indices
were then linearly combined to calculate a total index for each summer season based on a weighting scheme that considers the public and agricultural impact of each parameter. Lastly, final corrected
indices were qualitatively rated from very dismal too extremely oppressive based on the number of standard deviations the average of combined parameters was from the mean.
Five year running mean data generated from the final indices showed that summer severity conservatively averaged "very oppressive" during the mid and late 1930s and "oppressive" during the mid 1950s.
Outside these periods, the five-year running mean remained in the average and moderate categories.
I would like to thank Roy Freiburger, HMT, John Kwiatkowski, Goodland SOO, for the general review and Preston Leftwich, Central Region Science Officer, for the statistical review of the paper.
7. REFERENCES
Erikson, C., 1983: Hemispheric Anomalies of 700 mb Height and Sea Level Pressure Related to Mean Summer Temperatures over the United States. Mon. Wea. Rev., 112 545-561.
Namias, J., 1982: Anatomy of Great Plains Protracted Heat Waves. Mon.Wea. Rev., 110, 824-838.
Nouhan,V., 1992: A Winter Severity Index for Pittsburgh, Pennsylvania. DOC. NOAA, NWS, Winter Weather Conference, Portland, 330-344.
Ott, L., 1977: Introduction to Statistical Methods and Data Analysis, Wadsworth, Belmont, California, 658pp.
Wilks, D.S., 1995: Statistical Methods in the Atmospheric Sciences, Academic Press, New York, 467pp. | {"url":"http://www.crh.noaa.gov/crh/?n=arp18-03","timestamp":"2014-04-20T15:04:43Z","content_type":null,"content_length":"27175","record_id":"<urn:uuid:0fd9a40a-0f2c-42bc-bb3b-2d1445d8b590>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00137-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pi Day
From Wikipedia, the free encyclopedia
Pi Day
Significance 3, 1, and 4 are the three most significant figures of π
Celebrations Pie eating, discussions about π^1
Date March 14
Next time 14 March 2015
Frequency annual
Related to Pi Approximation Day
Pi Day is an annual American celebration commemorating the mathematical constant π (pi). Pi Day is observed on March 14 (or 3/14 in the month/day date format), since 3, 1, and 4 are the first three
significant digits of π in decimal form. In 2009, the United States House of Representatives supported the designation of Pi Day.^2
Pi Approximation Day is observed on July 22 (or 22/7 in the day/month date format), since the fraction ^22⁄[7] is a common approximation of π.^3
The earliest known official or large-scale celebration of Pi Day was organized by Larry Shaw in 1988 at the San Francisco Exploratorium,^4 where Shaw worked as a physicist,^5 with staff and public
marching around one of its circular spaces, then consuming fruit pies.^6 The Exploratorium continues to hold Pi Day celebrations.^7
On March 12, 2009, the U.S. House of Representatives passed a non-binding resolution (HRES 224),^2 recognizing March 14, 2009 as National Pi Day.^8
For Pi Day 2010, Google presented a Google Doodle celebrating the holiday, with the word Google laid over images of circles and pi symbols.^9
The entire month of March 2014 (3/14) has been observed by some to be "Pi Month".^10^11
In the year 2015, Pi Day will have special significance on 3/14/15 at 9:26:53 a.m. and p.m., with the date and time representing the first 10 digits of π.^12
Pi Day has been observed in many ways, including eating pie, throwing pies and discussing the significance of the number π.^1 Some schools hold competitions as to which student can recall Pi to the
highest number of decimal places.^13^14
The Massachusetts Institute of Technology (MIT) has often mailed its application decision letters to prospective students for delivery on Pi Day.^15 Starting in 2012, MIT has announced it will post
those decisions (privately) online on Pi Day at exactly 6:28 pm, which they have called "Tau Time", to honor the rival numbers Pi and Tau equally.^16^17
The town of Princeton, New Jersey, hosts numerous events in a combined celebration of Pi Day and Albert Einstein's birthday, which is also March 14.^18 Einstein lived in Princeton for more than
twenty years while working at the Institute for Advanced Study. In addition to pie eating and recitation contests, there is an annual Einstein look-alike contest.^19
See also
External links
Wikimedia Commons has media related to Pi Day.
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• Saint Patrick's Day
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• Good Friday (CT, NC, PR, Religious, sometimes in April)
• Holi (Religious, CA, NY)
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• Election Day (CA, DE, HI, KY, MT, NJ, NY, OH, PR, WV)
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November • Native American Heritage Day (MD)
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• Hanukkah (Religious)
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• Native American Indian Heritage Month (Month)
(Federal) = Federal holidays, (State) = State holidays, (Religious) = Religious holidays, (Week) = Weeklong holidays, (Month) = Monthlong holidays, (36) = Title 36 Observances and Ceremonies
Bolded text indicates major holidays that are commonly celebrated by Americans, which often represents the major celebration of the month.[1][2]
See also:
Lists of holidays
Hallmark holidays
, public holidays in the
United States
Puerto Rico
United States Virgin Islands
New Jersey
New York | {"url":"http://www.territorioscuola.com/wikipedia/en.wikipedia.php?title=Pi_Day","timestamp":"2014-04-18T06:55:57Z","content_type":null,"content_length":"121546","record_id":"<urn:uuid:66948a1c-ed9e-4e76-bb5d-97892d3219eb>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00310-ip-10-147-4-33.ec2.internal.warc.gz"} |
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English definition of “angle”
noun [C] (SPACE BETWEEN LINES) /ˈæŋ.ɡl̩/
C1 the space between two lines or surfaces at the point at which they touch each other, measured in degrees: The interior angles of a square are right angles or angles of 90 degrees. The boat settled
into the mud at a 35° angle/at an angle of 35°.Parts of geometrical shapes at an angle C1 not horizontal or vertical, but sloping in one direction: The picture was hanging at an angle. He wore his
hat at a jaunty angle.Parts of geometrical shapes › the corner of a building, table, or anything with straight sidesParts of geometrical shapesEdges and extremities of objectsSurfaces of objects
Focus on the pronunciation of angle | {"url":"http://dictionary.cambridge.org/dictionary/british/angle_1?topic=parts-of-geometrical-shapes","timestamp":"2014-04-18T10:57:02Z","content_type":null,"content_length":"78849","record_id":"<urn:uuid:a21a3964-67da-416e-858a-9f9ce9ecbf99>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00383-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Posts by liv
Total # Posts: 25
chemistry PLEASE HELP
How many milligrams of sodium azide will produce 8.35 Liters of N2 at STP???
The cup becomes positively charged and I know this 100% the right answer.
1 ml of whole blood is collected and centrifuged to obtain a plasma sample. The volume of the plasma sample is 600 microlitres. A 1 in 4 dilution of the plasma is made and analysed using a
cholesterol assay like the one you performed in the laboratory. 10 microlitres of both t...
1 ml of whole blood is collected and centrifuged to obtain a plasma sample. The volume of the plasma sample is 600 microlitres. A 1 in 4 dilution of the plasma is made and analysed using a
cholesterol assay like the one you performed in the laboratory. 10 microlitres of both t...
the phone company has two plans for cell phones. the first plan calls for a monthly fee of 35.00 with unlimited calling. The other plan is a monthly fee of 15.00, with a 0.50 fee for every minute
over 100 used in a month. What is the greatest number of minutes that you can use...
first do what is in perenthises then do exponints then multiply then divide then add then subtract. you can remember this by doing p e m d a s pemdas!!!!!!!!!!
you should reasearch some real interior designers and look up all of your questions
6/?=18/?=?/12 (fractions)
the answer is letter c think of food food is chemical energy
40 multiplyed by 12 equalls area
math help needed
thanks CON
math help needed
what is 5 percent of 40
plus wouldnt the answer be 36 because there both square and there distance isnt far
whats a ratino and then ill help because i have no idea what that means
i know but thanks anyway
What is 5 percent of 40
you dot need a calculator unless you memorized the area and you dont need tape unless its used to hold down string
A triangular plate with a non-uniform areal density has a mass M=0.700 kg. It is suspended by a pivot at P and can oscillate as indicated below. Its center of mass is a distance d=0.200 m from the
pivot axis and its moment of inertia about an axis through the CM and parallel t...
5,25,50,10 what are the next 3 numbers?
How many grams of KOH will be needed to boil a liter of cold water?
Point P lies in plane M. How many circles are there in plane M that have center P and a circumference of 6π inches? a) none b) one c) two d) four e) more than four
AP World History
I am analyzing the same DBQ, is this a good thesis? Both Han China and the Roman Empire were highly advanced in the development of technology, and both empires had very practical, organized and
precise mechanized skills, yet the Romans valued their technology for its beauty an...
What would the answer be to 4m+(-3n)+n? -3n squared +4m 4m+(-3n)+n = 4m + n(-3 +1) = 4m +(-2n) = 2(2m -n) | {"url":"http://www.jiskha.com/members/profile/posts.cgi?name=liv","timestamp":"2014-04-19T21:04:56Z","content_type":null,"content_length":"10065","record_id":"<urn:uuid:03643f4f-6a63-48fd-86f3-94f17049bccd>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00100-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Dekkers Algorithm
Dekker's algorithm solves the mutual exclusion defined, does not specify which of the suspended What we defined earlier is a general semaphore.
Dekkers Algorithm is described in multiple online sources, as addition to our editors' articles, see section below for printable documents, Dekkers Algorithm books and related discussion.
Suggested Pdf Resources
Suggested Web Resources
Great care has been taken to prepare the information on this page. Elements of the content come from factual and lexical knowledge databases, realmagick.com library and third-party sources. We
appreciate your suggestions and comments on further improvements of the site. | {"url":"http://www.realmagick.com/dekkers-algorithm/","timestamp":"2014-04-18T20:52:22Z","content_type":null,"content_length":"20023","record_id":"<urn:uuid:ad163dc1-32c2-4289-9019-ca7740466dd1>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00120-ip-10-147-4-33.ec2.internal.warc.gz"} |
Do we need to deal with ‘big data’ in R?
November 22, 2011
By Luis
David Smith at the Revolutions blog posted a nice presentation on “big data” (oh, how I dislike that term). It is a nice piece of work and the Revolution guys manage to process a large amount of
records, starting with a download of 70GB and ending up with a series of linear regressions.
I’ve spent the last two weeks traveling and finishing marking for the semester, which has somewhat affected my perception on dealing with large amounts of data. The thing is that dealing with hotel
internet caps (100MB) or even with my lowly home connection monthly cap (5GB) does get one thinking… Would I spend several months of internet connection just downloading data so I could graph and
plot some regression lines for 110 data points? Or does it make sense to run a linear regression with two predictors using 100 million records?
My basic question is why would I want to deal with all those 100 million records directly in R? Wouldn’t it make much more sense to reduce the data to a meaningful size using the original database,
up there in the cloud, and download the reduced version to continue an in-depth analysis? There are packages to query external databases (ROracle, RMySQL, RODBC, …, pick your poison), we can sample
to explore the dataset, etc.
We can deal with a rather large dataset in our laptop but is it the best that we can do to deal with the underlying modeling problem? Just wondering.
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FOM: infinity of the universe...
Allen Hazen a.hazen at philosophy.unimelb.edu.au
Sat Nov 4 07:59:21 EST 2000
A propos Holmes's post on the infinity of the universe.
Suppose the universe IS infinite in extent (for every natural number
there is a region of space of nore than that number of cubic light-years in
volume). It doesn't automatically follow that interesting mathematical
assertions, such as Goldbach's conjecture, have physical interpretations:
for that we need not only lots of physical entities (to take the place of
the numbers) but also physical RELATIONS to interpret the mathematical
predicates (& function-expressions), and it isn't obvious to me that these
will be available.
Simple example. Suppose that the infinite physical universe is like the
American Mid-west: a vast and boring plain stretching out farther than the
eye can see. And suppose the objects chosen to represent the numbers are
the (indistinguishable except by their position) fence-posts in a fence
that stretches off to infinity. Now, one of the mathematical predicates we
will want to interpret is DIVIDES. This is a relation relating any number
to others arbitrarily much larger, and (if the first number is >1) not
relating it to their immediate successors. So we want a physical relation
that relates a given fencepost DIFFERENTIALLY (YES to this one, NO to that)
to posts arbitrarily far away. Without cheating and defining the relation
in MATHEMATICAL terms, and thinking of PHYSICAL relations as those whose
holding or failing to hold between two objects can, at least in principle,
be ascertained by observational and experimental techniques, are there
likely to be such relations? I have an ugly feeling that, if the physics
of my imagined hyper-Iowa is at all like that of the real world, that for
any specified PHYSICAL relation there will be a distance such that it will
relate a given post to either all or to none of the posts more than this
distance past it. And similarly for physical relations in the perhaps
spatially (& temporally) infinite real world. But this is just a suspicion
on my part.
(Mathematical-- model-theoretic, to be precise-- point strengthening my
suspicion: there are infinite structures in whose First-Order theories
First Order arithmetic cannot be interpreted. I see no A PRIORI reason for
confidence that the "structure" having the set of physical "objects" as its
domain and "physical" relations as its relations isn't one of them.)
Moral. I don't feel attracted to the sort of ultra-finitism and
anti-platonism Kanovei and Sazonov have been arguing for, but I don't think
it is easy to use physical cosmology to establish the meaningfulness
(independent of proof) of infinitistic mathematical assertions.
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/2000-November/004531.html","timestamp":"2014-04-20T03:11:18Z","content_type":null,"content_length":"4963","record_id":"<urn:uuid:455549c9-f05b-46ce-9e6d-056c220903b8>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00658-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sine: Series representations
Series representations
Generalized power series
Expansions at z==z[0]
For the function itself
Expansions at z==0
For the function itself
For powers of the function
For the second power
For the third power
For symbolical integer power
For symbolical power
Expansions at z==Pi/2
For the function itself
For powers of the function
For the second power
For the third power
For symbolical integer power
Exponential Fourier series
Asymptotic series expansions
Other series representations
Residue representations
Dual Taylor series representations | {"url":"http://functions.wolfram.com/ElementaryFunctions/Sin/06/ShowAll.html","timestamp":"2014-04-18T13:06:46Z","content_type":null,"content_length":"75968","record_id":"<urn:uuid:5de48af2-c7e7-40a5-b426-a4c2ef31e526>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00523-ip-10-147-4-33.ec2.internal.warc.gz"} |
Stripping again [Archive] - OpenGL Discussion and Help Forums
06-03-2003, 02:41 AM
I procedurally generate a variable level of detail geo-sphere with number of triangles ranging from 8 to 8192, and output an array of vertices, normals, and the list of CCW triangle indices.
Using NVTriStrip: "GenerateStrips" (with strip stitching set to false) on my 8192 triangle (4098 vert) sphere, I get out the following -
Num Strips: 490. Max num. strip indices: 26. Min num. strip indices: 3. Avg num. strip indices: 19.
Num Strips: 844. Max num. strip indices: 18. Min num. strip indices: 3. Avg num. strip indices: 12.
What I'd like to know is-
Why are there a number of strips with only 3 indices?
Why aren't there any long strips on what I'd have thought would be a mesh suited to really long strips?
My large list of triangle indices for my geo-sphere is ordered in such a way that each consecutive triangle index group of three in the list will never/rarely be next to each other on the actual
sphere itself, so does the order/coherence of the triangle indices passed in to the stripping function affect the result?
Perhaps someone could also explain to me the exact details relating to the cache size and how this affects the resultant strips. | {"url":"http://www.opengl.org/discussion_boards/archive/index.php/t-131427.html","timestamp":"2014-04-21T09:49:46Z","content_type":null,"content_length":"4932","record_id":"<urn:uuid:523599e6-8f9f-4502-a7cb-1e97874e8171>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00225-ip-10-147-4-33.ec2.internal.warc.gz"} |
Use of Calculators on the CAAP Mathematics Test
You may use a calculator on the CAAP Mathematics Test but not on any of the other CAAP tests. You are not required to use a calculator. All problems on the Mathematics Test can be solved without a
WARNING: You are responsible for knowing if your calculator is permitted. If testing staff find that you are using a prohibited calculator or are using a calculator on any test other than the
Mathematics Test, you will be dismissed and your answer document will not be scored. If ACT determines later that you used a prohibited calculator or that you used a calculator on a test other than
the Mathematics Test, your scores will be cancelled.
Using the TI-89 is the most common reason students are dismissed from the ACT for prohibited calculator use.
If you choose to use a calculator, you are responsible for bringing it to the test center and making sure it works properly. Testing staff will not provide backup calculators or batteries. You may
not share a calculator with another examinee. You may bring a backup calculator, but you may have only one calculator on your desk or in operation at a time. If you need to use your backup
calculator, it must first be checked by a member of the testing staff.
You may use your calculator only while you are working on the Mathematics Test. At all other times, it must be turned off and put away. You may use only the mathematics functions of your
calculator—if your calculator has other functions (such as games) you may not use those functions. If you finish the Mathematics Test before time is called, and have rechecked your work on that test,
you must turn your calculator off and wait quietly.
Permitted Calculators
You may use any four-function, scientific, or graphing calculator, unless it has features described in the Prohibited Calculators list. For models on the Calculators Permitted with Modification list,
you will be required to modify some of the calculator's features.
Prohibited Calculators
The following types of calculators are prohibited:
• calculators with built-in computer algebra systems Prohibited calculators in this category include:
□ Texas Instruments: All model numbers that begin with TI-89 or TI-92 and the TI-Nspire CAS—Note: The TI-Nspire (non-CAS) is permitted.
□ Hewlett-Packard: HP 48GII and all model numbers that begin with HP 40G, HP 49G, or HP 50G
□ Casio: Algebra fx 2.0, ClassPad 300 and ClassPad 330, and all model numbers that begin with CFX-9970G
• handheld, tablet, or laptop computers, including PDAs
• electronic writing pads or pen-input devices—Note: The Sharp EL 9600 is permitted.
• calculators built into cell phones or any other electronic communication devices
• calculators with a typewriter keypad (letter keys in QWERTY format)—Note: Letter keys not in QWERTY format are permitted.
Calculators Permitted with Modification
The following types of calculators are permitted, but only after they are modified as noted:
• calculators with paper tape—Remove the tape.
• calculators that make noise—Turn off the sound.
• calculators with an infrared data port—Completely cover the infrared data port with heavy opaque material such as duct tape or electrician's tape (includes Hewlett-Packard HP 38G series, HP 39G
series, and HP 48G).
• calculators that have power cords—Remove all power/electrical cords. | {"url":"http://www.act.org/caap/sample/calculators.html","timestamp":"2014-04-16T16:03:41Z","content_type":null,"content_length":"41043","record_id":"<urn:uuid:3c1f911b-bd77-499e-914e-29c12ee7f940>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00314-ip-10-147-4-33.ec2.internal.warc.gz"} |
Electronics 101 - Fundamentals of Electricity - Lesson 31 - ELI the ICE man
ELI the ICE man
In a previous lesson, we covered the fact that two alternating currents can be either in phase, or out of phase with respect to each other. We also discussed the addition of two sine waves of
differing phase by using VECTOR ADDITION. I am fairly certain that you were hoping you would never see this again. Sorry, but you were SO wrong. We are soon going to get into the practical
applications of vector addition. You are about to learn that in electronics, the capacitor and the inductor are exact opposites. The reason for this is because they BOTH store electricity, but in
different ways. In a purely resistive circuit, there is no change in the phase from one component to another. When we add an inductor or capacitor into the circuit, however, the game changes
completely, and the rules to the game are written with vectoral math.
[L] ? ) and a resistor we would have to plot a graph like the one above and to the left. This combined resistance is called IMPEDANCE, which is the TOTAL RESISTANCE TO THE FLOW of current. Note that
Impedance is the TOTAL resistance to the flow, which includes "pure resistance" (from resistors), capacitive reactance, inductive reactance. The symbol for impedance is Z.
If you have ever studied trigenometry, or even basic geometry, you may recall the formula for finding the hypotenuse of a right triangle ( A^2+B^2=C^2). This will come in handy, as you compare it to
the formula for impedance:
This can be re-written as
^2 would be 3^2 which equals 9. X[L]^2 would be 4^2 which would be 16. 9+16=25. The square root of 25 = 5, so the impedance of the circuit would be Z=5. Sometimes we might say that the "complex
representation" of Z = R+Xj. In this case it would be 3+4j. This comes in handy as we begin adding capacitors into the circuit. Capacitors are like the opposite of inductors in a circuit. Whereas
inductors are added ( Z = R + Xj ).... capacitors are subtracted (Z = R - Xj ). I know this all sounds confusing, but it will become clear as mud shortly.
Recall the formula for Inductive Reactance?
X[L] = 2πfL
How could you forget? Well, CAPACITIVE REACTANCE is its opposite, and should also be memorized. Ready for this one?
X[C] = ----------------
WOW! It's almost the same formula! The only difference is that we substituted the L's for C's, and we reciprocated the formula (divided 1 by the formula). In the great scheme of things, that makes
this formula not too difficult to remember, assuming you did memorize the formula for inductive reactance when I told you to. If you didn't, take time now to memorize both formulas. Your survival in
electronics depends on them. Notice that I have flashed lots of formulas by you, but I have only asked you to memorize 3 of them... Ohm's Law, and the formula's for inductive and capacitive
reactance. That is because you will use them over, and over again.
^2 would be 3^2 which equals 9. X[C]^2 would be 4^2 which would still be 16. 9+16=25. The square root of 25 = 5, so the impedance of the circuit would once again be Z=5.
But there is a catch - this time, because the circuit is CAPACITIVE, we would have a complex representation of impedance being equal to 3 - 4j. What exactly does this mean? It means that instead
of plotting our graph in the POSITIVE direction along the Y axis of our graph, we would plot it in the NEGATIVE direction. Instead of our plotted point being (3,4) it would be located at (3,-4).
I realize, of course, this is a lot of math to remember, but unless you are designing radio frequency, or other resonant circuits, you probably won't be using these formulas on a daily basis. You
should be familiar with them though, and you SHOULD memorize the formulas I have pointed out thus far.
One important point to keep in mind, is that when current flows through a purely resistive circuit, the voltage and current arrive at the same point at the same time. In other words, Voltage and
Current are in phase in a purely resistive circuit. In a circuit which contains inductance or capacitance though this is not so. In an inductive circuit, the voltage leads the current by 90 degrees
(assuming a purely inductive circuit). Likewise, in a capacitive circuit, the current leads the voltage by 90 degrees. Which leads which is easy to remember. Just think "Eli the Ice man".
E=Voltage I=Current... L=Inductor......C=Capacitor
• ELI Inductive circuit...... Voltage arrives before Current .
• ICE Capacitive circuit... Current arrives before Voltage. | {"url":"http://www.electronicstheory.com/html/e101-31.htm","timestamp":"2014-04-19T02:06:01Z","content_type":null,"content_length":"22305","record_id":"<urn:uuid:931c96fd-8198-4dfe-ab7b-dd20b3240963>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00461-ip-10-147-4-33.ec2.internal.warc.gz"} |
Arbitrage Pricing Theory
November 15th 2010, 12:34 AM #1
Junior Member
Oct 2010
Arbitrage Pricing Theory
I have a question about Arbitrage Pricing Theory that I cannot solve or understand.
The formula I am using is
R(i) = (1 - b1 - b2)R(f) + b1(F1) + b2(F2)
The question I am asked says:
Explain how to make a risk-free return of 3.5% if R(i) = 4%.
I am confused by the wording of the question. I also have figures for the b1,b2,F1,F2 from the previous part of the question but I don't know if they are needed here. I will supply them anyway.
b1 = b2 = 1/4, R(f) = 1%, F1 = 2%, F3 = 3%
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Is {Ø,{Ø},{Ø,{Ø}}, ... } the only known universe?
up vote 13 down vote favorite
In the first pages of SGA4 I read
[...] Cependant le seul univers connu est l'ensemble des symboles du type {Ø,{Ø},{Ø,{Ø}}, ... } etc. (tous les éléments de cet univers sont des ensembles finis et cet univers est dénombrable). En
particulier, on ne connaît pas d'univers qui contienne un élément de cardinal infini. [...]
(the sole known universe is like {Ø,{Ø},{Ø,{Ø}}, ... }, and we don't know any universe with a infinite cardinal).
Mais, c'est vrai? I wonder if during all these years somebody discovered a universe "bigger" than that exhibited by Grothendieck.
ag.algebraic-geometry set-theory lo.logic
4 I bring Grothendieck into this because the quotation I wrote comes from SGA... I'm only asking if, from 1963 to now, someone found out a universe different from {Ø,{Ø},{Ø,{Ø}}, ... }. –
tetrapharmakon Jul 31 '10 at 10:02
3 The question looks very interesting to me. I don't see Ryan Budney's point. – Pierre-Yves Gaillard Jul 31 '10 at 10:24
4 This question is very vague. What do you mean by "discover"? I think it is likely that you're taking Grothendieck's quotation a bit too literally. Sure there are other universes, for example a
model of ZFC. Or the initial topos. Or the effective topos. OR a model of ZFC + measurable cardinals. And there are permutation models. And so on. – Andrej Bauer Jul 31 '10 at 10:26
2 You should understand Grothendieck as saying "I am not sure infinite sets actually exist". This of course is a matter of opinion, but most mathematicians don't have a problem with the existence of
the set of natural numbers. – Andrej Bauer Jul 31 '10 at 10:32
I don't understand where all these querulous comments are coming from. Like PYG, I think the question is perfectly clear. And indeed, it seems to be understood and answered correctly below (modulo
8 some trivial quibbling about the empty set). But also the sentence "I assume you are referring to Grothendieck universes" in the answer seems to suggest that there is some doubt in the matter --
but the OP refers to a specific passage from SGAIV and this pasage is talking about [what are now called] Grothendieck universes. So what's the problem? – Pete L. Clark Jul 31 '10 at 11:45
show 9 more comments
5 Answers
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The universe that Grothendieck intends to suggest by his notation is known in set theory as HF, the class of hereditarily finite sets, the sets that are finite and have all elements
finite and elements-of-elements, and so on (the transitive closure should be finite). The set HF is the same as $V_\omega$ in the Levy hiearchy, and can be built by starting with the
emptyset and iteratively computing the power set, collecting everything together that is produced at any finite stage. This is the smallest nonempty transitive set that is closed under
power set. It satisfies all the Grothendieck universe axioms, except that it doesn't have any infinite elements, since none appear at any finite stage of this consrtruction.
There is an interesting presentation of this universe by a simple relation on the natural numbers. Namely, define $n\ E\ m$ if the $n^{\rm th}$ bit in the binary expansion of $m$ is $1$.
The structure $\langle\mathbb{N},E\rangle$ is isomorphic to $\langle HF,{\in}\rangle$ by the map $\pi(n)=\{\pi(m)\,|\,m\,E\,n\}$, which set-theorists will recognize as the Mostowski
collapse of $E$.
up vote 17
down vote Since HF doesn't have any infinite elements, it is a rather impoverished universe for many applications of that concept. And so we naturally seek larger universes. But the difficulty is
accepted that we cannot prove they exist. The difficulty is not one of "discovery," but rather just that we can prove that the hypothesis of the existence of a univese containing infinite sets is
too strong for us to prove from our usual axioms. The reason is, as has been remarked in some of the other answers and comments, all other Grothendieck universes have the form $H_\kappa$,
the hereditarily size less than $\kappa$ sets, for an inaccessible cardinal $\kappa$. So this is just like HF, which is $H_\omega$, but on a higher level, and in this sense, these higher
universes are not so mysterious. They are intensely studied in set theory, a part of the research effort in large cardinals.
In this MO answer, I mention a number of weaker universe concepts that we can prove exist, and which I believe serve most of the uses of the universe concept in category theory, if one
wanted to care more about such set theoretic issues.
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Let me rephrase part of what Joel David Hamkins and Anon already said, but without mentioning inaccessible cardinals:
A Grothendieck universe strictly bigger than the one in the question would be a model of ZFC. (More precisely, it would become a model once we interpret the membership symbol of ZFC as
up vote 13 actual membership.) So the existence of such a Grothendieck universe would imply the consistency of ZFC. G"oedel's second incompleteness theorem implies that ZFC cannot prove the
down vote consistency of ZFC. Therefore ZFC cannot prove the existence of a Grothendieck universe.
Clear and concise, I understood what was the "problem". Thanks. – tetrapharmakon Jul 31 '10 at 22:17
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I assume you are referring to Grothendieck universes.
The existence of a bigger Grothendieck universe is equivalent to the existence of an inaccessible cardinal, which cannot be proved from ZFC, because it implies the consistency of ZFC.
up vote 6
down vote There is a smaller example of a Grothendieck universe: the empty set. This is the only other Grothendieck universe that can be proven to exist in ZFC.
1 A universe is nonempty by definition. – Pierre-Yves Gaillard Jul 31 '10 at 11:00
1 Thank you. I think it should be noticed that in SGA4 an universe is defined nonempty, but in the sequent Appendix (redige' par N. Bourbaki, pages 185--...) the empty set is accepted
as a universe... It seems like a problem, isn't it?! – tetrapharmakon Jul 31 '10 at 12:32
1 Dear tetrapharmakon: I just looked at the article you took the quotation from. I'd have been surprised if Grothendieck and Verdier had overlooked the empty set. I don't know why
Grothendieck and Verdier excluded the empty set, nor why Bourbaki included it. – Pierre-Yves Gaillard Jul 31 '10 at 12:50
Thanks for your answer. I don't know why Grothendieck and Verdier excluded the empty set, nor why Bourbaki included it. Excuse me but I can't understand what is your point: what
"philosophical" position shall I adopt about the empty set? Must I follow Grothendieck&Verdier or Bourbaki? – tetrapharmakon Jul 31 '10 at 17:56
3 Dear tetrapharmakon: Is it better to include the non-emptiness condition into the definition of a universe, or not to include it? I don't have the slightest idea. I don't think it's
an important question. – Pierre-Yves Gaillard Jul 31 '10 at 19:27
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Just to make perfectly sure: Grothendieck is absolutely not questioning the existence of infinite sets in this quotation. (He had, and has, some eccentricities, but not in this direction!)
Remember that "universe" is a technical term for a certain type of set, essentially one which has maximally nice closure properties. The universe he is talking about corresponds to the
up vote cardinal $\aleph_0$, a countably infinite set whose elements may themselves be identified with the finite cardinals (as is a standard operating procedure since von Neumann, although those who
4 down don't think that much about infinite sets can and often do safely forget this point). He is not discussing universes as models of formal set theory or anything like that, so the idea that
vote "internally" in this countably infinite universe, infinite sets do not exist, is not at all what he is getting at. Rather, since he has written down an example of an infinite set, we can
conclude (from this passage alone, notwithstanding the rest of his work) that he believes in and is comfortable working with infinite sets.
add comment
There exist plenty of other universe. Recall that most of the times one proves the (relative) consistency of some axiom independent from ZF, one actually builds a model that satisfy that
axiom. So for instance, the method of forcing invented by Cohen enables to list a infinite number of (elementary) different universes. Of course the universe you mentioned is much more
up vote -1 tangible and intuitive then the one built with forcing, but if you accept AC, then they have the same dignity.
down vote
Once again, I believe you are confusing the notion of universe studied in set theory with the (strictly narrower) notion of universe Grothendieck is discussing here, since called
Grothendieck universes. See en.wikipedia.org/wiki/Grothendieck_universe and en.wikipedia.org/wiki/Universe_(mathematics). – Pete L. Clark Jul 31 '10 at 13:13
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James Clerk Maxwell: A Treatise on Electricity and Magnetism
20 Quaternion Equations
James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field" ( Royal Society Transactions, Vol. CLV, 1865, p 459 ); Orally read Dec. 8, 1864. [ MS-Word.doc ]
Andre Waser : On the Notation of Maxwell's Field Equations [ PDF ]
The 1873 edition of A Treatise on Electricity & Magnetism contains the 20 Quaternion Equations that later were rewritten --- censored --- by Oliver Heaviside, et al.. These equations reconcile
relativity with modern quantum physics and help to explain "free energy" and anti-gravity.
Table of Contents:
Volume 1: 1 ~ 2 ~ 3 ~ 4 ~ 5 ~ 6 ~ 7 ~ 8 ~9 ~ 10 ~ 11 ~ 12 ~ 13
Volume 2: 1 ~ 2 ~ 3 ~ 4 ~ 5 ~ 6 ~ 7 ~ 8 ~ 9 ~ 10 ~ 11 ~ 12 ~ 13 ~ 14 ~ 15 ~ 16 ~ 17 ~ 18 ~ 19
Links to the complete copies in the Posner Collection at Carnegie Mellon University:
The complete copies also are included on Rex Research website CD.
Maxwell's Quaternion Equations
Col. Tom Bearden
Maxwell's original theory was published as:
James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Royal Society Transactions, Vol. CLV, 1865, p 459. The paper was orally read Dec. 8, 1864.
[ MS-Word.doc ]
It is also published in The Scientific Papers of James Clerk Maxwell, 2 vols. bound as one, edited by W. D. Niven, Dover, New York, 1952, Vol. 1, p. 526-597. Two errata are given on the
unnumbered page prior to page 1 of Vol. 1.
In this paper Maxwell presented his seminal theory of electromagnetism, containing 20 equations in 20 unknowns. His equations of the electromagnetic field are given in Part III, General Equations
of the Electromagnetic Field, p. 554-564. On p. 561, he lists his 20 variables. On p. 562, he summarizes the different subjects of the 20 equations, being three equations each for magnetic force,
electric currents, electromotive force, electric elasticity, electric resistance, total currents; and one equation each for free electricity and continuity. In the paper, Maxwell adopts the
approach of first arriving at the laws of induction and then deducing the mechanical attractions and repulsions.
A copy of the original Maxwell paper can easily be obtained for about $15 from Amazon etc. It is:
James Clerk Maxwell, The Dynamical Theory of the Electromagnetic Field, edited by Thomas F. Torrance, Wipf and Stock Publishers, Eugene, Oregon, 1996. This booklet, which sells for about $15,
contains Maxwell's original 1865 dynamical theory paper and some additional commentaries.
Here's what Barrett --- a nationally known electrodynamicist and one of the co-founders of ultrawideband radar --- has to say about Maxwell's theory:
"In the case of electromagnetism, the theory was first simplified before being frozen. Maxwell expressed electromagnetism in the algebra of quaternions and made the electromagnetic potential the
centerpiece of his theory. In 1881 Heaviside replaced the electromagnetic potential field by force fields as the centerpiece of electromagnetic theory. According to him, the electromagnetic
potential field was arbitrary and needed to be "assassinated" (sic). A few years later there was a great debate between Heaviside and Tate about the relative merits of vector analysis and
quaternions. The result was the realization that there was no need for the greater physical insights provided by quaternions if the theory was purely local, and vector analysis became
The vast applications of electromagnetic theory since then were made using vector analysis. Although generations of very effective students were trained using vector analysis, more might be
learned physically by returning, if not to quaternions, to other mathematical formulations in certain well-defined circumstances. As examples, since the time when the theoretical design of
electromagnetism was frozen, gauge theory has been invented and brought to maturity and topology and geometry have been introduced to field theory. Although most persons view their subject matter
through the filter of the mathematical tools in which they are trained, the best mathematical techniques for a specific analysis depend upon the best match between the algebraic logic and the
underpinning physical dynamics of a theoretical system." [Terence W. Barrett and Dale M. Grimes, Preface, p. vii-viii, in Advanced Electromagnetism: Foundations, Theory and Applications, Terence
W. Barrett and Dale M. Grimes (eds.), World Scientific, Singapore, 1995.]
Maxwell died in 1879 of stomach cancer.
In the 1880s, several scientists --- Heaviside, Gibbs, Hertz etc. --- strongly assaulted the Maxwellian theory and dramatically reduced it, creating vector algebra in the process. Then circa 1892
Lorentz arbitrarily symmetrized the already seriously constrained Heaviside-Maxwell equations, just to get simpler equations easier to solve algebraically, and thus to dramatically reduce the
need for numerical methods (which were a "real bear" before the computer). But that symmetrization also arbitrarily discarded all asymmetrical Maxwellian systems - the very ones of interest to us
today if we are seriously interested in usable EM energy from the vacuum.
So anyone seriously interested in potential systems that accept and use additional EM energy from the vacuum, must first violate the Lorentz symmetry condition, else all his efforts are doomed to
failure a priori.
We point out that quaternion algebra has a higher group symmetry than either vector algebra or tensor algebra, and hence it reveals much more EM phenomenology and dynamics than does EM in vector
or tensor form.
Today, the tremendously crippled Maxwell-Heaviside equations --- symmetrized by Lorentz --- are taught in all our universities in the electrical engineering (EE) department. Note that the EE
professors still dutifully symmetrize the equations, following Lorentz, and thus they continue to arbitrarily discard all asymmetrical Maxwellian systems. Hence none of them has the foggiest
notion of how to go about developing an "energy from the vacuum" system, which is asymmetrical a priori.
The resulting classical electromagnetics and electrical engineering (CEM/EE) model taught in all our university EE departments also contains very serious falsities. Most of modern physics, such
as special and general relativity, quantum field theory, etc., has been developed since the 1880s and 1890s fixating of the symmetrized Maxwell-Heaviside equations. A paper gathering together a
listing these serious flaws and giving proper citations, is T. E. Bearden, "Errors and Omissions in the CEM/EE Model," available for free downloading at:
http://www.cheniere.org/techpapers/CEM%20Errors%20-%20final%20paper%20complete%20w%20longer%20abstract4.doc .
This paper also shows a magnetic Wankel engine (suppressed from the world market) that can be built by any electrical engineering department or physics department, and then tested at COP>1.0 to
one's heart's content. The magnetic Wankel system is also easily close-looped for self-powering (where all its input energy is freely furnished by the vacuum, and the operator need furnish none
of the input energy at all --- thus providing fuel free, continuous use of the energy from the vacuum, at will.
In the hard physics literature, rigorous proof that eliminating the arbitrary Lorentz condition provides systems having free additional energy currents from the vacuum is given by M. W. Evans et
al., "Classical Electrodynamics without the Lorentz Condition: Extracting Energy from the Vacuum," Physica Scripta, Vol. 61, 2000, p. 513-517. Evans' own O(3) model is very advanced, and it also
directly specifies mechanisms for an EM system receiving and using excess energy freely from the vacuum.
Fortunately, today some scientists have turned again to higher group symmetry algebras in which EM is expressed. These higher group symmetry electrodynamics theories then show far more EM
phenomenology than the standard CEM/EE model used in electrical power engineering.
Anyway, that gives you a brief overview of the Maxwell theory, and the rather sharp curtailment of it that has become the accepted but very crippled model for electrical engineering.
Specifically, it is that crippled model and its continued propagation and use that is directly responsible for the increasing energy crisis worldwide, and our dependence on conventional fuels
We do point out that the original Maxwell quaternion and quaternion-like theory of 1865 also contained errors, by the physics that has been learned since then. One of those errors was Maxwell's
assumption of the material ether, an ether which was falsified experimentally in 1887 after Maxwell was already dead. But the present CEM/EE model still assumes that same old material ether, more
than a century later.
Also, after Maxwell published the first edition of his famous "Treatise.", not much happened. He was soundly criticized for using the quaternion approach, and even his own editor chastised him
rather unmercifully for it. His attachment to the potentials as primary was also roundly criticized, since almost all theorists of the day believed that the potentials were simply mathematical
conveniences having no physical reality whatsoever. To them, the force fields were the only physical reality in Maxwell's theory. Today, of course, we know in the quantum theory that it is the
potentials that are primary, and the fields are derived from changes in the potentials.
The history of Maxwell's famous treatise is as follows: The publications are James Clerk Maxwell, A Treatise on Electricity and Magnetism, Oxford University Press, Oxford, 1873, Second Edition
1881 (Maxwell was already dead), Third Edition, Volumes 1 and 2, 1891. Foreword to the second edition was by Niven, who finished the work as Maxwell had dramatically rewritten the first nine
chapters, much new matter added and the former contents rearranged and simplified. Maxwell died before finishing the rest of the second edition. The rest of the second edition is therefore
largely a reprint from the first edition. The third edition edited by J. J. Thomson was published in 1892, by Oxford University Press, and later was published unabridged, Dover Publications, New
York, 1954. J. J. Thomson finished the publication of the third edition, and wrote a "Supplementary Volume" with his notes. A summary of Maxwell's equations is given in Vol. II, Chapter IX of the
third edition. However, Maxwell had gone (in his second edition) to some pains to reduce the quaternion expressions himself, and not require the students to know the calculus of quaternions (so
stated on p. 257). We note that Maxwell did not finish the second edition, but died before that. He actually had no hand at all in the third edition as to any further changes. The Second edition
(unfinished by Maxwell) was later finished by Niven by simply adding the remaining material from the previous first edition approved by Maxwell to that part that Maxwell had revised. The printing
of the first nine chapters of the third edition was already underway when J. J. Thomson was assigned to finish the editing of the manuscript.
Indeed, as an example of a major error in the present CEM/EE model, we know today that matter is a component of force, and therefore the EM force fields prescribed in matter-free space by Maxwell
and his followers (and by all our electrical engineering departments today), do not exist. The EM field in massless space is force-free, and is a "condition of space" itself, as pointed out by
Feynman in his three volumes of sophomore physics. Specifically, speaking of the electric field Feynman states:
"...the existence of the positive charge, in some sense, distorts, or creates a "condition" in space, so that when we put the negative charge in, it feels a force. This potentiality for producing
a force is called an electric field." [Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading, MA, Vol. 1, 1964, p. 2-4].
He further states:
"We may think of E(x, y, z, t) and B(x, y, z, t) as giving the forces that would be experienced at the time t by a charge located at (x, y, z), with the condition that placing the charge there
did not disturb the positions or motion of all the other charges responsible for the fields." [ibid, vol. II, p. 1-3.]
But the CEM/EE texts still teach that old force field in empty space. However, Jackson --- a superb classical electrodynamicist of international reknown --- at least points out that this dramatic
error in the model is just ignored. Jackson states:
"Most classical electrodynamicists continue to adhere to the notion that the EM force field exists as such in the vacuum, but do admit that physically measurable quantities such as force somehow
involve the product of charge and field." [J. D. Jackson, Classical Electrodynamics, Second Edition, Wiley, 1975, p. 249].
Jackson does admit it and point out that this logical problem is just ignored, for which he is to be highly commended. Most textbooks simply do not even discuss it.
So at his death in 1879, Maxwell had already laboriously simplified some 80% of his "Treatise" himself, to comply with the severe demands of the publisher. The second edition of his book thus has
the first 80% considerably changed by Maxwell himself. The third edition contained the same theory as the second edition essentially, but just with additional commentary. It is this third edition
that is widely available and usually referred to as "Maxwell's theory".
Today, there is still a widespread belief that the third edition represents Maxwell's original EM work and theory, in pristine form just as created originally by Maxwell.
It doesn't.
Best wishes,
Tom Bearden | {"url":"http://www.rexresearch.com/maxwell.htm","timestamp":"2014-04-20T08:15:09Z","content_type":null,"content_length":"18072","record_id":"<urn:uuid:83d26ee0-55c1-4b83-922b-6f4aff6b7020>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00649-ip-10-147-4-33.ec2.internal.warc.gz"} |
s Last Theorem: n=4
Fermat’s Last Theorem: n=4
We prove Fermat’s Last Theorem for this case by showing $x^4 + y^4 = w^2$ has no solutions in the positive integers.
Suppose there is a solution. Then let $x,y,w$ be a solution with the smallest possible $w$. First note $x^2, y^2, w$ form a Pythagorean triple. Without loss of generality assume $x$ is odd, so write
\[x^2 = m^2 - n^2, y^2 = 2m n, z^2 = m^2 + n^2\]
for coprime $m,n$ that are not both odd.
Then the first equation implies that $x, n, m$ also form a Pythagorean triple with $x$ odd, so we may write
\[ x = r^2 - s^2, n = 2r s, m = r^2 + s^2 \]
for coprime integers $r,s$ that are not both odd.
The last of these three equations implies $r,s,m$ are pairwise coprime (otherwise $r,s$ could not be coprime) and from $y^2 = 4 r s m$ we deduce that $r = a^2, s = b^2, m = c^2$ for some integers
But substituting these in the equation for $m$ implies that $a^4 + b^4 = c^2$, contradicting the minimality of $w$. | {"url":"http://crypto.stanford.edu/pbc/notes/numberfield/fermatn=4.html","timestamp":"2014-04-18T02:58:22Z","content_type":null,"content_length":"4712","record_id":"<urn:uuid:5167236e-4b81-48d1-bed7-9eb603d8db11>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00541-ip-10-147-4-33.ec2.internal.warc.gz"} |
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2. Find a number whose product with 6 is the same as its sum with 60. (Hint: you may want to try guess and check or a table.) Show all steps or reasoning. Be sure to check your answer for
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let the number be x question says... \[6*x = 60+x\] solve for x..
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The number is 12. Follow @theloser 's idea to get your answer :)
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thankyou people!! im tutoring freshman on alg and we had an issue
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Appendix N: Procedures for Estimating Highway Capacity
An update of the manual is available! - HPMS Field Manual, September 2010
HPMS Field Manual
Appendix N: Procedures for Estimating Highway Capacity
Signalized Intersection Capacity
Sections that meet the criteria in the hierarchy for signalized intersection density (> .5 /mile) are analyzed using this procedure. For both rural and urban section where Left Turning Lanes (Data
Item 88) and Right Turning Lanes (Data Item 89) are missing, a simplified procedure is used (see page N-23). Data Items 88 and 89 are used to identify lane groups, a critical step in the signalized
intersection capacity procedure.
Typical Percent Green Time (Data Item 91) and Peak Parking (Data Item 61) are considered optional for the coding of rural sections. However, default values for these data are provided so that if Data
Items 88 and 89 are present, the urban procedure may be used.
Urban Procedure
Signalized Intersection Approach Capacity
For HPMS, the capacity of the entire approach is required, including all movements, primarily for consistency with the speed/delay procedures. Intersection approach capacity is based on HCM Equation
This approach is based directly on the H
C[A] = intersection approach capacity
s[i] = saturation flow rate for lane group i
= effective green ratio for lane group i
CM 2000 principles. When operational analysis is conducted using capacity computed in this way, the HCM 2000 method takes into account the volumes assigned to each lane group. In HPMS, turning
movement volumes are not available; only total intersection approach volume is computed from AADT. Therefore, the volume-to-service flow ratio for HPMS is computed using the total intersection
approach capacity and volume, rather than considering the operational characteristics of each lane group individually. The HPMS method therefore assumes that traffic is distributed roughly
proportionate to the available lane group capacity (i.e., lane groups are neither under-utilized nor over-utilized). While it is possible that this may result in an overestimation of capacity in some
instances, there is currently no basis upon which a realistic adjustment can be made. Further, because lane groups are based on the existence of exclusive turning lanes, there is no reason to suspect
some lane groups are underutilized at the expense of others; turning lanes are implemented to handle high turning volumes.
Determining Lane Groups
The HCM 2000 methodology for signalized intersections is based on determining capacity for individual lane groups. Lane groups take into account intersection geometry and turning movements. Separate
lane groups should be identified for:
• Continuous LT lane or 1 + exclusive LT lanes - LT only lane group
• Continuous RT lane or 1 + exclusive RT lanes - RT only lane group
In addition to possible exclusive turning lane groups, it is always assumed that through movements occur at the intersection. This assumption does not account for T-intersections, but no intersection
geometry data exist in HPMS that would allow this determination. The maximum number of lane groups for HPMS purposes is three. Table 9 is used in developing lane groups, based on the coding of left
and right turning lanes.
Table 9. Determining Lane Groups in HPMS
Data Item 88 Data Item 89 No. Peak Lanes Lane Groups No. Lanes in Group
Exclusive LT 2
1 Exclusive Through N
N/A Exclusive RT 2
Exclusive LT 2
2, 3 Exclusive Through N
Exclusive RT 1
1 1, 2 Exclusive LT 2
Shared Through/RT N
4 Exclusive LT 2
3+ Exclusive Through N- 1
Shared Through/RT 1
5 N/A Exclusive LT 2
Exclusive Through N
Exclusive LT 1
1 Exclusive Through N
N/A Exclusive RT 2
Exclusive LT 1
2, 3 Exclusive Through N
Exclusive RT 1
2,3 1, 2 Exclusive LT 1
Shared Through/RT N
4 Exclusive LT 1
3+ Exclusive Through N- 1
Shared Through/RT 1
5 N/A Exclusive LT 1
Exclusive Through N
1, 2 Shared LT/Through Data Item 87
Exclusive RT 2
1 Shared LT/Through 1
3+ Exclusive Through N- 1
Exclusive RT 1
1, 2 Shared LT/Through N
Exclusive RT 1
2, 3 Shared LT/Through 1
4 3+ Exclusive Through N- 1
Exclusive RT 1
1, 2 Shared LT/Through/RT N
4 Shared LT/Through 1
3+ Exclusive Through N- 2
Shared RT/Through 1
1, 2 Shared LT/Through N
5 3+ Shared LT/Through 1
Exclusive Through N- 1
1 N/A Exclusive Through N
Exclusive RT 2
2, 3 N/A Exclusive Through N
5 Exclusive RT 1
1, 2 Shared RT/Through N
4 3+ Shared RT/Through 1
Exclusive Through N- 1
5 N/A Exclusive Through N
Where: N = Data Item 87
Determining Saturation Flow Rate
A slightly modified version of HCM Equation 16-4 is used to determine saturation flow rate:
s=s[o] N f[w] f[HV] f[g ]f[p] f[bb ]f[a ]f[LU ]f[LT ]f[RT ]f[Lpb ]f[Rpb ]PHF
S = saturation flow rate for subject lane group, expressed as a total for all lanes in lane group (vph);
S[o] = base saturation flow rate per lane (pcphpl);
N = number of lanes in lane group;
f[w] = adjustment factor for lane width;
f[HV] = adjustment factor for heavy vehicles in traffic stream;
f[g] = adjustment factor for approach grade;
f[p] = adjustment factor for existence of a parking lane and parking activity adjacent to lane group;
f[bb] = adjustment factor for blocking effect of local buses that stop within intersection area;
f[a] = adjustment factor for area type;
f[LU] = adjustment factor for lane utilization;
f[LT] = adjustment factor for left turns in lane group;
f[RT] = adjustment factor for right turns in lane group;
f[Lpb] = pedestrian-bicycle adjustment factor for left-turn movements;
f[Rpb] = pedestrian-bicycle adjustment factor for right-turn movements; and
PHF = Peak Hour Factor.
Note that the Peak Hour Factor is included in the calculation of saturation flow rate. The HCM 2000 adjusts volume with this factor, but for HPMS purposes, it is used to adjust saturation flow rate.
Base Saturation Flow Rate, S[O]
The base saturation flow rate is set at 1,900 pcphpl.
Adjustment Factor for Lane Width, f[W]
The lane width adjustment factor is based directly on the HCM 2000 procedure:
W = Lane Width (Data Item 54); minimum of 8, maximum of 16
Adjustment for Heavy Vehicles, f[HV]
The heavy vehicle adjustment factor is based directly on the HCM 2000 procedure, assuming 2 passenger car equivalents for heavy vehicles:
HV = percent heavy vehicles
E[T] = 2.0 passenger car equivalents
The percent heavy vehicles factor is the sum of peak combination and single unit trucks (Data Items 81 and 83).
Adjustment for Grade, f[g]
For HPMS purposes, f[g] is set to 1.0 because of the lack of grade information on urban minor arterials and collectors.
Adjustment for Parking, f[p]
The calculation of the parking adjustment factor is:
f[p] = parking adjustment factor
N = number of lanes in group
N[m] = number of parking maneuvers per hour
= 6 for two-way streets with parking one side
= 12 for two-way streets with parking both sides or one-way streets with parking one side
= 24 for one-way streets with parking on both sides (based on HCM Exhibit 10-20).
The parking factor is applied only to lane groups that are immediately adjacent to parking spaces. For two-way streets or one-way streets with parking on one side, this is assumed to occur in the
right-most lane group; this will be either an exclusive right-turn lane or a lane with shared movements. For one-way streets with parking on both sides, the parking factor adjustment is made for the
left-most and the right-most lane groups.
When parking is not allowed or unavailable (Data Item 61 = 3), f[p] is set to 1.0. It is also set to 1.0 if Data Item 61 = 0.
Adjustment for Bus Blockage, f[bb]
For HPMS, f[bb] is set to 1.0. No data exist in HPMS as to the occurrence of bus routes. Further, the default values in the HCM 2000 for bus maneuvers lead to adjustment factors close to 1.0.
Type of Area Adjustment, f[a]
Area type is no longer coded in HPMS. An analysis of 1998 HPMS showed that 9 percent of urban signalized intersections were located in CBDs. The HCM 2000 indicates that f[a] should be 0.9 in CBDs,
1.0 elsewhere. Weighting these values with the findings from the 1998 HPMS data provides a value of 0.991 for f[a ] for urban conditions. A value of 1.0 is used for rural conditions.
Lane Utilization Adjustment, f[LU]
The HCM 2000 states that: "As demand approaches capacity, the analyst may use lane utilization factors [close] to 1.0, which would indicate a more uniform use of the available lanes and less
opportunity for drivers to freely select their lane" (HCM page 10-26). Because the purpose of this factor in HPMS is to estimate capacity rather than perform an analysis of an individual
intersection, a lane utilization adjustment factor of 1.0 is used. This will avoid underestimating capacity in investment analysis where demand volumes are forecast for a long time horizon (usually
20 years).
Adjustment for Left Turns, f[LT]
Left turns are a major determinant of intersection approach capacity, yet only limited data are available in HPMS for conducting capacity analyses. The HCM 2000 identifies six cases, as shown in
Table 10. The adjustment factor for left turns is applied only if left turns are made from the lane group; this determination is made by checking the coding of Data Item 88, also as shown in Table
Table 10. HCM 2000 Left Turn Adjustment Cases as Applied to HPMS
LT Adjustment Case Data Item 88 Functional Class f[LT]
1. Protected Phasing, Exclusive LT 1, 2, 3 All Arterials; Rural Major Collectors f[LT]=0.95
2. Permitted Phasing, Exclusive LT 1, 2, 3 Rural Minor Collectors; Urban Collectors (see below)
3. Protected/Permitted Phasing, Exclusive LT N/A N/A N/A
4. Protected Phasing, Shared LT 4 Principal Arterials (P[LT]= 0.10)
5. Permitted Phasing, Shared LT 4 Minor Arterials; (see below)
All Collectors
6. Protected/Permitted Phasing, Shared LT N/A N/A N/A
A major assumption here is that the phasing is determined by functional class; higher order facilities are assumed to have protected phasing.
Note that Protected/Permitted Phasing is assumed to not exist because of lack of information on signal phasing in HPMS. For Case 4, P[LT] is the proportion of left turns in the lane group. As
recommended in the HCM 2000, a default value of 10 percent is used for HPMS (HCM page 10-19).
If left turns are not made from the lane group, the left turn adjustment factor is set to 1.0.
Special Procedure for Permissive Left Turns
The permissive left turn procedure in the HCM 2000 is highly complex and dependent on many variables not present in the HPMS data (Appendix C, HCM Chapter 16). The new procedure is substantially more
complex than those in previous versions of the HCM. For HPMS purposes, the procedure from the 1994 edition of the HCM is used because it is more compatible with the available data in HPMS:
f[LT] = (1400-V[o])/[(1400-V[o])+(235+0.435*V[o])*P[LT]] for V[o ]<= 1,220 vph
= 1/(1+(4.525*P[LT])) for V[o ]>1,220 vph
V[o] = AADT * K * (1 - D); this is the opposing flow in the off peak direction
AADT = annual average daily traffic, (Data Item 33)
K = K-factor, (Data Item 85) expressed as a decimal
D = the directional factor for the peak direction, (Data Item 86) expressed as a decimal
P[LT = proportion of left turns; assume proportion of left turns is 10 percent
Adjustment for Right Turns, f[RT]
The adjustment factor for right turns is applied only if right turns are made from the lane group. The right turn adjustment factor is based on the coding of HPMS Data Item 89 as follows:
f[RT] = 0.85 (for Data Item 89 = 1, 2, or 3)
= 1.0 - 0.15P[RT ](for Data Item 89 = 4)
P[RT] = proportion of right turns in the lane group
= 0.10 (HCM, page 10-19)
If right turns are not made from the lane group, the right turn adjustment factor is set to 1.0.
Adjustment for Pedestrian-Bicycle Blockage on Left Turns, f[Lpb]
As with permissive left turns, calculation of the adjustment factor for pedestrians and bicycles is complex and requires extensive inputs, most of which are unavailable in HPMS. Further, pedestrian/
bicycle blockage is likely to have a major impact only in densely developed areas and these conditions can no longer be determined from HPMS data. Therefore, for the purpose of HPMS, f[Lpb] is set to
Adjustment for Pedestrian-Bicycle Blockage on Right-Turns, f[Rpb]
Based on the same conditions as for pedestrian-bicycle blockage on left turns, f[Rpb ]is set to 1.0.
Peak Hour Factor (PHF)
As discussed on HCM page 10-8, a default value of 0.92 is used for the PHF for urban sections. For rural sections, the PHF is set to 0.88.
Effective Green Ratios (g[i]/C) for Lane Groups
The following values are used for the various types of lane groups. They are based on the coded value for HPMS Data Item 91 (Typical Peak Percent Green Time), which applies to the through movement
only. For all lane groups except for exclusive left turn lane groups, the value for Data Item 91 is used for g[i]/C. (This assumes that an exclusive signal phase for right turns coincides with the
through movement phase.) For Exclusive Left Turn Lane Groups, the green ratio is a function of the green ratio for the through movement (Data Item 91). A distinction is made for low and high values
for through green ratio:
For g/C <= 0.65
g[LT ]/C = 0.35*g/C for principal arterials
= 0.25*g/C for all other highway types
For g/C > 0.65
g[LT ]/C = 0.5*(1 - g/C)(all highways)
g/C = green ratio for through movement (Data Item 91).
If Data Item 91 is missing, g/C default values of 0.45 for arterials and 0.40 for collectors are used (NCHRP 387, Table 9-3).
<< Previous Contents Next >> | {"url":"http://www.fhwa.dot.gov/ohim/hpmsmanl/appn5.cfm","timestamp":"2014-04-21T09:45:25Z","content_type":null,"content_length":"34556","record_id":"<urn:uuid:db2134c8-a9af-4376-a8a6-1de736db2485>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00391-ip-10-147-4-33.ec2.internal.warc.gz"} |
internal hom of chain complexes
Homological algebra
Monoidal categories
With symmetry
With duals for objects
With duals for morphisms
With traces
Closed structure
Special sorts of products
In higher category theory
A natural internal hom of chain complexes that makes the category of chain complexes into a closed monoidal category.
Let $R$ be a commutative ring and $\mathcal{A} = R$Mod the category of modules over $R$. Write $Ch_\bullet(\mathcal{A})$ for the category of chain complexes of $R$-modules.
For $X,Y \in Ch_\bullet(\mathcal{A})$ any two objects, define a chain complex $[X,Y] \in Ch_\bullet(\mathcal{A})$ to have components
$[X,Y]_n := \prod_{i \in \mathbb{Z}} Hom_{R Mod}(X_i, Y_{i+n})$
(the collection of degree-$n$ maps between the underlying graded modules) and whose differential is defined on homogeneously graded elements $f \in [X,Y]_n$ by
$d f := d_Y \circ f - (-1)^{n} f \circ d_X \,.$
This defines a functor
$[-,-] : Ch_\bullet(\mathcal{A})^{op} \times Ch_\bullet(\mathcal{A}) \to Ch_\bullet(\mathcal{A}) \,.$
The collection of cycles of the internal hom $[X,Y]$ in degree 0 coincides with the external hom functor
$Z_0([X,Y]) \simeq Hom_{Ch_\bullet}(X,Y) \,.$
The chain homology of the internal hom $[X,Y]$ in degree 0 coincides with the homotopy classes of chain maps.
By Definition 1 the 0-cycles in $[X,Y]$ are collections of morphisms $\{f_k : X_k \to Y_k\}$ such that
$f_{k+1} \circ d_X = d_Y \circ f_k \,.$
This is precisely the condition for $f$ to be a chain map.
Similarly, the boundaries in degree 0 are precisely the collections of morphisms of the form
$\lambda_{k+1} \circ d_X + d_Y \circ \lambda_k$
for a collection of maps $\{\lambda_k : X_k \to Y_{k+1}\}$. This are precisely the null homotopies.
A standard textbook account is | {"url":"http://www.ncatlab.org/nlab/show/internal+hom+of+chain+complexes","timestamp":"2014-04-19T09:27:20Z","content_type":null,"content_length":"38728","record_id":"<urn:uuid:0324746b-d20c-440c-aed2-3ad55b5ceaf0>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00573-ip-10-147-4-33.ec2.internal.warc.gz"} |
The DT_SUBTRACT function decrements an array of IDLDT date/time variables by a constant amount. Each value is subtracted from the corresponding component, then the result is renormalized with
JUL_TO_DT.The result is an array of IDLDT date/time structures, decremented by the desired amount.
PRINT, DT_SUBTRACT( TODAY(), DAY=1, YEAR=3)
; Takes 1 from today's dayand 3 from today's year. | {"url":"http://www.astro.virginia.edu/class/oconnell/astr511/IDLresources/idl_5.1_html/idl85.htm","timestamp":"2014-04-17T03:48:54Z","content_type":null,"content_length":"4786","record_id":"<urn:uuid:09a157ef-b8d3-439e-985c-a94318d31d14>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00152-ip-10-147-4-33.ec2.internal.warc.gz"} |
PhET Teacher Ideas & Activities: Wave Unit
written by the The PhET Project and Trish Loeblein
This is an instructional unit on the topic of waves, created for use in the high school physics classroom. It was designed to be used with interactive simulations developed by PhET, the Physics
Education Technology project. Included are detailed lessons for integrating labs, simulations, demonstrations, and concept questions to introduce students to properties and behaviors of waves.
Specific topics include frequency and wavelength, sound, the wave nature of light, geometric optics, resonance, wave interference, Doppler Effect, refraction, thin lenses, wave addition, and more.
Activities are aligned to AAAS Benchmarks.
Subjects Levels Resource Types
Education Practices
- Active Learning
= Modeling
Optics - Collection
- Geometrical Optics - Instructional Material
Oscillations & Waves = Activity
- Oscillations - High School = Best practice
= Simple Harmonic Motion - Lower Undergraduate = Curriculum support
= Springs and Oscillators - Upper Undergraduate = Instructor Guide/Manual
- Wave Motion = Laboratory
= Doppler Effect = Lesson/Lesson Plan
= Interference and Diffraction = Unit of Instruction
= Longitudinal Pulses and Waves
= Transverse Pulses and Waves
= Wave Properties of Sound
Intended Users Formats Ratings
- application/pdf
- Educators - image/gif
- Learners - image/jpeg
- text/html
Access Rights:
Free access
© 2006 The PhET Project
Fourier, Fourier Analysis, PHET, Phet, clicker questions, homework problems, labs, lesson plans, sound waves, wave superposition, waves
Record Cloner:
Metadata instance created April 17, 2008 by Caroline Hall
Record Updated:
October 1, 2012 by Caroline Hall
Last Update
when Cataloged:
March 31, 2008
Other Collections:
AAAS Benchmark Alignments (2008 Version)
4. The Physical Setting
4F. Motion
• 6-8: 4F/M2. Something can be "seen" when light waves emitted or reflected by it enter the eye—just as something can be "heard" when sound waves from it enter the ear.
• 6-8: 4F/M4. Vibrations in materials set up wavelike disturbances that spread away from the source. Sound and earthquake waves are examples. These and other waves move at different speeds in
different materials.
• 6-8: 4F/M5. Human eyes respond to only a narrow range of wavelengths of electromagnetic waves-visible light. Differences of wavelength within that range are perceived as differences of color.
• 6-8: 4F/M6. Light acts like a wave in many ways. And waves can explain how light behaves.
• 6-8: 4F/M7. Wave behavior can be described in terms of how fast the disturbance spreads, and in terms of the distance between successive peaks of the disturbance (the wavelength).
• 9-12: 4F/H5ab. The observed wavelength of a wave depends upon the relative motion of the source and the observer. If either is moving toward the other, the observed wavelength is shorter; if
either is moving away, the wavelength is longer.
• 9-12: 4F/H6ab. Waves can superpose on one another, bend around corners, reflect off surfaces, be absorbed by materials they enter, and change direction when entering a new material. All these
effects vary with wavelength.
• 9-12: 4F/H6c. The energy of waves (like any form of energy) can be changed into other forms of energy.
11. Common Themes
11B. Models
• 6-8: 11B/M4. Simulations are often useful in modeling events and processes.
• 9-12: 11B/H3. The usefulness of a model can be tested by comparing its predictions to actual observations in the real world. But a close match does not necessarily mean that other models would
not work equally well or better.
11D. Scale
• 6-8: 11D/M3. Natural phenomena often involve sizes, durations, and speeds that are extremely small or extremely large. These phenomena may be difficult to appreciate because they involve
magnitudes far outside human experience.
Common Core State Standards for Mathematics Alignments
High School — Functions (9-12)
Interpreting Functions (9-12)
• F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship.^?
• F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.^?
• F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
• F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
Building Functions (9-12)
• F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Trigonometric Functions (9-12)
• F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.^?
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@misc{ Author = "The PhET Project and Trish Loeblein", Title = {PhET Teacher Ideas & Activities: Wave Unit}, Volume = {2014}, Number = {18 April 2014}, Month = {March 31, 2008}, Year = {2006} }
%Q The PhET Project
%A Trish Loeblein
%T PhET Teacher Ideas & Activities: Wave Unit
%D March 31, 2008
%U http://phet.colorado.edu/en/contributions/view/3023
%O application/pdf
%0 Electronic Source
%A The PhET Project,
%A Loeblein, Trish
%D March 31, 2008
%T PhET Teacher Ideas & Activities: Wave Unit
%V 2014
%N 18 April 2014
%8 March 31, 2008
%9 application/pdf
%U http://phet.colorado.edu/en/contributions/view/3023
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PhET Teacher Ideas & Activities: Wave Unit:
Is Part Of PhET Teacher Ideas & Activities: Browse Contributions
This is the full collection of teacher-created lesson plans and labs designed to be used with specific PhET simulations. Each resource has been approved by the PhET project, and may be freely
relation by Bruce Mason
Supplements PhET Simulation: Sound
One of the PhET simulations which the author intended to be used with this particular unit of instruction for high school physics students.
relation by Caroline Hall
See details...
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Conceptual Understanding of Fractions
From UCI Wiki hosted by EEE
Title Conceptual Understanding of Fractions
All Grade Levels
• [Current Contributors: June Choi, Gina Lee, Jenny Nguyen, Doris Ni, and Poonam Patel; Previous Contributors: Kaitlin Franks, Jane Kim, Lori Labuzetta, Cynthia Leon, and Cheryl Panado]
August 2008
Cognition and Learning Background
Fractions are a major concept that students are first introduced to in second grade. This concept continues throughout their mathematical education. Teachers struggle to teach this concept
effectively, while students struggle to gain a conceptual understanding. The goal of fraction instruction is for students to be able to visualize fractional amounts accurately.
Why Teach Conceptually
It is important for teachers to ensure that students have a firm grasp on the concepts of fractions. Once these concepts are established, students can better visualize and understand the value of a
fraction. Students with a conceptual understanding will have fewer problems in the future when they are introduced to fraction operations and algorithms (especially when adding and subtracting with
unlike denominators).
Brizuela (2005) explored the kinds of notations young children make for fractional numbers. Exploring childrens' notations reveals their thinking and knowledge of fractions. According to the study,
most children in the study did not make conventional notations for fractions. The relationship between conceptual understanding and in making notations is a complex area. Brizuela argues that
children’s notational competencies and conceptual understandings are intertwined. Thus, it is vital to integrate notations as students are exposed to learning and understanding of fractional numbers.
Introducing fraction concepts to children at a young age builds their ability to conceptualize fractional numbers.
Theories of Cognition and Learning Frameworks
• Zone Proximal Development
Lev Vygotsky theorizes that every individual has a Zone of Proximal Development (ZPD) that varies for each concept. The ZPD is the difference between what people can achieve independently versus
what they can achieve with guidance from a more knowledgeable source. When children are working within their ZPD, learning occurs. The teacher’s responsibility is to move a child from their
current state of knowledge to the next level of development.
When students enter their classrooms, they bring with them knowledge of partitioning, fair share, and equality . The teacher’s challenge is to base fraction instruction on prior knowledge in
order to make learning occur more efficiently. Teachers must first identify a child’s prior knowledge or current level of development. Instruction is then designed to guide children through their
ZPD, to reach their potential level of development.
Many theorists propose that meaningful learning occurs when students construct their own knowledge; this is referred to as constructivism. Social constructivism relies on a group of people
collectively generating ideas and understandings of a concept, while individual constructivism focuses on individuals interacting with the content to make sense of it in their own way .
When introducing fractions, teachers should allow students to interact with materials to discover patterns and develop concepts before the teacher formally introduces the topic through direct
instruction . Teachers should integrate a variety of cooperative learning groups as well as independent work in order to foster individual and social constructivism.
Fraction Proficiency & Success in Higher Level Math
In relation to achieving success in algebra, students need to gain fluency in understanding fractions early on in their educational endeavors. Since elementary algebra is built on a foundation of
basic arithmetic concepts, a solid mastery of these concepts is essential towards generalizing their prior knowledge in symbolic representation of numbers (as cited in Brown & Quinn, 2007). According
to a study that investigates the relationship between fraction proficiency and success in algebra, inability to successfully manipulate and perform basic operations on common fractions has led to
negative performance in algebraic reasoning (Brown & Quinn, 2007). Suffice to say, promoting conceptual understanding of fractions leads to beneficial gains in higher level mathematics.
Cognitive Obstacles and Common Misconceptions
Causes for Low Performances in Fractions:
• Complexity of fraction concepts (Bezuk & Cramer, 1989; Streefland, 1991)
• Incorrect knowledge of rote procedures (Hiebert & Wearne, 1985; Mack, 1990)
• Difficulties with identification of units (Hunting, 1983, 1986), teaching computation before understanding the meanings of fractions (Aksu, 1997)
• More emphasis on teaching procedures than on teaching conceptual meanings (Moss & Case, 1999)
• Interference from whole-number knowledge (Lukhele, Murray, & Olivier, 1999)
• Influence of multiple representations on initial learning of fraction ideas (Cramer, Post, & delMas, 2002)
Complex Nature of Fractions vs. Rote Teaching and Learning Practices
• Complex nature of fractions
□ fraction concepts and sophisticated fraction arithmetic can be confusing to students who are only familiar with whole-number systems
□ students experience substantial conceptual leaps in symbolic representation, intrinsic meaning of a number, identification of a unit, and computational schemes when switching from whole
numbers to fractions
□ students tend to regard fractions as two or three whole numbers (Newstead & Murray, 1998; van Niekerk, Newstead, Murray, & Olivier, 1999)
□ easily confusing the multiple roles that fractions play in real-life situations such as measuring, quotient, ratio, operator, and so on (Miemi, 1995)
□ the hardest part of learning fractions is understanding that “what looks like the same amount might actually be represented by different numbers,” (Lamon, 1999, p. 22)
□ the ability to hinder students’ learning fraction arithmetic
□ students tend to apply a familiar algorithm (regardless of correctness) from whole number arithmetic in fraction arithmetic Lukhele et al., 1999; Niemi, 1995)
Student misconceptions are further noticed through the use of incorrect or inefficient strategies. It is often noticed that students with mathematical disabilities frequently utilize less mature
strategies that their peers; they also “have more difficulty retrieving an efficient strategy, even when it has been taught to them” (as cited in Parmar, 2003). A specific example of an incorrect
algorithm applied towards fractions is when students “invert the dividend and the divisor or make other related errors, which appear to be based on students’ rote learning of an algorithm without
corresponding understanding” (Parmar, 2003). In addition, many students are perplexed with the concept of understanding that a fraction involves a part to whole relationship, which may further hinder
their mathematical fluency and adeptness in working with fractions.
Three Main Explanatory Frameworks on Children's Interpretation and Understanding of Numerical Values of Fractions
1. Fraction consists of two independent numbers
2. Fractions as parts of a whole
3. Fractions can be smaller, equal, or even bigger than the unit
• The results agree with previous studies showing a large number of students adopt the second explanatory framework “Fractions as part of a unit,” which claim that students believe that a fraction
with a numerator equal to the denominator is equivalent to the unit.
• Stafylidou and Vosniadou (2004) agree with previous research on the need for more practice on partitioning and measurement activities. These activities can help children develop the concept of a
unit, which is best developed through activities that relate the unit to fractions.
Student Challenges
• Apply whole number knowledge to fractions
□ Fractions are not arranged in a successive order as are whole counting numbers, such as 0, 1, 2, etc. Fractions with larger numbers do not imply larger quantities as they do in counting
numbers, for example, > even through 3 > 2. The value of a unit varies.
• Depending on the situation and problem, the quantity of the unit varies.
□ For example: One-half can be visually represented in different ways depending on the size of the unit.
• Rote memorization of procedures results in operational errors.
□ Without a deep conceptual understanding of fractions, students incorrectly apply rote operational procedures when presented with addition problems.
Dealing with Student Errors
• Student errors should be used as a starting point for student inquiry into mathematics
• Discourse initiated around errors helped generate student inquiry into understanding student errors
□ Builds on cognitive processes
□ Teaches students’ to reflect on their work
□ Sometimes it helps students to hear it from other students (in peer language)
□ Provide students with diagrams
Studies Related to Common Misconceptions and Obstacles
Study 1: Research on fraction division has reported that teachers can confuse situations that call for dividing by a fraction with ones that call for dividing by a whole number or multiplying by a
fraction (Armstrong & Bezuk, 1995; Ball, 1990; Borko, Eisenhart, Brown, Underhall, Jones, & Agard, 1992; Ma, 1999; Sowder, Philipp, Armstrong, & Schappelle, 1998). Based on the findings of this
study, it is important that teachers are confident int heir teaching abilities and skills to properly explain and conceptually teach fractions so that students are not left with doubts on how to work
with fractions. In this study, teachers provided students with diagrams to teach students division of fractions, but the diagrams were utilized ineffectively and students walked away from the lesson
full of misconceptions and doubts. Teachers must know how to use available resources to enhance the lesson and not be use resources as a crutch in a procedural lesson.
Study 2: Jigyel and Afamasaga-Fuata’i (2007) found that Australian students (equivalent to grade 4, 6, and 8 in the U.S.) perceive the numerator and denominator of a fraction as two unrelated whole
numbers. Students were able to identify equivalent fractions when they were presented geometrically (particularly circle models compared to rectangular ones). However, misconceptions occurred when
students were presented with equivalent fractions that were presented numerically (e.g. 4/6 and 2/3). Most students responded that 4/6 and 2/3 are not equal and that 4/6 is double of 2/3. The authors
conclude that these misconceptions require a development and consolidation of students’ understanding fractions using multiple models that clearly represent the relationship between the numerator and
the denominator. In addition, students need to understand the number of parts and size of parts when comparing fractions (Jigyel & Afamasaga-Fuata’i, 2007).
Study 3: According to a study on notations that young children form about fractions, a greater percentage of children in first grade interpreted fractions or halves as “little bits.” (Brizuela, 2005)
Students have already been exposed to lessons regarding fractions and how to make notations for them. It may be such instruction that may lead students to certain misconceptions about fractions.
The best way to meet the needs of students is to categorize them into ability levels to understand their deficiencies and be able to teach how to conceptually deal with fractions.
Pedagogical Tools and Strategies
Ineffective Strategies
(Meagher, 2002; Lamon, 2005)
• Emphasize procedures over concept development
• Adult-centered rather than child-centered approach
• Rational number instruction is limited to the part-whole component of rational numbers
• Not addressing students’ prior knowledge of fraction concepts
• Not addressing student misconceptions throughout instruction
Models and Visual Imagery
An important component of fraction concept development is the use of manipulatives or models (Cramer & Henry, 2002). Extended use of multiple manipulatives and models will result in a transfer to
visual representations or mental images. Visual representations will lead to a more concrete understanding of fraction concepts (Cramer, Post, & Del Mas, 2002).
Types of Models
□ Physical
☆ Area models: circular pie models, paper folding
☆ Length models: Cuisenaire rods, fraction strips, number line
☆ Set models: the whole is a set of objects, and the subset of the whole make up the fractional parts (e.g., counters, base ten blocks, colored beans)
Develop Rational Number Concepts
Rational numbers encompass part-whole comparison, measures, operators, quotients, and ratios and rates (Lamon, 2005). These rational numbers are all represented in a symbolic notation of $\frac{a}{b}
$. Students must understand that the fractional notation can mean many things beyond a part-whole relationship. Teachers should expose students to each branch of rational numbers because students
need to be exposed to the broad range of meanings that the fraction notation symbolizes.
Types of Rational Numbers
□ Part-whole comparison
☆ Example: Jake has 10 balls. 5 of the balls are baseballs or $\frac{1}{2}$ of the balls are baseballs.
□ Measures
☆ Example: It takes Sue 30 minutes to drive to work or $\frac{1}{2}$ hour
□ Operators
☆ Problems that involve increasing, decreasing, shortening, enlarging, and resizing. Example: Jessica had an image on her computer. She enlarged it by 2 (200%). However, she is unhappy with
this size and would like it to be the original size again. What fraction of the resized image should she input in the computer in order to get the original sized image? $\frac{1}{2}$
□ Quotients
☆ Example: There are two people evenly sharing a pizza. What portion of the pizza with they each get? In this problem, the rational number $\frac{a}{b}$ is formed with the meaning a pizzas
are shared with b people. The answer would be $\frac{1}{2}$.
□ Ratios and rates
☆ Example: For every 1 pant Timmy buys, he buys 2 shirts. The ratio of pants to shirts is 1:2 or $\frac{1}{2}$
Equal Partitioning and Equal Whole
According to Yoshida and Sawano (2002), students in the experimental group, where there was a great emphasis on equal partitioning and equal whole strategies, displayed significantly better
comprehension of the equal-partitioning of fractions and the representation of fraction sizes compared to students in the control group, where traditional textbook instruction was implemented.
□ Equal Partitioning
☆ Comment on the process of equal parts during the teaching process. Have students particpate in a task that requires composing fractions from parts divided unequally. Afterwards, have the
students compare the given task with fractions composed from parts divided equally and emphasize the possibility of comparing fractions by dividing the whole equally (Yoshida & Sawano,
□ Equal Whole:
☆ Stress the importance of having equal wholes (same sizes) when comparing the order of 2/5 with 2/3 in the textbook, for example. Intentionally, in comparing two fractions, use incorrect
figures in which the sizes of the wholes are unequal and have the students discover and emphasize that it would not be possible to compare fractions if the sizes of the wholes were
different (Yoshida & Sawano, 2002)
Student Discourse
Empson (1999) found in her study of first graders that children who generated their own strategies and were involved in sharing, comparing, and justifying their mathematical thinking through
teacher-guided discussion provided the foundation for developing fraction concepts.
• Frequent classroom discussions of fraction problems and using familiar resources to solve fraction problems help children make sense of fraction concepts.
• The interactions of fractions assist students in developing the necessary skills to give appropriate explanations and justifications of their fraction understandings.
• Children may further benefit in learning by coming up with their own strategies and actively participating in rich discussion with their peers to share multiple ways of solving a problem, thus
facilitating student’s mathematical reasoning skills (Empson, 1999).
Integrated Mathematics Assessment (IMA)
Saxe, Gearhart, and Nasir’s (2001) study found that the professional development program, The Integrated Mathematics Assessment (IMA), was effective in enhancing teachers understandings of fractions,
knowledge of students’ mathematical thinking, and students’ motivation. The program primarily focused on problem solving and conceptual understanding of fractions. Students who were taught by
teachers participating in IMA achieved greater post test scores on the conceptual understanding of fractions compared to the two other groups of teachers, one group of teachers who only discussed
strategies and the other group who received no professional development support and relied on textbooks. Teachers who participated in the IMA were taught principles from the following mathematical
• Cognitively Guided Instruction program (Carpenter et al., 1989, as cited in Saxe, Gearhart & Nasir, 2001)
• Problem-Centered Mathematics Project (Cobb, Wood et al., 1991, as cited in Saxe, Gearhart & Nasir, 2001)
• The Educational Leaders in Mathematics Project (Simon & Schifter, 1993, as cited in Saxe, Gearhart & Nasir, 2001)
Manipulatives - Fraction Tiles
Martin and Schwartz (2005) stated that Physically Distributed Learning (PDL), or the adaptation and reinterpretion processes through active manipulation of fraction tiles, enhances students’
conceptual understanding of fractions compared to pie pieces.
• Teachers should not demonstrate how to use a manipulative to solve a math problem for children, but rather it is important to provide children with multiple opportunities to self-discover using
materials by generating their own structures and interpretations.
• Having students actively engage with manipulatives is important as it promotes deeper learning of fraction concepts.
• Students should physically rearrange fraction tiles rather than pie pieces because it helps prepare them to "learn how to use new materials" (Martin and Schwartz, 2005).
• Principles of adaptation and reinterpretation in distributed learning, or rearranging fraction tiles, play an important role in developing students’ understanding of fraction concepts (Martin and
Schwartz, 2005).
Manipulatives - Measurement Models
Pearn (2007) recommends using the measurement model, particularly paper folding, fraction walls, and number lines to develop students’ understanding of fractions, fractional language, and the
flexibility between everyday language and fractional symbols. For instance, while students fold paper strips in fractional parts, teachers can lead a discussion based on the following questions that
ask students to observe and predict the fractional parts:
• How many parts will there be? How many folds will there be? What do we call each part? Show me (one quarter) of the paper strip. Which is larger? How do you know?
Then, the paper strips are used to mark fraction walls showing a whole, half, third, fourth...tenths.
• Students will mark each of the folded parts with appropriate fraction units.
• Fraction walls help develop concepts of equivalent fractions by allowing students to compare fractions from zero to one.
• The author suggests that after work with folded paper strips, students can use Microsoft Word to make their own fraction walls using the Split Cells function.
After that, students can be provided number lines marked 0 to 1 and asked to estimate and write the given fraction units on the number line (which may change in the markings: 0 and 2, 1 and 2, 0 and
1/2 , etc.).
• Students need to provide reasoning for their estimation and placement of the fractions
The author also emphasizes explicit instruction of academic language of fractions during these activities.
• She recommends providing opportunities of practice where students express fraction units (with operations) in words and symbols.
In addition, Reys et al. (1999) recommends that teachers use benchmarks of zero, one-half, or one on the number line to help students develop their fraction number sense. Benchmarks allow student's
to compare the relative sizes of fractions, through estimating, ordering, and placing them on a number line.
Graphical Partitioning Model (GPM) - A Web-Based Cognitive Tool (CT)
Kong's (2007) study on examining the learning outcomes based on the use of a cognitive tool for teaching fractions found that students were enthusiastic about using the cognitive tool as an
educational tool. In addition, students found the cognitive tool to be an engaging resource. Thus, the cognitive tool has potential for further development in the learning of fractions and promoting
collaborative learning in the classroom.
• The CT is a web-based learning tool for students to develop their concept of fractions.
• It aims to support student learning about fraction equivalence and addition and subtraction of fractions with like and unlike denominator (Kong 2007).
• Graphical Partitioning Model (GPM), a web-based CT, represents both the part-whole perspective and the measure perspective of fractions using a rectangular bar. (DIAGRAM)
• The teacher in the study claimed that the use of the GPM encouraged constructive teacher–student interactions and student–student discussions in the classroom setting.
• GPM addresses the diverse abilities of students and serves as a helpful tool for students to learn and understand the key mathematical concepts of fractions in an exploratory manner.
Part-Whole Theoretical Model
Kieren first established that the concept of fractions is not a single construct, but rather consists of five interrelated subconstructs: part-whole, ratio, operator, quotient, and measure
(Charalambous and Pitta-Pantazi, 2007). However, Charalambous and Pitta-Pantazi's (2007) study on a theoretical model links the five subconstructs to the operations of fractions, fraction
equivalence, and problem solving. The authors found the following benefits of using this theoretical model:
• The part-whole interpretation of fractions has a significant role in developing understanding of fractions.
• Part-whole subconstruct overlooks and relates to the other four subcontructs (ratio, operator, quotien, measure) DIAGRAM.
• Understanding of the different interpretations of fractions can improve students' performance on tasks involving the operations of fractions and fraction equivalence.
Curricula and Technological Resources
These curricula and resources were chosen because they support the development of a conceptual understanding of fractions rather than procedural algorithms. Researchers recommend that teachers
promote a quantitative understanding of fractions in their instructions. For example, students should understand that 5/6 + 7/8 is equal to about 2 because 5/6 is almost a whole and 7/8 is almost
a whole. The following resources can be applied to build on student informal knowledge, resulting in a more meaningful conceptual understanding. The presented curricula resources can be useful
for teachers in lesson planning of fractions. The provided technological resources can further aid students’ understanding in another interactive visual environment. Last, but not least, the
literary resources can help students explore fraction ideas in a more open and informal context.
Curricula Resources
National Council of Teachers of Mathematics (NCTM) Illuminations
Three units on fractions with 5 to 6 lessons each that teachers can adapt to their classrooms. They address various fraction models, such as regions, lengths, and sets, to develop a conceptual
understanding of fractions.
Rational Number Project
As a cooperative research and development project, the Rational Number Project offers a book of sequential fraction lessons to help teachers facilitate students’ conceptual understanding of
fractions. The proposed fraction lessons reflect the theoretical framework by Jean Piaget, Jerome Bruner, and Zoltan Dienes. Basing on these theoretical ideas, Richard Lesh, a Rational Number
Project director, design an instructional model that illustrates how a teacher should organize his/her instruction to help students actively construct their learning. Thus, reflecting this
instructional model, this book offers teachers instructional strategies that engage students in learning through the use of manipulatives and classroom discourse. Assessment tools are also
available to provide teachers insight to students’ thinking for further planning of instruction (Cramer, Berh, Post, & Lesh, 1997).
• New Rational Number Project Units of Instruction (2009)
New Zealand Maths
The Numeracy Professional Development Projects present a book on teaching fractions, decimals, and percents that focuses on students' mental strategies. The activities are designed to meet the
learning needs of students at different strategy stages (i.e., emergent, counting all, advanced additive part-whole,...,advanced proportional part-whole). Each activity composes of suggested
instructional approaches that help develop students' strategies between and through the phases of Using Materials, Using Imaging, and Using Number Properties. (Numeracy Professional Development
Projects, 2006).
Van de Walle, J. A. (2007). Elementary and Middle School Mathematics: Teaching Developmentally 6th ed.. United States of America: Pearson Education, Inc.
• Chapters 16 addresses key learning issues in developing fraction concepts and provides activities as a guide to instructional planning.
Lamon, S. J. (2005). Teaching Fraction and Ratios for Understanding: Essential Content Knowledge and Instruction Strategies for Teachers.
• This book provides background knowledge on fractions and rational numbers. The end of each chapter includes activities that address various components of fractions and rational numbers.
Burns, M. (2001). Teaching Arithmetic: Lessons for Introducing Fractions, grades 4-5. Sausalito, CA: Math Solutions Publications.
• This book offers well-designed lessons with numerous details, sample student dialogue, and Blackline Masters. Lessons present introductory ideas of fraction concepts, including looking at
one-half as a benchmark.
Litwiller, B. (Editor). Making Sense of Fractions, Ratios, and Proportions (2002 Yearbook). National Council of Teachers of Mathematics.
• NCTM's yearbook offers a variety of articles that provide insights to students' thinking about fractions, rations, and proportions. The Classroom Companion booklet suggests different activities
that teachers can use in their classrooms.
Fraction PowerPoint Presentations
The following websites offer various PowerPoint Presentations that teachers can use to instruct students on fraction concepts. It is advised that teachers should be conscious at selecting these
PowerPoint slides according to the provided theoretical frameworks and to meet the learning needs of their own students.
Technological Resources On-Line
Visual Fractions: http://www.visualfractions.com/
• This website allows students to see a variety of fractions represented in multiple ways (circles, bars, etc.). This resource has segments focused on identifying fractions, comparing fractions,
and operating with fractions (addition, subtraction, multiplication, and division). Games are also included as possible extension activities.
No Matter What Shape Your Fractions Are In: http://www.math.rice.edu/~lanius/Patterns/
• Going beyond the basic circle and/or bar fraction representations, this website uses pattern blocks to explore fractions. This contains lessons and activities where fractions are represented in
different and unique ways.
Flashcards: http://www.aplusmath.com/Flashcards/fractions-mult.html
• Once students become familiar with operating on fractions, this website can be used to reinforce students’ skills and knowledge. This website presents students with virtual flashcards, in which
students must type in the answer (numerator, then denominator) to various fraction problems.
National Council of Teachers of Mathematics Illuminations Fraction Games:
National Library of Virtual Manipulatives: http://nlvm.usu.edu/en/nav/index.html
• This website contains teacher lesson plans for students in Pre-kindergarten through 12th grade. This resource contains applets and lessons involving: Numbers & Operations, Algebra, Geometry,
Measurement, and Data Analysis & Probability. Lessons allow students to work with virtual base-ten blocks, abacuses, charts, and counters.
Fraction Track from National Library of Virtual Manipulate: http://standards.nctm.org/document/eexamples/chap5/5.1/index.htm
• In this game, students position fractions on number lines with different denominators. They will have the opportunity to develop their conceptual understanding of fractions, including
equivalent-fraction concepts.
Fraction Bars from National Library of Virtual Manipulate: http://nlvm.usu.edu/en/nav/frames_asid_203_g_1_t_1.html
• Showing a similar model to Cuisenaire rods (except that bars are segmented), this virtual manipulation allow students to create and compare multiple bars (representing fraction units) of any size
from 1 to 10 in any color.
Fraction Pieces from National Library of Virtual Manipulate: http://nlvm.usu.edu/en/nav/frames_asid_274_g_2_t_1.html?open=activities
• Students can use this open-ended manipulation to explore fraction pieces through a circular model and a rectangular model. Students learn the different fraction units making up a whole.
Fractions—Comparing from National Library of Virtual Manipulate: http://nlvm.usu.edu/en/nav/frames_asid_159_g_2_t_1.html
• Through using visuals of circular pieces, students find common denominators and compare two fractions. Then, they locate these fractions on a number line and identify a new fraction between these
Fractions—Visualizing from National Library of Virtual Manipulate: http://nlvm.usu.edu/en/nav/frames_asid_103_g_1_t_1.html
• In this applet, students use the given rectangular area model to subdivide the area into sections and color in the appropriate sections to corresponding the given fraction.
Fraction Flag: http://www.oswego.org/ocsd-web/games/fractionflags/fractionflags.html
• This game allows students to explore the concept of equal in quantity, not necessarily in size.
MIND Research Institute: Jiji Cycle: http://www.mindresearch.net/media/edu/demoFolder/demo/games/JiJiCycle/jijicycle.html
• This math game offers a visual way to develop the player’s conceptual understanding of fraction through estimating a fraction unit on a number line (through the use of an animated scenario about
a penguin named Jiji). Students can choose to estimate fractions and fraction addition through a focus on spatial temporal (language free) or symbolic (language integration).
Fraction Pointer: http://www.shodor.org/interactivate/activities/FractionPointer/?version=1.6.0_03&browser=MSIE&vendor=Sun_Microsystems_Inc.
• This applet connects an area model with number line. Students create area models for two fractions indicated on the number line and then create a new fraction between the first two.
Who Wants Pizza?: http://math.rice.edu/~lanius/fractions/index.html
• This is a series of interactive, straightforward lessons on fractions that include visual and written explanation on fraction concepts as well as practice problems with immediate feedbacks. The
lessons cover topics on the meaning of fractions, equivalent fraction, adding fractions, and multiplying fractions.
Fraction Games: http://www.gamequarium.com/fractions.html
• This website provides various interactive games for all grade levels to use that reinforce multiple fraction concepts, such as learning halves, comparing fractions with like denominators,
equivalent fractions, simplifying fractions, as well as adding, subtracting, multiplying and dividing fractions.
Fresh Baked Fractions: http://www.funbrain.com/fract/index.html
• Students can choose from four difficulty levels (easy, medium, hard, super brain) on identifying the fraction that is not equivalent from the others.
Fraction Four: http://www.shodor.org/interactivate/activities/FractionFour/
• This is a Connect Four version on fractions. Students are able to place a piece on the board by answering questions about simplifying fractions. There are also more challenging questions that
involve converting fractions to decimals and percents, and algebra with fractions.
Comparing Fractions: http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/fractions/simplifyingfractions/flash2.shtml
• This game allows students to compare two fractions to see which one is larger or if they are the same. They can also choose from four different ways to see what the fraction looks like by slicing
a pizza, looking at the number of people in a group, filling up a jug, and dividing up a chocolate bar.
Dolphin Racing: http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/fractions/comparingfractions/flash1.shtml
• This game has two different levels (easy and hard) and offers different colored dolphins that students can choose from. The goal of the game is to choose the biggest fraction to move the dolphin
the farthest from the other dolphins in a time limit.
Pizza Party: http://www.primarygames.com/fractions/start.htm
• A fun website that teaches fractions by using pizzas that is targeted towards younger students who are just beginning to grasp basic fraction concepts.
Fraction Bar from New Zealand Maths: http://www.nzmaths.co.nz/LearningObjects/FractionBar/index.swf
• In this activity, students learn to visualize and solve problems involving fractions, including fractions greater than 1 where the answer is a whole number. This technology resource is
incorporated into lesson from New Zealand Maths: http://www.nzmaths.co.nz/number/Operating%20Units/LO/fractionbar/fractionbar.aspx
Technology Resources for Purchase:
MathKeys: Unlocking Fractions and Decimals, Grades 3-5
• This for-purchase software includes bar models, counter models, and a circle model for developing fraction concepts.
Literary Resources
The Doorbell Rang by Hutchins
• Sharing requires partitioning so everyone would have equal amounts of cookie.
Gator Pie by Mathews
• A group of alligators attempt to split a pie so that everyone gets a piece.
Discovering Math: Fractions by David Stienecker
• This book has different activities and games that help students develop conceptual understanding of fractions (i.e. Dividing Shapes, Fraction Blocks, Folding Fractions)
The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan
• A man uses his remarkable mathematical skills to settle conflicts and give wise advice. Through his journey, he shares insights from the minds of great mathematicians.
Fraction Fun by David A. Adler
• Fraction concepts are introduced by looking at a pizza pie that is divided, studied, compared, and eaten. Coins are weighed to determine how many make one ounce, and what the fractional value of
each coin is.
Fraction Action by Loreen Leedy
• Balancing drawings of geometric shapes divided into sections and familiar concepts of half of a sandwich and a pie cut into four pieces makes it easy for children to visualize the meanings of
various fractional quantities.
Apple Fractions by Jerry Pallotta
• Playful elves demonstrate how to divide apples into halves, thirds, fourths, and more.
Hershey's Fractions by Jerry Pallotta
• A Hershey's Milk Chocolate Bar, made up of 12 little rectangles, is used to teach fractions
Eating Fractions by Bruce McMillan
• Real-life photos illustrate the budding of fractions and food
Jump, Kangaroo, Jump by Stuart J. Murphy
• Kangaroo and his friends divide in (fractional) teams at field day.
Polar Bear Math: Learning about Fractions from Klondike and Snow by Ann Whitehead Nagda
• Young readers learn about fractions through following baby polar bears, Klondike and Snow, in their development from newborns to mature bears.
How Many Ways Can You Cut a Pie? by Jane Belk Moncure
• A squirrel decides of ways to cut a pie to give to her animal friends when she wins the pie contest.
Annotated References
Brooks, J. & Brooks, M. (1999). Becoming a constructivist teacher. In search of understanding: the case for constructivist classrooms. Virginia: Association for Supervision and Curriculum
Brown, G. & Quinn, R. J. (2007). Investigating the relationship between fraction proficiency and success in algebra. Australian Mathematics Teacher, 63(4), 8-15.
Brizuela, B. (2005) YOUNG CHILDREN’S NOTATIONS FOR FRACTIONS. Educational Studies in Mathematics 62: 281–305.
Charalambous C. & Pitta-Pantazi D. (2006). Drawing on a theoretical model to study students' understandings of fractions. Educational Studies in Mathematics, 64(3), 293-316.
Cramer, K & Bezuk, N. (1989). Teaching about fractions: what, when, and how? In National Council of teachers of Mathematics 1989 Yearbook: New Directions for Elementary School Mathematics (pp
156-167): Virginia.
Cramer, K & Henry, A. (2002). Using manipulative models to build number sense for addition of fractions. National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios,
and Proportions (pp. 41-48). Virginia: National Council of Teachers of Mathematics.
Cramer, K., Post, T., del Mas, R. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the
rational number project curriculum. Journal for Research in Mathematics Education 33 (2), 111-144.
Empson, S. B. (1999). Equal sharing and shared meaning: the development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17(3), 282-342.
Jigyel, K., & Afamasaga-Fuata'i, K. (2007). Students' conceptions of models of fractions and equivalence. Australian Mathematics Teacher, 63(4), 17-25. Retrieved from http://www.aamt.edu.au/
Kong, S. (2008) The development of a cognitive tool for teaching and learning fractions in the mathematics classroom: A design-based study. Computers & Education 51, 886–899.
Lamon, S. (2005). Teaching fraction and ratios for understanding. New Jersey: Lawrence Erlbaum Associates.
Mack, N. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education 21(1), p16-32.
Martin, T. & Schwartz, D. L. (2005). Physically distributed learning: adapting and reinterpreting physical environments in the development of fraction concepts. Cognitive Sciences 29(2005), 587-625.
Mayer, R. (2003). Learning and instruction. New Jersey: Merrill Prentice Hall.
Meagher, M. (2002). Teaching fractions: New methods, new resources. Clearinghouse for Science, Mathematics, and Environmental Education. ERIC Digest.
Ormond, J. (2004). Human learning. New Jersey: Pearson Education Inc.
Parmar, R.S. (2003). Understanding the concept of “division” assessment considerations. Exceptionality, 11(3), 177-189.
Pearn, C. A. (2007). Using paper folding, fraction walls, and number lines to develop understanding of fractions for students from years 5-8. Australian Mathematics Teacher, 63(4), 31-36. Retrieved
from http://www.aamt.edu.au/
Reys, B. J., Kim, O., & Bay, J. M. (1999). Establishing fraction benchmarks. Mathematics Teaching in the Middle School, 4, 530-532.
Saxe, G. B., Gearhart, M. & Nasir, N. S. (2001). Enhancing students’ understanding of mathematics: a study of three contrasting approaches to professional support. Journal of Mathematics Teacher
Education, 4, 55-79.
Stafylidou, S., & Vosniadou, S. (2004). The development of students' understanding of the numerical value of fractions. Learning and Instruction, 14(5), 503-518. Retrieved from http://dx.doi.org/
Van de Walle, J. A. (2007). Elementary and Middle School Mathematics: Teaching Developmentally 6th ed. United States of America: Pearson Education, Inc.
Yoshida, H. & Sawano, K. (2002). Overcoming cognitive obstacles in learning fractions: Equal-partitioning and equal-whole. Japanese Psychological Research, 44(4), 183-195. | {"url":"https://wiki.eee.uci.edu/index.php/Conceptual_Understanding_of_Fractions","timestamp":"2014-04-18T10:34:25Z","content_type":null,"content_length":"74210","record_id":"<urn:uuid:bbce7c47-cbb6-45d9-bec4-b4363865a983>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00517-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Results
In order to catalog the important mode shapes, I am using the modal designation which is most commonly used to describe the vibration of systems with cylindrical symmetry (cylinders, bells, drums,
baseball bats). Each mode shape is designated by a pair of integers, (n,m) where n represents the number of nodal diameters and m represents the number of nodal circles. A node a location (a line on
a 2-D surface, or a plane in a 3-D volume) where there is no vibration for a particular mode shape. A nodal diameter indicates that there are two lines down the length of the bottle (on opposite
sides of the bottle) which do not move while the rest of the bottle is vibrating. So, for a mode shape with n=2 if you traced a path around the circumference of the bottle you would encounter 4=2n
lines (2 diameters = 2 pairs of lines on opposite sites of the bottle) where no vibration occurs. On one side of a nodal line the bottle will be vibrating outwards, while on the other side of the
nodal line the bottle is vibrating inwards. The m integers representing nodal circles are evidenced by a line around the circumference - (a circle) at a fixed location along the length of the bottle
- that does not move while the bottle is vibrating. Actually, the integer m counts the number of anti-nodes (locations of maximum amplitude) so that the number of nodal circles is really m-1. Looking
at the measured mode shapes below might help. The mode shapes identified in red correspond to the most prominent peaks in the frequency spectrum above. (n=2,m=1) - 3200 Hz
The first prominent vibrational mode of the bottle is essentially a breathing mode, similar to what is observed in a bell, wineglass, or baseball bat. The mode shape designation (n=2,m=1) means that
there are two nodal diameters (moving around the circumference you would find 2n=4 lines with no motion) and as you moved along the length of the bottle you would find zero (m-1) nodal circles. As
the bottle vibrates it looks like it has two regions which alternate between bulging outwards and compressing inwards.
(n=3,m=1) - 4910 Hz
This mode, which has three nodal diameters (6 nodal lines around the circumference) and zero nodal circles, does not radiate sound nearly as well as the n=2 modes. It shows up as a peak right near
5000 Hz in the frequency spectrum at the top of the page, but has an amplitude about 25dB lower than the two most prominent peaks. As the bottle vibrates it looks like it has three regions which
alternate between bulging outwards and compressing inwards.
(n=4,m=1) - 8520 Hz
This mode has four nodal diameters (8 nodal lines around the circumference) and zero nodal circles. As the bottle vibrates it looks like it has four regions which alternate between bulging outwards
and compressing inwards. This mode is a very poor radiator of sound, and thus does not show up prominently in the microphone frequency spectrum.
(n=5,m=1) - 13,500 Hz
This mode has five nodal diameters (10 nodal lines around the circumference) and zero nodal circles. As the bottle vibrates it looks like it has five regions which alternate between bulging outwards
and compressing inwards. This mode is an extremely poor radiator of sound, and does not figure prominently in the sound spectrum.
(n=2,m=2) - 5360 Hz
The second prominent vibrational mode of the bottle is the second member of the n=2 family, with two nodal diameters and one nodal circle located about half-way up the bottle. Because this mode
involves significant motion of the neck, it shows up very strongly in the sound spectrum when the bottle is struck on the neck (as would happen when clinking two bottles together to make a toast).
(n=3,m=2) - 7437 Hz
This mode shape is the second member of the n=3 family, with three nodal diameters and one nodal circle (about 3 inches or 7 dots from the bottom of the barrel).
(n=4,m=2) - 10,330 Hz
This mode shape is the second member of the n=4 family, with four nodal diameters and one nodal circle (about 3 inches or 7 dots from the bottom of the barrel).
This mode shape is the second member of the n=5 family, with five nodal diameters and one nodal circle (about 3 inches or between the 7th and 8th dot from the bottom of the barrel).
1st Bending - 4440 Hz
This mode is essentially a flexural bending vibration in which the entire bottle behaves like a free-free beam. Because this mode involves significant motion of the neck, it shows up very strongly in
the sound spectrum when the bottle is struck on the neck (as would happen when clinking two bottles together to make a toast).
(n=3,m=3) - 11,610 Hz
This mode is the third member of the n=3 family and has three nodal diameters and two nodal circles, one about 2-inches (5 dots) from the bottom and the other about 4-inches (8 dots) from the bottom. | {"url":"http://www.acs.psu.edu/drussell/Demos/BeerBottle/beerbottle.html","timestamp":"2014-04-20T17:42:10Z","content_type":null,"content_length":"12708","record_id":"<urn:uuid:a0eabbfa-d13a-4aec-8644-a234bbe78317>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00072-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Here are some mathematical puzzles that I have enjoyed. Most of them are of the kind that you can discuss and solve at a dinner table, usually without pen and paper. So as not to spoil your fun, no
solutions are given on this page, but for some problems I have provided some hints.
Picking the larger of two cards
Someone picks, at their will, two cards from a deck of cards. The cards have different numbers, so one is higher than the other. (In other words, the person picks two distinct numbers in the
inclusive range 1 through 13.) The cards are placed face down on a table in front of you. You get to choose one of the cards and turn it face up. Now, you will select one of the two cards (one of
whose face you can see, the other one you can't). If you select the highest card, you win. Design a card-selection strategy for which your chance of winning is strictly greater than 50%.
Enclosing land by fence pieces
[I got this puzzle from Serdar Tasiran.]
You are given one 44-meter piece of fence and 48 one-meter pieces of fence. Assume each piece is a straight and unbendable. What is the large area of (flat) land that you can enclose using these
fence pieces?
Planar configuration of straight connecting lines
[Radu Grigore told me this problem. I think may have heard it 20 years earlier from Jan van de Snepscheut.]
Given an even number of points in general positions on the plane (that is, no three points co-linear), can you partition the points into pairs and connect the two points of each pair with a single
straight line such that the straight lines do not overlap?
Reducing nearby enemies
[I got this puzzle from Jason Koenig.]
You are given an irreflexive symmetric (but not necessarily transitive) "enemies" relation on a set of people. In other words, if person A is an enemy of a person B, then B is also an enemy of A. How
can you divide up the people into two houses in such a way that every person has at least as many enemies in the other house as in their own house?
[Hint: As Radu Grigore pointed out to me, solving the Planar configuration of straight connecting lines puzzle may provide a hint to solving this puzzle.]
Transporting bananas
[Rupak Majumdar told me this intriguing puzzle.]
You have 3000 bananas that you want to transport a distance of 1000 km. The transport will be done by a monkey. The monkey can carry as many as 1000 bananas at any one time. With each kilometer
traveled (forward or backward), the money consumes 1 banana. How many bananas can you get across to the other side?
Car and key hide-and-seek
[A puzzle Aistis Simaitis gave me inspired this puzzle.]
In a room are three boxes that on the outside look identical. One of the boxes contains a car, one contains a key, and one contains nothing. You and a partner get to decide amongst yourselves to each
point to two boxes. When you have made your decision, the boxes are opened and their contents revealed. If one of the boxes your partner is pointing to contains the car and one of the boxes you are
pointing to contains the key, then you will both win. What strategy maximizes the probability of winning, and what is the probability that you will win?
Handshakes at a dinner party
[Pamela Zave shared this problem with me.]
Hilary and Jocelyn are throwing a dinner party at their house and have invited four other couples. After the guests arrive, people greet each other by shaking hands. As you would expect, a couple do
not shake hands with each other and no two people shake each other's hands more than once. At some point during the handshaking process, Jocelyn gets up on a table and tells everyone to stop shaking
hands. She also asks each person how many hands they have shaken and learns that no two people on the floor have shaken the same number of hands. How many hands has Hilary shaken?
Rectifying a pill mistake
[This is a slight rewording of a problem I got from Phil Wadler, who said he read the problem on xkcd.]
A man has a medical condition that requires him to take two kinds of pills, call them A and B. The man must take exactly one A pill and exactly one B pill each day, or he will die. The pills are
taken by first dissolving them in water.
The man has a jar of A pills and a jar of B pills. One day, as he is about to take his pills, he takes out one A pill from the A jar and puts it in a glass of water. Then he accidentally takes out
two B pills from the B jar and puts them in the water. Now, he is in the situation of having a glass of water with three dissolved pills, one A pill and two B pills. Unfortunately, the pills are very
expensive, so the thought of throwing out the water with the 3 pills and starting over is out of the question. How should the man proceed in order to get the right quantity of A and B while not
wasting any pills?
[Clark Barrett told me this problem.]
Given is a (possibly enormous) rectangular chocolate bar, divided into small squares in the usual way. The chocolate holds a high quality standard, except for the square in the lower left-hand
corner, which is poisonous. Two players take turns eating from the chocolate in the following manner: The player whose turn it is points to any one of the remaining squares, and then eats the
selected square and all squares positioned above the selected square, to the right of the selected square, or both above and to the right of the selected square. Note, although the board starts off
rectangular, it may take on non-rectangular shapes during game play. The object of the game is to avoid the poisonous square. Assuming the initial chocolate bar is larger than 1x1, prove that the
player who starts has a winning strategy.
Hint: To my knowledge, no efficient strategy for winning the game is known. That is, to decide on the best next move, a player may need to consider all possible continuations of the game. Thus, you
probably want to find a non-constructive proof. That is, to prove that the player who starts has a winning strategy, prove just the existence of such a strategy; in particular, steer away from proofs
that would construct a winning strategy for the initial player.
Alternating T-shirt colors
[I received this puzzle from Vladislav Shcherbina, but I changed gloves into T-shirts to emphasize the people rather than the spaces between them.]
Ten friends walk into a room where each one of them receives a hat. On each hat is written a real number; no two hats have the same number. Each person can see the numbers written on his friends'
hats, but cannot see his own. The friends then go into individual rooms where they are each given the choice between a white T-shirt and a black T-shirt. Wearing the respective T-shirts they
selected, the friends gather again and are lined up in the order of their hat numbers. The desired property is that the T-shirts colors now alternate.
The friends are allowed to decide on a strategy before walking into the room with the hats, but they are not otherwise allowed to communicate with each other (except that they can see each other's
hat numbers). Design a strategy that lets the friends always end up with alternating T-shirt colors.
Digit sums of multiples of 11
[I got this one from Madan Musuvathi.]
Some multiples of 11 have an even digit sum. For example, 7*11 = 77 and 7+7 = 14, which is even; 11*11 = 121 and 1+2+1 = 4, which is even. Do all multiples of 11 have an even digit sum? (Prove that
they do or find the smallest that does not.)
Weighing piles of coins
[I got this puzzle from Dave Detlefs, who read it in an MIT Alumni magazine. This puzzle is a bit more involved than most puzzles on my page, so you may want a paper and pen (and some tenacity) for
this one. Once you get into it, though, it's a hard puzzle to put aside until you've solved it.]
There are two kinds of coins, genuine and counterfeit. A genuine coin weighs X grams and a counterfeit coin weighs X+delta grams, where X is a positive integer and delta is a non-zero real number
strictly between -5 and +5. You are presented with 13 piles of 4 coins each. All of the coins are genuine, except for one pile, in which all 4 coins are counterfeit. You are given a precise scale
(say, a digital scale capable of displaying any real number). You are to determine three things: X, delta, and which pile contains the counterfeit coins. But you're only allowed to use the scale
Guessing each other's coins
[I got this puzzle from Raphael Reischuk, who also has a little puzzle collection.]
You and a friend each has a fair coin. You can decide on a strategy and then play the following game, without any further communication with each other. You flip your coin and then write down a guess
as to what your friend's coin will say. Meanwhile, your friend flips her coin and writes down a guess as to what your coin says. There's a third person involved: The third person collects your
guesses and inspects your coins. If both you and your friend correctly guessed each other's coins, then your team (you and your friend) receive 2 Euros from the third person. But if either you or
your friend (or both) gets the guess wrong, then your team has to pay 1 Euro to the third person. This procedure is repeated all day. Assuming your object is to win money, are you happy to be on your
team or would you rather trade places with the third person?
The duck and the fox
[I got this problem from Koen Claessen.]
A duck is in circular pond. The duck wants to swim ashore, because it wants to fly off and this particular duck is not able to start flying from the water. There is also a fox, on the shore. The fox
wants to eat the duck, but this particular fox cannot swim, so it can only hope to catch the duck when the duck reaches the shore. The fox can run 4 times faster than the duck can swim. Is there
always a way for the duck to escape?
Dropping eggs
[I think I've heard some version of this problem before, but heard this one from Sophia Drossopoulou and Alex Summers.]
There's a certain kind of egg about which you wonder: What is the highest floor of a 36-story building from which you can drop an egg without it breaking? All eggs of this kind are identical, so you
can conduct experiments. Unfortunately, you only have 2 eggs. Fortunately, if an egg survives a drop without breaking, it is as good as new--that is, you can then conduct another dropping experiment
with it. What is the smallest number of drops that is sure to determine the answer to your wonderings?
Age of children
[I got this problem from the book In code: a mathematical journey by Sarah Flannery and David Flannery.]
A (presumed smart) insurance agent knocks on a door and a (presumed smart) woman opens. He introduces himself and asks if she has any children. She answers: 3. When he then asks their ages (which for
this problem we abstract to integers), she hesitates. Then she decides to give him some information about their ages, saying "the product of their ages is 36". He asks for more information and she
gives in, saying "the sum of their ages is equal to our neighbors' house number". The man jumps over the fence, inspects the house number, and the returns. "You need to give me another hint", he
begs. "Alright", she says, "my oldest child plays the piano". What are the ages of the children?
Capturing a pirate ship
[Howard Lederer sent me this problem.]
You're on a government ship, looking for a pirate ship. You know that the pirate ship travels at a constant speed, and you know what that speed is. Your ship can travel twice as fast as the pirate
ship. Moreover, you know that the pirate ship travels along a straight line, but you don't know what that line is. It's very foggy, so foggy that you see nothing. But then! All of a sudden, and for
just an instant, the fog clears enough to let you determine the exact position of the pirate ship. Then, the fog closes in again and you remain (forever) in the thick fog. Although you were able to
determine the position of the pirate ship during that fog-free moment, you were not able to determine its direction. How will you navigate your government ship so that you will capture the pirate
3-person duel
[I got this problem from Johannes Kinder, who said he heard it from his brother. I reformulated the setting.]
A particular basketball shootout game consists of a number of duels. In each duel, one player is the challenger. The challenger chooses another player to challenge, and then gets one chance to shoot
the hoop. If the player makes the shot, the playing being challenged is out. If the player does not make the shot, or if the player chooses to skip his turn, then the game continues with the next
duel. A player wins when only that player remains.
One day, this game is played by three players: A, B, and C. Their skill levels vary considerably: player A makes every shot, player B has a 50% chance of making a shot, and player C has a 30% chance
of making a shot. Because of the difference in skill levels, they decide to let C begin, then B, then A, and so on (skipping any player who is out of the game) until there is a winner. If everyone
plays to win, what strategy should each player follow?
[For this follow-up question, it will be helpful to have a paper and pen--not because the calculations are hard, but because it helps in remembering the numbers.]
If A, B, and C follow their winning strategies (as determined above), which player has the highest chance of winning the game?
The genders of the neighboring family's children
[This problem was inspired by some basic probability questions mentioned in a lecture by Eric Hehner (and that can be formalized and solved by calculation using this Probability Perspective), and
some subsequent discussions with him, Itay Neeman, Jim Woodcook, Ana Cavalcanti, and Leo Freitas.]
The house next door has some new neighbors. They have two children, but you don't know what mix of boys and girls they are. One day, your wife tells you "At least one of the children is a girl". What
is the probability that both are girls?
Your wife then tells you "The way I found out that at least one of the children is a girl is that I saw one of the children playing outside, and it was a girl". Now, what is the probability that both
are girls?
Finding a hermit
[Claude Marché told me this puzzle. At first, it seems quite similar to "Catching a spy" (below), but the solution is entirely unrelated.]
There are five holes arranged in a line. A hermit hides in one of them. Each night, the hermit moves to a different hole, either the neighboring hole on the left or the neighboring hole on the right.
Once a day, you get to inspect one hole of your choice. How do you make sure you eventually find the hermit?
Witches at a coffee shop
[I got this puzzle from Alex Pintilie.]
Two witches each makes a nightly visit to an all-night coffee shop. Each arrives at a random time between 0:00 and 1:00. Each one of them stays for exactly 15 minutes. On any one given night, what is
the probability that the witches will meet at the coffee shop?
Lemmings on a ledge
[Vladislav Shcherbina told me this problem.]
A ledge is 1 meter long. On it are N lemmings. Each lemming travels along the ledge at a constant speed of 1 meter/minute. A lemming continues in the same direction until it either falls off the
ledge or it collides with another lemming. If two lemmings collide, they both immediately change their directions. Initially, the lemmings have arbitrary starting positions and starting directions.
What is the longest time that may elapse before all lemmings have fallen off the ledge?
[Michael Jackson suggested the following variation of the problem: Suppose the ledge is not horizontal, but is leaning. A lemming now travels up the ledge at a speed of 1 meter/minute and travels
down the ledge at a speed of 2 meters/minute. What is the longest time before all lemmings have fallen off?]
Poisoned wine
[I got this problem from Vladislav Shcherbina.]
You have 240 barrels of wine, one of which has been poisoned. After drinking the poisoned wine, one dies within 24 hours. You have 5 slaves whom you are willing to sacrifice in order to determine
which barrel contains the poisoned wine. How do you achieve this in 48 hours?
Dropping 9-terms from the harmonic series
[Rajeev Joshi told me this problem.]
The harmonic series--that is, 1/1 + 1/2 + 1/3 + 1/4 + ...--diverges. That is, the sum is not finite. This is in difference to, for example, a geometric series--like ½^0 + ½^1 + ½^2 + ½^3 + ...--which
converges, that is, has a finite sum.
Consider the harmonic series, but drop all terms whose denominator represented in decimal contains a 9. For example, you'd drop terms like 1/9, 1/19, 1/90, 1/992, 1/529110. Does the resulting series
converge or diverge?
[More generally, you may consider representing the denominator in the base of your choice and dropping terms that contain a certain digit of your choice.]
[Here is a follow-up question suggested by Gary Leavens.]
Consider again the harmonic series, but drop a term only if the denominator represented in decimal contains two consecutive 9's. For example, you'd drop 1/99, 1/992, 1/299, but not 1/9 or 1/909. Does
this series converge or diverge?
Finding a restaurant in a park
[Michael Jackson told me this problem, as a variation of a problem he got from Koen Claessen.]
A park contains paths that intersect at various places. The intersections all have the properties that they are 3-way intersections and that, with one exception, they are indistinguishable from each
other. The one exception is an intersection where there is a restaurant. The restaurant is reachable from everywhere in the park. Your task is to find your way to the restaurant.
The park has strict littering regulations, so you are not allowed to modify the paths or intersections (for example, you are not allowed to leave a note an intersection saying you have been there).
However, you are allowed to do some bookkeeping on a pad of paper that you bring with you at all times (in the computer-science parlance, you are allowed some state). How can you find the restaurant?
You may assume that once you enter an intersection, you can continue to the left, continue to the right, or return to where you just came from.
[As an alternative puzzle, suppose you must continue through an intersection, turning either left or right, but not turning around to exit the intersection the way you just entered it.]
Opening boxes in a prison courtyard
[I got this problem from Clark Barrett, who got it from Robert Nieuwenhuis. Just when you thought you had heard all variations of prisoners and boxes...]
100 prisoners agree on a strategy before playing the following game: One at a time (in some unspecified order), each of the prisoners is taken to a courtyard where there is a line of 100 boxes. The
prisoner gets to make choices to open 50 of the boxes. When a box is opened, it reveals the name of a prisoner (the prisoners have distinct names). The names written in the boxes are in 1-to-1
correspondence with the prisoners; that is, each name is found in exactly one box. If after opening 50 boxes, the prisoner has not found his own name, the game is over and all the prisoners lose. But
if the prisoner does find the box that contains his name among the 50 boxes he opens, then the prisoner is taken to the other side of the courtyard where he cannot communicate with the others, the
boxes are once again closed, and the next prisoner is brought out into the courtyard. If all prisoners make it to the other side of the courtyard, they win.
One possible strategy is for each prisoner to randomly select 50 boxes and open them. This gives the prisoners 1 chance out of 2
^100 to win--a slim chance, indeed. But the prisoners can do better, using a strategy that for a random configuration of the boxes will give them a larger chance of winning. How good a strategy can
you develop?
Frugal selection of weights to weigh a thing
[Matthew Parkinson told me this problem, or a slight variation thereof.]
You are given a balance (that is, a weighing machine with two trays) and a positive integer N. You are then to request a number of weights. You pick which denominations of weights you want and how
many of each you want. After you receive the weights you requested, you are given a thing whose weight is an integer between 1 and N, inclusive. Using the balance and the weights you requested, you
must now determine the exact weight of the thing.
As a function of N, how few weights can you get by requesting?
Initializing an array in constant time
[Unlike most problems on this page, this problem requires some computer science knowledge. Many years ago, I read a solution to this problem in one of Donald Knuth's books (I think). The algorithm
intrigued me and stuck in my mind.]
Consider the following array operations. Init(N,d) initializes an array of N elements so that each element has value d. Once Init has been called, the following two operations can be applied: For any
i such that 0 <= i < N, Get(i) returns the array element at position i and Set(i,v) sets the array element at position i to the value v.
Given any amount of memory you want, implement the three operations so that each operation has an O(1) time complexity.
Translation error in a cookbook
[This problem is a bit fuzzier than most on this page, just because I don't know for sure that the cookbook publisher made a translation error. But I hope you'll still enjoy the problem.]
Recently, I received a wonderful cookbook. In an appendix, it shows a table that converts oven temperatures between Celsius and Fahrenheit. (Side remark: Approximate oven temperatures are actually
really simple to convert in your head--just double the number of degrees Celsius to get the number of degrees Fahrenheit. For oven temperatures, this will be within 10 F of the exact answer.)
The table has a footnote that says "If your oven has a fan, reduce the recipe temperature by 68 F". I have a strong hunch that this footnote suffers from a translation error. How many degrees
Fahrenheit should it have said to reduce the temperature by? (No knowledge of convection ovens required.)
Making a square larger
[I got this problem from Radu Grigore.]
You are given four points (on a Euclidian plane) that make up the corners of a square. You may change the positions of the points by a sequence of moves. Each move changes the position of one point,
say p, to a new location, say p', by "skipping over" one of the other 3 points. More precisely, p skips over a point q if it is moved to the diametrically opposed side of q. In other words, a move
from p to p' is allowed if there exists a point q such that q = (p + p') / 2.
Find a sequence of moves that results in a larger square. Or, if no such sequence is possible, give a proof of why it isn't possible. (The new square need not be oriented the same way as the original
square. For example, the larger square may be turned 45 degrees from the original, and the larger square may have the points in a different order from in the original square.)
Catching a spy
[I got this problem from Radu Grigore.]
A spy is located on a one-dimensional line. At time 0, the spy is at location A. With each time interval, the spy moves B units to the right (if B is negative, the spy is moving left). A and B are
fixed integers, but they are unknown to you. You are to catch the spy. The means by which you can attempt to do that is: at each time interval (starting at time 0), you can choose a location on the
line and ask whether or not the spy is currently at that location. That is, you will ask a question like "Is the spy currently at location 27?" and you will get a yes/no answer. Devise an algorithm
that will eventually find the spy.
Average clan size
[I got this problem from Ernie Cohen, who I think had made it up. Itay Neeman suggested the use of "clans" instead of Ernie's original "families", because "clans" has a stronger connotation of the
groups being disjoint.]
The people in a country are partitioned into clans. In order to estimate the average size of a clan, a survey is conducted where 1000 randomly selected people are asked to state the size of the clan
to which they belong. How does one compute an estimate average clan size from the data collected?
Colored balls in boxes
[I got this little problem from Leonardo de Moura.]
There are three boxes, one containing two black balls, another containing two white balls, and another containing one black and one white ball. You select a random ball from a random box and you find
you selected a white ball. What is the probability that the other ball in the same box is also white?
Mixed up airplane seats
[I got this problem from Rajeev Joshi, who I think said he heard it from Jay Misra.]
An airplane has 50 seats, and its 50 passengers have their own assigned seats. The first person to enter the plane ignores his seat assignment and instead picks a seat on random. Each subsequent
person to enter the plane takes her assigned seat, if available, and otherwise chooses a seat on random. What is the probability that the last passenger gets to sit in her assigned seat?
Multiples in the Fibonacci series
[Carroll Morgan told me this puzzle.]
Prove that for any positive K, every K^th number in the Fibonacci sequence is a multiple of the K^th number in the Fibonacci sequence.
More formally, for any natural number n, let F(n) denote Fibonacci number n. That is, F(0) = 0, F(1) = 1, and F(n+2) = F(n+1) + F(n). Prove that for any positive K and natural n, F(n*K) is a multiple
of F(K).
Coins in a line
[Rajeev Joshi told me this problem. He got it from Jay Misra.]
Consider a game that you play against an opponent. In front of you are an even number of coins of possibly different denominations. The coins are arranged in a line. You and your opponent take turns
selecting coins. Each player takes one coin per turn and must take it from an end of the line, that is, the current leftmost coin or the current rightmost coin. When all coins have been removed, add
the value of the coins collected by each player. It is possible that you and your opponent end up with the same value (for example, if all coins have the same denomination). Develop a strategy where
you take the first turn and where your final value is at least that of your opponent (that is, don't let your opponent end up with coins worth more than your coins).
Determining the number of one hat
[This problem was told to me by Clark Barrett, who got it from his father. It may sound reminiscent of the Three hat colors problem, but it's different in many ways.]
N people team up and decide on a strategy for playing this game. Then they walk into a room. On entry to the room, each person is given a hat on which one of the first N natural numbers is written.
There may be duplicate hat numbers. For example, for N=3, the 3 team members may get hats labeled 2, 0, 2. Each person can see the numbers written on the others' hats, but does not know the number
written on his own hat. Every person then simultaneously guesses the number of his own hat. What strategy can the team follow to make sure that at least one person on the team guesses his hat number
Voting on how to distribute coins
[This problem was communicated to me by Sophia Drossopoulou.]
100 coins are to be distributed among some number of persons, referred to by the labels A, B, C, D, .... The distribution works as follows. The person with the alphabetically highest label (for
example, among 5 people, E) is called the chief. The chief gets to propose a distribution of the coins among the persons (for example, chief E may propose that everyone get 20 coins, or he may
propose that he get 100 coins and the others get 0 coins). Everyone (including the chief) gets to vote yes/no on the proposed distribution. If the majority vote is yes, then that’s the final
distribution. If there’s a tie (which there could be if the number of persons is even), then the chief gets to break the tie. If the majority vote is no, then the chief gets 0 coins and has to leave
the game, the person with the alphabetically next-highest name becomes the new chief, and the process to distribute the 100 coins is repeated among the persons that remain. Suppose there are 5
persons and that every person wants to maximize the number of coins that are distributed to them. Then, what distribution should chief E propose?
[This problem and its solution caused my niece Sarah Brown to send me the following article from The Economist, which considers a human aspect of situations like these.]
Subsequence of coin tosses
[I got this problem from Joe Morris, who created it.]
Each of two players picks a different sequence of two coin tosses. That is, each player gets to pick among HH, HT, TH, and TT. Then, a coin is flipped repeatedly and the first player to see his
sequence appear wins. For example, if one player picks HH, the other picks TT, and the coin produces a sequence that starts H, T, H, T, T, then the player who picked TT wins. The coin is biased, with
H having a 2/3 probability and T having a 1/3 probability. If you played this game, would you want to pick your sequence first or second?
Children and light switches
[I got this problem from Mariela Pavlova.]
A room has 100 light switches, numbered by the positive integers 1 through 100. There are also 100 children, numbered by the positive integers 1 through 100. Initially, the switches are all off. Each
child k enters the room and changes the position of every light switch n such that n is a multiple of k. That is, child 1 changes all the switches, child 2 changes switches 2, 4, 6, 8, …, child 3
changes switches 3, 6, 9, 12, …, etc., and child 100 changes only light switch 100. When all the children have gone through the room, how many of the light switches are on?
Finding a counterfeit coin
[Ernie Cohen sent me this puzzle, and I also heard it from a student who got it from Ernie at Marktoberdorf.]
You have 12 coins, 11 of which are the same weight and one counterfeit coin which has a different weight from the others. You have a balance that in each weighing tells you whether the two sides are
of equal weight, or which side weighs more. How many weighings do you need to determine: which is the counterfeit coin, and whether it weighs more or less than the other coins. How?
Cutting cheese
[I got this problem many years ago, likely from Jan van de Snepscheut.]
You're given a 3x3x3 cube of cheese and a knife. How many straight cuts with the knife do you need in order to divide the cheese up into 27 1x1x1 cubes?
Fair soccer championship
[I got this question from Sergei Vorobyov.]
The games played in the soccer world championship form a binary tree, where only the winner of each game moves up the tree (ignoring the initial games, where the teams are placed into groups of 4, 2
of which of which go onto play in the tree of games I just described). Assuming that the teams can be totally ordered in terms of how good they are, the winner of the championship will indeed be the
best of all of the teams. However, the second best team does not necessarily get a second place in the championship. How many additional games need to be played in order to determine the second best
Path on the surface of the Earth
[I must have heard this problem ages ago, but as I remembered it, one was always satisfied after finding just one solution. It was a math professor at the Kaiserslautern Technical University who
brought asking for all solutions to my attention.]
Initially, you're somewhere on the surface of the Earth. You travel one kilometer South, then one kilometer East, then one kilometer North. You then find yourself back at the initial position.
Describe all initial locations from which this is possible.
Random point in a circle
[I heard this nice problem from Sumit Gulwani.]
You're given a procedure that with a uniform probability distribution outputs random numbers between 0 and 1 (to some sufficiently high degree of precision, with which we need not concern ourselves
in this puzzle). Using a bounded number of calls to this procedure, construct a procedure that with a uniform probability distribution outputs a random point within the unit circle.
Mixing vinegar and oil
[I read this problem in a puzzle book I have.]
You have two jars. One contains vinegar, the other oil. The two jars contain the same amount of their respective fluid.
Take a spoonful of the vinegar and transfer it to the jar of oil. Then, take a spoonful of liquid from the oil jar and transfer it to the vinegar jar. Stir. Now, how does the dilution of vinegar in
the vinegar jar compare to the dilution of oil in the oil jar?
Psycho killer
[I got this problem from Carroll Morgan.]
A building has 16 rooms, arranged in a 4x4 grid. There is a door between every pair of adjacent rooms ("adjacent" meaning north, south, west, and east, but no diagonals). Only the room in the
northeast corner has a door that leads out of the building.
In the initial configuration, there is one person in each room. The person in the southwest corner is a psycho killer. The psycho killer has the following traits: If he enters a room where there is
another person, he immediately kills that person . But he also cannot stand the site of blood, so he will not enter any room where there is a dead person.
As it happened, from that initial configuration, the psycho killer managed to get out of the building after killing all the other 15 people. What path did he take?
A special squarish age
[Chuck Chan created this little puzzle. Greg Nelson suggested introducing the name "squarish" in order to simplify the problem statement.]
Let's say that a number is squarish if it is the product of two consecutive numbers. For example, 6 is squarish, because it is 2*3.
A friend of mine at Microsoft recently had a birthday. He said his age is now squarish. Moreover, since the previous time his age was a squarish number, a squarish number of years has passed. How
many years would he have to wait until his age would have this property again?
Passing alternating numbers of coins around
[This problem appears as a sample question on the web page for the Putnam exam.]
A game is played as follows. N people are sitting around a table, each with one penny. One person begins the game, takes one of his pennies (at this time, he happens to have exactly one penny) and
passes it to the person to his left. That second person then takes two pennies and passes them to the next person on the left. The third person passes one penny, the fourth passes two, and so on,
alternating passing one and two pennies to the next person. Whenever a person runs out of pennies, he is out of the game and has to leave the table. The game then continues with the remaining people.
A game is terminating if and only if it ends with just one person sitting at the table (holding all N pennies). Show that there exists an infinite set of numbers for which the game is terminating.
The exact batting average
[I heard this problem from Bertrand Meyer, who had heard it was once given on the Putnam exam.]
At some point during a baseball season, a player has a batting average of less than 80%. Later during the season, his average exceeds 80%. Prove that at some point, his batting average was exactly
Also, for which numbers other than 80% does this property hold?
Summing pairs of numbers to primes
[I got this problem from Jay Misra, who got it from Gerard Huet.]
For any even number N, partition the integers from 1 to N into pairs such that the sum of the two numbers in each pair is a prime number.
Hint: Chebyshev proved that the following property (Bertrand's Postulate) holds: for any k > 1, there exists a prime number p in the range k < p < 2*k.
Right triangle with a 23
[Madan Musuvathi asked me this question.]
Find two positive integers that together with 23 are the lengths of a right triangle.
[Madan first told me 17, which I could solve right away, because I had just finished reading Mark Haddon's novel The curious incident of the dog in the night-time, which toward the end mentions that
a triangle with sides n^2+1, n^2-1, and 2*n, where n>1, is a right triangle. Madan then asked me about 23.]
[A follow-up question.]
There's a simple technique that, given any odd positive integer, allows you to figure out the other two integer sides of a right triangle in your head (or with pen and paper if the numbers get too
large). Find this technique.
Hint: Think of every number as a multiset of prime factors, so that multiplying the prime factors gives you the number. Move one of the addends of the Pythagorean Theorem to the other side and factor
it (a technique I recently learned from Raymond Boute).
The worm and the rubber band
[Jay Misra told me this problem.]
A rubber band (well, a rubber string, really) is 10 meters long. There's a worm that starts at one end and crawls toward the other end, at a speed of 1 meter per hour. After each hour that passes,
the rubber string is stretched so as to become 1 meter longer than it just was. Will the worm ever reach the other end of the string?
Placing coins on a table
[I got this problem from Amit Rao.]
Two players are playing a game. The game board is a circular table. The players have access to an ample supply of equal-sized circular coins. The players alternate turns, with each turn adding a
single coin to the table. The coins are not allowed to overlap. Once a coin is placed on the table, it is not allowed to be moved. The player who has no place to put his next coin loses. Develop a
winning strategy for the player who starts. (The table is large enough to accommodate at least one coin.)
Determining a hidden digit
[I got this problem from Olean Brown, who had pointed me to the following web page that claims to read your mind.]
Think of a positive integer, call it X. Shuffle the decimal digits of X, call the resulting number Y. Subtract the smaller of X,Y from the larger, call the difference D. D has the following property:
Any non-zero decimal digit of D can be determined from the remaining digits. That is, if you ask someone to hide any one of the non-zero digits in the decimal representation of D, then you can try to
impress the other person by figuring out the hidden digit from the remaining digits. How is this done? Why does it work?
Boris and Natasha
[I got this problem from Todd Proebsting.]
Boris and Natasha live in different cities in a country with a corrupt postal service. Every box sent by mail is opened by the postal service, the contents stolen, and the box never delivered. Except
: if the box is locked, then the postal service won't bother trying to open it (since there are so many other boxes whose contents are so much easier to steal) and the box is delivered unharmed.
Boris and Natasha each has a large supply of boxes of different sizes, each capable of being locked by padlocks. Also, Boris and Natasha each has a large supply of padlocks with matching keys. The
padlocks have unique keys. Finally, Boris has a ring that he would like to send to Natasha. How can Boris send the ring to Natasha so that she can wear it (without either of them destroying any locks
or boxes)?
Burning ropes to measure time
[I first got this problem from Ernie Cohen. Apparently, it has appeared as a Car Talk Puzzler, but I've been unable to find it on their web site.]
Warm-up: You are given a box of matches and a piece of rope. The rope burns at the rate of one rope per hour, but it may not burn uniformly. For example, if you light the rope at one end, it will
take exactly 60 minutes before the entire rope has burnt up, but it may be that the first 1/10 of the rope takes 50 minutes to burn and that the remaining 9/10 of the rope takes only 10 minutes to
burn. How can you measure a period of exactly 30 minutes? You can choose the starting time. More precisely, given the matches and the rope, you are to say the words "start" and "done" exactly 30
minutes apart.
The actual problem: Given a box of matches and two such ropes, not necessarily identical, measure a period of 15 minutes.
Flipping cards
[I have heard several versions of this problem. I first heard it from Bertrand Meyer, who got the problem from Yuri Gurevich.]
You're given a regular deck of 52 playing cards. In the pile you're given, 13 cards face up and the rest face down. You are to separate the given cards into two piles, such that the number of face-up
cards in each pile is the same. In separating the cards, you're allowed to flip cards over. The catch: you have to do this in a dark room where you cannot determine whether a card is face up or face
Three hat colors
[I think I got this puzzle from Lyle Ramshaw, who I think got it from some collection of problems or maybe the American Mathematical Monthly.]
A team of three people decide on a strategy for playing the following game. Each player walks into a room. On the way in, a fair coin is tossed for each player, deciding that player's hat color,
either red or blue. Each player can see the hat colors of the other two players, but cannot see her own hat color. After inspecting each other's hat colors, each player decides on a response, one of:
"I have a red hat", "I had a blue hat", or "I pass". The responses are recorded, but the responses are not shared until every player has recorded her response. The team wins if at least one player
responds with a color and every color response correctly describes the hat color of the player making the response. In other words, the team loses if either everyone responds with "I pass" or someone
responds with a color that is different from her hat color.
What strategy should one use to maximize the team's expected chance of winning?
For example, one possible strategy is to single out one of the three players. This player will respond "I have a red hat" and the others will respond "I pass". The expected chance of winning with
this strategy is 50%. Can you do better? Provide a better strategy or prove that no better strategy exists.
[Here's a related problem, which I got from Jim Saxe.]
In this variation, the responses are different. Instead of "red", "blue", "pass", a response is now an integer, indicating a bet on having the hat color red. Once the responses have been collected,
the team's score is calculated as follows: Add the integer responses for those players wearing red hats, and subtract from that the integers of those players wearing blue hats. For example, if the
three players respond with 12, -2, -100 while wearing blue, red, blue, respectively, then the team's score is (-2) - (12) - (-100) = 90. The team wins if and only if its score is strictly positive.
For example, any strategy used in the first game can be used with this second game by replacing "I have a red hat" with 1, "I have a blue hat" with -1, and "I pass with 0". Such a strategy wins
anytime the strategy would have produced a win in the first game; plus, this strategy may win in some cases where the strategy would not produce a win in the first game. For example, for hat colors
red, red, red, the strategy "red", "red", "blue" loses in the first game, whereas 1, 1, -1 still wins in the second game. Hence, playing this second game can only increase the team's expected chance
of winning.
[Further generalizations.]
Of course, you can generalize these two problems from 3 players to N players. The solution to the first problem with N players may require more mathematical sophistication than the solution to the
second problem with N players.
The line of persons with hats
[Ernie Cohen told me this problem.]
100 persons are standing in line, each facing the same way. Each person is wearing a hat, either red or blue, but the hat color is not known to the person wearing the hat. In fact, a person knows the
hat color only of those persons standing ahead of him in line.
Starting from the back of the line (that is, with the person who can see the hat colors of all of other 99 persons), in order, and ending with the person at the head of the line (that is, with the
person who can see the hat color of no one), each person exclaims either "red" or "blue". These exclamations can be heard by all. Once everyone has spoken, a score is calculated, equal to the number
of persons whose exclamation accurately describes their own hat color.
What strategy should the 100 persons use in order to get as high a score as possible, regardless of how the hat colors are assigned? (That is, what strategy achieves the best worst-case score?)
For example, if everyone exclaims "red", the worst-case score is 0. If the first 99 persons exclaim the color of the hat of the person at the head of the line and the person at the head of the line
then exclaims the color he has heard, the worst-case score is 1. If every other person exclaims the hat color of the person immediate in front and that person then repeats the color he has just
heard, then the worst-case score is 50. Can you do better?
[Here's a generalization of the problem.]
Instead of using just red and blue as the possible hat colors and exclamations, use N different colors.
The prisoners and the switch
[Tom Ball told me (a close variation of) this problem. The problem has been featured as a Car Talk Puzzler under the name Prison Switcharoo (beware: the Car Talk problem page also contains an
N prisoners get together to decide on a strategy. Then, each prisoner is taken to his own isolated cell. A prison guard goes to a cell and takes its prisoner to a room where there is a switch. The
switch can either be up or down. The prisoner is allowed to inspect the state of the switch and then has the option of flicking the switch. The prisoner is then taken back to his cell. The prison
guard repeats this process infinitely often, each time choosing fairly among the prisoners. That is, the prison guard will choose each prisoner infinitely often.
At any time, any prisoner can exclaim "Now, every prisoner has been in the room with the switch". If, at that time, the statement is correct, all prisoners are set free; if the statement is not
correct, all prisoners are immediately executed. What strategy should the prisoners use to ensure their eventual freedom?
(Just in case there's any confusion: The initial state of the switch is unknown to the prisoners. The state of the switch is changed only by the prisoners.)
As a warm-up, you may consider the same problem but with a known initial state of the switch.
Table with four coins
[This problem has crossed the desk of Jan van de Snepscheut, but it may have been due to someone else.]
A square table has a coin at each corner. Design an execution sequence, each of whose steps consists of one of the following operations:
• ONE: The operation chooses a coin (possibly a different one with each execution of the operation) and flips it.
• SIDE: The operation chooses a side of the table and flips the two coins along that side.
• DIAG: The operation chooses a diagonal of the table and flips the two coins along that diagonal.
such that at some point during the execution (not necessarily at the end), a state where all coins are turned the same way (all heads or all tails) obtains.
Stabilizing nodes from an anchor
[I got this problem from Jay Misra, who had received it from Edsger Dijkstra. This particular description of the problem borrows from a description given by Michael Jackson.]
In a finite, undirected, connected graph, an integer variable v(n) is associated with each node n. One node is distinguished as the anchor. An operation OP(n) is defined on nodes:
if node n is the anchor, then do nothing,
else set v(n) to the value 1 + min{v(m)}, where m ranges over all neighbors of n that are distinct from n.
An infinite sequence of operations <OP(n),OP(m), ...> is executed, the node arguments n, m, ... for the operations being chosen arbitrarily and not necessarily fairly. Show that eventually all v(n)
stabilize. That is, that after some finite prefix of the infinite sequence of operations, no further operation changes v(n) for any node n.
Points on a circle
[This problem was told to me by Dave Detlefs, who got it from the following impressive collection of math problems.]
Given N points randomly distributed around the circumference of a circle, what is the probability that all N points lie on the same semi-circle?
The electrician problem
[I learned about this problem from Greg Nelson, who phrased the problem in terms of a tall building and elevator rides. Here, I instead say a mountain and helicopter rides, which is the way Lyle
Ramshaw had heard the problem and which more forcefully emphasizes the price of each ride. As a computer scientist, I like this problem a lot, because of the variety of solutions of different
computational complexities.]
You're an electrician working at a mountain. There are N wires running from one side of the mountain to the other. The problem is that the wires are not labeled, so you just see N wire ends on each
side of the mountain. Your job is to match these ends (say, by labeling the two ends of each
wire in the same way).
In order to figure out the matching, you can twist together wire ends, thus electrically connecting the wires. You can twist as many wire ends as you want, into as many clusters as you want, at the
side of the mountain where you happen to be at the time. You can also untwist the wire ends at the side of the mountain where you're at. You are equipped with an Ohm meter, which lets you test the
connectivity of any pair of wires. (Actually, it's an abstract Ohm meter, in that it only tells you whether or not two things are connected, not the exact resistance.)
You are not charged [no pun intended] for twisting, untwisting, and using the Ohm meter. You are only charged for each helicopter ride you make from one side of the mountain to the other. What is the
best way to match the wires? (Oh, N>2, for there is no solution when N=2.)
The hidden card
[I learned about this problem from Lyle Ramshaw. See also puzzles 19 and 20 on the following large collection of mathematical puzzles.]
In this problem, you and a partner are to come up with a scheme for communicating the value of a hidden card. The game is played as follows:
• Your partner is sent out of the room.
• A dealer hands you 5 cards from a standard 52 card deck.
• You look at the cards, and hand them back to the dealer, one by one, in whatever order you choose.
• The dealer takes the first card that you hand her and places it, face up, in a spot labeled "0"'. The next three cards that you hand her, she places, similarly, in spots labeled "1", "2", and
"3". The last card that you hand her goes, face down, in a spot labeled "hidden". (While you control the order of the cards, you have no control over their orientations, sitting in their spots;
so you can't use orientation to transmit information to your partner.)
• Your partner enters the room, looks at the four face-up cards and the spots in which they lie and, from that information (and your previously-agreed-upon game plan), determines the suit and value
of the hidden card.
Question: What is the foolproof scheme that you and your partner settled on ahead of time?
As a follow-up question, consider the same problem but with a 124-card deck.
Knight, knave, commoner
[Communicated to me by Carroll Morgan.]
A king has a daughter and wants to choose the man she will marry. There are three suitors from whom to choose, a Knight, a Knave, and a Commoner. The king wants to avoid choosing the Commoner as the
bridegroom, but he does not know which man is which. All the king knows is that the Knight always speaks the truth, the Knave always lies, and the Commoner can do either. The king will ask each man
one yes/no question, and will then choose who gets to marry the princess. What question should the king ask and how should he choose the bridegroom?
[A follow-up question posed by Lyle Ramshaw.]
Suppose the three suitors know each other (an assumption that's not needed in the original problem). Then find a new strategy for the king where the king only needs to ask a question of any two of
the three suitors in order to pick the bridegroom.
[Another variation of the problem.]
Find a strategy for the king where the king can ask questions of only one suitor, but can ask two questions of that suitor.
[And another (at first sight incredible) follow-up question communicated to me by both Jim Saxe and Pierre Nallet.]
Find a strategy for the king where the king can ask only one yes/no question and only of one suitor.
Back to the homepage of Rustan Leino. | {"url":"http://research.microsoft.com/en-us/um/people/leino/puzzles.html","timestamp":"2014-04-17T15:48:13Z","content_type":null,"content_length":"73168","record_id":"<urn:uuid:8f260e43-37ab-448b-ae5b-51e2a7e6eb08>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00264-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculating velocity from acceleration
December 8th 2008, 08:15 AM #1
Dec 2008
Calculating velocity from acceleration
Hi everyone.
I'm working with a 1 dimensional accelerometer which records body movements and I'm outputting the information into a graph in Excel.
But now I am trying to figure out velocity from this information and I'm having real trouble coming up with an equation that will translate variable acceleration into velocity in my spreadsheet.
Any help would be greatly appreciated.
It depends if you "know" things like : is it uniformly accelerated? etc.
In general, you would plot acceleration vs. time.
Then fit a polynomial to your data set.
Compute the integral to know the area under the curve.
To find the constant, you have to know initial velocity.
Acceleration is not uniform, it is variable and the initial velocity is 0 as the motion I am tracking is a single jump on the spot, starting and finishing at rest.
Then you must do as I told you, that is you would plot acceleration vs. time.
You want to know the area under the curve.
The best way should be to fit a polynomial to your data set.
Compute the integral to know the area under the curve.
December 8th 2008, 08:28 AM #2
December 8th 2008, 08:34 AM #3
Dec 2008
December 8th 2008, 08:38 AM #4 | {"url":"http://mathhelpforum.com/calculus/63928-calculating-velocity-acceleration.html","timestamp":"2014-04-19T12:25:03Z","content_type":null,"content_length":"38475","record_id":"<urn:uuid:8e79ca3e-d79e-4568-a4fe-71b65bd743c3>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00222-ip-10-147-4-33.ec2.internal.warc.gz"} |
Infinite Universe?
That's the problem with infinities, you get swamped with paradoxes. Fortunately, our universe has a temporal cutoff, otherwise known as the finite age of the observable universe, which spares us such
indignities. We are causally disconnected from any regions beyond this boundary.
Thank you Hubble. One thing you need to keep in mind is that the further The Hubble Telescope looks out into space, the further back in time it is looking too. The furthest Hubble has seen is a
galaxy that is roughly 13 billion light years away.This means it took the electromagnetic wave (light wave) that Hubble is receiving 13 billion years to arrive. We are in fact seeing that Galaxy as
it was 13 billion years ago.
With that said, we would now have to review Hubble's Law. Hubble's Law says that we are sitting in a uniformly expanding universe and the expansion looks the same, regardless of your location. Also
Hubble's Law states that the Speed at which a galaxy is traveling away from is is proportional to its distance. This is proven using the Doppler Effect. All galaxies in the universe (except for
Andromeda and any other galaxy that is gravitationally locked with the Milky Way) will be redshifted.
Apparently, nothing can travel faster than the speed of light right? Well there is a theory that the expansion of space itself can travel faster. Which would make sense giving the fact that the
expansion of space is not an actual particle that has mass.
I would think that if this is true, then galaxies that are far enough away would eventually get to the point were it was traveling faster than the speed of light away from us, Canceling out the
velocity of light itself.
Imagine being in the back of a pick up truck, traveling 50 kph in one direction. If you were to throw a baseball in the opposite direction, for that baseball to have any advancement in the opposite
direction, you would have to throw that ball faster than 50 kph. If you throw the ball exactly the same speed as the truck but in the opposite direction, shouldn't it not move position from spot of
release from your hand?
I would think that the expansion of the universe would work the same way. Anything traveling the speed of light way from us would cancel out the velocity of light traveling towards us. This would
make it to where were would see nothing at all because there are no light waves traveling towards us at this point.
So I would think that the OBSERVABLE Universe is not the same size as the ACTUAL Universe itself. Please correct me if I am wrong at any point of this. Thanks | {"url":"http://www.physicsforums.com/showthread.php?s=3b3c70515fcc33dccc73bd3f651d4277&p=4550128","timestamp":"2014-04-20T21:26:15Z","content_type":null,"content_length":"75356","record_id":"<urn:uuid:74b6dd43-f0f8-4143-a002-2aa9cf15dc0e>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00610-ip-10-147-4-33.ec2.internal.warc.gz"} |
Polynomial Fit :: Data Analysis and Statistics (Mathematics)
Polynomial Fit
A first try in fitting the census data might be a simple polynomial fit. Two MATLAB functions help with this process.
Curve Fitting Function Summary
┃ Function │ Description ┃
┃ polyfit │ Polynomial curve fit. ┃
┃ polyval │ Evaluation of polynomial fit. ┃
The MATLAB polyfit function generates a "best fit" polynomial (in the least squares sense) of a specified order for a given set of data. For a polynomial fit of the fourth-order
The warning arises because the polyfit function uses the cdate values as the basis for a matrix with very large values (it creates a Vandermonde matrix in its calculations - see the polyfit M-file
for details). The spread of the cdate values results in scaling problems. One way to deal with this is to normalize the cdate data.
Preprocessing: Normalizing the Data
Normalization is a process of scaling the numbers in a data set to improve the accuracy of the subsequent numeric computations. A way to normalize cdate is to center it at zero mean and scale it to
unit standard deviation:
Now try the fourth-degree polynomial model using the normalized data:
Evaluate the fitted polynomial at the normalized year values, and plot the fit against the observed data points:
Another way to normalize data is to use some knowledge of the solution and units. For example, with this data set, choosing 1790 to be year zero would also have produced satisfactory results.
Case Study: Curve Fitting Analyzing Residuals
© 1994-2005 The MathWorks, Inc. | {"url":"http://matlab.izmiran.ru/help/techdoc/math/datafun14.html","timestamp":"2014-04-19T14:38:22Z","content_type":null,"content_length":"5420","record_id":"<urn:uuid:c658e7e7-1c98-43a6-b088-301a82285948>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00549-ip-10-147-4-33.ec2.internal.warc.gz"} |
Where Did Our Numbers Come From?
You can imagine how hard it must have been to do business in ancient times, before numbers were invented. To show a quantity, tradesmen used their fingers or bags of stones.
For instance, a person who wanted to buy five sheep might put five stones in a bag. The word calculate, in fact, comes from a Latin word for “stone.”
Ancient people also scratched lines in the dirt or on stone, one line for each object to be counted. These lines became the earliest numbers, around the time that writing began.
Since the fingers were used in counting, it’s not surprising that number systems came to be based on the number ten. The number systems of the ancient Egyptians, Greeks, and Romans were based on ten.
But systems based on other numbers, such as five, 12, or 20, were used in some places.
The number system we use today, which is also based on the number ten, came originally from India, and was brought to Europe by the Arabs. Therefore our numbers are known as Hindu-Arabic numbers. The
earliest known use of these numbers in Europe was in the 10th century.
Some primitive tribes in South America still have no numbers at all!
1. Megan says
I think I learned this in Social Studies in 8th grade. I like Roman Numerals. Once you look at them and study them, it’s easy to pick up on. Like M is 1000. We’re in 2000, so it would be MM then.
Since it’s 2010, it would be MMX. God I love Roman Numerals!!
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Projections used in State Plane Coordinate Systems
• "To convert geodetic positions of a portion of the Earth's surface to plane rectangular coordinates, points are projected mathematically from the ellipsoid to some imaginary developable surface
-a surface that can conceptually be developed or 'unrolled and laid out flat' without distortion of shape or size. A rectangular grid can be superimposed on the developed plane surface and the
positions of points in the plane specified with respect to X and Y grid axes. A plane grid developed using this mathematical process is called a map projection.... Today, two of the most commonly
used mapping projections are the Lambert conformal conic and the Transverse Mercator projections." (Elementary Surveying, 12th Edition by Ghilani and Wolf, p. 580)
Types of Projections
State Plane Coordinate System (SPCS)
• "...points couldn't be projected from the ellipsoid to developable surfaces without introducing distortions in the lengths of lines or the shapes of areas. However, these distortions are held to
a minimum by selected placement of the cone or cylinder secant to the ellipsoid, by choosing a conformal projection (one that preserves true angular relationships around points in a small
region), and also by limiting the zone size or extent of coverage on the Earth's surface for any particular map projection. If the width of zones is held to a maximum of 158 mi (254 km), and if
two-thirds of this zone width is between the secant lines, distortions (differences in line lengths on the two surfaces) are kept to 1 part in 10,000 or less. The NGS intended this accuracy in
its development of the state plane coordinate systems." (Elementary Surveying, 12th Edition by Ghilani and Wolf, p. 581-2)
• Clark County uses StatePlane Coordinate- Nevada East Zone (NV-E) along with other local agencies (e.g. City of Las Vegas, Henderson, North Las Vegas, and so on)
• Map of all Stateplane coordinate zones is included with ArcMap, depending upon where you installed the program, e.g. C:\Program Files\ArcGIS\Reference Systems\usstpln83.shp (download zip of
shapefiles from UNLV, usstpln83.zip - Geographic Coordinate System - GCS_WGS_1984)
• State Plane Coordinate System of 1983 NOAA Manual NOS NGS 5 by James E. Stem
□ "A new figure of the Earth, the Geodetic Reference System of 1980 (GRS 80), which approximates the Earth's true size and shape, supplied a better fit than the Clarke 1866 spheroid, the
reference surface used with NAD 27." (ibid p. 2)
□ "The ellipsoid that forms the basis of NAD 83, and consequently the SPCS 83, is identified as the Geodetic Reference System of 1980 (GRS 80). GRS 80 provides an excellent global approximation
of the Earth's surface. The Clarke spheroid of 1866, as used for NAD 27 approximated only the conterminous United States. Because the geoid separation at point MEADES RANCH was assumed equal
to zero, a translation exists between ellipsoids. The ellipsoid change is the major contributor of the coordinate shift from NAD 27 to NAD 83." (ibid p. 12)
• NAD 27 - U.S. coast and Geodetic Survey (USC&GS) Special Publication 235 - The State Coordinate Systems
□ Shapefile of NAD27 Zones (usstpln27.zip)
□ Shapefile of NAD83 Zones (usstpln83.zip)
□ The State Coordinate Systems (A Manual for Surveyors) older version but better quality (The State Coordinate Systems)
□ State Zone: Nevada East
□ Grid: Transverse Mercator
□ Central meridian = 115°35' (-115.583333°)
□ Scale Ratio = 1:10,000
□ Origin: Lat=34°45' (34.75°) Long=115°35'
□ False Easting x(ft)=500,000
□ False Northing y(ft)=0
□ x' of lines of exact scale (ft) = 295,700
• U.S. Coast and Geodetic Survey. Manual of Traverse Computation on the Transverse Mercator Grid by Oscar S. Adams, Senior Mathematician and Charles N. Claire, Associate Mathematician. GPO,
Washington, DC, 1935. 199 pages. Special Publication No. 195.
State Plane Coordinates Presentation by Dr. Ghilani
National Geospatial-Intelligence Agency Geodesy and Geophysics
• Geodesy for the Layman ( TM8358_1.pdf or download a copy from UNLV Geo4lay.pdf)
• TM8358.1: Datums, Ellipsoids, and Grid Reference Systems ( TM8358_1.pdf or download a copy from UNLV TM8358_1.pdf)
Fundamentals of the State Plane Coordinate Systems by Joseph F. Dracup, Sept 1974. National Geodetic Survey
Map Projections: A Working Manual by John P. Snyder. USGS Professional Paper 1395. Washington, D.C.: USGS, 1993.
• GRS 80
□ Equatorial Radius/Semiaxis, a = 6,378,137 meters
□ Polar Radius/Semiaxis, b = 6,356,752.3 meters
□ Flattening, f = 1/298.257
□ "in computations if the ellipsoid is assumed a sphere, its radius is usually taken such that its volume is the same as the reference ellipsoid. It is computed from (a^2b)^1/3. For the GRS80
ellipsoid, its rounded value is 6,371,000 meters." (Elementary Surveying, 12th Ed, Ghilani and Wolf, p. 523)
• WGS 84
□ Equatorial Radius, a = 6,378,135 meters
□ Polar Radius, b = 6,356,750.5 meters
□ Flattening, f = 1/298.26
• Nevada East Zone Map Projectsion: A Working Manual, p. 53 and 374
□ Transverse Mercator Projection
□ Central meridian = 115°35' West
☆ ESRI uses -115.583333333333300000 decimal degrees = -115°35'
□ Scale reduction = 1:10,000
☆ ESRI uses a scale factor of 0.999900000000000010
☆ "Lines of contact. Any single meridian for the tangent project. For the secant projection, two almost parallel lines equidistant from the central meridian...Accurate scale along the
central meridian if the scale factor is 1.0. If it is less than 1.0, there are two straight lines with accurate scale equidistant from and on each side of the central meridian." (ESRI
ArcGIS Transerve Mercator)
□ Origin (latitude) = 34°45' North
☆ ESRI uses 34.7500 decimal degrees = 34°45'
□ Coordinates of Origin (meters): False Easting x=200,000 and False Northing y=8,000,000
□ "State plane coordinate systems are generally designed to have a scale error maximum of about 1 unit in 10,000. Suppose you calculated the Cartesian distance (using the Pythagorean theorem)
between two points represented in a state plane coordinate system to be exactly 10,000 meters. Then, with a perfect tape measure, pulled tightly across an idealized planet, you would be
assured that the measured result would differ by no more than 1 meter from the calculated one. The possible error with the UTM coordinate system may be larger: 1 in 2500." (Introducing
Geographic Information Systems with ArcGIS, 2nd Edition by Michael Kennedy, p. 18)
□ Datum: NAD 1983 (Conus) (Mol) is used on the Trimble TSC2
□ GPS Course Lesson 6: Two-Coordinate Systems and Heights by Jan Van Sickle, Senior Lecturer
□ "...the projection of points from the Earth's surface onto a reference ellipsoid and finally onto flat maps..." (ibid)
□ Stateplane Coordinates in USA use Secant Projections to minimize distortion by providing 2 lines of intersection instead of one line with the Tangent case. Secant Projection intersect the
ellipsoid at two areas and these two lines are of exact scale (also known as standard lines) to the ellipsoid.
□ Ellipsoidal lengths = geodetic distances
□ Lengths on the map projection surface = grid distances
□ Grid North is parallel to the Central Meridian. Convergence is the angle between Grid North and Geodetic North
• False Easting and Northing
□ False easting is a linear value applied to the origin of the x coordinates.
□ False northing is a linear value applied to the origin of the y coordinates.
□ False easting and northing values are usually applied to ensure that all x and y values are positive.
☆ False easting and false northing adjustments by Margaret M. Maher
○ "False easting and false northing values are sometimes inserted into a projection file in order to make all the x- and y-coordinate values across the area of the data positive
numbers. False easting and false northing values can also be used to adjust the position of the data in either the east-west or north-south direction in order to align the data.
Making the false easting value in the projection file larger will adjust the position of the data to the west, moving the data to the left in the ArcMap display. Making the false
easting value in the project file smaller will adjust the position of the data east, to the right in the ArcMap display.
Adjustments to the false northing value in the projection file will move the data display north or south, though they are not as intuitive as the false easting adjustments. Making the
false northing value in the projection file larger will move the data display south in the ArcMap window. Making the false northing value smaller will move the data display north in
the ArcMap window.
Keep these adjustments in mind while creating the custom projection file to align your data in ArcMap." (Lining Up Data in ArcGIS by Margaret M. Maher, p. 55, ISBN 978-1-58948-249-4)
☆ "A constant E[0] is adopted to offset the N grid axis from the central meridian and make E coordinates of all points positive. Similarly, a constant N[b] can be adopted to offset the E
grid axis from the southern edge of the projection." (Elementary Surveying, 12th Edition by Ghilani and Wolf, p. 587)
□ You can also use the false easting and northing parameters to reduce the range of the x or y coordinate values. For example, if you know all y values are greater than 5,000,000 meters, you
could apply a false northing of -5,000,000.
□ ESRI ArcGIS Desktop 10 - Projection parameters
• False Northing - NAD27 vs NAD83
□ According to David Doyle with NGS, metadata is hard to obtain on current surveying work but was extremely difficult in the 1980s. So, the State of Nevada wanted to ensure when surveyors where
working with the two different Datums (NAD27 and NAD83) that a surveyor could easily tell from the coordinate values, which Datum is being used. So the NAD83 coordinates have a 8,000,000
meters added to the Northing (Y) coordinate values.
□ NAD27 - Nevada East Zone uses a False Northing of 0 ft
□ NAD83 - Nevada East Zone uses a False Northing of 8,000,000 meters (26,246,666.66666666 feet)
□ Notice how the same data (city boundaries in Clark County NV) do not overlay, that is the NAD83 is shifted 8,000,000 meters north of the NAD27 layer
• Conversion from NAD83/27 to Geodetic/Geographic
• Central Meridian (Longitude)
□ Notes from Earl F. Burkholder, PS, PE with Global COGO, Inc., Las Cruces NM 88003
☆ Central Meridian (Longitude) is a north/south line and secant lines in a transverse Mercator projection are parallel to the central meridian. Problem is Longitude lines are not parallel
because the all converge at the poles.
☆ The easting of a secant line will vary slightly from the south end of the state to the north end.
☆ secant line in state plane coordinates is always a constant distance from the central meridian.
How to draw the SPCS origin
• Step 0: download the World GeoReference Lines
• Step 1: Identify the SPCS Defining Parameters
□ If you have ArcGIS installed on your computer, view the projection file (.prg)
C:\Program Files\ArcGIS\Coordinate Systems\Projected Coordinate Systems\State Plane\NAD 1983\NAD 1983 StatePlane Nevada East FIPS 2701.prj
□ Elementary Surveying, 12th Edition by Ghilani and Wolf, Appendix F (and definitions on p. 586) U.S. State Plane Coordinate System Defining Parameters
• Step 2: Create a shapefile using WGS84 geographic coordinates (e.g. GCS_WGS_1984)
□ Use ArcCatalog to create a new shapefile. Assign it the GCS_WGS_1984 projection
• Step 3: create gratitcules
□ Using an ArcMap edit session, add the line features using Absolute X,Y to enter the latitude and longitude values
Scale Factors
• scale factors move distances from the stateplane grid to the ellipsoid
• "After a distance has been reduced to its ellipsoidal equivalent, it must then be scaled to its grid equivalent. This is accomplished by multiplying the ellipsoidal length of the line by an
appropriate scale factor." (Elementary Surveying, 12th Ed, Ghilani and Wolf, p. 599)
• Scale, Elevation, Grid, and Combined Factors Used in Instrumentation. Professional Surveyor Magazine - Feb 2006
Coordinate Systems (Geographic and Projected) in ArcMap
When adding data to ArcMap, will sometimes get a warning message "One or more layers is missing spatial reference information, Data from those layers cannot be projected"
● "If you add a layer that is in a projected coordinate system to ArcMap, and the coordinate system information is missing" will get that message. Much of the time this is not a problem. You can
still display and work with this data as long as ArcMap does not need to project it on the fly. ArcMap will not be able, however, to align this data with data in a different coordinate system."
[Ormsby 01, p. 340]
Map Projections
● Mathematical transformation of a model of the earth's shape (i.e. Oblate Spheroid) to a flat surface (grid). [Ormsby 01, p. 324]
● Can distort shape, area, distance, and direction
● Geographic Coordinates System (GCS)
□ based on a curved surface
☆ Sphere - less accurate approximation of the shape of the earth
☆ Spheroid - more accurate approximation of the shape of the earth (most widely used)
☆ Geoid - most accurate model of the shape of the earth
□ a measurement of a location on the earth's surface expressed in degrees of latitude and longitude. Tend to have a "taffy-pull appearance" when displaying [Ormsby 01, p. 331]
□ GCS includes: angular unit of measure (e.g. degrees), prime meridian (i.e. line of zero longitude which passes through Greenwich England), and a datum (e.g. position of spheroid relative to
the center of earth, typically use North American Datum of 1983 a.k.a. NAD83)
□ Latitude: horizontal lines (e.g. equator) and also known as parallels. Measurement values range from -90 to 90 degrees
□ Longitude: vertical lines, also known as merdians. Measurement values range from -180 to 180 degrees.
□ Degrees - 1/360th of a cirle
□ Minutes - 1/60th of a degree, or 60 minutes = 1 degree
□ Seconds - 1/60th of a minute, or 60 seconds = 1 minute
□ See ESRI Virutal Campus - Learning ArcGIS 9, Module 3 for more details
● Projected Coordinates
□ based on a flat surface
□ does NOT use spheriods, spheres, or geoids since these are approximations of the shape of the earth
□ also known as planar coordinates
□ a measurement of a location on the earth's surface expressed in a two-dimensional system that locates features based on their distance from an origin (0,0) along two axes.
● Map projections transform latitude and longitude to x,y coordinates in a projected coordinate system.
● Latitude and Longitude can located exact locations on the earth, but no uniform units of measurement (see figure on [Ormsby 01, p. 326])
● If all your GIS data is using the same coordinate system, don't have to worry about projections
● Empty data frames inherit the projection of the first layer added to it. [Ormsby 01, p. 333]
● On-the-fly Projections, [Ormsby 01, p. 328, 336]
● ArcMap determines if the coordinate system is geographic or projected by comparing the coordinates. Lat/Long values will be in the tens (Lat=36 degrees) and hundreds (Long=-115 degrees), where as
Stateplane coordinates hundred thousand (e.g. x=800,000) and tens of million (y=26,750,000)[Ormsby 01, p. 340]
□ On-the-fly projections are less mathematically rigorous than permanent projections done using the ArcToolbox Projection Wizard. [Ormsby 01, p. 330]
□ On-the-fly projections are defined by the Layer Properties. Note this doesn't change the actual file. Projection only applied to data frame. [Ormsby 01, p. 330]
□ "... a coordinate system is a framework for locating features on the earth's surface using either latitude-longitude or x,y values."
□ Works well when the data has the same geographic coordinate system (GCS). [Ormsby 01, p. 329]
□ To transform the coordinate location of a CAD file using coordinate values in ArcMap, see ESRI Article Number 20860
● Projection info is assigned to the feature dataset, not the geodatabase. Note all feature classes in a feature dataset must have the same projection. Doesn't appear that all feature datasets need
to have the same projection in a geodatabase. Remember a feature class can be contained in a feature dataset, which will ensure it has the same projection info, or can be a standalone feature
● ESRI software does not support vertical datums. Only reads the z-value as is, you must perform any pre-processing/corrections to the vertical data before entering into ArcGIS. Appears the
projection metadata doesn't allow you to enter any additional z-value related data (for example NAVD88 datum, elevation units of feet, and so on).fs
How to Project Geodatabases and Shapefiles
• Projections in ArcMap
□ Can project the data frame, not the actual feature class, shapefile, or coverage.
□ Can export the layer with the same projection as the data frame, so in a sense your actually reprojecting the layer.
□ ArcMap will not project data on the fly if the coordinate system for the data set has not been defined.
□ Additional info, see ESRI Article ID 24893, How to identify an unknown coordinate system using ArcMap.
□ ESRI Article ID 20837, how to align vector data in ArcMap
• Projections in ArcCatalog
□ Can only define a projection for a layer, not reproject it.
□ Data frame will inherit the projection of the first layer added to it.
□ ArcCatalog: select a geodatabase feature, right mouse click to bring up the context menu, Properties -> Fields tab, select Shape, then at the bottom of that window, click the ellispe (...)
and either Select or Import.
• Projections in ArcToolbox
□ Will reproject the layer permentently
□ ArcToolbox: Data Management -> Projections -> Project Wizard (shapefiles, geodatabase)
• Reference: see ESRI Article ID 21447 how to project shapefiles or geodatabase feature classes with the ArcToolbox Project wizard
Define a Shapefile's Projection
• Using ArcCatalog
□ Problem: metadata, spatial reference property says "unknown" or "assumed geographic" projection.
□ File -> Properties -> Fields tab: click Shape column. In Properties list below, select ellipses button to open the Spatial Reference Properties window. Click Select... button. Browse through
Projected Coordinate Systems folder -> State Plane folder -> NAD 1983 (Feet) folder -> NAD 1983 StatePlane Nevada East FIPS 2701 (Feet).prj
□ Metadata should now say the projected coordinate system name.
□ Shapefile's coordinate system parameters are stored in the same location and name as the shapefile but with a .prj extension
□ see ArcGIS Desktop Help -> ArcCatalog -> Working with shapefiles -> Defining a shapefile's coordinate system
• Using ArcToolbox
□ ArcToolbox -> Data Management Tools -> Projections -> Define Projection Wizard (shapefiles, geodatabase). Then give same inputs as the "Using ArcCatalog" solution above.
Define a ArcInfo Coverage's Projection
Define a GeoDatabase feature class Projection
Common Coordinate Systems used in Clark County NV
● StatePlane Coordinate System (SPCS)
□ Projection used by local agencies (e.g. Clark County, City of Las Vegas, Henderson, North Las Vegas, and so on)
□ Map of all Stateplane coordinate zones is included with ArcMap, depending upon where you installed the program, c:\arcgis\arcexe83\Reference Systems\usstpln83.shp or C:\Program Files\ArcGIS\
Reference Systems\usstpln83.shp (download shapefiles from UNLV, usstpln83.zip - Geographic Coordinate System - GCS_WGS_1984)
□ Clark County uses StatePlane Coordinate- Nevada East Zone (NV-E)
□ ArcGIS Resource Center - view of USSTPLN83.shp
● Universal Transverse Mercator, UTM
□ Earth is divided into 60 zones (each zone 6 degrees of longitude)
□ Origin for each zone is the Equator and its central meridian (3 degrees west and 3 degrees east). To eliminate negative coordinates, a false easting of 500,000 is applied
□ Typically used for statewide datasets
□ Map of all UTM zones is included with ArcMap, depending upon where you installed the program, c:\arcgis\arcexe83\Reference Systems\utm.shp or c:\Program Files\ArcGIS\Reference Systems\utm.shp
(download shapefiles from UNLV, utm.zip - Geographic Coordinate System - GCS_WGS_1984) and overlay with the USA Counties Layer countyp020.zip from the National Atlas
□ Best way to show two datasets that are in different UTM zones, is to project one into the other zone.
□ State of Nevada uses UTM Zone 11
□ Margaret Maher (mmaher@esri.com) with ESRI Tech Support - specialize in Map Projections and Symbology
● Local/Surface Coordinates
□ Used extensively for small development projects by surveyors
□ referred to as ground distances by surveyors
□ different origin for each design project
□ To project into another coordinate system, need 2 points and have coordinate values in both systems.
Define Local/Surface Coordinate Projection in ArcMap
• Objective is to create a projection file so ArcMap can project on the fly from local/surface coordinates to stateplance coordinates. The projection file (.prj) will be similar to a shapefile file
.prj file but for the AutoCAD .dwg, example anyfilename.dwg and anyfilename.prj (note cannot have any spaces in the filename for the .dwg and .prj files). Then ArcMap will automatically project
the dwg.
• Most difficult step is determining the local/surface coordinate parameters
• ArcMap Data Frame Properties -> New -> Projected Coordinate System
• Projection name cannot contain spaces
• Custom Projection File options for 7 local/surface projections
□ Local
☆ Parameters
○ False_Easting
○ False_Northing
○ Scale_Factor
○ Azimuth
○ Longitude_Of_Center
○ Latitude_Of_Center
☆ Linear Unit = Foot_US
☆ Datum is defined by Select... button under Geographic Coordinate System, select North America folder
○ North American Datum 1983.prj
○ North American 1983 HARN.prj (use if survey done to HARN accuracy)
○ North American 1983 (CSRS98).prj is for Canada
□ Hotine_Oblique_Mercator_Azimuth_Center
□ Hotine_Oblique_Mercator_Azimuth_Natural
□ Hotine_Oblique_Mercator_Two_Point_Center
□ Hotine_Oblique_Mercator_Two_Point_Natural
□ Rectified_Skew_Orthomorphic_Center
□ Rectified_Skew_Orthomorphic_Natural_O (has a rotation parameter)
Alternative method is to Define a Projection using ArcToolbox
Indepth Discussion on Projections | {"url":"https://faculty.unlv.edu/jensen/html/Projections/project.htm","timestamp":"2014-04-20T08:29:06Z","content_type":null,"content_length":"40145","record_id":"<urn:uuid:c2bd1366-e64f-450e-9f78-490edac4d277>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00225-ip-10-147-4-33.ec2.internal.warc.gz"} |
Commutative Properties
This rule for rearrangement is almost so obvious it isn't even worth mentioning. But, we are suckers for overkill, so here goes. The commutative property says that the order in which we add or
multiply numbers doesn't matter.
x + y = y + x
xy = yx
This move, changing the order in which you perform the summation or multiplication of terms in an expression, is one of the most basic ways we can rearrange an expression to find an equivalent
expression. Note that we don't say you can also do this move with subtraction or division. That's because you can't. Addition and multiplication are a little more go-with-the-flow; subtraction and
division are not quite as easy-going and could probably stand to chillax a bit.
Sample Problem
Consider the expression - 4y^2x + x^3. We can use the commutative property of addition to write - 4y^2x + x^3 = x^3 - 4y^2x.
We can then use the commutative property of multiplication to write - 4y^2x = - 4xy^2
Putting it together, we can rearrange - 4y^2x + x^3 using commutativity to find that x^3 = 4xy^2. We are left with an expression that doesn't feature any fewer terms than the original, but at least
it doesn't start out with a negative sign. That really chaps our patootie. | {"url":"http://www.shmoop.com/algebraic-expressions/commutative-properties-help.html","timestamp":"2014-04-16T21:00:54Z","content_type":null,"content_length":"34431","record_id":"<urn:uuid:b53a91b2-3ab1-4cd8-943e-a92526deaa2a>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00268-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Weston, MA Trigonometry Tutors | {"url":"http://www.purplemath.com/Weston_MA_Prealgebra_tutors.php","timestamp":"2014-04-21T00:15:25Z","content_type":null,"content_length":"24195","record_id":"<urn:uuid:837cda3b-d16b-4804-8627-7fec6ab3e6f9>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00255-ip-10-147-4-33.ec2.internal.warc.gz"} |
the first resource for mathematics
On the order of vanishing of modular L-functions at the critical point.
(English) Zbl 0719.11029
Let $f\left(z\right)={\sum }_{n=1}^{\infty }{a}_{n}{e}^{2\pi inz}$ be a Hecke eigenform, newform of weight 2 for ${{\Gamma }}_{0}\left(N\right)$. Its L-series $L\left(s\right)={\sum }_{n=1}^{\infty }
{a}_{n}{n}^{-s}$ is also the L-function of an elliptic curve E over $ℚ$. A condition for the finiteness of the group of rational points of E is ${L}^{"}\left(1,{\chi }_{d}\right)e 0$ for a certain
quadratic character ${\chi }_{d}=\left(\frac{-d}{y}\right)·$
In § 2 of [D. Bump, S. Friedberg and J. Hoffstein, Bull. Am. Math. Soc., New Ser. 21, 89-93 (1989; Zbl 0699.10038)] the theorem is announced that ${L}^{"}\left(1,{\chi }_{d}\right)e 0$ holds for
infinitely many ${\chi }_{d}$ associated to imaginary quadratic number fields. Their proof goes along the same lines as in their earlier paper [Ann. Mat., II. Ser. 131, 53-127 (1990; Zbl 0699.10039
)]. The same result follows from the main theorem in V. K. Murty [Proc. Conf. on Automorphic Forms and Analytic Number Theory, Montréal, June 1989, 89-113 (1990)].
In the paper under review a more quantitative statement is proved: ${L}^{"}\left(1,{\chi }_{d}\right)e 0$ for at least ${Y}^{2/3-ϵ}$ primitive quadratic characters with $d<Y$, for Y large enough.
This follows from the estimates
$\sum _{d\le Y}{|{L}^{"}\left(1,{\chi }_{d}\right)|}^{4}\ll {Y}^{2+ϵ},\phantom{\rule{1.em}{0ex}}\sum _{d}{L}^{"}\left(1,{\chi }_{d}\right)F\left(d/Y\right)={\alpha }_{F}YlogY+{\beta }_{F}Y+O\left({Y}
with ${\alpha }_{F}e 0$; the test function F is smooth with compact support, and d runs over a set of squarefree numbers.
The proof uses an integral representation for the L-series, the symmetric square L-series associated to f, the large sieve inequality, and other techniques of analytic number theory. It is quite
different from the proof of Bump, Friedberg and Hoffstein, which uses automorphic forms more heavily.
11F67 Special values of automorphic $L$-series, etc
11F11 Holomorphic modular forms of integral weight
11F66 Langlands L-functions; one variable Dirichlet series, etc.
11G05 Elliptic curves over global fields
11M41 Other Dirichlet series and zeta functions
11N36 Applications of sieve methods | {"url":"http://zbmath.org/?q=an:0719.11029","timestamp":"2014-04-16T04:26:06Z","content_type":null,"content_length":"26652","record_id":"<urn:uuid:39651286-a536-4bbe-9884-c04497399cc3>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00426-ip-10-147-4-33.ec2.internal.warc.gz"} |
Consumer choice involving risk, Microeconomics
The traditional theory of consumer behaviour does not include an analysis of uncertain situation. Von Neumann and Morgenstern showed that under some circumstances it is possible to construct a set of
numbers for a particular consumer that can be used to predict her choices in uncertain situations. However, there is a great controversy that has centered around the question of whether the resulting
utility index is ordinal or, cardinal. It will be shown that Von Neumann - Morgenstern utilities possess at least some cardinal properties.
It has been pointed out above that consumer behaviour analysis is unrealistic in the sense that it assumes actions the consumer are followed by determinate consequences which are knowable in advance.
For instance, all automobiles of the same model and produced in the same factory will not always have the same performance characteristics. As a result of random accidents in the production process,
some substandard automobiles could be occasionally produced and sold. The consumer has no way of knowing ahead of time whether the particular automobile, which she purchased, is of standard
quality or not.
Let A represent the situation in which the consumer possesses a standard quality automobile and B be a situation in which she does not. Again, let there be C, in which she possesses a substandard
automobile. Assume that the consumer prefers A to B and B to C. That is, not having a car is assumed preferable to owning a substandard one because of the nuisance and expense involved in its uptake.
Present her with a choice between two alternatives: (1) She can maintain the status quo and have no car at all. This is a choice with certain outcome i.e., the probability of the outcome equals
unity. (2) She can obtain a lottery ticket with a chance of winning either a satisfactory automobile (alternative A) or an unsatisfactory one (alternative C). The consumer may prefer to retain her
income (or money) with certainty, or she may prefer the lottery ticket with dubious outcome, or she may be indifferent between them. Her decision will depend upon the chances of winning or losing in
this particular lottery. If the probability of C is very high, she might prefer to retain her money with certainty; if the probability of A is very high, she might prefer the lottery ticket. The
triplet of numbers (P, A, B) is used to denote a lottery offering outcome A with probability 0
Posted Date: 10/26/2012 4:01:13 AM | Location : United States
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Processors of aseptically packaged juice-based beverages must adequately heat their product before packaging it in order to be sure that they have “killed” the microorganisms which
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MathGroup Archive: December 2010 [00707]
[Date Index] [Thread Index] [Author Index]
Re: newbie list question
• To: mathgroup at smc.vnet.net
• Subject: [mg115012] Re: newbie list question
• From: dr DanW <dmaxwarren at gmail.com>
• Date: Tue, 28 Dec 2010 06:50:43 -0500 (EST)
• References: <if70bt$330$1@smc.vnet.net>
Your first lesson in functional programming. First, if you are
serious about learning Mathematica, you should pick up a book that
covers the fundamentals, like The Mathematica Cookbook from O'Reilly.
I've been using Mathematica for over 20 years, and I still read one of
these books every so often because there is always something I have
missed. I would suggest reading the Doc Center, but alas it has
become so fragmented over the last few versions that it does not read
as a book (a linear introduction of concepts) which would really help
the new user.
Here are your lists:
listA = {4, 5, 8, 2, 6, 4};
listB = {8, 4, 2};
When facing a problem like this you want to build up your expression.
Only the very gifted can write an elegant Mathematica expression in
one pass. First, consider the question "how do I find the position of
one specific element in listA?" After reading tutorial/
TestingAndSearchingListElements, you realize that Position[] is the
function you need:
Position[listA, 4]
Out[3]= {{1}, {6}}
well, that gave you every position of 4 in listA, presented as a list
(which makes more sense if you are looking for elements in a matrix,
but is done this was for a list for consistency). So, use Part[]
(also done as double brackets [[ ... ]] ) to extract the value you
Position[listA, 4][[1,1]]
Out[4]= 1
Great, so you can do it for one value, but you have a list. This is
the functional programming part. In functional languages, once you
have a function to do something once, there are functions that you can
use to work that function over elements of a list (or matrix, or more
deeply nested structures.) In this case, you need Map[].
(Position[listA, #1][[1,1]] & ) /@ listB
Out[5]= {3, 1, 4}
Where is Map? This is the part that throws a lot on new users, the
use of operators instead of function names looks like the cat walked
across the keyboard. Once again, break it down. First you need to
make a Function[]. Completely spelled out, this would be
Function[{x}, Part[ Position[listA, x], 1, 1 ] ]
in operator shorthand, it becomes
(Position[listA, #1][[1,1]] & )
where the #1 indicates the first slot of the function (a generic tag
for the first argument in the function call) and the & is the operator
for Function[]. In Mathematica, operators can be placed before
(prefix) in the middle (infix) or after (postfix) the arguments. This
sounds confusing at first, but it makes sense as you use it. Finally,
the operator /@ is an infix operator for Map[]. Map[ list, fun ] (or
fun /@ list ) throws the function fun at every element of list and
returns a list of the results.
Practice with this. Experiment with it in Mathematica, see what
happens when you change things in the expression, or have a more
deeply nested structure you are looking in. | {"url":"http://forums.wolfram.com/mathgroup/archive/2010/Dec/msg00707.html","timestamp":"2014-04-17T04:04:27Z","content_type":null,"content_length":"27869","record_id":"<urn:uuid:156a5f64-08e0-45ce-8a4a-a24e7faf9e50>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00397-ip-10-147-4-33.ec2.internal.warc.gz"} |
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S.: Algebraic approaches to nondeterminism: An overview
Results 1 - 10 of 20
- Applied Categorical Structures , 1999
"... . We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of
the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particula ..."
Cited by 37 (24 self)
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. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of the
algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are one-to-one with the arrows of the free
gs-monoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r),
providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit
sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gs-monoidal categories, symmetric monoidal categories, term graphs.
Mathematical Subject Clas...
, 2000
"... Multi-algebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a
functorial presentation of various categories of multi-algebras and partial algebras, analogous to the classica ..."
Cited by 14 (7 self)
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Multi-algebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial
presentation of various categories of multi-algebras and partial algebras, analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to
Set. We introduce two dierent notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors from them to Rel or Pfn, the categories of sets and relations
or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines for the choice of syntactical notions for a class of algebras, and as an application we argue
that the natural generalization of usual terms are \conditioned terms" for partial algebras, and \term graphs" for multi-algebras. Contents 1 Introduction 2 2 A short recap on multi-algebras 4 3...
, 1998
"... . The paper characterises compositional homomorphims of relational structures. A detailed study of three categories of such structures -- viewed as multialgebras -- reveals the one with the most
desirable properties. In addition, we study analogous categories with homomorphisms mapping elements to s ..."
Cited by 11 (3 self)
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. The paper characterises compositional homomorphims of relational structures. A detailed study of three categories of such structures -- viewed as multialgebras -- reveals the one with the most
desirable properties. In addition, we study analogous categories with homomorphisms mapping elements to sets (thus being relations). Finally, we indicate some consequences of our results for partial
algebras which are special case of multialgebras. 1 Introduction In the study of universal algebra, the central place occupies the pair of "dual" notions of congruence and homomorphism: every
congruence on an algebra induces a homomorphism into a quotient and every homomorphism induces a congruence on the source algebra. Categorical approach attempts to express all (internal) properties
of algebras in (external) terms of homomorphisms. When passing to relational structures, however, the close correspondence of these internal and external aspects seems to get lost. The most common,
and natural, gene...
- Recent Trends in Algebraic Development Techniques, volume 1589 of LNCS , 1998
"... . Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial
presentation of various categories of multi-algebras and partial algebras, analogous to the classical pre ..."
Cited by 6 (4 self)
Add to MetaCart
. Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial
presentation of various categories of multi-algebras and partial algebras, analogous to the classical presentation of algebras over a signature \Sigma as cartesian functors from the algebraic theory
of \Sigma , Th(\Sigma), to Set. The functors we introduce are based on variations of the notion of theory, having a structure weaker than cartesian, and their target is Rel, the category of sets and
relations. We argue that this functorial presentation provides an original abstract syntax for partial and multi-algebras. 1 Introduction Nondeterminism is a fundamental concept in Computer Science.
It arises not only from the study of intrinsically nondeterministic computational models, like Turing machines and various kinds of automata, but also in the study of the behaviour of deterministic
, 2000
"... Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial
presentation of multi-algebras based on gs-monoidal theories, we argue that speci cations for multi-algebras ..."
Cited by 5 (4 self)
Add to MetaCart
Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial
presentation of multi-algebras based on gs-monoidal theories, we argue that speci cations for multi-algebras should be based on the notion of term graphs instead of on standard terms. We consider the
simplest case of (term graph) equational specification, showing that it enjoys an unrestricted form of substitutivity. We discuss the expressive power of equational specification for multialgebras,
and we sketch possible extensions of the calculus.
, 2006
"... This paper introduces a framework for rapid prototyping of object oriented programming languages and corresponding analysis tools. It is based on formal definitions of language features in
rewrite logic, a simple and intuitive logic for concurrency with powerful tool support. A domain-specific front ..."
Cited by 4 (1 self)
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This paper introduces a framework for rapid prototyping of object oriented programming languages and corresponding analysis tools. It is based on formal definitions of language features in rewrite
logic, a simple and intuitive logic for concurrency with powerful tool support. A domain-specific front-end consisting of a notation and a technique, called K, allows for compact, modular, expressive
and easy to understand and change definitions of language features. The framework is illustrated by first defining KOOL, an experimental concurrent object-oriented language with exceptions, and then
by discussing the definition of JAVA. Generic rewrite logic tools, such as efficient rewrite engines and model checkers, can be used on language definitions and yield interpreters and corresponding
formal program analyzers at no additional cost.
"... This paper is an attempt to bring some order into this chaos. Instead of listing and defending new definitions we hope that approaching the problem from a more algebraic perspective may bring at
least some clarification. Section 2 addresses the question of composition of homomorphisms. In subsection ..."
Cited by 2 (2 self)
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This paper is an attempt to bring some order into this chaos. Instead of listing and defending new definitions we hope that approaching the problem from a more algebraic perspective may bring at
least some clarification. Section 2 addresses the question of composition of homomorphisms. In subsection 2.1 we give a characterization of relational homomorphisms which are closed under composition
-- in fact, most of the suggested defintions, like most of those in table 1, do not enjoy this property which we believe is crucial. We also characterize equivalences associated with various
compositional homomorphisms. Then, in section 3 we introduce multialgebras which are relational structures with composition of relations (1.2) reflecting the traditional way of composing functions.
Subsection 3.1 sketches the relation between multialgebras and their quotients by (congruence)
"... Summary. We introduce a geometrical 3 setting which seems promising for the study of computation in multiset rewriting systems, but could also be applied to register machines and other models of
computation. This approach will be applied here to membrane systems (also known as P systems) without dyn ..."
Cited by 2 (1 self)
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Summary. We introduce a geometrical 3 setting which seems promising for the study of computation in multiset rewriting systems, but could also be applied to register machines and other models of
computation. This approach will be applied here to membrane systems (also known as P systems) without dynamical membrane creation. We discuss the rôle of maximum parallelism and further simplify our
model by considering only one membrane and sequential application of rules, thereby arriving at asynchronous multiset rewriting systems (AMR systems). Considering only one membrane is no restriction,
as each static membrane system has an equivalent AMR system. It is further shown that AMR systems without a priority relation on the rules are equivalent to Petri Nets. For these systems we introduce
the notion of asymptotically exact computation, which allows for stochastic appearance checking in a priori bounded (for some complexity measure) computations. The geometrical analogy in the lattice
N d 0, d ∈ N, is developed, in which a computation corresponds to a trajectory of a random walk on the directed graph induced by the possible rule applications. Eventually this leads to symbolic
dynamics on the partition generated by shifted positive cones C + p, p ∈ N d 0, which are associated with the rewriting rules, and their intersections. Complexity measures are introduced and we
consider non–halting, loop–free computations and the conditions imposed on the rewriting rules. Eventually, two models of information processing, control by demand and control by availability are
discussed and we end with a discussion of possible future developments. 1
"... Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a simple
inequational deduction system, based on term graphs, for inferring inclusions of derived relations in a multi-algeb ..."
Cited by 2 (1 self)
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Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a simple inequational
deduction system, based on term graphs, for inferring inclusions of derived relations in a multi-algebra, and we show that term graph rewriting provides a sound and complete implementation of it.
- ACM Trans. Comput. Logic , 2001
"... Exactly solving first-order constraints (i.e., first-order formulas over a certain prede ned structure) can be a very hard, or even undecidable problem. In continuous structures like the real
numbers it is promising to compute approximate solutions instead of exact ones. However, the quantifiers of ..."
Cited by 1 (0 self)
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Exactly solving first-order constraints (i.e., first-order formulas over a certain prede ned structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers
it is promising to compute approximate solutions instead of exact ones. However, the quantifiers of the first-order predicate language are an obstacle to allowing approximations to arbitrary small
error bounds. In this paper we solve the problem by modifying the first-order language and replacing the classical quantifiers with approximate quantifiers. These also have two additional advantages:
First, they are tunable, in the sense that they allow the user to decide on the trade-off between precision and efficiency. Second, they introduce additional expressivity into the first-order
language by allowing reasoning over the size of solution sets. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=488271","timestamp":"2014-04-18T11:46:56Z","content_type":null,"content_length":"39089","record_id":"<urn:uuid:2d7a1009-d07c-4881-8c3f-83905562c75b>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00151-ip-10-147-4-33.ec2.internal.warc.gz"} |
Passing the Tech Test
As many of you know, I teach One-Day Tech classes. At the start of each class, I go over the following to help focus students on what to keep in mind when taking the test.
Technical Topics
For the Tech test, the focus is on rules and safety. It is not very technical. Having said that, there are three technical topics that you need to know:
• Ohm’s Law,
• how to calculate power, and
• the relationship between frequency and wavelength.
Ohm’s Law
The basic formula for Ohm’s Law is voltage (E) equals current (I) times resistance (R), or E = I x R. On the test, there are several questions where they give you two of the values and ask you to
calculate the third. If you’re asked to calculate the current, you use the formula, I = E / R. If you need to calculate the resistance, use the formula R = E / I.
How to Calculate Power
The formula for calculating power is power (P) = voltage (E) times current (I), or P = E x I. To calculate the current drawn, when given the power being consumed and the voltage applied to the
circuit, use the formula I = P / E.
Relationship Between Frequency and Wavelength
There are several questions that require you to calculate the wavelength of a signal or some fraction of the wavelength. The reason for this is that antennas are often a fraction of a wavelength.
The formula that describes the relationship between frequency and wavelenght is wavelength in meters = 300 / frequency in MHz. One question asks for the approximate length of a quarter-wavelength
vertical antenna for 146 MHz. To figure that out, you first calculate the wavelength:
wavelength = 300/146 = 2.05 m or about 80 inches
One quarter of 80 inches is 20 inches, and the antenna will actually be a little bit shorter than that because radio travels more slowly in wire than it does in free space. The correct answer to this
question is 19 inches.
That’s all there is to the technical part of the test!
There are lots of questions on the test about operating safely and being safe when working on antennas. My advice when answering these questions is to always choose the most conservative answer. The
two exceptions are when asked what is the lowest voltage and current that can hurt you. For these questions, the correct answer is the second lowest choices.
There are lots of questions about what to do in emergencies. There are two things to keep in mind when answering these questions:
• You should do whatever you can to help someone who is in an emergency situation.
• You can even break the rules to help someone in an emergency situation. This includes operating on frequencies you are normally not allowed to operate on and communicating with other stations in
other radio services.
Miscellaneous Tips
Here are a couple of other miscellaneous tips:
• The answer is ‘D.’ If one of the answers to a question is, “D. All of these answers are correct,” chances are that is the correct answer. There are 18 questions with this option, and of those 18
questions, there are only two questions–T3B06 and T5B03–where that is not the correct answer.
• Long-Answer Rule. Where one answer is a lot longer than the other options, chances are that this is the correct answer. I haven’t done an exhaustive study of this, but when one answer is very
long, take a good, hard look at it.
That’s all I have. What tips do you have for passing the Tech test?
1. Howard says:
I would suggest studying the material so that you understand it, so that you won’t have to rely on suggestions like “pick D”.
2. Tom says:
Wish I’d read this overview before I studied for the tech exam! I bought the manual, read it twice, and started taking test exams online. As I began to identify where I was missing questions, I
reviewed those sections of the manual again, and kept taking exams. When I was down to missing only four or less questions consistently, I took the exam and did very well.
Your overview would have provided more context while I studied, and probably would have decreased my stress.
I did not take a class, nor did I talk with others while I studied. It was purely a solo exercise, over a period of about three weeks.
3. Jim says:
Do you really mean to do this? Hams can easily place themselves and
others in potentially life-threatening situations (high voltage/current,
high levels of RF, etc.). Giving someone a statistical solution that avoids
learning the fundamentals seems seriously dangerous.
Jim N9GTM (VE)
4. Mark Morgan says:
thanks I suddenly find myself asked to teach someone the materail for the Tech since I had not looked at the new pool (having passed the 2 exam way in 1987) thanks for the hints
5. Dan KB6NU says:
These are very interesting responses. Howard and Jim are taking me to task for passing along these tips, while Tom and Mark thank me for the advice.
Howard and Jim: You’re assuming that by learning the answers to the specific questions on the test, the students aren’t learning the material. I don’t think that’s the case. First of all, I think
that a lot of people learn by doing, rather than by reading some material in a book. With a license, they can do more, and thereby learn more. Also, I think that the process of taking practice
tests also helps people learn the material. If these tips help them think through a question, then I see no harm in them.
Since Jim brings up the topic of safety, note that I said, “Always choose the most conservative answer.” That advice also holds for situations where you have a potentially life-threatening
situation for real. Perhaps I should add that that advice applies to real-life as well as to taking the test. Hopefully, though, the students are smart enough to realize that for themselves.
Speak Your Mind Cancel reply
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non-Euclidean geometry (mathematics) :: Hyperbolic geometry
non-Euclidean geometry
Article Free Pass
The first description of hyperbolic geometry was given in the context of Euclid’s postulates, and it was soon proved that all hyperbolic geometries differ only in scale (in the same sense that
spheres only differ in size). In the mid-19th century it was shown that hyperbolic surfaces must have constant negative curvature. However, this still left open the question of whether any surface
with hyperbolic geometry actually exists.
In 1868 the Italian mathematician Eugenio Beltrami described a surface, called the pseudosphere, that has constant negative curvature. However, the pseudosphere is not a complete model for hyperbolic
geometry, because intrinsically straight lines on the pseudosphere may intersect themselves and cannot be continued past the bounding circle (neither of which is true in hyperbolic geometry). In 1901
the German mathematician David Hilbert proved that it is impossible to define a complete hyperbolic surface using real analytic functions (essentially, functions that can be expressed in terms of
ordinary formulas). In those days, a surface always meant one defined by real analytic functions, and so the search was abandoned. However, in 1955 the Dutch mathematician Nicolaas Kuiper proved the
existence of a complete hyperbolic surface, and in the 1970s the American mathematician William Thurston described the construction of a hyperbolic surface. Such a surface, as shown in the figure,
can also be crocheted.
In the 19th century, mathematicians developed three models of hyperbolic geometry that can now be interpreted as projections (or maps) of the hyperbolic surface. Although these models all suffer from
some distortion—similar to the way that flat maps distort the spherical Earth—they are useful individually and in combination as aides to understand hyperbolic geometry. In 1869–71 Beltrami and the
German mathematician Felix Klein developed the first complete model of hyperbolic geometry (and first called the geometry “hyperbolic”). In the Klein-Beltrami model (shown in the figure, top left),
the hyperbolic surface is mapped to the interior of a circle, with geodesics in the hyperbolic surface corresponding to chords in the circle. Thus, the Klein-Beltrami model preserves “straightness”
but at the cost of distorting angles. About 1880 the French mathematician Henri Poincaré developed two more models. In the Poincaré disk model, the hyperbolic surface is mapped to the interior of a
circular disk, with hyperbolic geodesics mapping to circular arcs (or diameters) in the disk that meet the bounding circle at right angles. In the Poincaré upper half-plane model, the hyperbolic
surface is mapped onto the half-plane above the x-axis, with hyperbolic geodesics mapped to semicircles (or vertical rays) that meet the x-axis at right angles. Both Poincaré models distort distances
while preserving angles as measured by tangent lines.
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Patent application title: COUNTING DEVICE AND COUNTING METHOD
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A counter counts the run lengths of a binarized signal. A counting result correcting portion generates frequency distributions for run lengths for first run lengths, which are from a rising edge to a
falling edge of the signal, and second run lengths, which are for a falling edge to a rising edge of the signal, calculates a total number of first run lengths of lengths that are no less than 0
times and less than 1 times a representative value for the first run lengths, calculates a total number of second run lengths of lengths that are no less than 0 times and less than 1 times a
representative value for the second run lengths, calculates a total number of first run lengths, calculates a total number of second run lengths, and corrects the counting results.
A counting device counting signals wherein there is a linear relationship between a specific physical quantity and the number of signals and wherein the signal has essentially a single frequency when
the specific physical quantity is constant, comprising: a binarizing device binarizing an inputted signal; a signal counter counting a number of run lengths of the binarized signal outputted from the
binarizing device during a prescribed counting interval; a run length measuring device measuring the run lengths of a binarized signal during the counting interval each time a run length worth of a
signal is inputted; a frequency distribution generator generating a frequency distribution for the run lengths of the binarized signals during the counting interval, from the measurement results by
the run length measuring device, for first run lengths, which are from a rising edge until the subsequent falling edge of the binarized signal, and second run lengths, which are from a falling edge
to the subsequent rising edge of the binarized signal; a representative value calculator calculating a representative value T
for the distribution of the first run lengths from the first run length frequency distribution and calculating a representative value T
for the distribution of the second run lengths from the second run length frequency distribution; and a corrected value calculator calculating a total Ns
number of first run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Ns
number of second run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Nw
H number of first run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and a total Nw
L number of second run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and calculating a number of inputted signals by correcting the counting results by the signal counting means based on these frequencies Ns
, Ns
, and Nw
H, and Nw
The counting device as set forth in claim 1, wherein: the corrected value calculator calculates a post-correction calculation result N' through: N ' = 1 2 [ N - ( Ns H + Ns L ) + n = 1 n max { 2 n ×
( Nw nH + Nw nL ) } ] n max ≦ ( T H + T L ) max T H + T L ##EQU00004## when the counting result by the signal counter is defined as N, and the maximum value that can be assumed by the sum of a first
run length and a second run length is defined as (T
The counting device as set forth in claim 1, wherein: the representative values T
and T
are each a median value, a modal value, a mean value, a bin value wherein the product of the bin value and the frequency is a maximum, or a bin value wherein the product of the bin value raised to
the a power (where 0<a<1) and the frequency is a maximum.
The counting device as set forth in claim 1 wherein: a threshold value for calculating the total Ns
number of first run lengths is no less than 0 times and less than
5 times the representative value T
or no less than 0 times the representative value T
and less than (T
)/4; and a threshold value for calculating the total Ns
number of second run lengths is no less than 0 times and less than
5 times the representative value T
or no less than 0 times the representative value T
and less than (T
4. 5.
A counting method counting signals wherein there is a linear relationship between a specific physical quantity and the number of signals and wherein the signal has essentially a single frequency when
the specific physical quantity is constant, comprising the steps of: a binarizing step binarizing an inputted signal; a signal counting step counting a number of run lengths of the binarized signal
outputted from the binarizing step during a prescribed counting interval; a run length measuring step measuring the run lengths of a binarized signal during the counting interval each time a run
length worth of a signal is inputted; a frequency distribution generating step generating a frequency distribution for the run lengths of the binarized signals during the counting interval, from the
measurement results by the run length measuring step, for first run lengths, which are from a rising edge until the subsequent falling edge of the binarized signal, and second run lengths, which are
from a falling edge to the subsequent rising edge of the binarized signal; a representative value calculating step calculating a representative value T
for the distribution of the first run lengths from the first run length frequency distribution and for calculating a representative value T
for the distribution of the second run lengths from the second run length frequency distribution; and a corrected value calculating step calculating a total Ns
number of first run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Ns
number of second run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Nw
H number of first run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and a total Nw
L number of second run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and for calculating a number of inputted signals by correcting the counting results by the signal counting step based on these frequencies Ns
, Ns
, and Nw
H, and Nw
The counting method as set forth in claim 5, wherein: the corrected value calculating step calculates a post-correction calculation result N' through: [ Equation 2 ] ##EQU00005## N ' = 1 2 [ N - ( Ns
H + Ns L ) + n = 1 n max { 2 n × ( Nw nH + Nw nL ) } ] n max ≦ ( T H + T L ) max T H + T L ##EQU
2## when the counting result by the signal counting step is defined as N, and the maximum value that can be assumed by the sum of a first run length and a second run length is defined as (T
The counting method as set forth in claim 5, wherein: the representative values T
and T
are each a median value, a modal value, a mean value, a bin value wherein the product of the bin value and the frequency is a maximum, or a bin value wherein the product of the bin value raised to
the a power (where 0<a<1) and the frequency is a maximum.
The counting method as set forth in claim 5, wherein: a threshold value calculating the total Ns
number of first run lengths is no less than 0 times and less than
5 times the representative value T
or no less than 0 times the representative value T
and less than (T
)/4; and a threshold value calculating the total Ns
number of second run lengths is no less than 0 times and less than
5 times the representative value T
or no less than 0 times the representative value T
and less than (T
The present application claims priority under 35 U.S.C, §119 to Japanese Patent Application No. 2011-083425, filed Apr. 5, 2011, which is incorporated herein by reference.
FIELD OF TECHNOLOGY [0002]
The present invention relates to a counting device and a counting method for counting a number of signals.
BACKGROUND [0003]
Conventionally, laser measuring devices of a wavelength modulating type have been proposed that use the self-coupling effect of semiconductor lasers (See Japanese Unexamined Patent Application
Publication 2006-313080 ("JP '080")). The structure of this type of laser measuring device is illustrated in FIG. 9. The laser measuring device of FIG. 9 includes a semiconductor laser 201 for
emitting a laser beam at an object 210; a photodiode 202 for converting into an electric signal the optical power of the semiconductor laser 201: a lens 203 that focuses a beam from the semiconductor
laser 201 to direct it to an object 210, and that focuses a return beam from the object 210 to inject it into the semiconductor laser 201; a first laser driver 204 for repetitively alternating
between a first emission interval over which the emission wavelength of the semiconductor laser 201 increases continuously and a second emission interval over which the emission wavelength decreases
continuously; an electric current/voltage converting/amplifying portion 205 for converting the outputted electric current from the photodiode 202 into a voltage, and then amplifying; a signal
extracting circuit 206 for performing double differentiation on the outputted voltage of the electric current/voltage converting/amplifying portion 205; a counting device 207 for counting the number
of mode hope pulses (hereinafter termed "MHPs") included in the outputted voltage of the signal extracting circuit 206; a calculating device 208 for calculating the distance to the object 210 and the
speed of the object 210; and a displaying device 209 for displaying the results of the calculations by the calculating device 208.
The laser driver 204 provides, as an injected electric current into the semiconductor laser 201, a triangle wave driving current that repetitively increases and decreases at a constant rate of change
in respect to time. Doing so drives the semiconductor laser 201 so as to repetitively alternate between a first emission interval, wherein the emission wavelength increases continuously at a constant
rate of change, and a second emission interval over which the emission wavelength is reduced continuously at a continuous rate of change. FIG. 10 is a diagram illustrating the change, in respect to
time, of the emission wavelength of the semiconductor laser 201. In FIG. 10, P1 is the first emission interval, P2 is the second emission interval, λa is the minimum value of the emission wavelength
in each of the intervals, λb is the maximum value for the emission wavelength in each of the intervals, and Tt is the period of the triangle wave.
The laser beam that is emitted from the semiconductor laser 201 is focused by the lens 203, to be incident on the object 210. The beam that is reflected from the object 210 is focused by the lens 203
to be injected into the semiconductor laser 201. The photodiode 202 converts the optical power of the semiconductor laser 201 into an electric current. The electric current/voltage converting/
amplifying portion 205 converts the outputted electric current from the photodiode 202 into a voltage, and then performs amplification, and the signal extracting circuit 206 performs double
differentiation on the outputted voltage from the electric current/voltage converting/amplifying portion 205. The counting device 207 counts the number of MHPs included in the outputted voltage from
the signal extracting circuit 206 in the first emission interval and the second emission interval P2, separately. The calculating device 208 calculates the distance of the object 210 and the speed of
the object 210 based on the minimum emission wavelength λa and the maximum emission wavelength λb the semiconductor laser 201, the number of MHPs in the first emission interval P1, and the number of
MHPs in the second emission interval P2. The use of this self-coupled laser measurement device technology makes it possible to measure the number of MHPs to calculate a vibration frequency for the
object from the numbers of MHPs.
The laser measuring device as described above has a problem in that there will be error in the number of MHPs that are counted by the counting device when, for example, counting noise such as
external light as MHPs or when there are MHPs that are not counted due to missing signals, producing error in the physical quantities that are calculated, such as the distance and the vibrational
Given this, a counting device was proposed that is able to eliminate the effects of undercounting or overcounting at the time of counting through measuring the period of the MHPs during the counting
interval, producing a distribution of the counts of the periods within the counting interval from the measurement results, calculating representative values for the periods of the MHPs from the
frequency distribution, calculating, based on the frequency distribution, a total Ns of the frequencies in each bin that is no more than a first specific multiple of the representative value and
calculating a total Nw of the frequencies of the bins that are no less than a second specific multiple of the representative value, and correcting the result for counting the MHPs based on these
frequencies Ns and Nw (See Japanese Unexamined Patent Application Publication 2009-47676 ("JP '676")).
The counting device disclosed in JP '676 is able to perform generally good correction insofar as the SN (signal-to-noise ratio) is not extremely low.
However, in the counting device disclosed in JP '676, in some cases a large number of signals with periods that are about one half of the actual period of the MHP, or signals with short periods, may
be produced through the occurrence of chattering due to noise at frequencies higher than those of the MHPs near to a threshold value of binarization of the signals inputted into the counting device
when, in a measurement of a short distance, the signal strength is extremely strong when compared to the hysteresis width. In this case, a period that is shorter than the actual period of the MHP
will be used as the representative value for the distribution of periods, making it impossible to correct the MHP counting result properly, and thus there is a problem in that the MHP counting result
may be, for example, several times larger than the actual value.
Given this, another counting device is proposed that is able to correct counting error even in a case wherein high-frequency noise is produced continuously in the input signal (See Japanese
Unexamined Patent Application Publication 2011-33525 ("JP '525")). The measuring device disclosed in JP '525 counts the number of run lengths of input signals during the counting interval, measures
the run lengths of the input signal during the counting interval, constructs a frequency distribution of the run lengths of the input signals in the measurement period from the measurement results,
calculates a representative value for the distribution of run lengths in the input signal from this frequency distribution, calculates a total Ns for the number of run lengths that are less than 0.5
times the representative value and a total Nwn number of run lengths that are no less than 2n times the representative value and less than (2n+2) times the representative value (where n is a natural
number no less than 1), and corrects the MHP counting result based on these frequencies Ns and Nwn.
However, the waveform of an interference pattern such as an MHP is asymmetrical in respect to time due to the characteristics of the carrier wave-removing circuit and the state of the target object
(See Japanese Patent 3282746). FIG. 11(A) is a diagram illustrating an interference waveform that is asymmetrical in this way, and FIG. 11(B) is a diagram illustrating the result of binarization of
the waveform in FIG. 11(A). TH1 and TH2 in FIG. 11(A) are threshold values for binarization. When the interference waveform is asymmetrical in respect to time in this way, the duty ratio of the
binarized signal will not be 0.5. Given this, the counting device disclosed in JP '525 has a problem in that the accuracy with which the count is corrected will suffer.
The counting device disclosed in JP '525 is able to correct counting error even in cases wherein high-frequency noise is produced in the input signal.
However, the counting device disclosed in JP '525 has a problem in that the accuracy of the correction to the count will suffer because the duty ratio of the signal wherein the interference waveform
is binarized will not be 0.5 when the interference waveform is asymmetrical in respect to time.
The present invention was created in order to solve the problems set forth above, and the object thereof is to provide a counting device and counting method able to correct accurately the counting
result even in cases wherein the signal that is inputted into the counting device is asymmetrical in respect to time.
SUMMARY [0015]
Examples of the present invention are a counting device for counting signals wherein there is a linear relationship between a specific physical quantity and the number of signals and wherein the
signal has essentially a single frequency when the specific physical quantity is constant, having binarizing means for binarizing an inputted signal; signal counting means for counting a number of
run lengths of the binarized signal outputted from the binarizing means during a prescribed counting interval; run length measuring means for measuring the run lengths of a binarized signal during
the counting interval each time a run length worth of a signal is inputted; frequency distribution generating means for generating a frequency distribution for the run lengths of the binarized
signals during the counting interval, from the measurement results by the run length measuring means, for first run lengths, which are from a rising edge until the subsequent falling edge of the
binarized signal, and second run lengths, which are from a falling edge to the subsequent rising edge of the binarized signal; representative value calculating means for calculating a representative
value T
for the distribution of the first run lengths from the first run length frequency distribution and for calculating a representative value T
for the distribution of the second run lengths from the second run length frequency distribution; and corrected value calculating means for calculating a total Ns
number of first run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Ns
number of second run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Nw
H number of first run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and a total Nw
L number of second run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and for calculating a number of inputted signals by correcting the counting results by the signal counting means based on these frequencies NS
, Ns
, and Nw
H, and Nw
Moreover, in one example configuration of a counting device according to the present example the corrected value calculating means calculate a post-correction calculation result N' through:
[ Equation 1 ] N ' = 1 2 [ N - ( Ns H + Ns L ) + n = 1 n max { 2 n × ( Nw nH + Nw nL ) } ] n max ≦ ( T H + T L ) max T H + T L ##EQU00001##
when the counting result by the signal counting means is defined as N, and the maximum value that can be assumed by the sum of a first run length and a second run length is defined as (T
Moreover, in one example configuration of a counting device according to the present invention, the representative values T
and T
are each a median value, a modal value, a mean value, a bin value wherein the product of the bin value and the frequency is a maximum, or a bin value wherein the product of the bin value raised to
the a power (where 0<a<1) and the frequency is a maximum.
Moreover, in one example configuration of a counting device, a threshold value for calculating the total Ns
number of first run lengths is no less than 0 times and less than 0.5 times the representative value T
or no less than 0 times the representative value T
and less than (T
)/4, and a threshold value for calculating the total Ns
number of second run lengths is no less than 0 times and less than 0.5 times the representative value T
, or no less than 0 times the representative value T
and less than (T
Moreover, the present example is a counting method for counting signals wherein there is a linear relationship between a specific physical quantity and the number of signals and wherein the signal
has essentially a single frequency when the specific physical quantity is constant, including a binarizing step for binarizing an inputted signal; a signal counting step for counting a number of run
lengths of the binarized signal outputted from the binarizing step during a prescribed counting interval; a run length measuring step for measuring the run lengths of a binarized signal during the
counting interval each time a run length worth of a signal is inputted; a frequency distribution generating step for generating a frequency distribution for the run lengths of the binarized signals
during the counting interval, from the measurement results by the run length measuring step, for first run lengths, which are from a rising edge until the subsequent falling edge of the binarized
signal, and second run lengths, which are from a frilling edge to the subsequent rising edge of the binarized signal; a representative value calculating step for calculating a representative value T
for the distribution of the first run lengths from the first run length frequency distribution and for calculating a representative value T
for the distribution of the second run lengths from the second run length frequency distribution; and a corrected value calculating step for calculating a total Ns
number of first run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Ns
number of second run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Nw
H number of first run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and a total Nw
L number of second run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and for calculating a number of inputted signals by correcting the counting results by the signal counting step based on these frequencies Ns
, Ns
, and Nw
H, and Nw
Moreover, in one example configuration of a counting method according to the present example, the corrected value calculating step calculates a post-correction calculation result N' through:
[ Equation 2 ] N ' = 1 2 [ N - ( Ns H + Ns L ) + n = 1 n max { 2 n × ( Nw nH + Nw nL ) } ] n max ≦ ( T H + T L ) max T H + T L ##EQU00002##
when the counting result by the signal counting step is defined as N, and the maximum value that can be assumed by the sum of a first run length and a second run length is defined as (T
In the present examples, the input signal is binarized, the number of run lengths of the binarized signal outputted by the binarizing means during a specific counting interval is counted, the run
lengths of the binarized signals during the counting interval are measured, frequency distributions for the run lengths of the binarized signals during the counting interval are produced for the
measured results for first run lengths that are from a rising edge to the subsequent falling edge of the binarized signal and for second run lengths that are from a falling edge to the subsequent
rising edge of the binarized signal, a representative value T
is calculated, from the frequency distribution of the first run lengths, for the distribution of the first run lengths, and a representative value T
is calculated, from the frequency distribution of the second run lengths, for the distribution of the second run lengths, a total Ns
number of first run lengths that are lengths from 0 times to less than 1 times the representative value T
, a total Ns
number of second run lengths that are lengths from 0 times to less than 1 times the representative value T
, a total Nw
H number of first run lengths that are lengths from {T
)} to less than {T
)} and a total Nw
L number of second run lengths that are lengths from {T
)} to less than {T
)} are calculated, and the counting results of the signal counting means are corrected based on these frequencies Ns
, Ns
, and Nw
L, thus making it possible to correct with high accuracy the counting error even when high-frequency noise appears continuously in the signal inputted into the counting device and even if the input
signal waveform is asymmetrical in respect to time.
BRIEF DESCRIPTION OF THE DRAWINGS [0024]
FIG. 1 is a block diagram illustrating a structure for a counting device according to an example of the present invention.
FIG. 2 is a flowchart illustrating the operation of the counting device according to the example of the present invention.
FIG. 3 is a block diagram illustrating one example of a structure of a counting result correcting portion in a counting device according to another example.
FIG. 4 is a diagram for explaining the operation of a counter in a counting device according to further example of the present invention.
FIG. 5 is a diagram for explaining the operation of a run length measuring portion of a counting device according to a yet further example.
FIG. 6 is a diagram illustrating one example of a frequency distribution of mode hop pulse periods in the case of a high-frequency noise mixed into the signal that is inputted into the counting
FIG. 7 is a diagram illustrating one example of a frequency distribution of the mode hop pulse run lengths.
FIG. 8 is a diagram for explaining the principle of correcting the results of counting by the counting device according to an example.
FIG. 9 is a block diagram illustrating a structure for a conventional laser measuring device.
FIG. 10 is a diagram illustrating one example of a time series of emission wavelengths of a semiconductor laser in the laser measuring device according to FIG. 9.
FIG. 11 is a diagram illustrating an interference waveform that is asymmetrical in respect to time.
DETAILED DESCRIPTION [0035]
Examples for carrying out the present invention are explained below in reference to the figures. FIG. 1 is a block diagram illustrating a structure for a counting device according to an example of
the present invention, and FIG. 2 is a flowchart illustrating the operation of the counting device. The counting device 1 is structured from: a binarizing portion 11; a logical product calculating
portion (AND gate) 12; a counter 13; a counting result correcting portion 14; and a storing portion 15. The counter 13 structures signal counting means.
FIG. 3 is a block diagram illustrating one example of a structure for the counting result correcting portion 14. The counting result correcting portion 14 is structured from: a run length measuring
portion 140; a frequency distribution generating portion 141; a representative value calculating portion 142; and a corrected value calculating portion 143.
In the example set forth below, the explanation uses a case wherein the counting device 1 is applied to a self-coupled laser measuring device, such as illustrated in FIG. 9, to count the number of
mode hop pulses (MHPs) that are the self-coupled signals.
FIG. 4(A) through FIG. 4(D) are diagrams for explaining the operation of the counter 13, wherein FIG. 4(A) is a diagram illustrating schematically the waveform of the signal inputted into the
counting device 1, that is, is an MHP waveform; FIG. 4(B) is a diagram illustrating the output of the binarizing portion 11 corresponding to FIG. 4(A); FIG. 4(C) is a diagram illustrating a gate
signal GS that is inputted into the counting device 1; and FIG. 4(D) is a diagram illustrating a counting result by the counter 13 corresponding to FIG. 4(B).
First the binarizing portion 11 of the counting device 1 identifies whether the input signal illustrated in FIG. 4(A) is at the high (H) level or the low (L) level, and outputs the identification
result as illustrated in FIG. 4(B). At this time, the binarizing portion 11 identifies a high level when the voltage of the input signal rises to be at or above the threshold value TH1, and
identifies a tow level when the voltage of the input signal falls to be below a threshold value TH2 (where TH2<TH1), to perform the binarization.
An AND gate 12 outputs the result of the logical product of the output of the binarizing portion 11 and the gate signal GS, as shown in FIG. 4(C), and the counter 13 counts the rising edges and
falling edges of the output of the AND gate 12 (the binarized signal) (FIG. 4(D)). At this point, the gate signal GS is a signal that rises at the beginning of the counting interval and falls at the
end of the counting interval (which, for example, is a first emission interval P1 or a second emission interval P2 when, for example, the counting device 1 is applied to a self-coupled laser
measuring device). Consequently, the counter 13 counts the number of rising edges and the number of falling edges (that is, the number of MHP run lengths) outputted by the AND gate 12, during the
counting interval. Consequently, the counter 13 counts the number of rising edges and counts the number of falling edges (that is, the number of MHP run lengths) outputted by the AND gate 12 during
the counting interval (Step S100 in FIG. 2).
FIG. 5 is a diagram for explaining the operation of the run length measuring portion 140 in the counting result correcting portion 14. The run length measuring portion 140 measures the run lengths of
the MHPs during the counting interval (Step S101 in FIG. 2). That is, the run length measuring portion 140, during the counting interval, detects the rising of the output of the AND gate 12 through
comparing the output of the AND gate 12 to the threshold value TH3, and detects the falling of the output of the AND gate 12 through comparing the output of the AND gate 12 to the threshold value
TH4. Given this, the run length measuring portion 140 measures the time tud from the rising edge of the output of the AND gate 12 to the subsequent falling edge, and measures the time tdu from the
falling edge of the output of the AND gate 12 to the subsequent rising edge, to measure the run lengths of the output of the AND gate 12 during the measuring interval through measuring the time tdu
from the falling edge of the output of the AND gate 12, to the subsequent rising edge thereof to thereby measure the run lengths of the output of the AND gate 12 during the counting interval (that
is, to measure the MHP run lengths). In this way, the MHP run lengths are the times tud and tdu. The run length measuring portion 140 performs the measurement as described above each time there is a
rising edge in the output of the AND gate 12, each time there is a rising edge or a falling edge in the AND gate 12 output.
The storing portion 15 stores the containing results of the counter 13 and the measurement results of the run length measuring portion 140.
After the gate signal GS has fallen and the counting interval has been completed, then the frequency distribution generating portion 141 of the counting result correcting portion 14 generates
frequency distributions of the MHP run lengths tud and tdu for the counting interval from the measurement results of the run length measuring portion 140, which are stored in the storing portion 15
(Step S102 in FIG. 2). At this time, the frequency distribution generating portion 141 generates frequency distributions for the first run lengths tud of the rising edges through the subsequent
falling edges of the MHPs, and for the second run lengths tdu for the falling edges through the subsequent rising edges of the MHPs.
Following this, the representative value calculating portion 142 of the counting result correcting portion 14 both calculates a representative value T
for the first run lengths tud from the frequency distribution for the first run lengths tud, generated by the frequency distribution generating portion 141 and calculates a representative value T
for the second run lengths tdu from the frequency distribution for the second run lengths tdu, generated by the frequency distribution generating portion 141 (Step S103 in FIG. 2). Here the
representative value calculating portion 142 may use the mode value, median value, or mean value of the first run lengths tud as the representative value T
, or may use the bin value wherein the product of the bin value and the frequency is maximal as the representative value T
, or may instead use the bin value wherein the product of the a power (wherein 0<a<1) of the bin value and the frequency is maximal as the representative value T
. The representative value T
for the second run lengths tdu may be calculated in the same way as the representative value T
. Table 1 illustrates a numerical example of a frequency distribution, and the products of the bin values and the frequencies thereof in this example of numerical values.
-US-00001 TABLE 1 Example of Numerical Values in a Frequency Distribution Bin Value 1 2 3 4 5 6 7 8 9 10 Frequency 11 2 0 3 7 10 6 2 3 1 Product 11 4 0 12 35 60 42 16 27 10
In the example in Table 1, the value (the bin value) with the highest count, wherein the count was the maximum, is 1. In contrast, the bin value wherein the product of the device and the count was a
maximum was 6, a value that is different from the value with the highest frequency. The reason for using, for the representative values T
and T
, the bin value wherein the product of the bin value and the frequency is disclosed in JP '525, and thus the explanation is omitted here. The representative values T
and T
, calculated by the representative value calculating portion 142, are stored in a storing portion 15. The representative value calculating portion 142 calculates these representative values T
and T
each time a frequency distribution is generated by the frequency distribution generating portion 141.
The corrected value calculating portion 143 of the counting result correcting portion 14 calculates, from the results of measurements by the run length measuring portion 140 and the results of the
calculations by the representative value calculating portion 142, a total Ns
for the number of first run lengths tud wherein the lengths are no less than 0 times and less that 1 times the representative value T
, a total Ns
number of second run lengths tdu wherein the lengths are no less than 0 times and less than 1 times the representative value T
, a total Nw
H number of first run lengths tud wherein the lengths are no less than {T
)} and less than {T
)}, and a total Nw
L number of second run lengths tdu wherein the lengths are no less than {T
)} and less than {T
)}, and corrects the counting results by the counter 13 as in the following equation (Step S104 in FIG. 2).
3 N ' = 1 2 [ N - ( Ns H + Ns L ) + n = 1 n max { 2 n × ( Nw nH + Nw nL ) } ] n max ≦ ( T H + T L ) max T H + T L ( 1 ) ##EQU00003##
In equation 1, N is the number of MHP run lengths that is the result of counting by the counter 13, N' is the number of MHPs obtained after the correction, and (T
is the maximum number produced by a sum of a first run length tud and a second run length tdu. The threshold value for calculating the total Ns
number of first run lengths tud may be no less than 0 and less than 0.5 times the representative value T
, or may be no less than 0 times the representative value T
and less than (T
)/4. Similarly, the threshold value for calculating the total Ns
of second run lengths tdu may be no less than 0 and less than 0.5 times the representative value T
, or may be no less than 0 times the representative value T
and less than (T
)/4. The counting device 1 performs the processing as described above in each counting interval.
The principal for correcting the counting results of the counter 13 in the counting device 1 is explained next. The fundamental principle for correcting the counting results, illustrated in Equation
(1), is identical to the principal for correcting the counting results disclosed in JP '676. However, in the principal for correcting as disclosed in JP '676, there are cases wherein the counting
results by the counter 13 could not be corrected well when burst noise of a frequency higher than that of the MHPs is mixed into the signal that is inputted into the counting device.
FIG. 6 is a diagram illustrating one example of a frequency distribution of periods of MHPs when high-frequency noise is mixed into the signal that is inputted into the counting device. When
high-frequency noise is mixed into the input signal, the MHP period frequency distribution, as illustrated in FIG. 6, has, in addition to the distribution 170 that has a local maximum value for the
frequency at the conventional period Ta for the MHP, a distribution 171 that has a local maximum value for the distribution at a period that is approximately half the period Ta, and a short period
172 of the noise. Given this, because of the high-frequency noise that is mixed in, there is a tendency for the times to shift towards the short side when the frequencies have local maximum values.
This high-frequency noise may be mixed in continuously. In the conventional counting device disclosed in JP '676, it is not possible to adequately correct the MHP counting result when high-frequency
noise is mixed in continuously. This type of problem is described in detail in JP '525.
Given this, in the present example a representative value T0 for the MHP run lengths is used, in the same manner as in JP '525, rather than a representative value for the MHP periods, to correct the
counting results. An example of the MHP run length frequency distribution is illustrated in FIG. 7. As is clear from FIG. 7, when calculating the MHP run length frequency distribution, no local
maximum value appears for the frequency in the vicinity of 0.5 T0, even when high-frequency noise is mixed into the signal that is inputted into the counting device 1. That is, local maximum values
for the frequencies disappear in the vicinity of the threshold values for calculating the total Ns
and Ns
numbers for the run lengths, thus making it possible to calculate the aforementioned Ns
and Ns
correctly, to make it possible to control the error in the correction.
However, in the counting device disclosed in JP '525, if the MHP waveform is asymmetrical in respect to time, then the MHP run length frequency distribution will not appear as illustrated in FIG. 7
or FIG. 8(A), but rather will be in a form that has the mode value (the maximum frequency value) at T
, as illustrated in FIG. 8(B), or a shape that has a mode value at T
, as illustrated in FIG. 8(C). If the MHP waveform is symmetrical in respect to time, then when there is an omission in the MHP waveform due to noise, then a run length that is an odd multiple of T0
will be produced. This produces the local maximum values for the frequencies at, for example, 3T0 and 5T0 of FIG. 8(A).
On the other hand, when the MHP waveform is asymmetrical in respect to time, as illustrated in FIG. 11(A), then when there is an omission in the MHP waveform, a run length of a length that satisfies
a value of an integer multiple of (T
) will be produced for T
, and a run length of a length that satisfies a value that is an integer multiple of (T
) will be produced for T
. When there are omissions in the MHP waveform, then a run length wherein 2n+1 run lengths have become a single run length will, in the case of the first run lengths tud, be T
) and, in the case of the second run lengths tdu, will be T
). Because noise of a variety of different frequencies is superimposed on the MHPs, the run lengths will have a Gaussian distribution centered on T
) and a Gaussian distribution centered on T
). As a result, in the example in FIG. 8(B), there are local maximum values for the frequencies at T
) and T
), and there are local maximum values for the frequencies at T
) and T
Given this, in the present example, the threshold value for calculating the total Nw
H number of first run lengths tud is defined as no less than {T
)} and less than {T
)}, and the threshold value for calculating the total Nw
L number of second run lengths tdu is defined as no less than {T
)} and less than {T
)}. Doing so makes it possible to calculate (Nw
L) properly, making it possible to control the error in the correction.
The total Ns
number of first run lengths tud that are no less than 0 times and less than 1 times the representative value T
, and the total Nw
H number of first run lengths that are no less than {T
)} and less than T
)} are illustrated in the example in FIG. 8(D), and the total Ns
number of first run lengths tdu that are no less than 0 times and less than 1 times the representative value T
, and the total Nw
L number of second run lengths that are no less than {T
)} and less than T
)} are illustrated in the example in FIG. 8(E). The above is the principal for correcting the counting results shown in Equation (1). Note that the reason for the 1/2 times on the right side in
Equation (1) is because of the conversion of the number of MHP run lengths into the number of MHPs.
As described above, in the present example, the number of run lengths of MHPs in a counting interval is counted by the counter 13, the run lengths of the MHPs in the counting interval are measured,
frequency distributions of run lengths of MHPs during the counting interval are generated from the measurement results for first run lengths, which are from the rising edge of an MHP until the
subsequent falling edge and for second run lengths, which are from the falling edge of an MHP until the subsequent rising edge, a representative value T
for the distribution of the first run lengths is calculated from the frequency distribution of the first run lengths and a representative T
for the distribution of the second run lengths is calculated from the frequency distribution of the second run lengths, a total Ns
number of first run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Ns
number of second run lengths wherein the length is no less than 0 times and less than 1 times the representative value T
, a total Nw
H number of first run lengths wherein the lengths are no less than {T
)} and less than {T
)}, and a total Nw
L number of second run lengths wherein the lengths are no less than {T
)} and less than {T
)} are calculated, and the result of counting by the counter 13 is corrected based on these frequencies Ns
, Ns
, Nw
H, and Nw
L, thereby making it possible to correct, with high accuracy, the MHP counting error, even when there is continuous noise at a frequency higher than that of the MHPs in the signal that is inputted
into the counting device, even when the MHP waveform is asymmetric in respect to time.
Note that in the present example, the counting device 1 may be achieved through, for example, a computer that is provided with a CPU, a storage device, and an interface, and through a program that
controls these hardware resources. The program for operating such computer is provided in a state that is stored on a storage medium such as a floppy disk, a CD-ROM, a DVD ROM, a memory card, or the
like. A CPU writes to a storage the device a program that has been read, to thereby achieve the processes described in the present example following the program.
Moreover, while in the example the explanation was for a case wherein the counting device according to the present invention is applied to a laser measuring device, there is no limitation thereto,
but rather the counting device according to the present example can be applied also to other fields, such as photoelectric sensors. Cases wherein the counting device according to the present example
are useful include cases wherein there is a linear relationship between the number of signals to be counted and a specific physical quantity (which, in the present example, is the distance between
the semiconductor laser and the object, and a dislocation of the object), where if the specific physical quantity is a constant, the signal has essentially a single frequency. Moreover, the counting
device according to the present example is effective when, rather than the signal being a single frequency, it is essentially a single frequency in a case wherein the spread in the period
distribution is small, such as when the specific physical quantity is the speed of an object that is vibrating at a frequency that is adequately low when compared to the measuring interval, such as a
frequency that is no more than 1/10 the inverse of the measuring interval.
Moreover, while in the present example the explanation was for a case of a laser measuring instrument for calculating the distance to an object and the speed of the object from the results of
counting by a counting device such as disclosed in JP '080, as an example of a physical quantity sensor to which the counting device is applied, there is no limitation thereto, but rather the present
example may be applied to other physical quantity sensors. That is, the tension of an object may be calculated from the counting results by the counting device, or the vibrational frequency of an
object may be calculated from the counting results of the counting device. As is clear from the variety of different physical quantities calculated by the physical quantity sensor, the specific
physical quantity referenced above may be the same as the physical quantity calculated by the physical quantity sensor, or may be different.
Note that the input signal in the present example indicates events or wave motions (in the case of self-coupling, interference patterns) in quantities that change continuously (which, in the case of
self-coupling, are the self-coupled signals).
The present invention can be applied to counting devices for counting signals.
Patent applications by Tatsuya Ueno, Tokyo JP
Patent applications by YAMATAKE CORPORATION
Patent applications in class Measuring or testing
Patent applications in all subclasses Measuring or testing
User Contributions:
Comment about this patent or add new information about this topic: | {"url":"http://www.faqs.org/patents/app/20120257708","timestamp":"2014-04-24T11:51:18Z","content_type":null,"content_length":"80893","record_id":"<urn:uuid:cb4aa491-0fb2-40d8-882d-ce0ee9d2a4fc>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00014-ip-10-147-4-33.ec2.internal.warc.gz"} |
GAP Forum: Beginner's Problems
> < ^ Date: Thu, 25 Feb 1999 12:53:01 +0000
^ From: David Cruickshank <dc@maths.gla.ac.uk >
> ^ Subject: Beginner's Problems
Dear GAP Forum,
I'm very new to GAP, and would appreciate your help to find my way
around it. Given an n-by-m matrix, with entries from the integer
group ring ZG of a free abelian group G, I would like GAP to
calculate the determinants of all k-by-k square sub-matrices (k \leq
min(m,n)), and then define I_k to be the ideal of ZG generated by
these determinants. My problems are:
1) I don't see how to define the ring ZG in GAP - the closest I can
get is a Multivariate Polynomial Ring, or a Univariate Laurent
Polynomial Ring. Can GAP produce group rings?
2) GAP seems to assume that the entries in any matrix are in a field.
Is there any way to avoid this?
3) Once I have my ideal I_k, is there a way of getting GAP to give a
"nice" basis for it, so that I can decide membership? Can GAP
produce Grobner bases?
I'm grateful in advance for any help you can give.
David Cruickshank
> < [top] | {"url":"http://www.gap-system.org/ForumArchive/Cruicksh.1/David.1/Beginner.1/1.html","timestamp":"2014-04-19T17:01:14Z","content_type":null,"content_length":"2134","record_id":"<urn:uuid:8476da1d-3015-46a9-9653-1ae07cc59391>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00361-ip-10-147-4-33.ec2.internal.warc.gz"} |
Perimeter & Area
Mrs Clark's suggested problems Ⓟⓔⓡⓘⓜⓔⓣⓔⓡ ⓐⓝⓓ Ⓐⓡⓔⓐ Andie D, Amanda K, Caroline F, and Melissa L
Area of a Square
The area of a figure measures the size of the region enclosed by the figure. This is usually expressed in terms of some square unit. A few examples of the units used are square meters, square
centimeters, square inches, or square kilometers.
If l is the side-length of a square, the area of the square is l2 or l × l.
What is the area of a square having side-length 3.4?
The area is the square of the side-length, which is 3.4 × 3.4 = 11.56.
Area of a Trapezoid
If a and b are the lengths of the two parallel bases of a trapezoid, and h is its height, the area of the trapezoid is
1/2 × h × (a + b) .
What is the area of a trapezoid having bases 12 and 8 and a height of 5?
Using the formula for the area of a trapezoid, we see that the area is
1/2 × 5 × (12 + 8) = 1/2 × 5 × 20 = 1/2 × 100 = 50.
Area of a Triangle
Consider a triangle with base length b and height h.
The area of the triangle is 1/2 × b × h.
What is the area of the triangle below having a base of length 5.2 and a height of 4.2?
The area of a triangle is half the product of its base and height, which is 1/2 ×5.2 × 4.2 = 2.6 × 4.2 = 10.92..
Area of a Circle
The area of a circle is Pi × r2 or Pi × r × r, where r is the length of its radius. Pi is a number that is approximately 3.14159.
What is the area of a circle having a radius of 4.2 cm, to the nearest tenth of a square cm? Using an approximation of 3.14159 for Pi, and the fact that the area of a circle is Pi × r2, the area of
this circle is Pi × 4.22
The perimeter of a polygon is the sum of the lengths of all its sides.
What is the perimeter of a rectangle having side-lengths of 3.4 cm and 8.2 cm? Since a rectangle has 4 sides, and the opposite sides of a rectangle have the same length, a rectangle has 2 sides of
length 3.4 cm, and 2 sides of length 8.2 cm. The sum of the lengths of all the sides of the rectangle is 3.4 + 3.4 + 8.2 + 8.2 = 23.2 cm.source: http://www.mathleague.com/help/geometry/area.htm
Practice Problems:
(Andie's Question) 1.) Mrs. Clark wants to buy carpet for her room. Her room's measurements are below. With these measurements, How much carpeting does she need to buy to cover the entire floor?
(Caroline's Question) 2.) Mrs. Clark wants to make a jewelry box with the square mosaic pieces below. What is the area of the box's lid?
(Amanda's Question) 3.) Find the area to this triangle. (Melissa's question) 4.) What is the perimeter and the area to this trapezoid? ANSWERS: 1.) 140 ft. 2.) 24 in. 3.) 32 4.) perimeter- 54 area-
112 Games: http://www.mathplayground.com/area_perimeter.html http://www.funbrain.com/poly/index.html http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/perimeter_and_area/index.html
Worksheets: http://www.superteacherworksheets.com/perimeter.html http://www.atozteacherstuff.com/Printables/Math_Worksheets/Area___Perimeter/index.shtml Videos: http://videos.howstuffworks.com/hsw/
Emma R., Carrie H., Katherine K., Lauren L.Triangle--Emma. Circle--Carrie. Rectangle--Katherine. Lauren--Square.
Perimeter for a Triangle: add all of the sides together
Area for a Triangle: 1/2 x base x height
Practice Problem with Triangles:
Find the perimeter,
, of the triangle shown below.
2. formula being used to find the area of a triangle.
Solution: problem 1
So, the perimeter is 28 cm.
source: http://www.mathsteacher.com.au/year7/ch12_length/07_triangle/triangle.htm Formulas and Practice Problems For Circles http://www.efunda.com/math/areas/circlegen.http://www.efunda.com/math/
1. Calculate the perimeter (circumference) of the circle below.
2. Calculate the area of the circle below:
Answers For Circle Problems:
1. 75.4
2. 28.3
FORMULAS AND PRACTICE PROBLEMS FOR AREA AND PERIMETER OF A RECTANGLE AREA = Length x Width PERIMETER = 2length + 2width
Area: 12 x 5 =60
help with perimeter and area of a rectangle http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/perimeter_and_area/index.html http://easycalculation.com/area/learn-rectangle.php Square
Perimeter and Area -Lauren Langdon's questions A square is a four sided shape with equal sides. H, in the equation to the left, stands for the height of the square, and the B stands for the length of
the base. So you multiply B and H to equal the area of square. Area is the amount of space in an object. Then to find the perimeter, the outside edge of a shape, of a square use the formula 4xS. That
means you multiply four time S, or the length of one side. Find the area and the perimeter of each squares. Question 1: Question 2: Area: Area: Perimeter: Perimeter: Answers:
1. Area- 16 inches
Perimeter- 16 inches
2. Area- 64
Perimeter- 32
Perimeter and AreaHannah T., Emma A., Kate W., Rachel T.Friday Class: Period 2
Rectangle: Perimeter- Length is "L" and width is "W" Perimeter Formula- P=2L+2W Area- Length is "L" and width is "W" Area Formula- LW Practice Problems for Rectangles: 1) Mrs. Clark wants to buy wood
to fence in her yard. Calculate the perimeter of Mrs.Clark's yard to find the amount of wood needed.
http://www.atozteacherstuff.com/Printables/Math_Worksheets/Area_Perimeter/index.shtml 2) Joe wants to redo his kitchen floor. He wants to put tile flooring in the kitchen. Use the measurements below
to calculate the amount of tile flooring Joe needs in yards. http://www.mathgoodies.com/lessons/toc_vol1.html 3) With the measurements given below find the area and perimeter of the rectangle
http://www.analyzemath.com/Geometry/rectangle_problems.html 4) Calculate the area and the perimeter of the following rectangle. http://www.sparknotes.com/math/prealgebra/perimeterarea/problems.html
Answers: 1) P=160m 2) 28 square yards 3) P=22cm A= 24cm squared 4) P=18 A= 20 Useful Resources for Rectangles: Videos- Area
Games-Perimeter and Area 1
Perimeter and Area 2
Worksheets- Perimeter and Area 1
Perimeter and Area 2
In the following formulas: s = side Perimeter Formula: P = 4s Area Formula: A=s² Practice Problems for Squares: 1. Find the perimeter of a square if each side is 15m. 2. Find the area of a square
courtyard if each side is 50 feet. 3. Suzie Star wants to find the perimeter of a piece of square origami paper before she makes a crane. Find the perimeter if one side is 4 inches. 4. Mrs. Clark is
looking at a square coffee table for her living room. If one side of the table is 2m, what is the area of the table? Answers to Square Practice Problems: (1. 60m) (2. 2500 feet) (3. 16 inches) (4. 4
m) Circle:
• d stands for diameter, which is the full length of the circle from one side to the next
• r stands for radius, which is half of the full length of the circle, it goes from the middle to the side
• 3.14 stands for what is Pie
Area The formula for finding area is 3.14 x r x r Circumference The formulas for finding the circumference is 3.14 x d or 3.14 x r x 2 Extra Practice:
1. 1-6: Find the circumference and area of each circle using the correct formula.
#7: Mrs. McGinley needs to know how much fabric she will need for their new hot tub cover, find the area of the circle to figure out what the dimensions will be. Answers: 1) C= 9.42cm A= 28.26cm 2) C
= 251.2mm A= 20,096mm 3) C= 3.14cm A= 3.14cm 4) C= 12.8dm A= 52.8dm 5) C= 16.3m A= 84.9m 6) C= 24.8cm A= 195.9cm 7) 78.5 Perimeter of a Triangle
The formula to find the perimeter of a triangle is
P= a+b+c
1) Given that a=5, b=4 and c=3, what is the perimeter?2) What is the perimeter?3) What is the perimeter?4) What is the perimeter?ANSWERS1) 122) 163) 124) 24Area of a Triangle
The triangle area formula is the Base, multiplied by the height,
then multiplied by one half (1/2).
EXAMPLE 1:
= · (15 in) · (4 in)
=· (60 in2
= 30 in2
Example 2:
=· (6 cm) · (9 cm)
=· (54 cm2)
= 27 cm2
PRACTICE A) If the triangle has a height of 7 cm, and a base of 6 cm, what is the area? B) If the triangle has a height of 7 cm, and a base of 4 cm, what is the area? Answers: A) 21 cm B) 14 cm
Jordan A, Sara M, Serenity W, and Margie S. Period 1 Perimeter of a Square:Serenity Circumference of a Circle: C= 2∏ x r R is the radius of the circle. 2R can be translated into D. D is the diameter
of the circle. Problem: http://www.google.com/imgres?imgurl=http://www.mathsteacher.com.au/year9/ch14_measurement/02_circle/Image2518.gif&imgrefurl=http://www.mathsteacher.com.au/year9/
Find the circumference of the circle. Answer: C= 2(3.14) x 14 = 6.28 x 14 = 14.96 cm Worksheet links: http://www.dadsworksheets.com/v1/Worksheets/Basic%20Geometry/
Circle_Area_And_Circumference_V3.html http://www.dadsworksheets.com/v1/Worksheets/Basic%20Geometry/Circle_Area_And_Circumference_V4.html Area of a Circle: A= ∏ x r^2 (r squared) Problem: Radius= 15
inches Image of circle: http://www.google.com/imgres?imgurl=http://www.mathwarehouse.com/geometry/circle/images/circumference/ Find the area of the circle. Answer: A= (3.14) x 15^2 = (3.14) x 225 =
706.5 in. Worksheet links: http://www.dadsworksheets.com/v1/Worksheets/Basic%20Geometry/Circle_Area_And_Circumference_V1.html
Perimeter of a Rectangle: P = 2l + 2w Area of a Rectangle: A = lw http://www.google.com/imgres?imgurl=http://0.tqn.com/d/math/1/0/B/F/rectangler.gif&imgrefurl=http://math.about.com/od/formulas/ss/
areaperimeter_3.htm&usg= N8KWUp0nIe5gLcuAOjSYZIc3uHg=&h=330&w=355&sz=3&hl=en&start=0&zoom=1&tbnid=cvR8DTMhUPZsLM:&tbnh=155&tbnw=167&prev=/
iact=hc&vpx=144&vpy=77&dur=6454&hovh=216&hovw=233&tx=180&ty=172&ei=vIqHTOjzCYWFnQfzy7meDA&oei=SIqHTKmdG6W4nAfpz4m6Dg&esq=5&page=1&ndsp=8&ved=1t:429,r:0,s:0 At the park, a new brick path is being laid
out. If the width of the path is 35 m and the length is 45 m, what is the perimeter of the path? http://www.mathsteacher.com.au/year7/ch12_length/06_rect/rect.htm What is the area of a rectangle that
has a length of 9 ft. and a width of 3.5 ft.? 1.) 160 meters 2.) 31.5 squared feet Triangle Perimeter of a triangle= side+side+side Find the perim eter of the triangle. Answer: 9+11+10=30So the
perimeter of the triangle is 30 cm. Area of a triangle= 1/2 x base/ height Find the area of the triangle. Answer: 1/2 x 6 x 3= 9So the area of the triangle is 9 cm. | {"url":"http://stageometrych1.wikispaces.com/Perimeter+%26+Area?responseToken=02604dba276de4df174f761bfe559452c","timestamp":"2014-04-23T10:21:44Z","content_type":null,"content_length":"120351","record_id":"<urn:uuid:422b0e35-6d03-4cc4-9e31-afa19ff799e6>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00626-ip-10-147-4-33.ec2.internal.warc.gz"} |
weibull parameter
October 30th 2008, 11:21 AM #1
weibull parameter
Hi All..
I wanted to know how do you make the Weibull equation below into the form y=mx+c. The solution to this has the logs taken twice..
$P_f= 1 - exp\left[-\left(\frac{\sigma}{\sigma{_0}}\right)^m\right]$
Thank you
This is the cdf of the Weibull distribution. It's not clear to me what you're actually trying to do with it. Post the whole question, exactly as it's written.
Hi.. thanks for your reply.. there is no question just from notes where the weibull is in the form y=mx+c.. just wanted to know the process on how it is derived?
I will put the solution..
$\ln\ln\left(\frac{1}{1-P_f}\right) = m\ln\sigma - m\ln\sigma_0$
Hi dadon,
We have
Note that we can re-write this as
Taking logs to the base $e$ of both sides gives
Take the minus over to make it easier to deal with....
Now we know that $a\ln(b)\equiv\ln(b^a)$ thus we are left with
$\left(\frac{\sigma}{\sigma{_0}}\right)^m=\ln\left( \frac{1}{1-P_f}\right)$
Taking logs of base $e$ again and using that same principle leads to
$<br /> m\ln\left[\left(\frac{\sigma}{\sigma{_0}}\right)\right]=\ln\left[\ln\left(\frac{1}{1-P_f}\right)\right]<br />$
Using the fact that $\ln\left(\frac{a}{b}\right)\equiv \ln(a)-\ln(b)$ gives the required result.
Hope this helps.
Thank you.. that's exactly what I needed..You made it very clear line by line..
Last edited by dadon; October 30th 2008 at 01:44 PM.
October 30th 2008, 12:38 PM #2
October 30th 2008, 12:54 PM #3
October 30th 2008, 01:09 PM #4
Aug 2007
October 30th 2008, 01:19 PM #5 | {"url":"http://mathhelpforum.com/advanced-statistics/56620-weibull-parameter.html","timestamp":"2014-04-20T08:51:52Z","content_type":null,"content_length":"45587","record_id":"<urn:uuid:9418b820-ca99-4a4a-9f95-317828100890>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00159-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculate concentrations in air, water and octanol - WyzAnt Answers
A student was performing an extraction procedure of o-Xylene from water into octanol. The total mass of o-Xylene was 0.10 gram (molecular weight 106.2 g/mole). The volume of water was 990mL and
volume of octanol 10 mL. There is a head space of 200 mL above the solution and it is sealed. Temperature was 25°C. The Henry’s Law constant for o-Xylene is 0.2Matm-1 and logKOW is 3.12. Calculate
the concentrations of o-Xylene in the air above the solution in ?g/m3, concentration in water and octanol in mg/L.
Tutors, please sign in to answer this question.
3 Answers
Using Henry's Law, we get the partial pressure of the gas above the solution as
p = 0.2 (atm M^-1) * 0.1 (g) / 106.2 (g/mole) * 1/1 (L) , since total volume of liquid is 1L.
This value of p is transferred to the gas equation PV = (g/M)RT to determine the weight g of xylene in the headspace gas. The number of moles "n" has been replaced by weight (g) divided by molecular
weight (M).
I did not calculate the value of p in the above equation, because p * M = 0.02 and simplifies things as we go along.
g now becomes pVM/RT, or
g = 0.02 (atm*g*mol^-1) * 0.2 (L) * 1/298 (K) / 0.082 (L*atm mol^-1 K^-1)
After calculations, g = 1.63693 * 10^-4 g (in 200cc). Its concentration in g m^-3 is
1.63693 * 10^-4 (g) * 1/200 (cm^3) * 10^6 (cm^3 / m^3), or 0.8185 g m^-3.
Since g is ~ 1000 times less than 0.1g, we ignore this in the partition calculations. Total mass of xylene is 0.1 g (100mg) and log K[OW] = 3.12. Volume of water and xylene are 990 mL and 10mL
If x is the concentration of xylene in octanol, then (1-x) is the concentration in water, since extraction is from water to octanol, and assuming that 1M was the original concentration in water.
Original concentration of xylene in water = 100 (mg) /0.99 (L) = 101.01 mg L^-1.
log K[OW] = 3.12 , hence Conc in octanol / Conc in water is 10^3.12 = 1318.257.
1318.257 = x / (1-x). Solving for x, we get
x = 1318.257 / 1319.257 = 0.999242, and
1-x = 0.000758
If the original concentration of xylene in water was 101.01 mg L^-1, then
The new concentration of xylene in water is 0.07656 mg L^-1 , and the concentration of xylene in octanol is 100.9335 mg L^-1 . These values were obtained by multiplying (1-x) and x respectively with
101.01. The units of concentration did not make a difference, as we were using ratios of concentration in the partition coefficient equation.
Did the person who wrote this problem even try solving it? It's actually not solvable with the information given. When I tried it, I actually got more xylene than is actually present, when I added
the three amounts together.
Partly, this is because you were given the wrong Henry's constant. A quick Google search told me that the Henry's constant for o-xylene in water is 0.2 M/atm. This is irrelevant, however, since
octanol is the top layer.
Another problem is that Henry's Law relates the partial pressure of gas to the solubility of that gas in a liquid. Solubility is different from actual concentration. Thus, you can't use Henry's Law
for this problem. You should be using vapor pressure calculations, instead.
If we were ignoring the air concentration, the first answer would have almost the right procedure for solving the octanol and water concentrations, but there's an error. First, there's only 0.1 g of
xylene present. Second, the oil-water partition coefficient uses molar concentrations. Thus, the numerator is the concentration of xylene in octanol (we'll call this co), while the demoninator is the
concentration of xylene in water (cw).
The octanol concentration is moles of xylene (no) per liter of octanol (0.01 L), and moles of xylene is mass of xylene (mo) divided by its molar mass (106.2 g/mol). Thus, co = mo/0.01L/(106.2) g/mol
= mo/1.062 M.
We can do something similar for water concentration, and get cw = mw/95.6 M.
Now, going back to the earlier o-w partition coefficient,
1318 = co/cw = (mo/1.062) / (mw/95.6)
simplifying, we get mo/mw = 14.64.
Remembering that the total mass of xylene is 0.1 g,
mo + mw = 0.1, or
mw = 0.1 - mo,
we can substitute to get
14.64 = mo / (0.1-mo)
mo = 0.0936 g
mw = 0.0064 g
Or, if we're correctly following the sig fig rules,
mo = 0.09 g
mw = 0.01 g
Helllo Brook,
I expect that "logKOW" means the log of the octanol-water partition coefficient. Taking the inverse log of 3.12, we get 1318.256739. If before the extraction we have 0.10g of o-xylene in water, then
post extraction we have x grams of o-xylene in octanol, and 1-x is the grams of o-xylene left in the water. Set up an equation as follows:
1318.256739 = [x/10. mL] / [(1-x)/990. mL] = (x/10)(990/1-x) = (990x/10-10x)
Solving for x we get x=0.930 g o-xylene, and 1-x=0.0699 g water
Convert your grams to mg: 930 mg o-xylene and 69.9 mg water
The volume is 990. mL + 10. mL = 1000. mL = 1.00 L
I don't understand what "?g/m3" represents.
remove that '?' and you get grams per cubic meter, which would be the concentration of xylene in the air. | {"url":"http://www.wyzant.com/resources/answers/12381/calculate_concentrations_in_air_water_and_octanol","timestamp":"2014-04-18T06:03:50Z","content_type":null,"content_length":"47404","record_id":"<urn:uuid:f789259d-9e85-4706-92b8-47e0e9e56fc2>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00334-ip-10-147-4-33.ec2.internal.warc.gz"} |
The great run estimator shootout (part 2)
Last week, we left off with a bit of a cliffhanger. Hopefully we can resolve those issues now.
But first, I want to clarify the reason for doing this. There are a lot of different rate metrics of offense available out there, such as EqA, wOBA, and OPS+. (There are also quite a few measures of
runs above average and replacement in the world.) The idea is to capture which best measures a player’s contribution on offense. Now, onwards.
What’s wrong with RMSE?
I don’t mean to pick on root mean squared error—I actually like RMSE, in situations that call for it. I don’t mean to pick on correlation or mean absolute error either. They’re valuable tools in the
toolbox that we use to evaluate our results. But each tool has its limits, and we need to be careful about not using a tool past those limits.
Let’s check on an updated version of the chart from last week. There have been a few changes: We’re now looking at all games (rather than half-innings) from 1954 to 2008, plus the 1953 NL. (This is
commonly referred to as the “Retro-era”.) I have tweaked the way a few of the run estimators are applied, based upon feedback and some additional bugchecking I’ve done. The biggest change is in the
way I adjusted the run results. The formula to convert rates to runs is typically something along the lines of:
(2*(Rate/LgRate)-1) * PA * R_PA
Where Rate is the stat under consideration, LgRate is the average, and R_PA is the average runs per plate appearance. In other words, the formula first gives us the player’s production relative to
average and then is converted to total runs.
And this makes sense for a metric evaluating a position player because whenever a player can avoid making an out, he contributes additional runs to his team by creating more plate appearances. At the
team level (and this is true if you want to look at games, innings or seasons), though, a team is limited to a set number of outs regardless, and so outs is the correct measure of playing time. So,
for these purposes, we will be looking at runs per out, not runs per plate appearance from now on. And, the table:
RC BsR EqR OPS OPS+ GPA TA wOBA Reg House
R 0.86 0.87 0.85 0.84 0.84 0.84 0.85 0.87 0.86 0.87
MAE 1.86 1.78 1.90 1.93 1.92 1.91 1.94 1.79 1.82 1.79
RMSE 2.37 2.25 2.46 2.46 2.45 2.46 2.49 2.30 2.30 2.27
Again, we see little to differentiate one run estimator from the next. Going by MAE, we see a difference in only .16 runs per game from the best and the worst run estimators. This difference tends to
become even smaller at the team season level, because the spread of run environments becomes smaller. (BaseRuns, the undisputed king of the inning-level test, appears far more ordinary at the game
level for that very same reason—as the variation between runs scored becomes smaller, the opportunities for a run estimator to be more accurate grows smaller.)
Is there anything to be gained by such small gains in RMSE? Or is it fair to say that any measure of offense is good enough for the job, so well as it’s reasonably well designed?
In short, no.
If we look at how these metrics are designed, we can see a very real difference of opinion between them on important matters. A metric like Total Average, for instance, considers the walk to be as
valuable as a single. Any of our metrics based upon OBP and SLG, on the other hand, treat the walk as only about half as valuable as the single. Is this a big deal?
When we aggregate at the team level, this usually isn’t a big deal at all. Let’s consider the walk, including (for right now) the hit by pitch and the intentional walk. In 2008, for instance, if we
look at the Red Sox, who walked the most, and the Royals, who walked the least, we only find a difference of about 26 walks per 650 plate appearances. So long as the overall construction of the
metric is generally sound, this is not going to significantly impact the overall RMSE or correlation coefficient at the team level.
But what about individual players? Looking at qualified starters in 2008, we have Jack Cust as the absolute walkingest player there was and Yuniesky Betancourt as the least. Over the course of 650
plate appearances, there’s a difference of 104 walks between the two players.
The problem with our tests is that we are validating at the team level and then applying these values to individual players. But there is simply not enough variation between teams at the seasonal,
game or inning level to truly test the differences between hitters.
Two of every sort
What we cannot do is simply test every run estimator against individual players, as we would like to do. After all, if we could figure out how many runs a player contributes without run estimators,
we wouldn’t need to conduct any of this testing. So we need to come at the problem from a different angle. One idea is to look at how each run estimator values each event, relative to how much each
event is worth in runs.
So how can we estimate how well each run estimator handles, say, the walk? One way is by using matched pairs. What we do is look at a pair of games which had exactly the same number of all events
(such as singles, doubles and home runs) except for one game having an additional walk. Then we look at the average number of additional runs that score in our “plus one” games, as well as the number
of additional runs that our estimators say should have scored. This allows us to measure the accuracy of all of our run estimators at the individual event level.
(Why games instead of innings, as we were working with last week? It turns out that run scoring tends to cluster—you cannot score half runs or quarter runs, so in order to make everything balance out
the majority of innings are scoreless, roughly three-quarters in fact. When you do matched pairs, you end up with nearly 99 percent scoreless innings. Matching at the game level provides a more
natural distribution of run scoring.)
And now, for the results of that trial. “Num” indicates how many of each event there were over the entirety of the Retroera; Similarity is a measure of how close each run estimator is to the observed
runs, weighted by the number of events in the sample. A smaller similiarity is better.
Event Num R EqR OPS OPS+ GPA TA BsR RC wOBA Reg House
1B 1322772 0.46 0.45 0.50 0.52 0.50 0.30 0.42 0.50 0.46 0.53 0.46
2B 331713 0.72 0.76 0.86 0.86 0.76 0.59 0.73 0.79 0.75 0.61 0.73
3B 47125 0.86 1.10 1.25 1.22 1.03 0.90 1.03 1.11 1.04 1.23 1.04
HR 190386 1.36 1.42 1.63 1.57 1.31 1.19 1.44 1.37 1.39 1.45 1.39
BB 642069 0.32 0.29 0.25 0.28 0.32 0.30 0.28 0.22 0.31 0.34 0.31
HBP 52857 0.28 0.26 0.21 0.25 0.30 0.29 0.28 0.22 0.31 0.31 0.30
Similarity 0.04 0.11 0.10 0.04 0.13 0.05 0.07 0.03 0.09 0.03
On average, based upon our matched pairs sample, during the Retro-era a walk was worth an additional .32 runs on average. Our robust linear weights measures, like wOBA and the “house” weights, match
up very well here. Something like RC or OPS fares much more poorly here, with values of .22 and .25 for the walk respectively. Those measures are going to underrate our high-walk players and overrate
our low-walk players.
Something like TA, on the other hand, measures the walk about fine, but vastly underrates the single. That means that it will underrate high-average, low-slugging players like Ichiro and overrate
low-average, high-slugging players like Adam Dunn.
Going by similarity, we can see that measures like wOBA and the house weights are very close to the observed values; metrics like BsR, EqR and GPA also do very well here. (And bear in mind that BsR
is not tuned to the environment, which could improve the accuracy here.) In the middle of the pack are things like Basic RC and the regression-based linear weights. The worst contestants are the
OPS-derived measures and TA.
Some reservations
This test isn’t perfect. You’ll note that until now I haven’t addressed the issue of the triple. Three things could be true here:
1. Every serious run estimator ever devised overweights the triple significantly.
2. This is a sampling error caused by the low number of triples in general.
3. There is a selective sampling problem with the triple in doing a matched-pair study.
My personal feeling is that the problem is number three, but I have no evidence for this.
This is certainly true of the stolen base and the caught stealing, however. Teams tend to attempt steals much more frequently in close games, which unduly biases the sample. This is why those terms
are excluded from the study.
I don’t think these problems invalidate the study as a whole, but that’s only my opinion, and as with medicine you should probably seek a second one.
What’s the point?
There is nothing in the world requiring that you be entirely accurate or even as accurate as possible or is reasonable. If all you’ve ever used is and all you ever want to use is OPS, I can’t stop
you. And I can’t make sites that publish flawed run estimators correct their mistakes.
But if we want to be correct, or at least as correct as possible, we need to consider the potential biases of a run estimator, not only its accuracy on the whole. And after all, it is very easy to
figure out the very easy cases. We don’t need a run estimator to tell us that Albert Pujols is good at hitting baseballs or that Michael Bourn isn’t. It’s the more difficult cases where we need run
estimators. And those are the players most likely to test the biases of our estimates.
And if you are planning on testing your latest and greatest run estimation formula, please, spare us the same tired R and RMSE tests where only your formula gets the benefit of being tuned to the
environment at hand. Thanks.
References & Resources
For more reading on the difference between runs per PA and runs per out, read this.
The similarity measures in the article were inspired by PECOTA’s sim scores, and are calculated by using a derivation of the Pythagoeran Theorum (the one developed by Pythagoreas, not the one
developed by Bill James.)
While Equivelent Runs grades out very well here, there are some concerns about the way Equivelent Average is figured, totally apart from its accuracy as a run estimator.
Some notes on the modificaitons made since the last batch of tests. Since I changed the dataset in use, I ran another regression to estimate the regression weights, which are now:
0.53*1B + 0.61*2B + 1.23*3B + 1.46*HR + 0.34*BB + 0.31*HBP -.11*IBB + 0.18*SB – 0.05*CS -0.10*Outs
wOBA is not converted to runs the same way as the other rates; it is in fact not converted to runs at all. Instead, I took the base weights used to generate wOBA (based upon work by Tom Tango) and
applied them to the data, the same as the House weights. To clarify something: wOBA could be computed using the house or the regression weights as easily as the weights used; this is intended to
measure the accuracy of the values provided by a specific implementation (the one in use at Fangraphs), rather than the concept of wOBA in particular.
Some great help was provided in this thread at Tango’s blog. Special thanks to Tango, Patriot and terpsfan. Terpsfan corrected a key error in figuring the value for CS in the updated House weights.
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REFLECTION ON THE TEACHING OF “THE MULTIPLICATION ALGORITM OF THE 3rd GRADE OF PRIMARY SCHOOL” THROUGH VTR
By Marsigit
Recently study in Indonesia indicated that the use of VTR (Video Tape Recorder) in the teacher training program was perceived by the teachers as good and useful. There is a higher frequency to use
the VTR to promote teachers’ professional development in Japan and in developed countries; however, in Indonesia, it pops up like a jack-in-the-box. VTR for teacher education and reform movement in
Mathematics Education, specifically for developing lesson study has some benefits as: a) short summary of the lesson with emphasis on major problems in the lesson, b) components of the lesson and
main events in the class, and, c) possible issues for discussion and reflection with teachers observing the lesson (Isoda, M., 2006).
Katagiri, S (2004) listed the types of mathematical thinking as mathematical attitudes, mathematical thinking related to mathematical methods, and mathematical thinking related to mathematical
contents. This identification of mathematical thinking by Katagiri can be the starting point to reflect any mathematics teaching learning process at school as for to reflect the teaching of “the
multiplication algoritm of the 3rd grade of primary school” by Mr. Hideyuki Muramoto; then, the VTR of this lesson will be targeted for the series of activities: observation and reflection.
Characterizing The Lesson from Lesson Plan
The preliminary characteristics of Muramoto’s teaching are stipulated in the Lesson Plan such as follows:
a. Theme : Third grade mathematics lessons that foster students’ ability to use what they learned before to solve problems and make connections in order to solve problems in new learning situations
b. Method: Teaching “the Multiplication algorithm (1)” in a way that develops students who can use what they learned before to solve problems in new learning situations by making connections.
c. Goals of the Unit: To be able to think about how to carry out the calculation of a 2-digit number x a 1-digit number by using what was previously learned about multiplication (mathematical
d. Scenario of Teaching
1) Developing teaching that help students to become aware of the connection between what they learned before and what they are learning now and use previously learned knowledge to overcome obstacles
in a new situation.
2) Connections between previously learned knowledge and new learning
3) Representing a problem situation with diagrams based on the idea of “how many times as much as a unit quantity” consistently and helping students to understand the situation and solution of the
problem more clearly.
4) Developing lessons that incorporate this idea and help students to use the diagram to think
Characterizing the Lesson through VTR
a. The problem of video taping
- The quality of pictures are relatively good
- The single camera made the limitation of landscaping the class
- The small caption in the screen helps to catch more the picture of the class
b. The components of the lesson
- The whole class teaching has reduced the complexity of class interaction into the simple or linear pattern of interaction between teacher and students.
- Highlighting the certain ideas from certain student has ignored the other students’ ideas.
- Highlighting the certain aspect of mathematical thinking of a certain students endanger the total management of the class.
c. Encouraging and uncovering students’ mathematical thinking
- Teacher’s effort in encouraging and uncovering students’ mathematical thinking were effective enough.
- Teacher’s effort in serving individual students has not been effective yet.
- Some of the students were able to perform mathematical thinking
- Teacher was able to achieve the goal of the lesson
- Mathematical thinking of a certain students can be a model for others.
- Different students, in the same allocation of time, did similar problems by employing different methods to cultivate the similar results.
- Students’ discussion among themselves has not emerged yet.
- Students’ involvements in classroom management were still limited.
- Teacher has effectively employed the proper teaching aids.
The conclusion of the paper highlights some problems as follows:
- The problem of the reduction of the complexity of classroom interaction into the simple or linear pattern between teacher and his students.
- The problem of landscaping the whole classroom activities
- The negative correlation between focusing a certain aspect of students thinking and reducing the variant of their learning contexts.
- The problem of the pattern for the relation for promoting individual needs and the whole classroom management.
- The problem of the gap amongst teachers’ effort (including methods and media), students’ findings and the concept/understanding/rational of the vertical way of calculating 23 times 3.
- The problem of matching the theory of the concept of mathematical thinking and the factual condition of students’ mathematical thinking.
- The problem of mathematical thinking of the lower achievement students.
- The problem of exploring intrinsic, extrinsic and systemic of mathematical thinking.
Isoda, M. (2006). Reflecting on Good Practices via VTR Based on a VTR of Mr.
Tanaka's lesson `How many blocks? Draft for APEC-Tsukuba Conference in
Tokyo, Jan 15-20, 2006
Marsigit, (2006), Lesson Study: Promoting Student Thinking On TheConcept Of Least
Common Multiple (LCM) Through Realistic Approach In The 4th Grade Of
Primary Mathematics Teaching, in Progress report of the APEC project: “Colaborative Studies on Innovations for Teaching and Learning Mathematics in Diferent Cultures (II) – Lesson Study focusing on
Mathematical Thinking -”, Tokyo: CRICED, University of Tsukuba.
Shikgeo Katagiri (2004)., Mathematical Thinking and How to Teach It. in Progress
report of the APEC project: “Colaborative Studies on Innovations for Teaching
and Learning Mathematics in Diferent Cultures (II) – Lesson Study focusing on
Mathematical Thinking -”, Tokyo: CRICED, University of Tsukuba.
No comments: | {"url":"http://pbmmatmarsigit.blogspot.com/2008/12/reflection-on-teaching-of.html","timestamp":"2014-04-19T11:56:39Z","content_type":null,"content_length":"61013","record_id":"<urn:uuid:eb1d8376-ab79-4542-89aa-c0077a32cdb6>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00417-ip-10-147-4-33.ec2.internal.warc.gz"} |
Modular Exponentiation
Suppose we are asked to compute $3^5$ modulo $7$. We could calculate $3^5 = 243$ and then reduce $243$ mod $7$, but a better way is to observe $3^4 = (3^2)^2$. Since $3^2 = 9 = 2$ we have $3^4 = 2^2
= 4$, and lastly
\[ 3^5 = 3^4\times 3 = 4 \times 3 = 5 \pmod{7}. \]
The second way is better because the numbers involved are smaller.
This trick, known as repeated squaring, allows us to compute $a^k$ mod $n$ using only $O(\log k)$ modular multiplications. (We can use the same trick when exponentiating integers, but then the
multiplications are not modular multiplications, and each multiplication takes at least twice as long as the previous one.)
The Discrete Log Problem
Let us examine the behaviour of the successive powers of $3$ modulo $7$.
\[ 3^1 = 3 \] \[ 3^2 = 2 \] \[ 3^3 = 6 \] \[ 3^4 = 4 \] \[ 3^5 = 5 \] \[ 3^6 = 1 \]
Note we compute each power by mulitplying the previous answer by $3$ then reducing modulo $7$. Beyond this, the sequence repeats itself (why?):
\[ 3^7 = 3 \] \[ 3^8 = 2 \]
At a glance, the sequence $3, 2, 6, 4, 5, 1$ seems to have no order or structure whatsoever. In fact, although there are things we can say about this sequence (for example, members three elements
apart add up to 7), it turns out that so little is known about the behaviour of this sequence that the following problem is difficult to solve efficiently:
(The discrete log problem) Let $p$ be a prime, and $g, h$ be two elements of $\mathbb{Z}_p^*$. Suppose it is known that $g^x = h \pmod{p}$. Then what is $x$?
Example: One instance of the discrete log problem: find $x$ so that $3^x = 6 \pmod{7}$. (Answer: $x = 3$. Strictly speaking, any $x = 3 \pmod{6}$ will work.)
Recall when we first encountered modular inversion we argued we could try every element in turn to find an inverse, but this was too slow to be used in practice. The same is true for discrete logs:
we could try every possible power until we find it, but this is impractical.
Euclid’s algorithm gave us a fast way to compute inverses. However no fast algorithm for finding discrete logs is known. The best discrete log algorithms known are faster than trying every element,
but are not polynomial time.
Why don’t we bother studying the behaviour of nonunits under exponentiation?
First consider when $n = p^k$ for some prime $p$. Then $a\in\mathbb{Z}_n$ is nonunit exactly when $\gcd(a, n) \gt 1$, which in this case means $a = d p$ for some $d$.
We have $a^k = d^k p^k = 0$, thus in at most $k$ steps we hit zero, which is uninteresting (at least for our purposes).
In general, write $n=p_1^{k_1}...p_n^{k_m}$. By the Chinese Remainder Theorem we have
\[ \mathbb{Z}_n = \mathbb{Z}_{p_1^{k_1}} \times ... \times \mathbb{Z}_{p_m^{k_m}} \]
Thus an element $a\in\mathbb{Z}_n$ corresponds to some element $(a_1,...,a_m)$ on the right-hand side, and $a$ is a nonunit if at least one of the $a_i$ is a multiple of $p_i$. From above, this means
in at most $k_i$ steps, the $i$th member will reach zero, so in general, for some $k$, each $a_i^k$ is zero or a unit, hence we can restrict our study to units.
Note we have again followed an earlier suggestion: we handle the prime power case first and then generalize using the Chinese Remainder Theorem. | {"url":"http://crypto.stanford.edu/pbc/notes/numbertheory/exp.html","timestamp":"2014-04-20T19:08:26Z","content_type":null,"content_length":"7692","record_id":"<urn:uuid:42417dc3-4987-4c0b-8ca5-a6e936b6d8c5>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00391-ip-10-147-4-33.ec2.internal.warc.gz"} |
proofs for 1=0.(9)
Re: proofs for 1=0.(9)
Excellent Idea! Yes, we can moderate this heavily, and leave discussions elsewhere. SOME discussion is allowed of course, but please keep it closely to the point.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman | {"url":"http://www.mathisfunforum.com/viewtopic.php?pid=55990","timestamp":"2014-04-18T13:33:59Z","content_type":null,"content_length":"17852","record_id":"<urn:uuid:c9ba48b9-104b-47f3-b80b-2ca9bfa9ef03>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00019-ip-10-147-4-33.ec2.internal.warc.gz"} |
[FOM] First Order Logic
Richard Heck richard_heck at brown.edu
Sat Sep 7 10:49:05 EDT 2013
On 09/07/2013 06:46 AM, Arnon Avron wrote:
> So I do not see much point in repeating what I wrote then in several
> postings. I'll just repeat the words started that debate (in my
> posting from Fri Oct 20 19:32:16 EDT 2006): "For many years I maintain
> that the appropriate language for formalizing logic and mathematics is
> neither the first-order language nor the second-order one. The first
> is too weak for expressing what we all understand, the second involves
> too strong ontological commitments. The adequate language is something
> in the middle: what is called "ancestral logic" in Shapiro's book
> "Foundations without Foundationalism". This logic is equivalent to
> weak second-order logic (as is shown in Shapiro's book, as well as in
> his chapter in Vol. 1 of the 2nd ed. of the Handbook of Philosophical
> logic). However, I prefer the "ancestral logic" version, because the
> notion of "ancestor" is part of everybody's logic, 100% understood
> also by non-mathematicians."
For what it's worth, there is a very natural generalization of ancestral
logic, discussed in my paper "A Logic for Frege's Theorem" [1], that is
equivalent to the \Pi_1^1 fragment of second-order logic. The idea is
just to take the mechanism through which the ancestral is characterized
in ancestral logic, and then generalize it to cover any relation that is
defined through \Pi_1^1 comprehension. The suggestion is then that the
very same resources needed if one is to understand the ancestral give
one access to a much stronger logic. It's arguable, moreover, that the
resulting logic is still first-order, in the sense in which non-standard
quantifiers such as "most" are still first-order.
Indeed, it is arguable, and more or less has been argued by Aldo
Antonelli, that the alleged "uniqueness" of first-order logic has much
less to do with first-order-ness than it has to do with the restriction
to the simple quantifiers "every" and "some" [2], a restriction whose
motivations are not exactly obvious.
Richard Heck
[1] http://rgheck.frege.org/pdf/published/LogicOfFregesTheorem.pdf
[2] http://aldo-antonelli.org/Papers/genint.pdf
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University
Website: http://rgheck.frege.org/
Blog: http://rgheck.blogspot.com/
Amazon: http://amazon.com/author/richardgheckjr
Google+: https://plus.google.com/108873188908195388170
Facebook: https://www.facebook.com/rgheck
Check out my books "Reading Frege's Grundgesetze"
and "Frege's Theorem":
or my Amazon author page:
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/2013-September/017573.html","timestamp":"2014-04-17T07:41:46Z","content_type":null,"content_length":"5866","record_id":"<urn:uuid:7b9cf673-98db-4d6f-b42f-e20282279379>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00037-ip-10-147-4-33.ec2.internal.warc.gz"} |
The MacNeille Completion of the Poset of Partial Injective Functions
Renner has defined an order on the set of partial injective functions from $[n]=\{1,\ldots,n\}$ to $[n]$. This order extends the Bruhat order on the symmetric group. The poset $P_{n}$ obtained is
isomorphic to a set of square matrices of size $n$ with its natural order. We give the smallest lattice that contains $P_{n}$. This lattice is in bijection with the set of alternating matrices. These
matrices generalize the classical alternating sign matrices. The set of join-irreducible elements of $P_{n}$ are increasing functions for which the domain and the image are intervals.
Full Text: | {"url":"http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r62","timestamp":"2014-04-18T08:09:08Z","content_type":null,"content_length":"14612","record_id":"<urn:uuid:a6e930a4-c7fa-463e-92d5-46464fd64261>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00564-ip-10-147-4-33.ec2.internal.warc.gz"} |
Linear Kinematics
Let us consider the motion of an object. For simplicity, let us take an ideal object, which does not take up any real space, or interact with the rest of the world. To begin with, let us describe its
position. In order to say where an object is, we must first define a coordinate system. This is known as a frame of reference. Frames of reference are wholly man made. They are artifacts that allow
for easier description of the problem. Since we live in three dimensions, we describe the position of an object by stating how far away the object is from three separate coordinate axes. Again, for
simplicity, let us consider one dimension only. Then in order to describe the position, we need only know the point on a number line where the object is at. The easiest way of seeing this is to look
at a graph of the position as a function of time. If the object is not moving, then the graph would look like
We see that the slope of an object at rest is zero. What if the object were moving at a constant speed? Then a graph of the speed as a function of time would be
Similarly, intuition tells us that if something moves with constant speed, then it is changing position at a constant rate, and a graph of the position as a function of time looks like
Here the slope of the line is a constant, and in fact is equal to the speed. Lastly, we can also ask what would happen if we let the speed change at a constant rate? A change in the speed of the
object is called its acceleration. From before, we see that graphs of the speed and acceleration verses time would look like
Notice that again the acceleration is the slope of the graph of the speed verses time. What would a graph of the position verses time look like? We have seen that the speed is the slope of the
position, but now that slope is increasing at a constant rate. Thus, we see that the graph of the position must look like
We see that for many cases, the easiest way to analyze one dimensional motion is to look at it graphically. However, if the object has a complicated motion, or it moves in more than one dimension, it
is not always easy to graph it.
Algebraic Equations for Motion
Now let us denote the object's position by the vector x. We associate the unit meters with x. Now look at the graph of distance verses time. Since the y coordinate is the distance, and the x
coordinate is the time, the slope can be written as
We call v[ave] the average velocity of the object. If we let the time difference t[2] t[1] go to zero, then we get the definition of the instantaneous velocity
This last term may be unfamiliar to many of you. This is the definition of a derivative. In this course we will only use the average velocity.
Can we come up with a similar definition for acceleration? Looking at the graph of the velocity verses time, we recall that the slope of that line was the acceleration, so we can follow the same
approach as before to define the average acceleration
Similarly, we can define the instantaneous acceleration as
Notice from (2) and (4) that if position is in meters, then velocity is in meters per second and acceleration is in meters per second per second, or meters per second squared.
For now, lets work in one dimension. Suppose that t = t, v = v - v[0] and x = x - x[0]. Then we can rewrite (3) as
and (1) as
Now recall that v in (6) is the average velocity. In order to determine the average, remember that the average is defined as
Substituting (5) in for v, we get
If we substitute this into (6), we get the desired result
where I have dropped the average on a since we will usually take it to be a constant.
We can eliminate t by using (7) and (5). Solving (5) for t we get
Substituting this into (7), we get
A car is traveling east at 45 m/s when the streetlight turns red. If the car decelerates at 5 m/s^2, how long does it take to stop? How far away from the streetlight must the car start breaking in
order to stop in time?
We can find the time from (5)
We can get the distance from either (7) or (8). From (7) we get
From (8) we get
as before.
Ballistic Motion
An important example of this is when the acceleration is due to gravity. Galileo was the first one to realize that two objects will accelerate as they fall, and that the acceleration was independent
of the composition of the object. By performing experiments, we can determine the amount that gravity accelerates an object. This is
g = 9.8 m/s^2
in the downward direction. Also, notice that this result is an idealization. If we were to drop a baseball and a feather, we would see that the baseball reached the ground first. Was Galileo wrong?
No. In this case we also have the effect of the air on the object, and it does depend on the composition of the object. If we were to repeat the experiment in a vacuum we would find that they fall at
the same rate. The motion of an object due to gravity is known as ballistic motion. This is because it also describes the motion of a cannonball that is fired.
A student wants to play a practical joke on his poor physics professor. He climbs to the top of a 5 story (15.25 m) tall building and prepares to drop a water balloon on the professor as he walks
underneath. If the professor is walking by at a speed of 2.0 m/s, how far from "ground zero" should the professor be when the student drops the balloon?
In order to determine where the professor should be, we need to know how long it takes for the water balloon to fall. We get this from the position equation
Since v[0] = 0. Using y(t) = 0 and solving for t, get
To find the professor's position, we use the fact that his motion is uniform. Thus
or, since x(t) = 0, | {"url":"http://physics.tamuk.edu/~suson/html/1401/linear.html","timestamp":"2014-04-17T18:23:55Z","content_type":null,"content_length":"12105","record_id":"<urn:uuid:b1528692-143e-4bec-827b-e8fe27737421>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00089-ip-10-147-4-33.ec2.internal.warc.gz"} |
Counting Abelian Squares
An abelian square is a nonempty string of length $2n$ where the last $n$ symbols form a permutation of the first $n$ symbols. Similarly, an abelian $r$'th power is a concatenation of $r$ blocks, each
of length $n$, where each block is a permutation of the first $n$ symbols. In this note we point out that some familiar combinatorial identities can be interpreted in terms of abelian powers. We
count the number of abelian squares and give an asymptotic estimate of this quantity.
Full Text: | {"url":"http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r72","timestamp":"2014-04-21T13:16:54Z","content_type":null,"content_length":"14675","record_id":"<urn:uuid:bc24a08f-c941-4759-980d-3c5f9330104f>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00058-ip-10-147-4-33.ec2.internal.warc.gz"} |
What percentage of prize money will my horse get?
24th of September, year 264
What percentage of prize money will my horse get?
First place gets 20%, second 14%, third 10% The other 17 get the rest. If there are not 20 horses in the show, there will be some multiplier factor, up to 3 times. So if there is only one horse in
the show, it will get 60% of the prize.
How do I transfer horses to my second stable? | {"url":"http://www.avirtualhorse.com/faq/73/How%20do%20I%20put%20magic%20item%20on%20a%20horse%3F","timestamp":"2014-04-20T20:56:18Z","content_type":null,"content_length":"22367","record_id":"<urn:uuid:ab44a721-1def-4b7d-942d-3dbf0b601b4c>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00145-ip-10-147-4-33.ec2.internal.warc.gz"} |
Exercises in Classical Ring Theory 2nd edition by Lam | 9780387005003 | Chegg.com
Details about this item
Exercises in Classical Ring Theory: This useful book, which grew out of the author's lectures at Berkeley, presents some 400 exercises of varying degrees of difficulty in classical ring theory,
together with complete solutions, background information, historical commentary, bibliographic details, and indications of possible improvements or generalizations. The book should be especially
helpful to graduate students as a model of the problem-solving process and an illustration of the applications of different theorems in ring theory. The author also discusses "the folklore of the
subject: the tricks of the trade' in ring theory, which are well known to the experts in the field but may not be familiar to others, and for which there is usually no good reference." The problems
are from the following areas: the Wedderburn-Artin theory of semisimple rings, the Jacobson radical, representation theory of groups and algebras, (semi)prime rings, (semi)primitive rings, division
rings, ordered rings, (semi)local rings, the theory of idempotents, and (semi)perfect rings. Problems in the areas of module theory, category theory, and rings of quotients are not included, since
they will appear in a later book. T. W. Hungerford, Mathematical Reviews
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Basic Semiconductor Theory Ppt
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Author: rosemary | {"url":"http://ebookily.org/pdf/basic-semiconductor-theory-ppt","timestamp":"2014-04-16T13:04:27Z","content_type":null,"content_length":"40869","record_id":"<urn:uuid:ee7a1612-f29e-4e03-8834-08a8bdf5c371>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00276-ip-10-147-4-33.ec2.internal.warc.gz"} |
SAS Macro
1. Studying missing data patterns.
· The macro is designed to look at missing data in four ways: the proportion of subjects with each pattern of missing data, the number and percentage of missing data for each individual variable, the
concordance of missingness in any pair of variables, and possible unit nonresponse.
· The SAS macro is %missingPattern:
%missingPattern(datain=, varlist=, exclude=, missPattern1=, dataout1=, missPattern2=, dataout2=, missPattern3=, dataout3=, missPattern4=, dataout4=);
· Reference: T. Schwartz, Q. Chen, and N. Duan, (2011). Studying missing data patterns using a SAS macro, SAS Global Forum 2011 proceedings.
· Download: macro studying missing pattern.sas
2. MIANALYZE for survey weighted linear regression models from multiply-imputed data.
· This macro is designed to use the MIANALYZE procedure to combine the regression coefficient estimates of survey weighted linear regression models (fitted using PROC SURVEYREG) from multiply-imputed
· The SAS macro is %MI_SREG:
%MI_SREG(dset, outcome, var, catVar, strata, cluster, weight, output1, output2);
· Download: MI_SREG.sas
3. MIANALYZE for survey weighted logistic regression models from multiply-imputed data.
· This macro is designed to use the MIANALYZE procedure to combine the regression coefficient estimates of survey weighted logistic regression models (fitted using PROC SURVEYLOGISTIC) from
multiply-imputed data.
· The SAS macro is %MI_SLOGIT:
%MI_SLOGIT(dset, outcome, var, catVar, strata, cluster, weight, output1, output2);
· Download: MI_SLOGIT.sas
4. Forward stepwise selection for survey weighted logistic regression models using PROC SURVEYLOGISTIC
· The SAS macro is %SLOGIT_STEPWISE:
%SLOGIT_STEPWISE(dset, outcome, force, forceCat, varlist, catVarlist, strata, cluster, weight, alpha1, alpha2, output);
· Download: SLOGIT_STEPWISE.sas | {"url":"http://www.columbia.edu/~qc2138/SAS%20Macro.htm","timestamp":"2014-04-17T00:57:08Z","content_type":null,"content_length":"53177","record_id":"<urn:uuid:fa335148-c7b4-4f1f-82f8-43aba60d9adf>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00137-ip-10-147-4-33.ec2.internal.warc.gz"} |
Planet X aka NIBIRU - Club Conspiracy Forums
Greetings and Welcome to the Planet X Files! The
equations that follow are real. The numbers have not been changed to protect the innocent. The consequences will be real. I've done my very best to keep it simple so anyone can follow along and
understand. There are no "doomsday" predictions here, but this event will take place. Knowing this can save the life of you and yours. My personal advice should headlines start creeping around by
2010? By late
= stay away from the coasts.
Utilizing several timeline scales - pick what works for you:
- tya = thousand years ago
Age of Taurus
= 4500 BC = {4,286 - 2,143 bce} = (4-6 tya)
Age of Gemini
= 7500 BC = {6,429 - 4,286 bce} = (6-8 tya)
Age of Leo
= 11,500 BC = {10,715 - 8,572 bce} = (10-12 tya)
Age of Virgo
= 14,500 BC = {12,858 - 10,715 bce} = (12-15ya and on)
- Sumerian Definition 4200 BCish
Tzoltze ek'
- Mayan Definition 3800 BCish
Planet X
- Modern Astronomy Science
"modern day definition"
Planet X aka Nibiru
A large scale hypothetical planet that's part of our solar system with an orbit beyond that of Neptune. Was first hypothesized in 1841 when astronomers noticed Uranus did not move as predicted in
it's orbit. Based on these residuals, the search for Planet X began.
Please keep in mind,
I'm not a graphic artist. (doh) But most importantly, the dates illustrated below are just to ROUND OFF for purpose's of simplicity. The current figure generally accepted by science is that planet x/
nibiru is on a 3600 year elongated (elliptical) solar orbit.
My personal calculations put this at 4320.26 years <---!!!
(See Below). Since this is closest to 4000 years I thought it would be fair enough when referring to a chronological scale of human/earth history.
This means that
PlanetX/Nibiru is visible every 2000 (2,160) years during its orbital pass. ( Sumerian and Mayan text both state that Nibiru is clearly visible by day as well as night )
That being said,
all science arenas confirm that the below charted events took place in the past at around those periods. The debate of course, is when exactly they occurred, and what exactly caused them to occur
which isn't relevant for our purpose. You'll clearly see that the timeline's show something happens on a catastrophic scale every 4000 years (or so).
The sumerians
told us that Nibiru wreaks havoc with the earth's axis every
orbital pass. (Every 4000 years or so). Basically, Earth's axis precessed from a right tilt forward and probably 180 to the left in around 10 or 20 hours due to the gravational "jolt" that takes
Hmmmmm... Sumerian Mathmatics 101
The Sumerian sexigesimal system of numbers works with changing factors that increase by 6 and 10.
The formula equates as follows:
1, 10, 60, 600, 3,600, 36,000, 216,000 bla bla so on, etc.
Every other preceding number is multiplied by 6 or 10.
The number 5,000 is written as 12,320 in Sumerian.
3,600 600 60 10 1
---- thus,
1x3600 + 2x600 + 3x60 + 2x10 + 0x1 = 5,000. Still hanging with me?
If we look at the number 3,600 and write it
down the Sumerian way, we see that the result is 2,160.
3,600 600 60 10 1
N/A 3 6 0 0
---- thus,
3x600 + 6x60 + 0x10 + 0x1 = 2160.
The Sumerian
mathmatics system might seem odd at first, but it's actually ideal for geometry, calculation with fractions, and time. The hour was divided into 60 minutes of 60 seconds each by the Mesopotamians
using their sexagesimal system of counting.
state that Geometry and Astronomy was the language bestowed upon them by the gods (flesh and blood gods) and is still used by freemason architects today gaining knowledge on their Templar crusades in
the Middle East. (The Templars disbanded and later reappeared as Freemasons).
So, are the Sumerians lying and it's actually just a myth as modern religion would have you think? Only problem is, they have this documented 4000 years before Chirstians even exsisted. Anyway, its
clear I back up this claim. Me and that pesky science fella.
Various studies of Sumerian mathematics point out that the numerals are intimately connected to the precessional cycle. The unusual alternating structure of the Sumerian sexagesimal system throws
special emphasis on the number 12,960,000, which represents exactly 500 great precessional cycles of 25,920 years.
The lack of any connotations, other than astronomical, for the multiples of 25,920 and 2,160 can only suggest a deliberate design for astronomical purposes, yes? I think we can all agree on that.
my suspicions strongly indicate that the revolutionary orbit of this 12th planet Nibiru could not consist of 3,600 years, but of 4,320 years. Assuming that's correct, then what would the consequences
If a whole new calculation is performed from the time of the great flood, which according to Alford occurred in 10,983 BCish, with the new orbital pass every 2,160 years then we get a whole new
series of data, namely:
10,983 - 8,823 - 6,663 - 4,503 - 2,343 - 183 BC and 1977 AD. Alford
also describes the arguments between the two gods known as Thoth and his brother Marduk. These two conflicted over the fact as to when the precession of the Earth exactly started. Thoth was able to
convince Marduk that it occurred one and a half degrees later than what Marduk had calculated. A degree and a half of the precession cycle is 108 years (a very sacred number in itself!). If we add
this number to 1977 then we come up with the year 2085. Hmmm, No sweat, old coot here by then!
Still conflicted (nag nag), and after a couple solid sleepless nite weeks of continued research, I came across yet another precession article promoting facts that the current data on the precessional
cycle does not consist of 25,920 years as the Sumerians had recorded. (doh)!
In ancient times the precessional length was 25,920 years, but now, due to the increased speed of precession, it is now closer to 25,776 years.
The number 2,160 is the twelfth fraction of a precessional cycle that takes 25,920 years.
If we divide 25,776 by twelve we get 2,148 years.
We now conclude that the bi-orbital timeline of P-X is 2,148 years, not 3,600 or 2,160 years.
If we proceed to calculate in periods of 2,148 years starting at 10,983 BC,
then add another degree and a half of 25,776 to that:
{ meaning 107.4 years } thus/and/or
{ A degree and a half of 25,920 is 108 years},
- we then come up with a result of...
2012 AD!
... Yikes!
The Mayan's, Egyptians, Zulu, Hindu, Incas, Aztecs, Dogon (Africa), Cherokee, Pueblo, Tibetan's bla bla bla.. all have the same calander target year! We can safely take "coincidence" off the table.
I challange you to calculate the odds of 2012 being a target year for all these cultures as coincidental. It's a number you couldn't even define. But if you're mathmatical minded, have a go!
So, this is also the end of the
calendar, the end of the month Pisces (2148 years), and the end of the cycle known as the "Platonic Year or Great Year" which we now know lasts 25,776 years.
Keep in mind that the Maya and Sumerians we're the two "Dawn of Civilizations", that spawned all other cultures thereafter. And yep, they of course lived on different continents and had No Idea each
other existed.
Although Nibiru now passes every 2,148 years it only orbits close enough to cause a pole shift every other time. Unfortunalty the numbers indicate last time it passed beyond Jupiter or Saturn
resulting only in magnetic fluctuations and severe weather patterns. So this next pass should be between Mars and Jupiter. That can't be good folks. And the fact that both Maya and Sumer cultures
(and about ten others), claim that their Gods told them this exact same timeline?
Question mark indicates "what exactly" caused the above events, not if they actually took place - we know they did...
The last
'Major' passage, which caused noticeable catastrophe, was during the Age of Taurus, 4,286 - 2,143 bce - the two before that would have been in the Ages of Leo (10,500 BC.) and Scorpio (14,500 BC.) -
(see chart above for "years ago" timeline ).
The Gnostic authors of the Bible were also aware of these hallmark conjunctions, and wrote them into the New Testament. The four beasts of the Apocalypse, the Lion, Ox, Man and Eagle, correspond to
the four zodiac signs Leo, Taurus, Aquarius and Scorpio, in which the Grand Cross conjunction of Jupiter and Saturn takes place in 6,444 year intervals.
Even if the authors of the Bible didn't know what exactly would happen at this time (which they probably did) they appear to have used this event as a point for triangulating many prophetic
statements. The sign of Taurus, named by the Sumerians 'GU.ANNA', meaning the "Heavenly Bull", also represented the Sumerian God Ishkur, a.k.a Teshub the storm God!
So my conclusion
is that we experience catastrophe approximately every 4200 years (2 x 2,148 = 4200) and 3 times within a precessional time scale of 25,776 years. Mayan Elder, Hunbatz Men, also revealed that the
Mayans have known about Tzoltze ek' (Nibiru) for many years. They say "The planet has a period of 6,500 years, not 3,600, and visits us 4 times every 26,000 year precessional cycle" (Platonic Year or
Great Year, that ends on 21st December 2012). If we go back 6,444 years (3 x 2,148) from 2012 AD, then it brings us to 4,432 BC, when the Antarctic ice cap formed.
There is much controversy over the dating of the Biblical Flood, but if we look at archaeological data for 10,983 BC and 4,432 BC, we see evidence of major flooding and catastrophic climate change.
There is also evidence of a flood around 2,200 BC, leading bible scholars to conclude this was the Biblical Flood. But I suggest this flood was much more localized than the great flood, due to the
passage of Nibiru at a reasonably safe distance.
Just a coincidence that these other cultures have a 2012 target?
• Hopi Predict a 25yr period of purification followed by End of Fourth World and beginning of the Fifth.
• Mayans Call it the 'end days' or the end of time as we know it.
• Maoris Say that as the veils dissolve there will be a merging of the physical & spiritual worlds.
• Zulu Believe that the whole world will be turned upside down.
• Hindus Kali Yuga (end time of man). The Coming of Kalki & critical mass of Enlightened Ones.
• Incas Call it the 'Age of Meeting Ourselves Again'.
• Aztec Call this the Time of the Sixth Sun. A time of transformation. Creation of new race.
• Dogon Say that the spaceship of the visitors, the Nommo, will return in the form of a blue star
• Pueblo Acknowledge it'll be the emergence into the Fifth World
• Cherokee Their ancient calendar ends exactly at 2012 as does the Mayan calendar.
• Tibetan Kalachakra teachings are prophesies left by Buddha predicting Coming of the Golden Age.
• Egypt According to the Great Pyramid (stone calendar), present time cycle ends in year 2012 AD
for more go to Planet X Nibiru Projected Orbital Return - 2012 | {"url":"http://www.clubconspiracy.com/forum/showthread.php?p=67503","timestamp":"2014-04-16T11:36:39Z","content_type":null,"content_length":"72095","record_id":"<urn:uuid:5df83fbb-9d3e-4bb9-902c-fc11455dcd06>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00265-ip-10-147-4-33.ec2.internal.warc.gz"} |
Eclipse Community Forums - RDF feedsemantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commensRe: semantic quick fixes and commens
//www.eclipse.org/forums/ Eclipse Community Forums //www.eclipse.org/forums/index.php/mv/msg/376877/913732/#msg_913732 I am trying to use semantic modifications in quick fixes, and I have a
hard time figuring out where comments are reconciled.
I am performing an EMF 'move' operation on a list, and as a result all
comments disappear except one, but this comment is in the wrong place.
x => 1, // a
y => 2, // b
z => 3 // c
If I move the y => 2 to position 1 the end result is something like
y => 2,
x => 1, // b
z => 3
Where is the logic that reconciles the comments?
- henrik]]> Henrik Lindberg 2012-09-16T01:15:17-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914435/#msg_914435 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914185/#msg_914185 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914460/#msg_914460 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914210/#msg_914210 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914485/#msg_914485 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914235/#msg_914235 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914510/#msg_914510 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914260/#msg_914260 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914285/#msg_914285 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914035/#msg_914035 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914310/#msg_914310 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914060/#msg_914060 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914335/#msg_914335 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914085/#msg_914085 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914360/#msg_914360 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914110/#msg_914110 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914385/#msg_914385 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914135/#msg_914135 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914410/#msg_914410 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/914160/#msg_914160 > Hi,
> I am trying to use semantic modifications in quick fixes, and I have a
> hard time figuring out where comments are reconciled.
> I am performing an EMF 'move' operation on a list, and as a result all
> comments disappear except one, but this comment is in the wrong place.
> e.g
> x => 1, // a
> y => 2, // b
> z => 3 // c
> If I move the y => 2 to position 1 the end result is something like
> y => 2,
> x => 1, // b
> z => 3
> Where is the logic that reconciles the comments?
I think I can answer this myself... there is no special place, simply
serializing a model with an existing INode model.
First, my example had a typo. It should have been:
y => 2,
x => 1, // b
z => 3 // c
i.e. that the first comment disappears, and the remaining comments stay
grammatically where they were (i.e. between index 1 and 2, and between 3
and what follows.
The first comment disappears because at the point where it "should have
been found" the serializer is looking for comments between index 0 and
1, the the old comment is simply not there it was between the old 2 and 3.
(Bloody hell...)
Seems incredibly complex to figure out how to get the comments to move
with the moved object and to make sure the dangling comment appears
where a human expects it to appear.
Any tips? Has someone written something similar?
- henrik]]> Henrik Lindberg 2012-09-16T02:53:07-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/915085/#msg_915085
I would have assumed that
INode) can help to get that right, but unfortunately that one is never
called. Neither is the old ICommentAssociator used by the serializer. I
think Moritz or I will come back to that question.
Best regards,
Looking for professional support for Xtext, Xtend or Eclipse Modeling?
Go visit: http://xtext.itemis.com
Am 16.09.12 04:53, schrieb Henrik Lindberg:
> On 2012-16-09 3:15, Henrik Lindberg wrote:
>> Hi,
>> I am trying to use semantic modifications in quick fixes, and I have a
>> hard time figuring out where comments are reconciled.
>> I am performing an EMF 'move' operation on a list, and as a result all
>> comments disappear except one, but this comment is in the wrong place.
>> e.g
>> x => 1, // a
>> y => 2, // b
>> z => 3 // c
>> If I move the y => 2 to position 1 the end result is something like
>> y => 2,
>> x => 1, // b
>> z => 3
>> Where is the logic that reconciles the comments?
> I think I can answer this myself... there is no special place, simply
> serializing a model with an existing INode model.
> First, my example had a typo. It should have been:
> y => 2,
> x => 1, // b
> z => 3 // c
> i.e. that the first comment disappears, and the remaining comments stay
> grammatically where they were (i.e. between index 1 and 2, and between 3
> and what follows.
> The first comment disappears because at the point where it "should have
> been found" the serializer is looking for comments between index 0 and
> 1, the the old comment is simply not there it was between the old 2 and 3.
> (Bloody hell...)
> Seems incredibly complex to figure out how to get the comments to move
> with the moved object and to make sure the dangling comment appears
> where a human expects it to appear.
> Any tips? Has someone written something similar?
> - henrik
>]]> Sebastian Zarnekow 2012-09-16T11:52:10-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/915109/#msg_915109 > Hi Henrik,
> I would have assumed that
> org.eclipse.xtext.serializer.sequencer.HiddenTokenSequencer.getCommentsForEObject(EObject,
> INode) can help to get that right, but unfortunately that one is never
> called. Neither is the old ICommentAssociator used by the serializer. I
> think Moritz or I will come back to that question.
Thanks, I am (naturally ;-)) using the DOM based serializer and CSS
driven formatter that I developed for Geppetto so not too hard to change
the way it works (it already has a number of tweaks compared to the
Meanwhile, if you or Moritz would like to write a line or two about how
it is supposed to work that would help me (i.e. a high level
description). Meanwhile, will take a look at
HiddenTokenSequencer.getCommentsForEObject(...) and see if I can figure
out what it is supposed to do.
Other than the comments, semantic fixes seems to work well with my DOM
based serializer once I figured out an approach to serializing and
formatting the semantic replacements (it is kind of tricky in a language
where the semantic change may alter the formatting, and just inserting
it in raw, one-space-separated form makes it look like a train-wreck...
Wonder why I always get into the difficult stuff two weeks before
planned release...
- henrik]]> Henrik Lindberg 2012-09-16T13:33:31-00:00 //www.eclipse.org/forums/index.php/mv/msg/376877/915116/#msg_915116 > On 2012-16-09 13:52, Sebastian Zarnekow wrote:
>> Hi Henrik,
>> I would have assumed that
>> org.eclipse.xtext.serializer.sequencer.HiddenTokenSequencer.getCommentsForEObject(EObject,
>> INode) can help to get that right, but unfortunately that one is never
>> called. Neither is the old ICommentAssociator used by the serializer. I
>> think Moritz or I will come back to that question.
> Meanwhile, if you or Moritz would like to write a line or two about how
> it is supposed to work that would help me (i.e. a high level
> description). Meanwhile, will take a look at
> HiddenTokenSequencer.getCommentsForEObject(...) and see if I can figure
> out what it is supposed to do.
I looked at getCommensForEObject(EObject, INode), and that was very
straight forward, given an EObject and a node, it just produces a Set of
comments in the tree associated with the semantic object in/under the
given node.
The more interesting issue is - where and how can that be used? If the
semantic model and node model are in sync, the logic that finds
"comments between" is enough.
To solve this a look ahead is required - will all comments match a
"between" slot? If not they need to be associated with EObject. Can't
really wait until that comment is found as it should maybe already have
been emitted.
I was thinking that this could potentially be expensive to perform in
the serializer when doing more agressive semantic modifications, and
maybe this is something that should be done as part of applying a
semantic change (there it is known what changed, the comment nodes can
be found). This is still quite difficult and I wonder if it would be a
good idea to move the responsibility to the user (implementor of
semantic change, or grammar driven) so it can be configured if comments
are left or right associative - or do you think there is a general rule
that always works?
- henrik]]> Henrik Lindberg 2012-09-16T14:03:14-00:00 | {"url":"http://www.eclipse.org/forums/feed.php?mode=m&th=376877&basic=1","timestamp":"2014-04-17T09:46:45Z","content_type":null,"content_length":"56256","record_id":"<urn:uuid:85f353df-5473-4d2a-bf23-20f68d07e2de>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00124-ip-10-147-4-33.ec2.internal.warc.gz"} |
Solution 1
The length of the path traveled by P is 14pi/3.
I drew a bunch of pictures to figure this out. Basically I drew the triangle in the square as it rotated, with a new picture for each new position of the triangle. Then I drew in the paths (in
red) of how P moved. The picture shows that sometimes P moved a lot, sometimes it didn't move at all when the triangle rotated, and sometimes it moved a little.
I noticed that there were two different movements done by P. Both of them are arcs of a circle, since you are rotating a segment (a side of the triangle) around a point, which makes a circle. For
the big ones, P rotates 120 degrees (I know this because the angle of the triangle is 60, which leaves 120). 120 degrees is one third of a circle (which is 360), so those paths are 1/3 of the
circumference of the circle. There are three of them, so that gives us 1 circumference.
P also moves in small arcs. These are rotations of 30 degrees, which is 1/12 of a circle (360/30 = 12). There are two of those, so that is 2/12, or 1/6 of a circumference.
When we add them together, we have 7/6 of the circumference of the circle. This circle has a radius of 2 (since that is the length of the side of the triangle), so the circumference is 4pi
(circumference = 2 * pi * r, and r is 2). 7/6 of 4pi is 14pi/3.
Solution 2
The length of the path is (14pi)/3 inches. For the bonus, the answer is 40pi/3 inches.
I made a picture that shows all of the different positions of P, and then I drew in paths (the dotted red lines) each time P moves.
P is moving along the arc of a circle each time, because BP (or AP, depending on which rotation it is) is like the radius of a circle when you rotate around B. Since each arc will be part of a
circle, we can find the circumference of the circle, then figure out how long each arc is.
circumference = 2 * pi * r = 2 * pi * 2 = 4pi
The first time P moves, it rotates 120 degrees around point B. I know that it's 120 because angle PBA is 60, so angle PBX must be 120 (since the whole thing is 180). To find out what part of the
circle this is, we divide 360 by 120, which is 1/3. So the first time, it moves 1/3 of the circumference of the circle.
1 4pi
--- * 4pi = -----
The second time it moves it also goes 120 degrees, so that is another 4pi/3.
The third time it moves it only goes 30 degrees. I know that it's 30 because angle XYZ is 90 degrees and the the angle of the triangle is 60, so that leaves 30. We divide 360 by 30 to find that
this is 1/12 of the circumference of the circle. The length of this arc is 1/12 * 4pi, which is pi/3.
The fourth time it also moves 30 degrees, so that is another pi/3.
The fifth time it moves 120 degrees, so that's 4pi/3.
Now we have to add them all up to find the total length.
4pi 4pi pi pi 4pi 14pi
----- + ----- + ---- + ---- + ----- = ------
This is all in inches, so the final answer is 14pi/3 inches.
For the bonus, I found that it's 40pi/3.
To do this, I made two more pictures - you have to rotate the triangle around the inside of the square two more times to get the vertices of the triangle back in their original positions. I will
list out the amounts that P moves each time you rotate the triangle - some of them are 0 because that is when you are rotating the triangle around P, so it doesn't move. There are eight
"measurements" for P during each trip around the square because that's how many times you have to rotate the triangle to get it back where it started.
Rotation 1 (which goes with the picture in the first part):
4pi 4pi pi pi 4pi
----- + 0 ----- + ---- + 0 + ---- + ----- + 0
Rotation 2:
4pi pi pi 4pi 4pi pi
----- + ---- + 0 + ---- + ----- + 0 + ----- + ----
Rotation 3:
pi 4pi 4pi pi pi
0 + ---- + ----- + 0 + ----- + ---- + 0 + ----
Add all that up and you have 8 long arcs (4pi/3) and 8 short arcs (pi/ 3).
4pi pi 32pi 8pi 40pi
8 * ----- + 8 * ---- = ------ + ----- = ------ inches
It is interesting to look at the pattern of numbers. Each time P moves to or from a corner of the square, the arc is long. Each time it moves to or from a midpoint of the edge of the square, the
arc is short. | {"url":"http://mathforum.org/workshops/nctm2002/geopow.html","timestamp":"2014-04-16T13:24:16Z","content_type":null,"content_length":"11674","record_id":"<urn:uuid:03053281-269f-431a-af42-85ec2c92e511>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00309-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Surface w/ max volume and min surface area
Well, first of all, you should state your question more precisely:
Alternative 1:
Of all geometric objects with the same volume, which has minimal surface area?
Alternative 2:
Of all geometric objects with the same surface area, which encloses the maximal volume?
Under certain assumptions of niceness, you may solve problems like these with the calculus of variations.
Yhen, in both cases, the ball (solid sphere) will be your solution. | {"url":"http://www.physicsforums.com/showpost.php?p=990831&postcount=2","timestamp":"2014-04-20T14:12:10Z","content_type":null,"content_length":"7342","record_id":"<urn:uuid:07f3bf49-a09e-45dc-8240-f5d1a8182c09>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00142-ip-10-147-4-33.ec2.internal.warc.gz"} |
Flower Mound Precalculus Tutor
Find a Flower Mound Precalculus Tutor
...For Hubble observation statistics I used various Excel features such as writing cell formulas, generating plots, and doing import from/export to text files. I still use the formula capability
often. I do not have practice with pivot tables, but do have good knowledge of general techniques.
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...Admittedly, I am not that proficient in the biological aspects of it, but I am very much proficient in its statistical aspects and I have tutored a number of Applied Statistics courses. I have
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Physics II, both algebra and calculus based.
8 Subjects: including precalculus, calculus, physics, geometry | {"url":"http://www.purplemath.com/Flower_Mound_precalculus_tutors.php","timestamp":"2014-04-20T04:33:30Z","content_type":null,"content_length":"24176","record_id":"<urn:uuid:a9860ab4-2f74-44a7-b713-b1c0b128e43c>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00323-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Qubits on the Poincaré (Bloch) Sphere
The Poincaré (Bloch) sphere provides a geometric representation of a pure qubit (quantum bit) state space as points on the surface of the unit sphere . Any point of the surface represents some pure
qubit. The mixed qubit states can be represented by points inside of the unit sphere, with the maximally mixed state laying at the center. The red line from the center to the surface of the sphere
corresponds to the pure state and has unit length. For mixed qubit state the length of line must be less than 1.
This Demonstration neatly visualizes the common quantum information processing operations on single qubits.
The most general single qubit state has the spinor form , where and are spherical polar coordinates with and .
The states and correspond to the north and south poles of the sphere with , , . They are referred to as the (computational) basis states and are eigenvectors of the Pauli matrix with eigenvalues .
Two other special cases are the qubit state and with , and , ; they are eigenvectors of and form a diagonal basis. The Hadamard gate interchanges computational and diagonal bases.
The eigenvectors of are sometimes called the circular basis; they are and with , and , .
Altogether they depict six important points on the Poincaré sphere.
The Pauli spin operators are defined as , , . The three directions , , and correspond to the diagonal
and computational bases. The most general qubit state is an eigenvector of the operator with eigenvalue 1. The Bloch vector is a unit vector connecting the origin to a point with Cartesian
coordinates (, , ). | {"url":"http://demonstrations.wolfram.com/QubitsOnThePoincareBlochSphere/","timestamp":"2014-04-18T08:54:43Z","content_type":null,"content_length":"47766","record_id":"<urn:uuid:854b2d9d-4b69-425c-ad31-dae8cc14174a>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00356-ip-10-147-4-33.ec2.internal.warc.gz"} |
Parabolic Integration
From Math Images
{{Image Description |ImageName=Real Life Parabolas |Image=Golden Gate Bridge.jpg |ImageIntro=Parabolas are very well-known and are seen frequently in the field of mathematics. Their applications are
varied and are apparent in our every day lives. For example, the main image on the right is of the Golden Gate Bridge in San Francisco, California. It has main suspension cables in the shape of a
parabola. Of course there are many more examples of parabolic architecture such as roller coasters, flight paths, and probably the most recognized, the Golden Arches of McDonald's. With all of these
appearances in real life, have you ever wondered how to find the area under one? |ImageDescElem=Two methods for finding parabolic area exist. One is very accurate and the other is more of an
approximation method. This procedure for approximation is known as the Rectangle Method and is used by finding the area of rectangles that can fit in the parabola. The area of these rectangles are
added together, giving you the approximate area under or above the parabola.
For a detailed overview of parabolas, see the page, Parabola. However, we will provide a brief summary and description of parabolas below before explaining how to find the area beneath or above one.
Basic Definition
You may recall first learning about parabolas and your teacher telling you that it is a curve in the shape of a "u" and can be oriented to open upwards, downwards, sideways, or diagonally. To be a
little more mathematical, a parabola is a conic section formed by the intersection of a cone and a plane. Below is an image illustrating this.
When you were first introduced to parabolas, you learned that the quadratic equation, $y= a(x-h)^2+ k$ is its algebraic representation (where $h$ and $k$ are the coordinates of the vertex and $x$ and
$y$ are the coordinates of an arbitrary point on the parabola.
As you progressed in mathematics, you learned how to find the area of the space enclosed by the parabola. This can be accomplished in two different ways:
• Using Definite Integration
• Using the Rectangle method (Also referred to as finding the Riemann Sum)
We will explain both of these approaches by posing a problem and then solving it step by step. But first we are going to familiarize you with some parabolic architecture and occurrences found in the
real world.
|ImageDesc===Integral Approach to Determining Area== Typically, when attempting to find the area underneath a parabola, we take its integral. Below is a proposed problem with a numbered procedure of
individual steps for completion:
Find the area under the curve $y =4-x^2$ between $x = 0$ and $x = 2$ and the $x-axis$.
1.Graphing the function first will help you to visualize the curve. Below I have graphed the function using the mathematical program Derive, but you can easily graph it either using your calculator
or by hand.
2.Now we will algebraically evaluate the expression by taking its integral; doing so will give us an EXACT area. Integration is shown and calculated by: | {"url":"http://mathforum.org/mathimages/index.php?title=Parabolic_Integration&oldid=35693","timestamp":"2014-04-16T07:23:08Z","content_type":null,"content_length":"19534","record_id":"<urn:uuid:34dc31d9-5d40-4c8c-ae41-c69e157a66d5>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00189-ip-10-147-4-33.ec2.internal.warc.gz"} |
What are the various methods of measuring Elasticity of Demand?
Elasticity of demand is known as price-elasticity of demand. Because elasticity of demand is the degree of change in amount demanded of a commodity in response to a change in price. Price elasticity
of demand can be measured through three popular methods. These methods are:
1. Percentage method or Arithmetic method
2. Total Expenditure method
3. Graphic method or point method.
1. Percentage method:-
According to this method price elasticity is estimated by dividing the percentage change in amount demanded by the percentage change in price of the commodity. Thus given the percentage change of
both amount demanded and price we can derive elasticity of demand. If the percentage charge in amount demanded is greater that the percentage change in price, the coefficient thus derived will be
greater than one.
If percentage change in amount demanded is less than percentage change in price, the elasticity is said to be less than one. But if percentage change of both amount demanded and price is same,
elasticity of demand is said to be unit.
2. Total expenditure method
Total expenditure method was formulated by Alfred Marshall. The elasticity of demand can be measured on the basis of change in total expenditure in response to a change in price. It is worth noting
that unlike percentage method a precise mathematical coefficient cannot be determined to know the elasticity of demand.
By the help of total expenditure method we can know whether the price elasticity is equal to one, greater than one, less than one. In such a method the initial expenditure before the change in price
and the expenditure after the fall in price are compared. By such comparison, if it is found that the expenditure remains the same, elasticity of demand is One (ed=I).
If the total expenditure increases the elasticity of demand is greater than one (ed>l). If the total expenditure diminished with the change in price elasticity of demand is less than one (ed<I). The
total expenditure method is illustrated by the following diagram.
3. Graphic method:
Graphic method is otherwise known as point method or Geometric method. This method was popularized by method. According to this method elasticity of demand is measured on different points on a
straight line demand curve. The price elasticity of demand at a point on a straight line is equal to the lower segment of the demand curve divided by upper segment of the demand curve.
Thus at mid point on a straight-line demand curve, elasticity will be equal to unity; at higher points on the same demand curve, but to the left of the mid-point, elasticity will be greater than
unity, at lower points on the demand curve, but to the right of the midÂpoint, elasticity will be less than unity. | {"url":"http://www.preservearticles.com/201105307215/what-are-the-various-methods-of-measuring-elasticity-of-demand.html","timestamp":"2014-04-25T04:17:13Z","content_type":null,"content_length":"17653","record_id":"<urn:uuid:34150d7e-65c5-4d82-b246-93299f7bc65f>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00266-ip-10-147-4-33.ec2.internal.warc.gz"} |
What is a closed-form number
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone
would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not cl ..."
Cited by 4 (3 self)
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The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would
say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms
and also to give examples from modern research where the question of closure looms both important and elusive.
"... It is shown that the Lambert W function cannot be expressed in terms of the elementary, Liouvillian, functions. The proof is based on a theorem due to Rosenlicht. A related function, the Wright
ω function is similarly shown to be not Liouvillian. ..."
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It is shown that the Lambert W function cannot be expressed in terms of the elementary, Liouvillian, functions. The proof is based on a theorem due to Rosenlicht. A related function, the Wright ω
function is similarly shown to be not Liouvillian.
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone
would say that (π + log 2) is a closed form, but some of us would think that the Euler constant γ is not ..."
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The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would
say that (π + log 2) is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed
forms and also to give examples from modern research where the question of closure looms both important and elusive.
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone
would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not clo ..."
Add to MetaCart
The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would
say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms
and also to give examples from modern research where the question of closure looms both important and elusive.
"... Mathematics abounds in terms that are in frequent use yet are rarely made precise. Two such are rigorous proof and closed form (absent the technical use within differential algebra). If a
rigorous proof is “that which ‘convinces ’ the appropriate audience,” then a closed form is “that which looks ‘f ..."
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Mathematics abounds in terms that are in frequent use yet are rarely made precise. Two such are rigorous proof and closed form (absent the technical use within differential algebra). If a rigorous
proof is “that which ‘convinces ’ the appropriate audience,” then a closed form is “that which looks ‘fundamental’ to the requisite consumer. ” In both cases, this is a community-varying and
epoch-dependent notion. What was a compelling proof in 1810 may well not be now; what is a fine closed form in 2010 may have been anathema a century ago. In this article we are intentionally informal
as befits a topic that intrinsically has no one “right ” answer. Let us begin by sampling the Web for various approaches to informal definitions of “closed form”.
, 2004
"... Modifying the proof of a theorem of Wilkie, it is shown that if a one dimnsional set S is definable in an O minimal expansion of the ordered field of the reals, and if it is regularly
exponentially near to many integral points, then there is an unbounded set, which is R definable without parameters, ..."
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Modifying the proof of a theorem of Wilkie, it is shown that if a one dimnsional set S is definable in an O minimal expansion of the ordered field of the reals, and if it is regularly exponentially
near to many integral points, then there is an unbounded set, which is R definable without parameters, and which is exponentially near to S. 1 | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=3343688","timestamp":"2014-04-16T06:36:12Z","content_type":null,"content_length":"26812","record_id":"<urn:uuid:07d11a9a-4fc4-4262-974a-5237ffb6144d>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00080-ip-10-147-4-33.ec2.internal.warc.gz"} |
Gravitational and Inertial Mass in Newton’s Principia
In the opening paragraph of the Principia, Newton defines the term "mass" to be used in his law of inertia, and says
It [mass] can also be known from a body's weight, for - by making very accurate experiments with pendulums - I have found it to be proportional to the weight...
Thus the proportionality between inertia and gravitational attraction is, for Newton's theory, an independent empirical fact, not something that follows from the first principles of the theory.
Indeed, Newton himself observed (in Corollary 5, Proposition 6, Book 3) that this proportionality does not apply to forces in general, citing as an example the force of magnetism, which is not
proportional to the mass of the attracted body. Thus the proportionality in the case of gravity is an accidental fact. In contrast, the identity between inertial and gravitational mass is a necessary
and unavoidable feature of any theory (such as general relativity) that conceives of gravitational motion as nothing other than inertial motion in curved spacetime. In such a theory, inertial mass
and gravitational mass are not just accidentally numerically equal, they are ontologically identical. As a result, general relativity is far more exposed to falsification than Newtonian theory, which
is to say, general relativity is a much stronger theory.
Newton obviously recognized the importance of the absolute proportionality between inertia and weight, since he conducted many careful experiments to verify this fact. He discussed this
proportionality in each of the three books of the Principia. In Book 3 he wrote
Others have long since observed that the falling of all heavy bodies toward the earth (at least on making adjustment for the inequality of the retardation that arises from the very slight resistance
of the air) takes place in equal times, and it is possible to discern that equality of times, to a very high degree of accuracy, by using pendulums.
Book 2 contains Newton’s proof of the proposition that forms the basis of these pendulum experiments. Proposition 24 asserts that
In simple pendulums whose centers of oscillation are equally distant from the center of suspension, the quantities of matter are in a ratio compounded of the ratio of the weights and the squared
ratio of the times of oscillation in a vacuum.
To prove this, he notes that the second law of motion implies that a force F applied to an inertial mass m for an interval of time dt results in a change in velocity equal to dv = (F/m)dt. Now, since
the pendulum arms are of equal length, the ratio of forces applied to the pendulum bobs equals the ratio of the weights at any given angle from vertical. Therefore, the motive forces (at any given
point of the arc) are in proportion to the weights. In other words, letting F(q) denote the motive force at the point q on the arc, we have F[1](q)/F[2](q) = W[1]/W[2] for all q. Thus the ratio of
the equations of motion at any given q can be written as
Noting that dv[1] = (dq/dt[1])L and dv[2] = (dq/dt[2])L where L is the length of the pendulum, we can substitute into this equation and re-arrange terms to give
(The use of dq/dt[j] rather than the more familiar dq[j]/dt in the expression of dv[j] is nice example of Newton’s ingenuity.) This shows that the ratio of time increments for any given increment of
q is constant, so it equals the ratio of the total periods T[1]/T[2]. Making this substituting gives Newton’s Proposition 24, which is expressed symbolically as
Thus if we find (by experiment) that two simple pendulums of equal length swinging though equal arcs have equal periods, then it follows that the gravitational weights are proportional to the
inertial masses. Returning again to Book 3, Newton wrote
I have tested this with gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I got two wooden boxes, round and equal. I filled one of them with wood, and I suspended the same weight
of gold (as exactly as I could) in the center of oscillation of the other. The boxes, hanging by equal eleven-foot cords, made pendulums exactly like each other with respect to their weight, shape,
and air resistance. Then, when placed close to each other, they kept swinging back and forth together with equal oscillations for a very long time. Accordingly, the amount of matter in the gold (by
book 2, prop. 24, corols. 1 and 6) was to the amount of matter in the wood as the action of the motive force upon all the gold to the action of the motive force upon all the [added] wood—that is, as
the weight of one to the weight of the other. And it was so for the rest of the materials. In these experiments, in bodies of the same weight, a difference of matter that would be even less than a
thousandth part of the whole could have been clearly noticed.
Newton’s ability to prepare equal weights of those nine diverse substances (gold, silver, lead, glass, sand, salt, wood, water, and wheat) may owe something to his alchemical experiments, but it’s
surprising that he didn’t include mercury, which was one of the main ingredients in his alchemical recipies. Mostly likely he included wheat as an example of a highly pourous substance, but it may
also have been selected as an example of an organic substance (along with wood).
These pendulum experiments are presented as the main empirical basis for Newton’s claim that weight is always proportional to mass, but he also considers the astronomical evidence for this
proportionality. For example, he points out that if the moons of Jupiter were attracted to the Sun by a force that was disproportionate to their masses (in relation to the attraction of Jupiter
itself to the Sun), their orbits would not be stable.
Surprisingly, Newton does not refer to what is perhaps the most direct and obvious astronomical evidence for the proportionality of inertial and gravitational mass, namely, Kepler’s third law. Recall
that the centripetal force required to keep a body of inertial mass m[i] moving with speed v in a circle of radius r is
where w = v/r is the angular speed. This force is purely dependent on the inertia of the body, having nothing to do with gravity. Now, letting m[g] denote the gravitational mass of this body, the
gravitational force exerted on it by a large central body of gravitational mass M[g] is
Equating this with the force required to keep the revolving body in its orbit, we get
This is, of course, nothing but Kepler’s third law, which expresses the observed fact that for every satellite of a given central body (such as the planets orbiting the Sun), the square of the
angular speed times the cube of the orbital radius has the same value. It follows that the ratio m[g]/m[i] is the same for all the planets, despite the differences between their sizes, densities, and
compositions. Likewise, since the moons of Jupiter also satisfy Kepler’s third law, it follows that the ratio of m[g]/m[i] is the same for all those moons, which range in size from Ganymede (roughly
the size and mass of the smallest planet) to tiny boulders. We also know that small man-made satellites in orbit around the Earth have the same m[g]/m[i] as the Moon. Hence it’s clear that the
proportionality of inertial to gravitational mass applies at least from the smallest objects up to objects the size of Jupiter. (Needless to say, this places very strict limits on any theory of
gravity that implies shielding or saturation for sufficiently large or dense objects.)
Although the geometrical interpretation of general relativity seems to provide a compelling reason for the strict proportionality between weight and mass, this “explanation” has not been
whole-heartedly accepted by all modern physicists. For example, in his “Lectures on Gravity” Richard Feynman emphasized the field approach to gravitation based on the spin-two graviton as the carrier
of the force, and he wrote
It is one of the peculiar aspects of the theory of gravitation that it has both a field interpretation and a geometrical interpretation… the fact is that a spin-two field has this geometrical
interpretation; this is not something readily explainable – it is just marvelous. The geometric interpretation is not really necessary or essential to physics. It might be that the whole coincidence
might be understood as representing some kind of gauge invariance…
Steven Weinberg took a similar view in his “Gravitation and Cosmology”, where he regarded the equivalence principle itself as primary, and treated the geometric interpretation as incidental. He
…this geometrical analogy is an a posteriori consequence of the equations of motion derived from the equivalence principle, and plays no necessary role in our considerations.
Returning to Newton, it’s interesting that he immediately invoked the equality of inertial and gravitational mass to argue for the existence of vacuum. In Corollary 3 to Proposition 6 of Book 3 he
says “all spaces are not equally full”, meaning there cannot be an equal quantity of inertial mass in each volume of space, because this would imply equal weight, and therefore a rock would not fall
through the air. So some spaces must contain less matter than others. He continued “If the quantity of matter in a given space can, by any rarefaction, be diminished, why should it not be diminished
indefinitely?” In Corollary 4 he contends that “If all the solid particles of bodies are of the same density, and cannot be rarefied, then a void, space, or vacuum must be granted”.
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Dockweiler, CA Precalculus Tutor
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HOMEWORK ASSIGNMENT #1 (15 POINTS)
HOMEWORK ASSIGNMENT #1 (20 POINTS)
(Due in Class on Wed., June 25)
Instructions. Answer all parts of each question. Clearly label which part you are answering.
1. (5 points) (a) Choose your own example of two properties, A and B, such that A is a sufficient condition for B, but B is not a sufficient condition for A. This part of your answer must use the
words "sufficient condition". (b) Use the arrow to state the relation between the two properties A and B identified in part (a). (c) Explain what your answer to part (b) means. (You must use the
definition from the handout.)
2. (5 points) (a) Choose your own example of two properties, A and B, such that A is a necessary condition for B, but B is not a necessary condition for A. This part of your answer must use the
words "necessary condition". (b) Use the arrow to state the relation between the two properties identified in part (a). (c) Explain what your answer to part (b) means. (You must use the definition
from the handout.)
3. (10 Points) (a) (2 Points) Give an example of a property PD which is your best attempt to find a property that provides a purely descriptive sufficient condition for moral wrongness. (Your answer
must include the words "sufficient condition".) (b) (2 Points) Abbreviate the property you identified in part (a) and use the abbreviation to state your answer to part (a) in the form of an
implication (you may use the arrow to abbreviate "implies"). (c) (2 Points) What does your answer to part (b) mean? (You must use the definition from the handout.) (d) (2 Points) Logically, what
would be required for there to be a counterexample to your answer to part (b)? (e) (2 Points) Do you think there is a counterexample to it? Explain. | {"url":"http://faculty.washington.edu/wtalbott/phil240/hw1.htm","timestamp":"2014-04-19T01:59:33Z","content_type":null,"content_length":"8002","record_id":"<urn:uuid:8f59b8a0-cca8-4127-bc89-5299e9cd9d2b>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00259-ip-10-147-4-33.ec2.internal.warc.gz"} |
Inf of a mutivariate function
up vote 1 down vote favorite
Let $f(x_1,\ldots , x_n) = \frac{x_1}{x_2+x_3} + \frac{x_2}{x_3+x_4} + \cdots + \frac{x_n}{x_1+x_2}$, defined for $x_i>0$.
1. Is there $(x_1, \ldots ,x_n)\in {\mathbb{R}^*_+}^n$ such that $f(x_1,\ldots , x_n) < n/2$?
2. Can we find $\inf_{x_i>0}f(x_1,\ldots , x_n)$?
real-analysis tag-removed ca.analysis-and-odes
add comment
2 Answers
active oldest votes
This is discussed briefly as a generalization of Shapiro's cyclic sum inequality by J. Michael Steele in his book The Cauchy-Schwarz Master Class. He remarks that (1.) holds for
up vote 5 down vote $n\ge25$ and refers to this paper: P. J. Bushell, Shapiro’s “Cyclic Sums", Bull. L.M.S. (1994) 26, 564–574.
Thanks Robin, this closes my question – Portland Sep 5 '10 at 1:21
3 If you're satisfies with this response, you should accept the answer! – Cam McLeman Sep 7 '10 at 16:15
add comment
This is discussed in detail on the following MathWorld article: Shapiro's Cyclic Sum Constant . Detailed proofs of the main result (inequality holds only for even $n \le 12$ and odd
up vote 3 down $n \le 23$) can be found in the following note by Khrabrov.
Thanks FG, the MathWorld article is very useful. – Portland Sep 5 '10 at 1:22
add comment
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Electromagnetic Induction
Hello everyone,
How are you doing?
I have a doubt about electromagnetic induction, in three particular cases. I need to confirm that I have the right concepts, so I ask for your help.
The main problem:
Imagine that you have a permanent magnet, axially polarized and rotating on its axis with a constant angular speed. Surrounding this magnet, a coil (constant area section pointing in the same
direction of magnet polarization). The main question is: will there be induced voltage?
This is what I think:
1) We know that, for a constant Area, flux linkage ψ = B*A*cos θ.
In this case θ = 0°, so ψ = B*A.
And the induced voltage is ε = -N*dψ/dt = -N*A*dB/dt.
In this main case, I think that there will be no variation in B, because the rotation does not change it at all. So dB/dt = 0, thus ε = 0.
2) Let's suppose the magnet is now radially polarized, but keeping the surrounding coil. In this case, can I affirm that rotation still doesn't change B at all (actually it does change B, but if we
consider the whole thing it does not)? And not only because of this ε is zero, but θ = 90°, which implies ψ = 0.
3) Now suppose the coil doesn't fully surround the magnet. Let's say it covers only 270° of it (a little abstraction is needed, I know
Am I correct? Did I miss something?
Thank you, | {"url":"http://www.physicsforums.com/showthread.php?p=3872383","timestamp":"2014-04-21T04:40:25Z","content_type":null,"content_length":"38416","record_id":"<urn:uuid:043848ff-6780-43d8-9805-2a12efa56053>","cc-path":"CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00030-ip-10-147-4-33.ec2.internal.warc.gz"} |
Yahoo Groups
RE: hinting
Expand Messages
View Source
> > Since BFS by definition finds the shortest path between the
> > source and the
> > destination...
Actually patsolve doesn't use pure BFS since that suffers from exponential
blowup. The hybrid prioritized BFS/DFS strategy doesn't always find the
shortest solution, and it (empirically) is very sensitive to things like
pile order and so on.
I think you just have to keep a "current solution" that you regenerate
when the user deviates from it. One problem I just realized with that is
that it's quite possible for the user to get into a position that is
unsolvable, and the only hint is "undo". Maybe that's not such a problem,
since unsolvable positions are usually detected pretty quickly, especially
mid-game, and then you just suggest "undo" as the hint, which takes you
back to some precomputed solution... You just have to keep restarting the
solver when the user makes an unexpected move, which for patsolve means
rerunning the program (it was never designed to be restartable, because of
the way the position hash-store works).
A really clever hinter would not only generate "the" solution, but also
side solutions, so that the "expected move" would be two or more instead
of just one. But it starts to become a completely different algorithm,
since the solvers need to do a "self-avoiding walk" of the game tree,
which precludes consideration of most side solutions...
View Source
On Thu, 15 Feb 2001, Chen Shapira wrote:
> >
> > Actually, when using BFS it can be proved that such thing is
> > not possible.
> > Since BFS by definition finds the shortest path between the
> > source and the
> > destination than if a BFS on node A suggested a movement to
> > node B and if
> > the BFS on node B suggested a move to node A, then we could
> > have saved two
> > moves by staying at B, which contradicts the definition of BFS.
> But he talked about a case when a user moves NOT according to the hint.
Then, in that case the solver will give a completely different solution.
If you follow the hints all along you will eventually get to a
complete solution.
Besides, that paragraph is about Graph Theory, not really about practice.
Shlomi Fish
> To unsubscribe from this group, send an email to:
> fc-solve-discuss-unsubscribe@yahoogroups.com
Shlomi Fish
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The prefix "God Said" has the extraordinary logical property of
converting any statement that follows it into a true one.
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Kew Gardens Trigonometry Tutor
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pre/post increment / decrement operators
Author pre/post increment / decrement operators
I having a problem understanding pre and post operators. I can understand the following simple code:
Joined: Jan
30, 2002 still results 0 because it is increment after it is assigned to i. But take a look at this more complex version:
Posts: 16
every way I look at it, i keep getting different answers! How does one work out the results for this kind of question?
To me I would have thought that starting from the left, it would first (++n) to get a three, then n++ still leaves it three because it is added after, then --n to 2, and again --n to
get one, finally incrementing the earlier post operator n++ to get a four (because it was 3 at that point). Finally add all the results, 3+4+2+1 = 10, but the answer is 11?
Any help would be gratefully appreciated, especially any rules of how it works.
Also, how does one KNOW when the object is garbaged collected at a certain point?
Ranch Hand
you almost did it right:
Joined: Jan ++n + n++ + --n + --n;
13, 2002 ++n gives u 3 as u said.
Posts: 732 then at n++ it is still 3 u are correct so the answer until now is 6.
now: --n. u must remember that n at this point is 4 (bacsue in the last stage u did n++ so it was inceremnted after the addiotion to 6 u did).
so now n is 3 again so the answer is 9.
at last:
--n now n is 2 so the answer is 11.
3+3+3+2=11; !
Gold Digger
Sheriff Check this link out:http://www.javaranch.com/cgi-bin/ubb/ultimatebb.cgi?ubb=get_topic&f=24&t=014606
Joined: Aug
26, 2001 SCJP 5, SCJD, SCBCD, SCWCD, SCDJWS, IBM XML
Posts: 7610 [Blog] [Blogroll] [My Reviews] My Linked In
Joined: Jan oh, I see now! It is all crystal clear now!
30, 2002 Thanks all the info. Bye
Posts: 16
Ranch Hand
Joined: Nov Dareen, I wrote a POST couple of days ago on the post=fix operators:
10, 2000 http://www.javaranch.com/cgi-bin/ubb/ultimatebb.cgi?ubb=get_topic&f=24&t=014701
Posts: 165
subject: pre/post increment / decrement operators | {"url":"http://www.coderanch.com/t/236724/java-programmer-SCJP/certification/pre-post-increment-decrement-operators","timestamp":"2014-04-18T00:29:52Z","content_type":null,"content_length":"28428","record_id":"<urn:uuid:295eb5ae-e2f1-4b6e-bad5-47ef5bf66f00>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00571-ip-10-147-4-33.ec2.internal.warc.gz"} |
A rapidly-converging series of the Hasse–Weil L-function associated with an elliptic curve over rationals
up vote 0 down vote favorite
I know that for some L-series there is still a rapidly-converging series. My question is about the existence of a such a series for the Dirichlet series of the Hasse–Weil L-function associated with
an elliptic curve over rationals. A google search do not gives important answers.
1 Where do you want the series to converge rapidly? If it's the whole complex plane, such a series would be a proof of analytic continuation, which is as hard as modularity. Thus, you probably want
to find the associated modular form and look for a rapidly converging series for the L-function of that modular form? – Will Sawin Nov 20 '12 at 17:22
for any complex number s. I remamber that there is a rapidly converging series obtained by using Lavrik method, but I can't find it. – Shpigle Nov 20 '12 at 17:26
2 Yes, there is a rapidly exponentially converging series for L(E,s) which relies on the functional equation and thus on modularity of E. This has been implemented, see e.g. arxiv.org/abs/math/
0207280 If you're more interested in values of s with high imaginary part, then you might need other methods, like the double exponential method. See Pascal Molin's PhD thesis math.jussieu.fr/
~molinp/files/these.pdf – François Brunault Nov 20 '12 at 17:48
Thank you Francois. – Shpigle Nov 20 '12 at 18:01
add comment
2 Answers
active oldest votes
Possibly the answer would be the so-called "approximate functional equation" for the $L$-function. This of course takes as input the modularity of the Hasse-Weil zeta function, and
gives rapidly convergent series representing it at any point. I would expect Cremona's book on algorithms for modular elliptic curves to contain a description. Software like Pari/GP
up vote 2 down implements such algorithms (see the command elllseries).
vote accepted
Thank you Denis. I think that there is a rapidly converging series obtained by using Lavrik method, but I can't find it in any reference. – Shpigle Nov 20 '12 at 17:42
3 See the book Analytic Number Theory by Iwaniec and Kowalski – Stopple Nov 20 '12 at 18:29
add comment
Also look at my paper: Computational methods and experiments in analytic number theory, in Recent Perspectives in Random Matrix Theory and Number Theory. Available here: http://
arxiv.org/pdf/math/0412181v1.pdf See section 3.4, example 5 which deals with elliptic curves.
up vote 2
down vote Another reasonable reference is Akiyama and Tanigawa:http://www.jstor.org/discover/10.2307/2584959?uid=3739448&uid=2129&uid=2&uid=70&uid=3737720&uid=4&sid=21102659822167
add comment
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Why some fitness landscapes are fractal
Results 1 - 10 of 19
- FOUNDATIONS OF GENETIC ALGORITHMS , 1995
"... Holland's Schema Theorem is widely taken to be the foundation for explanations of the power of genetic algorithms (GAs). Yet some dissent has been expressed as to its implications. Here,
dissenting arguments are reviewed and elaborated upon, explaining why the Schema Theorem has no implications f ..."
Cited by 93 (3 self)
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Holland's Schema Theorem is widely taken to be the foundation for explanations of the power of genetic algorithms (GAs). Yet some dissent has been expressed as to its implications. Here, dissenting
arguments are reviewed and elaborated upon, explaining why the Schema Theorem has no implications for how well a GA is performing. Interpretations of the Schema Theorem have implicitly assumed that a
correlation exists between parent and offspring fitnesses, and this assumption is made explicit in results based on Price's Covariance and Selection Theorem. Schemata do not play a part in the
performance theorems derived for representations and operators in general. However, schemata re-emerge when recombination operators are used. Using Geiringer's recombination distribution
representation of recombination operators, a "missing" schema theorem is derived which makes explicit the intuition for when a GA should perform well. Finally, the method of "adaptive landscape"
analysis is exa...
, 1996
"... Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in
simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive const ..."
Cited by 89 (15 self)
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Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in
simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are
characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of
the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with
respect to a partition of the set of all vertex pairs.
, 1993
"... In this paper we view the folding of polynucleotide (RNA) sequences as a map that assigns to each sequence a minimum free energy pattern of base pairings, known as secondary structure.
Considering only the free energy leads to an energy landscape over the sequence space. Taking into account structur ..."
Cited by 70 (29 self)
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In this paper we view the folding of polynucleotide (RNA) sequences as a map that assigns to each sequence a minimum free energy pattern of base pairings, known as secondary structure. Considering
only the free energy leads to an energy landscape over the sequence space. Taking into account structure generates a less visualizable non-scalar "landscape", where a sequence space is mapped into a
space of discrete "shapes". We investigate the statistical features of both types of landscapes by computing autocorrelation functions, as well as distributions of energy and structure distances, as
a function of distance in sequence space. RNA folding is characterized by very short structure correlation lengths compared to the diameter of the sequence space. The correlation lengths depend
strongly on the size and the pairing rules of the underlying nucleotide alphabet. Our data suggest that almost every minimum free energy structure is found within a small neighborhood of any random
sequence. The...
- New Ideas in Optimization , 1999
"... Introduction The notion of fitness landscapes has been introduced to describe the dynamics of evolutionary adaptation in nature [40] and has become a powerful concept in evolutionary theory.
Fitness landscapes are equally well suited to describe the behavior of heuristic search methods in optimizat ..."
Cited by 58 (7 self)
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Introduction The notion of fitness landscapes has been introduced to describe the dynamics of evolutionary adaptation in nature [40] and has become a powerful concept in evolutionary theory. Fitness
landscapes are equally well suited to describe the behavior of heuristic search methods in optimization, since the process of evolution can be thought of as searching a collection of genotypes in
order to find the genotype of an organism with highest fitness and thus highest chance of survival. Thinking of a heuristic search method as a strategy to "navigate" in the fitness landscape of a
given optimization problem may help in predicting the performance of a heuristic search algorithm if the structure of the landscape is known in advance. Furthermore, the analysis of fitness
landscapes may help in designing highly effective search algorithms. In the following we show how the analysis of fitness landscapes of combinatorial optimization problems can aid in designing the
components of
, 1993
"... We present and study the behavior of a simple kinetic model for the melting of RNA secondary structures, given that those structures are known. The model is then used as a map that assigns
structure dependent overall rate constants of melting (or refolding) to a sequence. This induces a "landscape" ..."
Cited by 32 (13 self)
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We present and study the behavior of a simple kinetic model for the melting of RNA secondary structures, given that those structures are known. The model is then used as a map that assigns structure
dependent overall rate constants of melting (or refolding) to a sequence. This induces a "landscape" of reaction rates, or activation energies, over the space of sequences with fixed length. We study
the distribution and the correlation structure of these activation energies. 1. Introduction Single stranded RNA sequences fold into complex three-dimensional structures. A tractable, yet reasonable,
model for the map from sequences to structures considers a more coarse grained level of resolution known as the secondary structure. The secondary structure is a list of base pairs such that no
pairings occur between bases located in different loop regions. Algorithms based on empirical energy data have been developed to compute the minimum free energy secondary structure of an RNA sequence
- Computers Chem , 1993
"... The evolution of RNA molecules in replication assays, viroids and RNA viruses can be viewed as an adaptation process on a 'fitness' landscape. The dynamics of evolution is hence tightly linked
to the structure of the underlying landscape. Global features of landscapes can be described by statistical ..."
Cited by 31 (16 self)
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The evolution of RNA molecules in replication assays, viroids and RNA viruses can be viewed as an adaptation process on a 'fitness' landscape. The dynamics of evolution is hence tightly linked to the
structure of the underlying landscape. Global features of landscapes can be described by statistical measures like number of optima, lengths of walks, and correlation functions. The evolution of a
quasispecies on such landscapes exhibits three dynamical regimes depending on the replication fidelity: Above the "localization threshold" the population is centered around a (local) optimum. Between
localization and "dispersion threshold" the population is still centered around a consensus sequence, which, however, changes in time. For very large mutation rates the population spreads in sequence
space like a gas. The critical mutation rates separating the three domains depend strongly on characteristics properties of the fitness landscapes. Statistical characteristics of RNA landscapes are
, 1996
"... In many cases fitness landscapes are obtained as particular instances of random fields by assigning a large number of random parameters. Models of this type are often characterized reasonably
well by their covariance matrices. We characterize isotropic random fields on finite graphs in terms of thei ..."
Cited by 14 (6 self)
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In many cases fitness landscapes are obtained as particular instances of random fields by assigning a large number of random parameters. Models of this type are often characterized reasonably well by
their covariance matrices. We characterize isotropic random fields on finite graphs in terms of their Fourier series expansions and investigate the relation between the covariance matrix of the
random field model and the correlation structure of the individual landscapes constructed from this random field. Our formalism suggests to approximate landscape with known autocorrelation function
by a random field model that has the same correlation structure.
, 1997
"... A landscape is rugged if it has many local optima, if it gives rise to short adaptive walks, and if it exhibits a rapidly decreasing pair-correlation function (and hence if it has a short
correlation length). The "correlation length conjecture" allows to estimate the number of meta-stable states fro ..."
Cited by 10 (6 self)
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A landscape is rugged if it has many local optima, if it gives rise to short adaptive walks, and if it exhibits a rapidly decreasing pair-correlation function (and hence if it has a short correlation
length). The "correlation length conjecture" allows to estimate the number of meta-stable states from the correlation length, provided the landscape is "typical". Isotropy, originally introduced as a
geometrical condition on the covariance matrix of a random field, can be re-interpreted as maximum entropy condition that lends a precise meaning to the notion of a "typical" landscape. The
XY-Hamiltonian, which violates isotropy only to a relatively small extent, is an ideal model for investigating the influence of anisotropies. Numerical estimates for the number of local optima and
predictions obtained from the correlation length conjecture indeed show deviations that increase with the extent of anisotropies in the model.
- in Biosciences 116 , 1997
"... Understanding complex behavior requires a multidisplinary effort from the neurosciences, psychology, behavioral biology, and computer science. This paper gives an overview of the current state
of theoretical thinking in the field. The focus is on a behavior--oriented approach to cognition, i.e., not ..."
Cited by 8 (3 self)
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Understanding complex behavior requires a multidisplinary effort from the neurosciences, psychology, behavioral biology, and computer science. This paper gives an overview of the current state of
theoretical thinking in the field. The focus is on a behavior--oriented approach to cognition, i.e., not so much on the mental representations themselves, but on the behaviors that do require these
representations. It is the intention of the paper to support the exchange between the different disciplines involved. Examples of different types of models and explanations are discussed, but no
comprehensive review of all relevant work is attempted. In the second part, I collect a number of elements that in my view are essential to a future theory of cognitive behavior. Keywords: Cognition,
Perception and Action, Brain Theory, Computational Theory, Artificial Life, Virtual Reality Mallot: Behavior--Oriented Approaches to Cognition Page 2 1 Introduction 1.1 Perception, Action, and
Cognition The ...
- Discr. Math , 1994
"... The relationship of orthogonal functions associated with vertex transitive graphs and random walks on such graphs is investigated. We use this relations to characterize the exponentially
decaying autocorrelation functions along random walks on isotropic random fields defined on vertex transitive gra ..."
Cited by 7 (7 self)
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The relationship of orthogonal functions associated with vertex transitive graphs and random walks on such graphs is investigated. We use this relations to characterize the exponentially decaying
autocorrelation functions along random walks on isotropic random fields defined on vertex transitive graphs. The results are applied to a simple spin glass model. Motivation Recently "combinatory
landscapes" --- maps from the vertex set of some graph into the real or complex numbers --- have received increasing attention. The classic example from physics is a Hamiltonian that assigns an
energy value to a spin configuration (M'ezard et al. 1987). Combinatorial optimization problems, like the travelling salesman problem (Lawler et al. 1985), are of the same type. In evolutionary
biology maps assigning free energies or "fitness values" to biomolecules --- encoded as strings over a finite alphabet --- are of particular interest (Eigen et al. 1989, Fontana et al. 1991, 1992).
For each of these mo... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1770853","timestamp":"2014-04-17T01:57:49Z","content_type":null,"content_length":"39800","record_id":"<urn:uuid:b9ea6097-461f-421b-9530-4af310b2c874>","cc-path":"CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00472-ip-10-147-4-33.ec2.internal.warc.gz"} |
East Lake, CO Geometry Tutor
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I have taught students from kindergarten through eighth grade for 20 years in Boulder Valley School District, spending the majority of that time teaching middle school math and science. I have
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Big Buffer
High-Speed Digital Design Online Newsletter: Vol. 8 Issue 07
My latest movie "Power Integrity Basics" is now complete, and will be aired on the web for the first time November 9, 2005.
This movie explores the distributed nature of power and ground planes in your pcb. In the film I demonstrate the correct method for defining and measuring voltages within a distributed system,
discuss the placement and modeling of bypass capacitors, and show some cool animations of the distributed behavior of the actual power system on an Intel processor card. Thanks to Sigrity, Intel and
Tektronix for their help making this production possible.
For registration, see http://webcast.you-niversity.com/sigrity.
Big Buffer
Ed Hitt of Louisville, Colorado writes [edited for clarity]:
What is the purpose of using a buffer-driver (EG 74LCX125) to drive a circuit with a large capacitive load?
Is it merely to make the rise time faster by decreasing the output impedance of the driving circuit, thereby increasing the timing margin, or is the low current drive capability of the non-buffer
gate simply not adequate?
Is there any reason to think that the large current due to C dv/dt might damage a non-buffer type gate?
The output impedance and the current-driving capability of a driver are intimately related. Comparing two totem-pole CMOS drivers operated at the same power-supply voltage, a larger current-driving
capability almost always implies a smaller output resistance.
Figure 1 illustrates the complete form of the output voltage-current relationship for a typical CMOS totem-pole driver. This type of plot is called a V-I diagram (or sometimes an I-V diagram). The
vertical axis represents current emanating from your driver. The horizontal axis represents the output voltage. From this V-I diagram you can determine how the driver reacts to any static load.
The diagram displays two curves. The top curve (green) illustrates the relation of current to voltage for this driver when switched into the high state. This is a static curve, showing the behavior
at DC. It is derived by connecting the driver to a load that draws progressively more and more current from the driver, over a very slow scale of time, and making a record of the driver output
voltage at each particular current. Then the driver is switched to a low state, and the experiment repeated (purple curve).
This same V-I information may be encoded in a standard form called an IBIS model. An IBIS model holds, at a minimum, the high-state and low-state static V-I curves for a driver, plus some information
about how quickly the driver morphs from one state to the other. Issues you may have heard about related to the accuracy of IBIS modeling have to do with the form and quality of the information used
to morph the V-I curves from low to high (or vice versa) during a fast switching transient, the effects of packaging parasitics, and the lack of information within an IBIS model about cross-coupling
between drivers within the same package (SSO noise). Still, I like IBIS models and find them useful.
A typical driver specification calls out only a couple of points along the V-I curves. If we are talking about a CMOS totem-pole driver, you are expected to know that the gate switches fully rail to
rail in the absence of DC loading. In the V-I domain, this implies that the high-state curve includes the point [VCC,0], meaning the voltage floats all the way up to VCC when you draw zero current
from the driver. In the diagram, the green curve crosses the horizontal axis at the point [VCC,0].
Starting from the point [VCC,0] on the green curve, as you pull progressively more current from the driver (moving up) the output droops (moving to the left), with the result that the high-state V‑I
curves for CMOS totem-pole drivers always move up and to the left away from point [VCC,0].
Near VCC, the slope of the V-I curve (the localized delta-V divided by delta-I) is defined as the "output resistance" of the driver, R[ACTUAL]. For problems involving late reflections that return to
the source long after the source has switched to a voltage above VOH, the value R[ACTUAL] determines the behavior of the driver at that point in time.
A typical datasheet specification calls out only one point in the high state. That point, [VOH,IOH], guarantees that the high-state V-I curve, on its way up and to the left away from [VCC,0], will
always pass to the right side of the point [VOH,IOH] (marked "A" in the diagram).
The diagram illustrates the slope of the wimpiest possible driver (i.e., highest possible output resistance) that barely meets this specification. That resistance, marked R[MAX], can be calculated:
R[MAX] = (VCC–VOH) / IOH
When working through this formula use the minimum allowed value of VCC for your driver. The driver has to still meet VOH at the specified IOH current under this condition. The output resistance will
likely be no greater than that amount under other conditions of VCC.
The maximum output resistance in the low state is simply –(VOL / IOL). Watch out for the polarity of current definitions on the specification sheet—a "sinking current" of 8 mA means an IOL of
negative 8 mA.
Ed, your question had to do with the behavior of a driver when connected to a big clump of capacitance. I will assume the capacitance is all locally connected, at a distance of much less than one
rise time from the driver, so we may treat it as a lumped-element load.
To determine the rise time of a large capacitive load we have to know how much current comes out of the driver at voltages other than VOH. For example, it would be nice to know the short-circuit
current when switched to the high state. This current is the point where the high-state V-I curve crosses the vertical axis (VOUT=0). If you knew that, and assuming the V-I curve were at least
concave down (no undulations), then a straight-line approximation drawn from [0,ISHORT] to [VCC,0] would yield a 90% rise time of:
TR = 2.3*(VCC/ISHORT)*CLOAD
• VCC is the minimum allowed value of VCC,
• ISHORT is the short-circuit current in Amps at that power-supply voltage, and
• CLOAD is the total load capacitance in farads.
This formula models the high-state V-I curve as a simple straight line passing from [VCC,0] to the short-circuit point [0,ISHORT]. In other words, for the purpose of first-order modeling, you may
replace the driver with a single equivalent linear resistance having a value of VCC/ISHORT.
I happen to know that it takes 2.3 times the natural R-C time constant for the step response of an R-C circuit to pass from zero to the 90response point. The formula therefore reads, "2.3 times the
product of the source resistance (VCC/ISHORT) times the load capacitance (CLOAD)."
In a real circuit, the actual V-I curve for your driver exceeds the straight-line approximation everywhere (because a real high-state V-I curve is concave-down). The real driver therefore pumps out
more current than assumed by the straight-line approximation, so the actual response time in a real circuit will be less, sometimes considerably less, than the simple estimate shown above. Any
IBIS-model simulator can demonstrate that effect.
Now we come to an interesting facet of your question that concerns the maximum safe output current for a driver. First I will deal with the DC effects of massive current.
For many CMOS totem-pole drivers, you can short the outputs to ground for brief periods of time without damaging the driver. Damage to a driver under a static shorted condition is caused mostly by
heating due to the excessive current. As long as the short circuit does not persist, no harm is done. Because CMOS drivers go into a current-limiting state as you pull more current from them (i.e.,
the output curve is concave down, flattening out as you approach short-circuit conditions), the CMOS output stage is somewhat self-protecting.
The output stage of a PECL driver, on the other hand, has no current-limiting feature. Without and special protection circuitry, PECL outputs are easily destroyed by unintentional grounding.
Finally, let's look at the effects of large AC currents. These effects have to do with simultaneously switching output noise (SSO noise). If you are not familiar with that topic check out these
recent articles:
In brief, SSO noise is property of every IC package. When the outputs of your IC switch, self-coupling within the IC package couples a certain amount of noise back into your IC inputs. If the noise
is sufficiently large, it causes data errors at the inputs. The same SSO noise effect also produces undesirable glitches in your IC outputs.
The main factors that affect SSO noise are (1) the aggregate amount of current switched by your IC, (2) the rise time of that current, and (3) the number and quality of the power and ground
connections provided in the IC package. Most relevant to this discussion is the total amount of current—more capacitive loading enlarges that current, increasing the amplitude of SSO noise.
An IC manufacturer is supposed to guarantee that, as long as you live within the loading guidelines published for their IC, you can switch any combination of outputs at will without causing SSO
errors. This guarantee is enforced by incorporating an adequate number of power and ground pins, and possibly also solid plane layers, bypass capacitors and other features, into the IC package.
Do you suppose there is much SSO noise margin left in a typical IC package design? Can you safely exceed the loading guidelines without causing SSO errors? I doubt it. In a large package already
crammed with oodles of power and ground pins, if there were any margin left for SSO noise the manufacturer would likely have shaved off a few of the power and ground pins to reduce the size and cost
of the package. In fact, some packages carry specific limitations on how many outputs can be heavily loaded, or how many can be allowed to switch at any one given time.
If you exceed the loading guidelines, burdening each output with huge gobs of load capacitance, you will probably not damage the driver, but you may create excessive amounts of SSO noise, enough to
induce data errors on your inputs.
My comments here refer to the undesirable practice of connecting gobs of capacitance locally to your driver, all located within a small fraction of one rise time of the driver. If you spread your
many loads along a transmission structure with a uniform spacing of, say, 1/3 rise time between each load, then your driver need not charge all the loads at once. Spreading the loads in this manner
reduces the peak current required of the driver (and thus limits the SSO noise). That is the secret to successfully driving many, many loads.
Best Regards,
Dr. Howard Johnson | {"url":"http://www.sigcon.com/Pubs/news/8_07.htm","timestamp":"2014-04-20T01:18:03Z","content_type":null,"content_length":"21416","record_id":"<urn:uuid:6ef97087-bc6b-4f83-ab85-e2b446e72d39>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00447-ip-10-147-4-33.ec2.internal.warc.gz"} |
A quantity specified by a single number or value, as distinct from a vector, matrix, or array, which contain multiple values; it is the simplest form of tensor. Examples of scalars include mass,
speed, volume, and temperature.
A scalar field is an arrangement of scalar values distributed in a space.
Related category | {"url":"http://www.daviddarling.info/encyclopedia/S/scalar.html","timestamp":"2014-04-20T21:22:19Z","content_type":null,"content_length":"5722","record_id":"<urn:uuid:37ebfc0f-bd0a-4c74-9419-bcbffc45df1f>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00404-ip-10-147-4-33.ec2.internal.warc.gz"} |
MT: April 2000, Volume 93, Issue 4
Finger Math in Geometry
Don Warkentin
To gesture or not to gesture; is that really the question? Try out my technique in your geometry classrooms. You may be pleased to learn that the careful and conscious use of finger math will
reinforce meaning for many of your mathematics students.
Algebra for Students with Learning Disabilities
Ruth Feigenbaum
The learning environment created in the special algebra classes, combined with the teaching strategies developed, has produced positive results for students with learning disabilities.
Consul, the Educated Monkey
Sidney Kolpas, Gary Massion
"Consul", the Educated Monkey, is an outstanding, practical example of a plane linkage. In learning why the monkey works the way it does, students are required to review many important concepts from
plane geometry, algebra, and arithmetic. Making their own "monkey" linkage similar to Consul, which one of the authors has done with construction paper and paper fasteners, would give students
additional, hands-on experience with many important mathematical concepts.
Traveling toward Proof
Timothy Craine, Rheta Rubenstein
We have found that the extended metaphor of Aristotle Airlines is extremely helpful in supporting students' construction of proofs. It fits well with both flow and two-column forms, and it acts as a
stepping-stone to paragraph proofs, as well.
Messy Monk Mathematics: An NCTM Standards-Inspired Class
Larry Copes
A description of inquiry-based teaching and learning.
Using Financial Headlines and the Internet to Keep Statistics Classes Fresh
Marilyn Durkin
Stocks, market, and financial data, and their connection to mathematics.
A Geometry Solution from Multiple Perspectives
Phillip Nissen
It is perhaps informative for students to see that no one approach to geometry is the best. Students should be encouraged to try a variety of approaches when attempting a solution to any problem,
experimenting and discussing which method is helping them find an answer. For some problems, students could develop a hybrid method, which would allow them to experience the creative aspects of
Project Jacobean: A Mathematical Exploration of a Literary Era
Josi Binongo, M. Smith
In this investigation, differential calculus appears in a field where few would expect to see it: the study of literature, in particular, the attribution of an anonymous work. | {"url":"http://www.nctm.org/publications/toc.aspx?jrnl=MT&mn=4&y=2000","timestamp":"2014-04-20T21:59:28Z","content_type":null,"content_length":"46772","record_id":"<urn:uuid:edf7e36a-008d-4902-ae9e-46f04e85a6be>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00294-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pottstown Science Tutor
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A Methodology for Deriving Probabilistic Correctness from Recognizers
"... In several tasks from different fields, we are encountering sparse events. In order to provide with probabilities for such events, researchers commonly perform a maximum likelihood (ML)
estimation. However, it is well-known that the ML estimator is sensitive to extreme values. In other words, config ..."
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In several tasks from different fields, we are encountering sparse events. In order to provide with probabilities for such events, researchers commonly perform a maximum likelihood (ML) estimation.
However, it is well-known that the ML estimator is sensitive to extreme values. In other words, configurations with low or high frequencies are respectively underestimated or overestimated and
therefore nonreliable. In order to solve this problem and to better evaluate these probability values, we propose a novel approach based on the probabilistic logic (PL) paradigm. For a sake of
illustration, we focuss on this paper on events such as word trigrams (w 3 ; w 1 ; w 2 ) or word/pos-tag trigrams ((w 3 ; t 3 ); (w 1 ; t 1 ); (w 2 ; t 2 )). These latter entities are the basic
objects used in speech or handwriting recognition. In order to distinguish between for example: "replace the fun" and "replace the floor" an accurate estimation of these two trigrams is needed. The
ML estimation is equival... | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=8008753","timestamp":"2014-04-16T09:07:36Z","content_type":null,"content_length":"12900","record_id":"<urn:uuid:303c3670-af7b-4c95-adc7-2a6df1d75724>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
Need help with Higher derivative Euler method
May 9th 2013, 11:12 AM #1
Need help with Higher derivative Euler method
I am trying solve numerically the equation using the higher derivative Euler method with 3 terms:
y' = 2x +y
with y(0)=0
h = 0.1 and aiming to to estimate y(0.5)
Can anyone who is familiar with this method offer some help in formulating the terms?
I have worked out the following:
y' = 2x +y
y'' = 2
So, yi+1 ≈ yi + (0.1)(2)xi + (0.1)yi + (0.1)^2/2! (2)
Is this correct?
Re: Need help with Higher derivative Euler method
Hey zzizi.
Hint: When you are differentiating d/dx (2x + y) then you get y'' = d/dx (2x + y) = 2 + dy/dx = 2 + 2x + y.
Also what is your formula for updating y(x+h) given y(x) for your Euler Method update?
May 9th 2013, 10:50 PM #2
MHF Contributor
Sep 2012 | {"url":"http://mathhelpforum.com/differential-equations/218753-need-help-higher-derivative-euler-method.html","timestamp":"2014-04-19T15:40:53Z","content_type":null,"content_length":"32405","record_id":"<urn:uuid:050a27fd-943a-4c28-a979-5b880893ac9c>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00251-ip-10-147-4-33.ec2.internal.warc.gz"} |
Program and Degree Options
Program and Degree Options
We offer both a B.A. and a B.S. degree. As a mathematics major, you can choose from several programs of study:
To earn the B.A. degree in mathematics, you need to complete at least 40 hours in mathematics, including
Required core courses:
Calculus: MTH 141 or 151, MTH 152, MTH 153, and MTH 254
Mathematical Proofs: MTH 280
Algebra: MTH 300 and MTH 421
Real Analysis: MTH 461
Seminar: MTH 490
Problem Solving: MTH 375 (at least 1 credit hour)
Electives (select one group):
Group 1: At least four of the following courses: MTH 305, MTH 310, MTH 315, MTH 323, MTH 341, MTH 342, MTH 355, MTH 422, MTH 462, MTH 473
Group 2: (required for students seeking a supplemental major in secondary education) MTH 305, MTH 310, MTH 341, MTH 342, and completion of the secondary education supplemental major (see description
of Education Department programs).
To earn the B.S. degree in mathematics, you need to complete at least 44 hours in mathematics, including
Required core courses:
Calculus: MTH 141 or 151, MTH 152, MTH 153, and MTH 254
Mathematical Proofs: MTH 280
Algebra: MTH 300 and MTH 421
Complex Variables: MTH 323
Real Analysis: MTH 461
Seminar: MTH 490
Problem Solving: MTH 375 (at least 2 credit hours)
Choose at least two of: MTH 422, MTH 462, MTH 473
Choose at least one pair: MTH 341 and MTH 342, or MTH 315 and MTH/PHY 355
Required Support Courses for the B.S. degree in mathematics:
CSC 161 and an additional computer science course at the 200-level or above of at least 3.00 credit hours; and
a minor in biology, chemistry, computer science, economics, or physics, or
completion of the requirements for certification as a secondary teacher of
No course numbered below MTH 141 may be counted toward the B.A. or B.S.
degree in Mathematics. Students intending to go to graduate school in mathematics
are strongly encouraged to take MTH 323, MTH 422, MTH 462, and MTH 473.
To earn a Mathematics Minor, you need to complete at least 18 credit hours in
mathematics, with at least 9 credit hours numbered 200 or higher, including MTH 256,
and no more than 3 credit hours numbered below MTH 141. ECN 440 or ECN 445
may be included in the 18 credit hours required for the minor.
To earn the B.A. degree in Applied Mathematics, you need to complete the following requirements:
1. MTH 141 or 151, MTH 152, MTH 153, MTH 254, MTH 280, MTH 300, MTH 315, MTH 341, MTH 342, MTH/PHY 355 and MTH 461;
2. A minor or major outside of the math department;
3. Three-credit-hour capstone experience in
• a 400 level course with topics involving significant applications of mathematics in the other major/minor field -or-
• a 400 level independent study project involving significant applications of mathematics in the other major/minor.
The option needs to be approved by the chair of the mathematics department before the end of junior year.
To earn the B.S. degree in Applied Mathematics, you need to complete all of the B.A. requirements, plus CSC 161, a Computer Science course at the 200-level or above of least 3.00 credit hours, and
MTH 462.
To earn the B.S. degree in Actuarial Science, you need to complete 51 credit hours in Mathematics, Accounting, Economics, and Finance, including
MTH 141 or 151, MTH 152, MTH 153, MTH 254, MTH 300, MTH 341, MTH 342, ACC 201, ACC 202, ECN 250, ECN 252, ECN 360, ECN 445,
FIN 350, FIN 400, FIN 425 and MTH/FIN 365.
Required Support Courses:
CSC 161 plus an additional Computer Science course at the 200-level or above of least 3.00 credit hours.
Recommended Electives:
CSC 230, FIN 475, IFS 103, SPC 214, and SPC 230.
Areas of Emphasis
Applied Mathematics
Actuarial Science
David Schmitz
Associate Professor of Mathematics
It’s not unusual to find David Schmitz on campus until 10:30 p.m. This award-winning associate professor of math devotes his time to teaching calculus, tutoring and then playing piano for Concert
Choir and musical theatre rehearsals. “I love teaching but in the theatre, I can relax and have a different interaction with my students.” Marathon running, square dancing and city living round out
his busy life. “Teaching at North Central has been very rewarding, despite some sleepless nights.” | {"url":"http://northcentralcollege.edu/academics/dept-div-progs/mathematics/program-and-degree-options","timestamp":"2014-04-20T11:17:55Z","content_type":null,"content_length":"16890","record_id":"<urn:uuid:5785af90-1f66-4258-b271-ac507d6b7445>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00251-ip-10-147-4-33.ec2.internal.warc.gz"} |
LaTeX bug?
Looks to me that you found a bug in MathJax!
Spaces are not supposed to be significant in Latex.
They are only required to separate operators from operands if necessary, like \sin x, which is different from \sinx (which is an error:##\sinx##).
I googled for confirmation, but could not quickly find it.
Even letters with spaces between them are rendered as if the spaces were not there.
And operations on text without spaces, treat the letters as if they are separate,
like \frac ab versus \frac a b (both rendered as ##\frac ab## respectively ##\frac a b##).
Afaik the only other time when white space should be significant is within a \text environment.
Note that some latex engines have difficulty rendering long formulas without spaces (a bug).
So I believe it's a bug in MathJax that your fraction does not render correctly.
A quick check shows that it is rendered correctly by another engine.
If the space were supposed to be significant, it should have been escaped. | {"url":"http://www.physicsforums.com/showthread.php?s=27a22e17a889d868ee98b8bdc738568b&p=3949447","timestamp":"2014-04-23T12:31:37Z","content_type":null,"content_length":"49196","record_id":"<urn:uuid:c6046da7-fcb4-4a3e-9871-bc5866461c7b>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00606-ip-10-147-4-33.ec2.internal.warc.gz"} |
Standard Errors for Two-Step MLE Procedure for Computationally Intensive Likelihood Functions
up vote 1 down vote favorite
Suppose we have a likelihood function, $L(\theta_{1},\theta_{2},\theta_{3};X_{1},X_{2})$ where $\theta_{1}...\theta_{3}$ are sets of parameters and $X_{1}$ and $X_{2}$ are data. The model is fully
identified, but computational difficulties make it nearly impossible to maximize this function over $\theta_{1},$ $\theta_{2},$ and $\theta_{3}$ simultaneously, because the liklihood function is
time-consuming to compute and can only be computed with error. So with too many dimensions it takes any appropirate algorithm a very long time to converge. However, we can restrict the model to
estimate just some of the parameters consistently. More specifically, we can generate likelihood functions $L_{1}(\theta_{1};X_{1})$ and $L_{2}(\theta_{2};X_{2})$ to estimate $\theta_{1}$ and $\
theta_{2}$ consistently. If doing so gives us estimates $\hat{\theta_{1}}$ and $\hat{\theta_{2}},$ then maximizing $L(\hat{\theta_{1}},\hat{\theta_{2}},\theta_{3};X_{1},X_{2})$ gives a consistent--if
inefficient--estimate of $\theta_{3}$, which is not subject to the computational difficulties.
My question concerns the standard errors for $\hat{\theta_{3}}$, or confidence intervals obtained via, say, the liklihood ratio test. Is it appropriate to uses the usual methods for obtaining
standard errors in this case, or should I do something different to account for the odd way in which I obtained my estimate? If I do things the usual way, I should use approximations and/or
evaluations involving the function $L(\theta_{1},\theta_{2},\theta_{3};X_{1},X_{2})$ and not $L(\hat{\theta_{1}},\hat{\theta_{2}},\theta_{3};X_{1},X_{2})$, right?
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What (if anything) happened to Intersection Homology?
up vote 18 down vote favorite
In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined
initially by Goresky and MacPherson, this is a version of homology which agrees with ordinary homology on manifolds, but also retains crucial properties like Poincare Duality and Hodge Theory on
singular (non-)manifolds. The original definition was combinatorial, but it was later re-interpreted in sheaf-theoretic terms (perverse sheaves?).
Back then it certainly looked like an exciting new development. So, I'm curious - where does the field stand today? Is it still thriving, or has it been merged with something else, or just faded
gt.geometric-topology at.algebraic-topology sheaf-theory
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4 Answers
active oldest votes
Intersection homology and cohomology are still around, but as a topic they have just substantially been renamed. They are part of the theory of perverse sheaves, which are widely used
in the Langlands program, in algebraic geometry approaches to categorification, and elsewhere in algebraic geometry.
up vote 25 To the extent that intersection homology was intended for topology, it has stoked relatively less interest than in algebraic geometry. On the one hand, there has been a trend away from
down vote homological algebra in geometric topology. On the other hand, singularities are part of the structure of intersection homology. Singularies are more germane to geometric topology than
accepted to algebraic topology in the sense of homotopy theory. Both singularities and homological algebra are major aspects of algebraic geometry.
add comment
Intersection homology is alive and well in a large number of guises. It's true that a lot of the work trended to algebraic geometry, representation theory, and categorical constructions,
such as perverse sheaves, through the 90s, but there also continues to be work in the more topological settings by people such as me, Cappell, Shaneson, Markus Banagl, Laurentiu Maxim, and
many others. At least some of this work is dedicated to extending classical manifold invariants, such as characteristic classes, in a meaningful way to stratified spaces, such as algebraic
varieties, and there is a lot of recent interest (though slow progress) in figuring out how intersection homology might tie into various algebraic topology constructions. There are also
analytic formulations such as L^2 cohomology (initiated by Cheeger), and much more.
Here are some good references to get started in the area:
Books: An Introduction to Intersection Homology by Kirwan and Woolf (mostly concerned with telling the reader about the fancy early applications to algebraic geometry and representation
theory, but a great overview nonetheless)
Intersection Cohomology by Borel, et.al. This is a great serious technical introduction to the area and, to my mind, the canonical source for the foundations of the subject)
up vote
20 down Topological Invariants of Stratified Spaces by Markus Banagl (topological but mostly from the sheaf point of view)
For an overview of state-of-the-art in intersection homology and related fields, I'm co-editing a volume on Topology of Stratified Spaces that will be published in the MSRI series.
Unfortunately, it's not out yet, but look for it soon.
Papers: The original papers of Goresky and MacPherson are quite good.
Topological invariance of intersection homology without sheaves by Henry King is a good introduction to the singular version of the theory.
And for a whole pile of recent papers, I'll shamelessly plug my own web site: http://faculty.tcu.edu/gfriedman/ and Markus Banagl's: http://www.mathi.uni-heidelberg.de/~banagl/
And many further references can be found from these locations.
i was about to comment on Greg Kuperberg's above answer saying something like "hey, what about Greg Friedman's work, that's topological." anyway, the extent of my knowledge is literally
your midwest topology seminar talk... – Sean Tilson Jun 9 '10 at 15:55
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Intersection homology quickly found applications in representation theory, starting with the Kazhdan-Lusztig conjectures. Today, the theory of perverse sheaves is an important tool
up vote 6 down in geometric representation theory.
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There do seem to be people thinking about it. I heard Sylvain Cappell lecture a few weeks ago; he spoke about using intersection (co)homology as part of a generalization of
characteristic classes to singular varieties.
up vote 3 down
vote Here's a recent paper by Cappell-Maxim-Shaneson that he referred to: "Intersection cohomology invariants of complex algebraic varieties".
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The Principle of Relativity and its Application to some Special Physical Phenomena
Translation:The Principle of Relativity and its Application to some Special Physical Phenomena
From Wikisource
The Principle of Relativity and its Application to some Special Physical Phenomena (1910)
by , translated from German by Wikisource
The Principle of Relativity and its Application to some Special Physical Phenomena.
By H. A. Lorentz.
To discuss Einstein's principle of relativity here in Göttingen, where Minkowski has worked, appears to me as a particular welcomed task.
One can highlight the importance of this principle from different points of views. About the mathematical side of the question, which has found such a gleaming representation by Minkowski and which
was further developed by Abraham, Sommerfeld etc., shall not be spoken here. Rather (after some epistemological considerations concerning the concepts of space and time) those physical phenomena
shall be discussed, which could contribute to an experimental test of this principle.
The relativity principle asserts the following: When a physical phenomenon is described in reference system $x, y, z, t$ by certain equations, then a phenomenon which can be described in another
reference system $x', y', z', t'$ by the same equations, will exist as well. There, both reference systems are connected by relations in which the speed of light occurs, and which express that a
system is moving with uniform velocity relative to the other one.
If observer $A$ is in the first, $B$ in the second reference system, and if any of them is equipped with measuring rods and clocks at rest in his respective system, then $A$ will measure the values
of $x, y, z, t$, while $B$ measure the values of $x', y', z', t'$, where it is to be remarked, that $A$ and $B$ can use the same measuring rods and the same clocks as well. We have to assume, that
when measuring rods and clocks are somehow transfered from the first observer to the second one, then they take over the correct length and the correct rate by themselves, so that $B$ obtains the
values of $x', y', z', t'$ form his measurements. Both will now obtain the same value for the speed of light, and generally can make the same observations.
Provided that there is an aether, then under all systems $x, y, z, t$, one is preferred by the fact, that the coordinate axes as well as the clocks are resting in the aether. If one connects with
this the idea (which I would abandon only reluctantly) that space and time are completely different things, and that there is a "true time" (simultaneity thus would be independent of the location, in
agreement with the circumstance that we can have the idea of infinitely great velocities), then it can be easily seen that this true time should be indicated by clocks at rest in the aether. However,
if the relativity principle had general validity in nature, one wouldn't be in the position to determine, whether the reference system just used is the preferred one. Then one comes to the same
results, as if one (following Einstein and Minkowski) deny the existence of the aether and of true time, and to see all reference systems as equally valid. Which of these two ways of thinking one is
following, can surely be left to the individual.
In order to discuss the physical side of the question, we have to state the transformation formulas first, where we confine ourselves to the special form in which they were already used in the year
1887 by Voigt at investigations concerning Doppler's principle, namely:
$x'=x,\ y'=y,\ z'=az-bct,\ t'=at-\frac{b}{c}z;$
there, the constants $a > 0,\ b$ satisfy the relation
which cause the identity
$x'^{2} + y'^{2} + z'^{2} - c^{2}t'^{2} = x^{2} + y^{2} + z^{2} - c^{2}t^{2}\,$
The origin of system $x', y', z'$ moves towards system $x, y, z$ in the $z$-direction with velocity $\tfrac{b}{a}c$, which is always smaller than $c$. Generally, any velocity has to be assumed as
being smaller than $c$.
All state variables of any phenomenon, measured in one or the other system, are connected by certain transformation formulas. They read, e.g. for the velocity of a point:
$\mathfrak{v}'_{x}=\frac{\mathfrak{v}_{x}}{\omega},\ \mathfrak{v}'_{y}=\frac{\mathfrak{v}_{y}}{\omega},\ \mathfrak{v}'_{z}=\frac{a\mathfrak{v}_{z}-bc}{\omega}{,}$
We furthermore consider a system of pints, whose velocity is a steady function of the coordinates. Let $dS$ be a space element surrounding point $P(x,y,z)$ at time $t$; to this value $t$ and the
coordinates of $P$, a moment $t'$ is corresponding in another reference systems according to the transformation equations, and every point lying in $dS$ at time $t'$, has certain $x', y', z'$ for
this definite value of $t'$. Points $x', y', z'$ satisfy a space element $dS'$, which is connected with $ds$ as follows:
If we imagine an agent (matter, electricity etc.) as connected with these points, and if we assume that observer $B$ has reason to connect the same amount of that agent with every point as observer
$A$, then the space density must be inversely proportional to the volume elements, i.e.,
All of these relations are reciprocal, i.e., one can permute the primed and unprimed letters, when one simultaneously replaces $b$ by $-b$.
The fundamental equations of the electromagnetic field retain their form at the transformation, when one introduces the following magnitudes^[1]:
$\begin{array}{ccc} \mathfrak{d}'_{x}=a\mathfrak{d}_{x}-b\mathfrak{h}_{y}, & \mathfrak{d}'_{y}=a\mathfrak{d}_{y}+b\mathfrak{h}_{x}, & \mathfrak{d}'_{z}=\mathfrak{d}_{z},\\ \mathfrak{h}'_{x}=a\
mathfrak{h}_{x}+b\mathfrak{d}_{y}, & \mathfrak{h}'_{y}=a\mathfrak{h}_{y}-b\mathfrak{d}_{x}, & \mathfrak{h}'_{z}=\mathfrak{h}_{z}; \end{array}$
between these ones, and the transformed space density $\varrho'$, and the transformed velocity $\mathfrak{v}'$, the following equations hold in system $x', y', z', t'$:
$\begin{array}{l} \operatorname{div}\ \mathfrak{d}'=\varrho',\\ \operatorname{div}\ \mathfrak{h}'=0,\\ \operatorname{rot}\ \mathfrak{h}'=\frac{1}{c}\left(\mathfrak{\dot{d}}'+\varrho'\mathfrak{v}'\
right),\\ \operatorname{rot}\ \mathfrak{d}'=-\frac{1}{c}\mathfrak{\dot{h}}'. \end{array}$
In so far, the field equations of the theory of electrons satisfy the relativity principle; though we have to bring the equations of the electron themselves into accordance with this principle.
We will (somewhat more general) consider the motion of an arbitrary material point. At this occasion, the introduction of the concept of "proper time" (a nice invention of Minkowski) is useful.
According to this, every point is so to speak connected with its own time which is independent of the reference system chosen; its differential is defined by the equation:
The expressions formed by the aid of proper time
$\frac{d}{d\tau}\frac{dx}{d\tau},\ \frac{d}{d\tau}\frac{dy}{d\tau},\ \frac{d}{d\tau}\frac{dz}{d\tau}{,}$
which are linear homogeneous functions of the ordinary acceleration components, are denoted by us as components of the "Minkowski acceleration". We describe the motion of a point by the equations
$m\frac{d}{d\tau}\frac{dx}{d\tau}=\mathfrak{K}_{x}\text{, etc.,}$
where $m$ is a constant which we call the "Minkowski mass". Vector $\mathfrak{K}$ is denoted by us as "Minkowski force".
The transformation formulas for this acceleration and force can be easily derived; $m$ is left unchanged by us. Then one has
$\mathfrak{K}'_{x}=\mathfrak{K}_{x},\ \mathfrak{K}'_{y}=\mathfrak{K}_{y},\ \mathfrak{K}'_{z}=a\mathfrak{K}_{z}-\frac{b}{c}(\mathfrak{v}\cdot\mathfrak{K}).$
The essential thing is now as follows: The relativity principle requires, that (at an actual phenomenon) the Minkowski forces are in a certain way depending on the coordinates, velocities, etc. in
one reference system, and the transformed Minkowski forces in the other reference system are depending in the same way on the transformed coordinates, velocities etc. That is a special property,
which all forces of nature must have, when the relativity principle shall hold. If we presuppose this, then one can calculate the forces acting on moving bodies, when one know them for the case of
rest. If e.g. an electron of charge $e$ is moving, then we imagine a reference system in which it is momentarily at rest. The Minkowski force is acting upon the electron in this system:
from that it follows by application of the transformation equations for $\mathfrak{K}$ and $\mathfrak{d}$, that the Minkowski force acting upon an electron moving with velocity $\mathfrak{v}$ in an
arbitrary reference system, amounts to
This formula is not in agreement with the ordinary Ansatz of electron theory, due to the presence of the denominator. The difference stems from the fact, that one usually doesn't operate with our
Minkowski force, but with the "Newtonian force" $\mathfrak{F}$, and we see, that (for an electron) these two forces are connected as follows:
One will assume that this relation holds for arbitrary material points.
Thus one can treat the phenomena of motion in two different ways, either with the Minkowski- or with the Newtonian force. In the latter case, the equations of motions read:
here, $\mathfrak{j}_{1}$ means the ordinary acceleration into the direction of motion, and $\mathfrak{j}_{2}$ the ordinary normal acceleration, and the factors
$\begin{array}{rl} m_{1} & =\frac{m}{\sqrt{\left(1-\frac{\mathfrak{v}^{2}}{c^{2}}\right)^{3}}},\\ & \\ m_{2} & =\frac{m}{\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}} \end{array}$
are called the "longitudinal" and "transverse mass".
In the same way as the Minkowski force, also the Newtonian forces occurring in nature must satisfy certain conditions, when the relativity principle should be satisfied. This is e.g. the case, when
(independently from motion) a normal pressure of constant magnitude $p$ is acting per unit area; in the transformed system, a normal pressure of the same magnitude is acting upon the corresponding
surface element in motion.
Since we already recognized the invariance of the field equations, then the question, as to whether the motions in a system of electrons are in agreement with the relativity principle, only
tantamounts to the experimental test of the formulas for longitudinal and transverse mass $m_{1},\ m_{2}$; although the experiments of Bucherer and Hupka seem to confirm these formulas, we have not
arrived at a definite decision.
Regarding the mass of the electron, it is to be considered that they are of electromagnetic nature; thus it will depend on the distribution of the charges in the interior of the electron. Therefore,
the formulas for the mass can only then be correct, when the charge distribution and thus also the shape of the electron are variable with velocity in a certain way. One has to assume, that in
consequence of the translation of the electron, which is a sphere when at rest, the electron becomes an oblate ellipsoid in the direction of motion; the amount of oblateness is
If we assume, that shape and magnitude of the electron are regulated by inner forces, then they must (to be compatible with the relativity principle) have such properties by which this oblateness is
arising by itself. In relation to this, Poincaré has made the following hypothesis. The electron is a charged and expandable shell, and an inner normal stress of invariable magnitude is resisting to
the electric repulsions of the individual points. According to the above, such normal stresses indeed satisfy the relativity principle.
In the same way, all molecular forces acting in the interior of ponderable matter, as well as the quasi-elastic forces and resisting forces acting upon the electron, must satisfy certain conditions
in order to be in agreement with the relativity principle. Then, every moving body will be invariable for a co-moving observer, yet it will experience a change of dimensions for a stationary
observer, which is just a consequence of the change of molecular forces required by those conditions. From that, the contraction of the body – which was already imagined before to explain the
negative outcome of Michelson's interference experiment – follows by itself, and also the negative outcome of all similar experiments which should demonstrate an influence of Earth's motion upon
optical phenomena.
Regarding the rigid body (with which Born, Herglotz, F. Noether, Levi-Cività were dealing): the difficulties emerging during the consideration of rotations, will probably be solved by ascribing
rigidity to the effectiveness of particularly intensive molecular forces.
Eventually we want to turn our attention to gravitation. The relativity principle requires a modification of Newton's laws, above all it requires the propagation of this effect with the speed of
light. The possibility of a finite propagation velocity of gravitation was already discussed by Laplace, who imagined a fluid streaming against the sun as the cause of gravity, which pushes the
planets towards the sun. He found, that the speed $c$ of this fluid must be assumed to be at least 100 million times greater than the speed of light, so that the calculation remains in agreement with
the astronomical observations. The necessity of such a great value of $c$ stems from the fact, that the magnitude $\tfrac{v}{c}$ arises in its end formulas in the first power, where $v$ is the
planetary velocity. However, if the propagation speed $c$ of gravitation shall have the speed of light, as required by the relativity principle, then a contradiction with observations can only then
be avoided, when only magnitudes of second (or higher) order in $\tfrac{v}{c}$ arise in the expression for the modified law of gravitation.
If one confines oneself to magnitudes of second order, then a condition can easily be given on the basis of an obvious electron-theoretical analogy, which defines the modified law in a definite
If one namely considers the force acting upon an electron moving with velocity $\mathfrak{v}$:
then the vectors $\mathfrak{d}$ and $\mathfrak{h}$ also depend on velocities $\mathfrak{v}'$ of the electrons generating the field; therefore, in the vector product $[\mathfrak{v}\cdot\mathfrak{h}]$
there indeed arise products of the form $\mathfrak{vv'}$, yet not the square $\mathfrak{v}^{2}$ of the speed of the considered electron. If we accordingly assume, that the square of the velocity $\
mathfrak{v}_{1}^{2}$ of point 1 doesn't occur in the expression of the attraction acting upon point 1 and exerted by point 2, then this velocity must entirely drop in a reference system in which
point 2 is at rest $(\mathfrak{v}_{2}=0)$; the law in this system thus will be reduced to the ordinary Newtonian one. If one now passes to an arbitrary coordinate system by transformation, then one
finds that the force acting upon point 1 is composed of two parts, first an attraction in the direction of the connecting line of amount
$R+\frac{1}{c^{2}}\left\{ \frac{1}{2}\mathfrak{v}_{2}^{2}R+\frac{1}{2}\mathfrak{v}_{2r}^{2}\left(r\frac{dR}{dr}-R\right)-\left(\mathfrak{v}_{1}\cdot\mathfrak{v}_{2}\right)R\right\}{,}$
second a force in the direction $\mathfrak{v}_{2}$ of amount
here, $r$ means the distance between two simultaneous points, $\mathfrak{v}_{r}$ the component of $\mathfrak{v}$ towards the connection line drawn from 1 to 2, and $R$ that function of $r$ which
represents the attraction law in the case of rest ($R=\tfrac{k}{r^{2}}$ at Newtonian attraction, $R=kr$ at quasi-elastic forces). It is to be noticed, that under force we always have to understand
the "Newtonian force", not the Minkowski force. Incidentally, Minkowski has given a somewhat different expression than this one. In Poincaré's paper, both this one and also the one written above can
be found.
It is to be noticed, that the principle of equality of action and reaction is not satisfied in these laws of gravitation.
Now, the disturbances shall be discussed, which can arise by those supplementary terms of second order. Besides many short-periodic disturbances of no importance, there is a secular motion of the
perihelion of the planets. De Sitter calculates, that it amounts to 6,69" per century.^[2] Now, the perihelion anomaly of mercury of amount 44" per century is known since Laplace; although it has the
correct sign, it is much too great to be explained by those supplementary terms. It is rather explained by Seeliger as due to a disturbance of the carrier of the zodiacal light, whose mass one can
conveniently determine in a plausible way. From that, no decision can be gained as long as the precision of astronomical measurements is not essentially increased. At absolute precision, also the
difference between the "proper time" of earth and the time of the solar system has to be considered.
Another method to test the correctness of the modified law of gravitation, can be based upon a procedure proposed by Maxwell for the decision, as to whether the solar system is moving through the
aether. If this is the case, then the eclipses of the satellites of Jupiter, depending on the location of these planets with respect to Earth, must suffer earlinesses or delays.
If the distance Jupiter-Earth is $a$ and the velocity component of the solar system in the aether in the direction of the connecting line Jupiter-Earth is $v$, then the time $\frac{a}{c}$ which would
be required by light (in the case of rest) to traverse distance $a$, is transformed into $\frac{a}{c\pm v}$; thus an earliness or a delay occurs due to motion, which amounts to $\frac{av}{c^{2}}$ up
to terms of second order, and which attains different values, depending on the value of velocity component $v$, which indeed depends on the location of both planets. Now it is clear, that such a
dependency of the phenomena from the motion through the aether contradicts the relativity principle.
To solve this contradiction, we want to simplify the state of facts in a schematic way. We imagine, that sun $S$ shall have a mass which is infinitely great in relation to that of the planets. Let
the velocity of the solar system coincide with the $z$-axis, which we let go through the sun. The intersections of the orbit of the planets with the $z$-axis, is denoted by us as the upper and lower
transit $A$ and $B$.
We place the observer upon the sun. At every transit of the planet through the $z$-axis, a light signal shall be traveling to the sun. Let $T$ be the orbital period. When the sun is at rest, the time
between the upper and lower transit will amount to $\frac{1}{2}T$ (at a motion being presupposed as circular); the same is true for the time between the arrival of both light signals. However, if the
sun is moving in the $z$-direction, then the light signal of the upper transit must suffer an earliness of $\textstyle{\frac{av}{c^{2}}}$, and the one of the lower transit a delay of the same amount;
in case the uniform orbital motion (as presupposed by Maxwell as self-evident) remains conserved in an undisturbed way, then the time interval between the arrival of the light signal of two
successive transits would be alternately appear to be increased and diminished by $\tfrac{2av}{c^{2}}$. The conservation of uniform circular motion at a translation in the aether which is presupposed
here, however, is impossible according to the relativity principle. If we namely describe the process in a coordinate system that doesn't share the motion, then the modified law of gravitation is to
be applied, and this gives a non-uniformity of planetary motion, due to which the difference of the time intervals between the arrival of the light signals is exactly canceled.
The demonstration as to whether an earliness or delay of the eclipses actually occurs, can therefore be used for the decision in favor or against the relativity principle. However, the numerical
relations are quite unfavorable again. For example, Burton (who has 330 photometric observations at his disposal, which were undertaken at the Harvard-Observatory concerning the eclipses of the 1st
Jupiter-satellite) estimates the probable error of the final result for $v$ as being 50 km/s; on the other hand, one has observed star velocities of 70 km/s, and the velocity of the solar system with
respect to the fixed stars is estimated to be 20 km/s. The relativity principle is thus hardly supported by Burton's calculations, it can at most be disproved, for example, when eventually a value
would be given which exceeds 100 km/s.
Let us leave it undecided, whether or not the new mechanics will experience a confirmation by astronomical observations. Though we won't omit, to learn about its fundamental formulas.
If one defines work as the scalar product of "Newtonian force" and displacement, then the equations of motion give the energy principle in the ordinary form, i.e., that the work performed per unit
time is equal to the increase of energy $\varepsilon$:
There, the energy has the form:
which agrees for small velocities with the value of kinetic energy of ordinary mechanics
Furthermore, one can derive the Hamiltonian principle
$\int_{t_{1}}^{t_{1}}(\delta L+\delta A)dt=0$
from the equations of motion; here, $\delta A$ is the work of "Newtonian force" at virtual displacement, and $L$ the Lagrangian function which reads as follows:
From Hamilton's principle, one conversely can derive the equations of motion again. The quantities
$\frac{\partial L}{\partial\dot{x}},\ \frac{\partial L}{\partial\dot{y}},\ \frac{\partial L}{\partial\dot{z}}$
are to be denoted as components of momentum.
All of these formulas can be verified at the electromagnetic laws of motion of an electron; then one has to set the value for the "Minkowski mass" $m$
$m=\frac{e^{2}}{6\pi Rc^{2}}$,
and to add the energy of those inner stresses to the electric and magnetic energy, which (as we saw) determine the shape of the electron. By specializing on an electron, one can derive them from the
general principle of least action for arbitrary electromagnetic systems, which was discussed in the first lecture^[3], though the work of the inner stresses must be considered again.
Now we consider the equations of the electromagnetic field for ponderable bodies. Those are stated by Minkowski in a pure phenomenological way, and then it was shown by M. Born and Ph. Frank that
they can be derived from the concepts of the theory of electron; I also have by myself obtained in the latter way the equations, whose shape is formally somewhat different.
In order to obtain the relation between observable magnitudes, one has to blur the details of the phenomena stemming from the electrons, by formation of averages over great quantities of electrons.
In this way, one is led to the following equations (which are in agreement with those of ordinary Maxwellian theory):
$\begin{array}{l} \operatorname{div}\ \mathfrak{D}=\varrho_{l}{,}\\ \operatorname{div}\ \mathfrak{B}=0{,}\\ \operatorname{rot}\ \mathfrak{H}=\frac{1}{c}(\mathfrak{C}+\mathfrak{\dot{D}}){,}\\ \
operatorname{rot}\ \mathfrak{E}=-\frac{1}{c}\mathfrak{\dot{B}}. \end{array}$
Herein, $\mathfrak{D}$ is the dielectric displacement, $\mathfrak{B}$ the magnetic induction, $\mathfrak{H}$ the magnetic force, $\mathfrak{E}$ the electric force, $\mathfrak{C}$ the electric
current, $\varrho_{l}$ the density of the observable electric charges. If one indicates the average formation by overlines, then it is e.g.
$\mathfrak{E}=\mathfrak{\bar{d}}, \mathfrak{B}=\mathfrak{\bar{h}}{,}$
where $\mathfrak{d}$, $\mathfrak{h}$ have the earlier meaning; furthermore it is
$\begin{array}{l} \mathfrak{D}=\mathfrak{E}+\mathfrak{P}{,}\\ \mathfrak{H}=\mathfrak{B}-\mathfrak{M}-\frac{1}{c}[\mathfrak{P}\cdot\mathfrak{w}]{,} \end{array}$
where $\mathfrak{P}$ is the electric moment, $\mathfrak{M}$ the magnetization per unit volume, and $\mathfrak{w}$ the velocity of matter. In the derivation of these formulas, one separates the
electrons into three kinds. The first kind, the polarization electrons, produce the electric moment $\mathfrak{P}$ by their displacement; the second kind, the magnetization electrons, produce the
magnetic moment $\mathfrak{M}$ by their orbits; the third kind, the conduction electrons, are freely moving in matter and produce the observable charge density $\varrho_{l}$ and the current $\
mathfrak{C}$. The latter is still to be separated into two parts; if $\mathfrak{u}$ is the relative velocity of the electrons towards matter, then the total velocity of the electrons is $\mathfrak{v}
=\mathfrak{w}+\mathfrak{u}$, thus the current transported by them
$\bar{\varrho}$ is the observable charge $\varrho_{l}$, \$bar{\varrho}\mathfrak{w}$ the convection current, $\overline{\varrho\mathfrak{u}}$ the actual conduction current $\mathfrak{C}_{l}$.
Transformation formulas exist for all these magnitudes, of which some may be given:
$\mathfrak{C}'_{x}=\mathfrak{C}_{x},\ \mathfrak{C}'_{y}=\mathfrak{C}_{y},\ \mathfrak{C}'_{z}=a\mathfrak{C}_{z}-bc\varrho_{l},\ \varrho'_{l}=a\varrho_{l}-\frac{b}{c}\mathfrak{C}_{z}{,}$
$\begin{array}{l} \mathfrak{P}'_{x}=a\mathfrak{P}_{x}-\frac{b}{c}(\mathfrak{w}_{z}\mathfrak{P}_{x}-\mathfrak{w}_{x}\mathfrak{P}_{z})+b\mathfrak{M}_{y}{,}\\ \mathfrak{P}'_{y}=a\mathfrak{P}_{y}-\frac
{b}{c}(\mathfrak{w}_{z}\mathfrak{P}_{y}-\mathfrak{w}_{y}\mathfrak{P}_{z})-b\mathfrak{M}_{x}{,}\\ \mathfrak{P}'_{z}=\mathfrak{P}_{z}. \end{array}$
Furthermore, the following auxiliary vectors are useful:
$\begin{array}{cc} \mathfrak{H}_{1}=\mathfrak{H}-\frac{1}{c}[\mathfrak{w}\cdot\mathfrak{D}], & \mathfrak{B}_{1}=\mathfrak{B}-\frac{1}{c}[\mathfrak{w}\cdot\mathfrak{E}]{,}\\ \mathfrak{E}_{1}=\mathfrak
{E}+\frac{1}{c}[\mathfrak{w}\cdot\mathfrak{B}], & \mathfrak{D}_{1}=\mathfrak{D}+\frac{1}{c}[\mathfrak{w}\cdot\mathfrak{H}]. \end{array}$
The given field equations now are still to be supplemented by stating the relations, which exist between the vectors $\mathfrak{E}, \mathfrak{H}$ and $\mathfrak{D}, \mathfrak{B}$. One can derive
these relations in two ways.
The first phenomenological method follows that procedure: One considers an arbitrarily moving point of matter, and introduces a reference system in which it is at rest; then, in case the volume
element surrounding the point is isotropic in the rest system, e.g. the equations valid for resting systems hold between $\mathfrak{E}$ and $\mathfrak{D}$:
or also
because the auxiliary vectors $\mathfrak{D}_{1}$, $\mathfrak{E}_{1}$ are identical with $\mathfrak{D}$, $\mathfrak{E}$ for $\mathfrak{w}=0$. However, $\mathfrak{D}_{1}$ and $\mathfrak{E}_{1}$ are
transformed in the same way, and from that it follows, that the equation
remains valid in the initial reference system as well. Accordingly it is
As regards the conduction current, we only remark that it depends on $\mathfrak{E}_{1}$.
The second method is based upon the mechanics of the electrons. In the same way, as (for resting bodies) equation $\mathfrak{D}=\varepsilon\mathfrak{E}$ proves to be the consequence of the assumption
of quasi-elastic forces, which draw back the electron in their rest states, one will obtain the equation $\mathfrak{D}_{1}=\epsilon\mathfrak{E}_{1}$ at moving bodies, when one ascribes to those
quasi-elastic forces those properties, which are required by the relativity principle. The latter will be satisfied, when one uses the expression of the generalized attraction law for these forces,
where $R$ must be taken proportional to $r$.
The similar is valid for the explanation of the conduction resistance. A satisfying electron-theoretical explanation of the magnetic properties of the bodies is not present for the time being.
At last, the importance of the previous equations shall be shown at three remarkable cases.
The first remark is based on the equation
According to it, $\varrho'_{l}$ can vanish without the need of $\varrho_{l}=0$, as long as only current $\mathfrak{C}$ exists; i.e. an observer $A$ will declare the body as charged, while it is
uncharged for an observer $B$ moving relative to him. This can be understood when it is considered, that positive and negative electrons of same amount are present in every body, which compensate
themselves at uncharged bodies. If the body is moving with velocity $\mathfrak{w}$, then (when a conduction current is present) both kinds of electrons will obtain different total velocities, thus
also the quantity $\omega=a-b\tfrac{\mathfrak{v}_{z}}{c}$ will have different values for both kinds. Now, if an observer $B$ which is moving with the body, is calculating the average of charge
density $\overline{\varrho'}=\overline{\omega\varrho}$ for both kinds of electrons, then he can obtain the sum zero, even when for an observer $A$ (in whose reference system the body is moving) the
averages $\bar{\varrho}$ of the positive and negative electrons are not compensating themselves.
This circumstance causes a reminiscence of an old question. Around the year 1880, there was a great discussion among physicists concerning Clausius' fundamental law of electrodynamics. At that time,
it was tried to derive a contradiction between this law and the observations by concluding that according to this law, a current-carrying conductor on earth shall exert an action upon a co-moving
charge $e$ due to Earth's motion, which could possibly be detected. That the law actually doesn't require this action, was noticed by Budde; it stems from the fact, that the current is acting upon
itself due to Earth's motion, and is causing a "compensation charge" upon the traversed conductor, which exactly compensates the first action. The theory of electron leads to similar conclusions, and
I find for the density of the compensation charge, when the velocity has the direction of the $z$-axis,
this must be assumed as existent by an observer $A$ who doesn't share the motion with Earth, while it doesn't exist for a co-moving observer $B$. The given value exactly agrees with the formula
derived from the relativity principle; if $\varrho'_{l}=0$, then one finds from this formula
and since $\mathfrak{w}_{z}=\frac{bc}{a}$ is the mutual velocity of both reference systems according to the things previously said (p. 75), then one indeed finds
The second remark is based on the transformation equations for the electric moment $\mathfrak{P}$ (p. 84), which shows the impossibility (because the magnetization $\mathfrak{M}$ occurs in them) to
clearly distinguish between polarization and magnetization electrons. In a magnetized body $(\mathfrak{M}eq0)$, as seen from a reference system, it can rather be $\mathfrak{P}=0$, while $\mathfrak{P}
'$ is different from zero in another reference system. This shall now be applied to a special case, where we confine ourselves to magnitudes of first order. The considered body (e.g. a steel magnet)
shall contain only conduction electrons and such ones (when the body is at rest) which produce $\mathfrak{M}$, yet not $\mathfrak{P}$; it shall have the shape of an infinitely extended even plate,
bounded by two planes $a,b$: the middle plane is made by us to the $yz$-plane (Fig. 6). When it is at rest, a constant magnetization $\mathfrak{M}_{y}$ may exist in the $y$-direction, while $\
mathfrak{P}=0$. If the body acquires the velocity $v$ in the $z$-direction, then an observer not participating at the motion, is observing the electric polarization
Now we imagine two conductors $c, d$ at both sides of the body, which together with it are forming two equal condensers, and they shall be short circuited by a wire (from $c$ to $d$). When in motion,
$d$ charges will arise upon $c$ now, which can be calculated as follows. Since it is evidently impossible that a current exists in the $x$-direction, it is $\mathfrak{E}_{1x}=0$ or $\mathfrak{E}_{x}=
\frac{v}{c}\mathfrak{B}_{y}$. Since the process is stationary, it becomes $\mathfrak{\dot{B}}=0$; then the existence of a potential $\varphi$ follows from $\operatorname{rot}\ \mathfrak{E}=0$. If $\
Delta$ is the thickness of the plate, then one has
From the symmetry of the arrangement if evidently follows
and because the plates $c, d$ are short circuited, it must be
from that if follows
If $\gamma$ is the capacity of one of the two condensers, then the charge of the plate $d$ becomes equal to
and $c$ obtains the oppositely equal amount.
Now we compare this procedure with the inverse case, that the magnet $ab$ is at rests and plates $c, d$ are moving with opposite velocity. Then according to the relativity principle, everything must
be quite the same as in the first case. Indeed, one immediately finds from the ordinary law of induction, exactly the mount of charge upon plate $d$ previously given. But now this charge upon $d$
must produce the opposite equal charge upon plane $a$ of the resting magnet by electrostatic induction, and the corresponding must hold for $b$ and $c$. Since no current can flow $(\mathfrak{C}=0)$,
then in both cases (whether the magnet is moving and the plate are at rest, or vice versa) the same charges must be present upon the magnet. Thus we have to consider, as to how it comes that in the
first treated case, the opposite charge arises upon plane $a$ of the moving magnet, than upon plate $d$; this is only possible by the polarization $\textstyle{\mathfrak{P}_{x}=-\frac{v}{c}\mathfrak
{M}_{y}}$ emerging during the motion. Because one has
since $\mathfrak{P}$ (thus the term $[\mathfrak{P}\cdot\mathfrak{w}]$) is to be neglected in the velocity of first order, it becomes
though $\mathfrak{H}$ is zero, because the plate is assumed to be infinitely extended. From that if follows
no dielectric displacement arises in the moving plate, thus the charge upon $a$ agrees with that upon $d$, as required by the relativity principle.
The last remark concerns the circumstance, that the motion of Earth cannot have an influence upon electromagnetic processes according to the relativity principle. However, Liénard alluded to a
phenomenon, where such an influence (namely amounting to first order) shall be expected; also Poincaré has discussed this case in his book Electricité et Optique. It is about the ponderomotive force
upon an conductor. In order to determine it, one will make the obvious assumption for the force acting upon the conduction electrons per unit force:
then the force caused by Earth's motion upon the conductor in the direction of motion, amounts to
is the heat produced by the conduction current
, this expression is easily to be calculated numerically (however, an unobservable value will be obtained).
If one now asks oneself, as to how this result (which contradicts the relativity principle) can arise, then one sees that one hasn't actually calculated the force acting upon the matter of the
conductor, but the force which acts upon the electrons moving in the interior of the conductor. The latter forces are still to be transfered to matter by individually unknown forces, and this only
happens (without change of the magnitude) when equality between action and reaction exists for the forces between matter and electrons. However, for moving bodies, the action is not equal to the
reaction according to the relativity principle, and this circumstance exactly cancels the force of Liénard.
In summary one can say, that there is little prospect of testing the relativity principle by experiment; except some astronomic observations, only the measurement of the mass of electrons comes into
account. Though one shall not forget, that the negative outcome of different experiments such as Michelson's interference experiment and the experiments to demonstrate double refraction due to
Earth's motion, could only be explained by the relativity principle.
1. ↑ Regarding the notations, see Mathematische Encyklopädie Vol. 14.
2. ↑ This was a first approximation. By a new calculation, de Sitter found the value 7,15". (Monthly Notices of R. A. Sc. 71 (1911), p. 405).
3. ↑ Phys. Zeitschr. 11 (1910) p. 1235.
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1. Introduction2. Carrier Density, Mobility and Sheet Resistance3. Inter-Band Effects of the Magnetic Field4. Effects of Zero-Mode Landau Level on Transport5. ConclusionsAcknowledgmentsReferences
crystals Crystals Crystals Crystals 2073-4352 MDPI 10.3390/cryst2020643 crystals-02-00643 Review Transport Phenomena in Multilayered Massless Dirac Fermion System α-(BEDT-TTF)[2]I[3] Tajima Naoya *
Nishio Yutaka Kajita Koji Department of Physics, Toho University, Miyama 2-2-1, Funabashi-shi, Chiba JP-274-8510, Japan; Email: nishio@ph.sci.toho-u.ac.jp (Y.N.); kajita@ph.sci.toho-u.ac.jp (K.K.)
Author to whom correspondence should be addressed; Email: naoya.tajima@sci.toho-u.ac.jp; Tel.: +81-47-472-6990; Fax: +81-47-472-6991. 11 06 2012 06 2012 2 2 643 661 14 03 2012 28 05 2012 29 05 2012 ©
2012 by the authors; licensee MDPI, Basel, Switzerland. 2012
This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
A zero-gap state with a Dirac cone type energy dispersion was discovered in an organic conductor α-(BEDT-TTF)[2]I[3] under high hydrostatic pressures. This is the first two-dimensional (2D) zero-gap
state discovered in bulk crystals with a layered structure. In contrast to the case of graphene, the Dirac cone in this system is highly anisotropic. The present system, therefore, provides a new
type of massless Dirac fermion system with anisotropic Fermi velocity. This system exhibits remarkable transport phenomena characteristic to electrons on the Dirac cone type energy structure.
Dirac fermion transport phenomena α-(BEDT-TTF)[2]I[3] inter-band effects of the magnetic field zero-mode Landau level ν = 0 quantum Hall effect
Since Novoselov et al. [1] and Zhang et al. [2] experimentally demonstrated that graphene is a zero-gap system with massless Dirac particles, such systems have fascinated physicists as a source of
exotic systems and/or new physics. At the same time, the quasi-two-dimensional (2D) organic conductor α-(BEDT-TTF)[2]I[3] (BEDT-TTF = bis(ethylenedithio) tetrathiafulvalene) was found to be a new
type of massless Dirac fermion state under high pressures [3,4,5,6]. Originally, this material had been considered as a narrow-gap semiconductor [7,8,9]. In contrast to graphene, this is the first
bulk (multilayer) zero-gap material with a Dirac cone type energy dispersion. In this review, we describe the remarkable transport phenomena characteristic to electrons on the Dirac cone type energy
structure in α-(BEDT-TTF)[2]I[3] under high pressure.
The organic conductor α-(BEDT-TTF)[2]I[3] is a member of the (BEDT-TTF)[2]I[3] family [10]. All the crystals in this family consist of conductive layers of BEDT-TTF molecules and insulating layers of
I[3]^− anions as shown in Figure 1 [10,11,12,13]. The difference among them lies in the arrangement and orientation of BEDT-TTF molecules within the conducting plane, and this difference gives rise
to variations in the transport phenomena. Most of the members of this family are 2D metals with large Fermi surfaces, and some of them are superconducting with T[C] values of several Kelvin
[11,12,13]. On the other hand, α-(BEDT-TTF)[2]I[3] is different from the other crystals. According to the band calculation, this system is a semimetal with two small Fermi surfaces; one with electron
character and the other with hole character [8].
(a) BEDT-TTF molecule and I[3]^− anion; (b) Crystal structure of α-(BEDT-TTF)[2]I[3] viewed from a-axis; (c) Crystal structure viewed from c-axis. Reproduced with permission from [5].
When cooled, it behaves as a metal above 135 K, where it undergoes a phase transition to an insulator as shown in Figure 2 [10]. In the insulator phase below 135 K, the rapid decrease in the magnetic
susceptibility indicates that the system is in a nonmagnetic state with a spin gap [14]. According to the theoretical work by Kino and Fukuyama [15] and Seo [16] and the experimental investigation by
Takano et al. (nuclear magnetic resonance) [17], Wojciechowski et al. (Raman) [18] and Moldenhauer et al. [19], this transition is due to the charge disproportionation. In the metal state, the charge
distribution of each BEDT-TTF molecules is approximately 0.5e [18,20]. At temperatures below 135 K, however, horizontal charge stripe patterns for +1e and 0 have been formed as shown in the inset of
Figure 2.
Temperature dependence of the resistance under several hydrostatic pressures [5,8]. The inset shows a schematic picture of the arrangement of BEDT-TTF molecules in a conducting layer viewed from the
c-crystal axis and the charge pattern under ambient pressure. At temperatures below 135 K, the BEDT-TTF molecule layer forms horizontal charge stripe patterns for +1e and 0 [15,16,17,18,19].
Reproduced with permission from [5].
When the crystal is placed under a high hydrostatic pressure of above 1.5 GPa, the metal-insulator transition is suppressed and the metallic region expands to low temperatures as shown in Figure 2
[7,21,22]. This change in the electronic state is accompanied by the disappearance of the charge-ordering state as shown by the Raman experiment [18]. Note that in the high-pressure phase, the
resistance is almost constant over the temperature region from 300 to 2 K. It looks like a dirty metal with a high density of impurities or lattice defects. In dirty metals, the mobilities of
carriers are low and depend on temperature only weakly, and therefore, resistance is also independent of temperature. In the present case, however, the situation is different. Carriers with a very
high mobility approximately 10^5 cm/V·s exhibit extremely large magnetoresistance at low temperatures [5,6,7,8,21], indicating that the crystal is clean.
To clarify the mechanism of these apparently contradictory phenomena, the Hall effect was first examined by Mishima et al. [22]. However, the magnetic field of 5 T they used was too high to
investigate the property of carriers in the zero-field limit. This is because the carrier system at low temperatures is sensitive to the magnetic field and its character is varied even in weak
magnetic fields below 1 T. Thus, Tajima et al. reexamined the Hall effect using a much lower magnetic field of 0.01 T [8]. They also investigated the magnetoresistance and Hall resistance as a
function of the magnetic field up to 15 T and the temperature between 0.5 and 300 K. The experimental results they obtained are summarized as follows: (1) Under a pressure of above 1.5 GPa, the
resistivity is nearly constant between 300 and 2 K [5,8,22]. (2) In the same region, both the carrier (hole) density and the mobility change by about six orders of magnitude as shown in Figure 3. At
low temperatures below 4 K, the carrier system is in a state with a low density of approximately 8 × 10^14 cm^−3 and a high mobility of approximately 3 × 10^5 cm^2/V·s [5,8]. (3) The effects of
changes in the density and the mobility just cancel out to give a constant resistance [5,8,22]. (4) At low temperatures, the resistance and Hall resistance are very sensitive to the magnetic field
[5,7,8,21]. However, neither quantum oscillations nor the quantum Hall effect were observed in magnetic fields of up to 15 T [5]. (5) The system is not a metal but a semiconductor with an extremely
narrow energy gap of less than 1 meV [5,8].
Temperature dependence of the carrier density and the mobility for P= 1.8 GPa. The data plotted by close circle is the effective carrier density n[eff] and the mobility μ[eff]. The magnetoresistance
mobility μ[M] and the density n[M], on the other hand, is shown by open square from 77 K to 2 K. The results of two experiments agree well and the density obeys from 10 K to 50 K (indicated by broken
lines). Reproduced with permission from [5].
The mechanism of such anomalous phenomena, however, was not clarified until Kobayashi et al. performed an energy band calculation in 2004 based on the crystal structure analysis of α-(BEDT-TTF)[2]I
[3] under uniaxial strain by Kondo and Kagoshima [23] and suggested that this material under high pressures is in the zero-gap state [3,4]. According to their calculations, the bottom of the
conduction band and the top of the valence band are in contact with each other at two points (we call these “Dirac points”) in the first Brillouin zone. The Fermi energy is located exactly on the
Dirac point. In contrast to the case of graphene, in which the Dirac points are located at points in the k-space with high symmetry, the positions of the Dirac points in the present system are
unrestricted. Figure 4 depicts the energy structure around one of the contact points. In the vicinity of the Dirac point, a Dirac cone type linear energy dispersion exists. It is expressed as E = ±hν
[F](φ)(k− k[0]) where ν[F](φ) is the Fermi velocity and k[0] are the positions of the two contact points. As shown in Figure 4, the present Dirac cone is slanted and has strong anisotropy. Therefore,
the Fermi velocity ν[F](φ) depends on the direction of the vector (k− k[0]), which is denoted by φ in the above equation. These results were supported by first-principles band calculations [24].
The picture of the zero-gap system grow understanding of the remarkable transport phenomena of α-(BEDT-TTF)[2]I[3]. In this paper, we describe the interpretation of transport phenomena in this system
based on the zero-gap picture.
(a) Band structure; and (b) energy contours near a Dirac point. They are calculated using the parameters for P = 0.6 GPa in [4]. Note that the origin of the axes is taken to be the position of the
Dirac point. Reproduced with permission from [6].
First, let us examine the temperature dependence of the carrier density shown in Figure 3. Between 77 and 1.5 K, it decreases by about four orders of magnitude. This large reduction of the carrier
density is characteristic of semiconductors. However, the temperature dependence is different from those of the ordinary semiconductors. In ordinary semiconductors with finite energy gaps, the
carrier density depends exponentially on the inverse temperature. On the other hand, the data in Figure 3 show a power-law-type temperature dependence , with α ≈ 2. This temperature dependence of
carrier density is explained on the basis of 2D zero-gap systems with a linear dispersion of energy. When we assumed the Fermi energy E[F] is located at the contact point and does not move with
temperature, this relationship is derived as
where is the density of state for zero-gap structure, f(E) is the Fermi distribution function, C = 1.75 nm is the lattice constant along c-axis and is an average of ν[F](φ) defined by . The averaged
value of the Fermi velocity, is estimated to be approximately 3 × 10^4 cm/s. This value corresponds to the result obtained from realistic theories [4,25].
Carrier mobility, on the other hand, is determined as follows. According to Mott's argument [26], the mean free path l of a carrier subjected to elastic scattering can never be shorter than the
wavelength λ of the carrier, so l≤λ. For the cases in which scattering centers exist at high densities, l~λ. As the temperature is decreased, l becomes long because λ becomes long with the decreasing
energy of the carriers. As a result, the mobility of carriers increases in proportion to T^−2 in the 2D zero-gap system. Consequently, the Boltzmann transport equation gives the temperature
independent quantum conductivity as
where ν[xx] is the velocity of carriers on Dirac cone when the electric field is applied along x-axis and τ is the scattering lifetime, which is assumed to depend on the energy of the carriers.
In order to examine Equation 2, we refer to Figure 5 where the resistivity per layer (sheet resistance R[s]) for seven samples is plotted. Note that a conductive layer of this material is sandwiched
by insulating layers as shown in Figure 1, and therefore, each conductive layer is almost independent. Thus, the concept “resistivity per layer” is valid for this system. In this figure, it is shown
that the sheet resistance depends on the temperature very weakly except for below 7 K. Note that the sheet resistance is close to the quantum resistance, h/e^2 = 25.8 kΩ. It varies from a value
approximately equal to the quantum resistance at 100 K to about 1/5 of it at 7 K. The reproducibility of data was checked using six samples. Many realistic theories predict that the sheet resistance
of intrinsic zero-gap systems is given as R[s] = gh/e^2, where g is a parameter of order unity [27,28,29]. Thus, the constant resistance observed in this material is ascribed to a zero-gap energy
(a) Temperature dependence of R[s] for seven samples under a pressure of 1.8 GPa. The inset R[s] at temperatures below 10 K; (b) Temperature dependence of R[s] for Sample 6. It was examined down to
80 mK. Reproduced with permission from [30].
Here, we mention the resistivity below 7 K in Figure 5. The sample dependence due to the effects of unstable I[3]^− anions appears strongly. A rise of R[s] may be a symptom of localization because it
is proportional to log T as shown in the inset of Figure 5a. The unstable I[3]^− gives rise to a partially incommensurate structure in the BEDT-TTF layers. As for Sample 6, the log T law of R[s] was
examined down to 80 mK. The log T law of resistivity, on the other hand, is also characteristic of the transport of Kondo-effect systems. In graphene, recently, Chen et al. have demonstrated that the
interaction between the vacancies and the electrons give rise to Kondo-effect systems [31]. The origin of the magnetic moment was the vacancy. In α-(BEDT-TTF)[2]I[3], on the other hand, Kanoda et al.
detected anomalous NMR signals which could not be understood based on the picture of Dirac fermion systems at low temperatures [32]. We do not yet know whether, in the localization, the Kondo-effect
or other mechanisms are the answer for the origin of the log T law of R[s]. This answer, however, will answer the question as to why the sample with the higher carrier density (lower R[H] saturation
value in Figure 6) exhibits the higher R[s] at low temperature. Further investigation should lead us to interesting phenomena.
Temperature dependence of the Hall coefficient for (a) hole-doped-type; and (b) electron-doped-type samples. Note that in this figure, the absolute value of R[H] is plotted. Thus, the dips in (b)
indicate a change in the polarity. The inset in the upper part of (a) shows the configuration of the six electrical contacts. The schematic illustration of the Fermi levels for the hole-doped type
and the electron-doped type are shown in the inset in the lower part of (a) and the inset of (b), respectively. Reproduced with permission from [30].
Here we adduce other examples for the effect of unstable I[3]^− anions. It was also seen in the superconducting transition of the organic superconductors β-(BEDT-TTF)[2]I[3] [33] and θ-(BEDT-TTF)[2]I
[3] [34].
In conclusion of this section, α-(BEDT-TTF)[2]I[3] under high hydrostatic pressure is an intrinsic zero-gap system with Dirac type energy band E = ±hν[F](φ)(k − k[0]). In the graphite systems, a
monolayer sample (graphene) is inevitable to realize a zero-gap state. On the other hand, in the present system, bulk crystals can be two dimensional zero-gap systems. The carrier density depends on
temperature as . On the other hand, the resistance remains constant in the wide temperature region. The value of the sheet resistance coincide the quantum resistance h/e^2 = 25.8 kΩ within a factor
of 5.
A magnetic field gives us characteristic phenomena. According to the theory of Fukuyama, the vector potential plays an important role in inter-band excitation in electronic systems with a vanishing
or narrow energy gap [35]. The orbital movement of virtual electron-hole pairs gives rise to anomalous orbital diamagnetism and the Hall effect in a weak magnetic field. These are called the
interband effects of the magnetic field. This discovery inspired us to examine the interband effects of the magnetic field in α-(BEDT-TTF)[2]I[3]. In this section, we demonstrate that these effects
give rise to anomalous Hall conductivity in α-(BEDT-TTF)[2]I[3]. Bismuth and graphite are the most well-known materials that serve as testing ground for the interband effects of magnetic fields. To
our knowledge, however, α-(BEDT-TTF)[2]I[3] is the first organic material in which the inter-band effects of the magnetic field have been detected.
Realistic theory predicts that the interband effects of the magnetic field are detected by measuring the Hall conductivity σ[xy] or the magnetic susceptibility at the vicinity of the Dirac points
[35,36]. In order to detect these effects, we should control the chemical potential μ. In this material, however, the multilayered structure makes control of μ by the field-effect-transistor method
much more difficult than in the case of graphene. Hence, we present the following idea.
We find two types of samples in which electrons or holes are slightly doped by unstable I[3]^− anions. The doping gives rise to strong sample dependences of the resistivity or the Hall coefficient at
low temperatures (Figure 5 and Figure 6). In particular, the sample dependence of the Hall coefficient R[H] is intense. In the hole-doped sample as shown in the inset of the lower part of Figure 6a,
R[H] is positive over the whole temperature range (Figure 6a). In the electron-doped sample as shown in the inset of Figure 6b, however, the polarity is changed at low temperatures (Figure 6b). The
change in polarity of R[H] is understood as follows. In contrast to graphene, the present electron-hole symmetry is not good except at the vicinity of the Dirac points [24]. Thus, μ must be dependent
on temperature. Of significance is the fact that according to the theory by Kobayashi et al., when passes the Dirac point (μ = 0), R[H] = 0 [36]. Thus, R[H] at the vicinity of R[H] = 0 for
electron-doped samples must be determined to detect the inter-band effects of the magnetic field. The saturation value of R[H] at the lowest temperature, on the other hand, depends on the doping
density n[d], as n[d] = n[s]/C = 1/(eR[H]), where n[s] is the sheet density. n[d] = n[s]/C = 1/(eR[H]).
Note that the carrier density and the mobility in Figure 3 is the behavior for hole-doped type sample. However, the temperature dependence of μ is much weaker than the thermal energy and the doping
levels are estimated to be several ppm. Thus, the relationship of Equation 1 is valid which is derived when we assumed that μ locates at the Dirac point and does not move with the temperature, in
Section 2.
In this section, thus, to detect the inter-band effects of the magnetic field, we focused on the behavior of R[H] in which the polarity is changed (Figure 6b). It was examined as follows.
The first step is to examine the temperature dependence of μ. As mentioned before, we believe μ passes the Dirac point at the temperature T[0], shown as R[H] = 0 [36]. The sheet density n[s], on the
other hand, is approximately proportional to T[0]^2 as shown in Figure 7(a). This result suggests that μ is to be written as μ/k[B] = E[F]/k[B]− AT at T[0] approximately because , where A is the
fitting parameter depending on the electron-hole symmetry (). Thus, we obtain the E[F] versus T[0] curve in Figure 7b. E[F] is estimated from the relationship n[s] = E[F]^2/(4πh^2ν[F]^2), where the
averaged Fermi velocity ν[F], is estimated to be approximately 3.3 × 10^4 m/s from the temperature dependence of the carrier density. Note that the weak sample dependence of both R[s] and R[H] at
temperatures above 7 K (Figure 5 and Figure 6) strongly indicates that the ν[F] values of all samples are almost the same. When we assume that A is independent of E[F], A is estimated to be
approximately 0.24 from Figure 7b. Thus, we examine the temperature dependence of μ as μ/k[B] = E[F]/k[B]− AT with A ~ 0.24. This experimental formula reproduces well the realistic theoretical curve
by Kobayashi et al. [36], as shown in Figure 7(c). Our simple calculations, on the other hand, also reproduce well this curve when we assume , where and are the Fermi velocities for lower and upper
Dirac cones, respectively. This is the electron-hole asymmetry in this system. Actually, the result of the first principle band calculation by Kino and Miyazaki indicate that the electron-hole
symmetry is not complete [24].
(a) Sheet electron density n[s] for five samples plotted against temperature at R[H]. (b) E[F] was estimated from the relationship n[s] = E[F]^2/(4πh^2ν[F]^2) with ν[F] ~ 3.3 × 10^4 m/s. From this
curve, the temperature dependence of μ is written approximately as μ/k[B] = E[F]/k[B]− AT with A ~ 0.24 when we assume A is independent of E[F]. (c) Temperature dependence of μ for E[F] = 0. Our
experimental formula is quantitatively consistent with the theoretical curve of Kobayashi et al. [36] at temperature below 100 K. Reproduced with permission from [30].
The second step is to calculate the Hall conductivity as σ[xy] = ρ[yx]/(ρ[xx]ρ[yy] + ρ[yx]^2). In this calculation, we assume ρ[xx] = ρ[yy] for the following reasons. According to band calculation,
the energy contour of the Dirac cone is highly anisotropic [4]. In the galvano-magnetic phenomena, however, the anisotropy is averaged and the system looks very much isotropic. A simple calculation
indicates that the variation in the mobility with the current direction is within a factor of 2. Experimentally, Iimori et al. showed that the anisotropy in the in-plane conductivity is less than 2
Based on this assumption, we show the temperature dependence of σ[xy] for Samples 1 and 7 in Figure 5 and Figure 6 in Figure 8a as an example. We see the peak structure in σ[xy] at the vicinity of σ
[xy] = 0. This peak structure is the anomalous Hall conductivity originating from the interband effects of the magnetic field. The realistic theory indicates that the Hall conductivity without the
interband effects of the magnetic field has no peak structure [36].
In the last step, we redraw σ[xy] in Figure 8b as a function of μ using the experimental formula, μ/k[B] = E[F]/k[B] − AT with A = 0.24. It should be compared with the theoretical curve [36], at T =
0. Our experimental data are roughly expressed as , where g is a parameter that depends on temperature because the effect of thermal energy on the Hall effect is strong. Note that σ[xy] depends
strongly on temperature. The energy between two peaks is the damping energy that depends on the density of scattering centers in a crystal. The intensity of the peak, on the other hand, depends on
the damping energy and the tilt of the Dirac cones [36].
(a) Temperature dependence of the Hall conductivity for Samples 1 and 7 in Figure 5 and Figure 6; (b) Chemical-potential dependence of the Hall conductivity for Samples 1 and 7. Solid lines and
dashed lines are the theoretical curves with and without the interband effects of the magnetic field by Kobayashi et al., respectively [36]. Reproduced with permission from [30].
Lastly, we briefly mention the zero-gap structure in this material. The smooth change in the polarity of σ[xy] is also evidence that this material is an intrinsic zero-gap conductor. Nakamura
demonstrated theoretically that in a system with a finite energy gap, σ[xy] is changed in a stepwise manner at the Dirac points [38].
In conclusion of this section, we succeeded in detecting the interband effects of the magnetic field on the Hall conductivity when μ passes the Dirac point. Good agreement between experiment and
theory was obtained.
One of the characteristic features in Dirac fermion system is clearly seen in the magnetic field. In this section, the transport phenomena in the magnetic field are described. Note that we interpret
the transport phenomena based on the assumption which E[F] locates at the Dirac point because E[F] is lower than broadening energy of Landau levels.
In the magnetic field, the energy of Landau levels (LLs) in zero-gap systems is expressed as , where n is the Landau index and B is the magnetic field strength [39]. One important difference between
zero-gap conductors and conventional conductors is the appearance of a (n = 0) LL at zero energy when magnetic fields are applied normal to the 2D plane. This special LL is called the zero-mode.
Since the energy of this level is E[F] irrespective of the field strength, the Fermi distribution function is always 1/2. It means that half of the Landau states in the zero mode are occupied. Note
that in each LL, there are states whose density is proportional to B. The magnetic field, thus, creates mobile carriers.
For k[B]T < E[1LL], most of the mobile carriers are in the zero-mode. Such a situation is called the quantum limit. The carrier density per valley and per spin direction in the quantum limit is given
by D(B) = B/2φ[0], where φ[0] = h/e is the quantum flux. The factor 1/2 is the Fermi distribution function at E[F]. In moderately strong magnetic fields, the density of carriers induced by the
magnetic field can be very high. At 3 T, for example, the density of zero-mode carriers will be 10^15 m^−2. This value is by about 2 orders of magnitude higher than the density of thermally excited
carriers at 4 K, and, in the absence of the magnetic field, it is 10^13 m^−2. Therefore, carrier density in the magnetic field is expressed as B/2φ[0] except for very low fields.
This effect is detected in the inter-layer resistance, R[zz], in the longitudinal magnetic field. In this field configuration, the interaction between the electrical current and the magnetic field is
weak because they are parallel to each other. Hence, the effect of the magnetic field appears only through the change in the carrier density. In this regard, the large change in the density of
zero-mode carriers gives rise to remarkable negative magnetoresistance in the low magnetic field region, as shown in Figure 9 [40,41].
(a) Magnetic field dependence of interlayer resistance under the pressure about 1.7 GPa at 4 K. When B > 0.2 T, remarkable negative magnetoresistance is observed. As for negative magnetoresistance,
fitting is done with Equation 3, (red line); (b) Angle dependence of magnetoresistance. When θ = 0 and θ = 180°, the direction of the magnetic field is parallel to the 2D plane. θ = 90° is the
direction normal to the 2D plane. Fitting of the data measured at 1, 2, and 3 T was done using Equation 3. Reproduced with permission from [40].
Recently, Osada gave an analytical formula for interlayer magnetoresistance in a multilayer Dirac fermion system as follows:
where A = πh^3/2C't[c]^2ce^3 is a parameter that is considered to be independent of the magnetic field if the system is clean. B[0] is a fitting parameter that depends on the quality of the crystal,
C' is defined by using the spectral density of the zero-mode Landau level and ρ[0](E) satisfies [42].
Except for narrow regions around θ = 0 and θ = 180°, this formula can be simplified to R[zz] = A/(|B| + B[0]). Using this formula and assuming B[0] = 0.7 T, we tried to fit the curves in Figure 9.
This simple formula reproduces well both the magnetic field dependence and the angle dependence of the magnetoresistance at magnetic fields above 0.5 T as shown by solid lines in Figure 9, which
evidences the existence of zero-mode Landau carriers in α-(BEDT-TTF)[2]I[3] at high pressures.
Here, we briefly mention the origin of positive magnetoresistance around θ = 0 and θ = 180°. In the magnetic field in these directions, the Lorentz force works to bend the carrier trajectory to the
direction parallel to the 2D plane. It reduces the tunneling of carriers between neighboring layers so that the positive magnetoresistance is observed. Note that the Equation 3 for θ = 0 and θ = 180°
dose not correctly evaluate the effect of the Lorentz force and, thus, loses its validity. The value of the resistance peak depends weakly on the azimuthal angle. At 3 T, for example, the ratio of
maximum value to the minimum value is less than 1.3. According to the calculation of interlayer magnetoresistance by Morinari, Himura and Tohyama, the effect of Dirac cones with highly anisotropic
Fermi velocity is averaged and gives rise to this small difference [43].
Angle dependence of interlayer Hall resistivity ρ[zx] in the fixed magnetic field gives also us the characteristic phenomena in quantum limit state [44,45]. Figure 10 shows cotθ dependence of
inter-layer Hall resistivity ρ[zx] in the magnetic field below 7 T at 4.2 K. We find the relationship of ρ[zx]=Acotθ, where slope A is close to the quantum resistance, h/e^2 = 25.8 kΩ and is
independent of the magnetic field. This is simply understood as follows. In general, Hall voltage is proportional to cosθ. Thus, angle dependence of interlayer Hall resistivity is written as ρ[zx] =
ρ[0] cosθ = B/ne cosθ. The degeneracy of zero-mode, on the other hand, is proportional to the magnetic field, and then n = B/2φ[0] sinθ. Thus, we obtain the relationship ρ[zx] = h/2e^2 cotθ.
cotθ dependence of inter-layer Hall resistivity ρ[zx] at 4.2 K. When the magnetic field is applied along to 2D plane, θ = 0 or θ = 180°. Reproduced with permission from [44].
Here, let us return to Figure 9. An apparent discrepancy from Equation 3 of the data is also seen at both low and high magnetic fields normal to the 2D plane, because the model is oversimplified.
Equation 3 was derived based on the quantum limit picture in which only the zero-mode LL is considered. In fact, each LL has a finite width due to scattering. At a sufficiently low magnetic field,
the zero-mode LL overlaps with other Landau levels. In such a region, Equation 3 loses its validity. We can recognize this region in Figure 9 below 0.2 T, where positive magnetoresistance is
observed. This critical magnetic field, B[p], shifts to a lower field with decreasing temperature, where it almost saturates at about 0.04 K, as shown in Figure 11b,d.
(a) Magnetic field dependence of R[zz] below 4.1 K under pressure of approximately 1.7 GPa; (b) R[zz] in the low field region below 0.4 T; (c) Magnetic field dependence of R[zz] B; (d) Temperature
dependence of B[p] (solid triangles) and B[min] (solid circles). The solid line is the curve of E[1LL] with ν[F] ~ 4 ×10^4 m/s. Reproduced with permission from [41].
The overlap between the zero-mode and other LLs, primarily the n = 1 LL will be sufficiently small above B[p] and as a result, the negative magnetoresistance is observed there. Then, we have a
tentative relationship: E[1LL] ~ 2k[B]T[p] at B[p] [41,46]. In fact, E[1LL] with ν[F] ~ 4 × 10^4 m/s is reproduced well except in the temperature region below 2 K. This Fermi velocity corresponds to
that estimated from the temperature dependence of the carrier density [6]. The discrepancy of the data from the curve of E[1LL] below 2 K, on the other hand, suggests that thermal energy is
sufficiently lower than the scattering broadening energy г[0]. Thus, г[0] is roughly estimated to be approximately 2 K from the constant value of B[p] as at 0.1 T [41,46]. This scattering broadening
energy is much lower than that of graphene. In graphene, г[0] was estimated to be about 30 K.
The deviation from Equation 3 of data in the high field region of Figure 9 is much more serious. In this region, the resistance increases exponentially with increasing field. This phenomenon is
understood as follows.
In the above discussion, we did not consider the Zeeman effect. The Zeeman effect, however, should be taken into consideration because it has a significant influence on the transport phenomena at low
temperatures. In the presence of a magnetic field, each LL is split into two levels with energies E[nLL] = ±ΔE, where ΔE = μ[B]B is the Zeeman energy. This change in the energy structure gives rise
to a change in the carrier density in LLs. In particular, the influence on the zero-mode carrier density is the strongest, because the energy level is shifted from the position of the Fermi energy.
The value of the Fermi distribution function varies from f(E[F]) = 1/2 to f(ΔE) = 1/(exp(ΔE/k[B]T) + 1). At low temperatures where k[B]T < ΔE , this effect becomes important. It works to reduce the
density of zero-mode carriers and, thus, increases the resistance as (Figure 11 a,c) [38,39]. According to the theory of Osada, on the other hand, μ[B]B ~ г[0] at the magnetoresistance minimum [42].
The Zeeman energy when B = 1 T is about 1 K. Therefore, in the experiment performed at 1 K, the deviation of experimental results from Equation 3 is expected to start around 1 T. This is confirmed in
Figure 11a. We find a resistance minimum. At 1.8 K, for example, the deviation is prominent in fields above 2 T. This critical field shifts to about 0.3 T at 0.07 K.
Recently, Osada pointed out the possibility of ν = 0 quantum Hall effect (QHE) in this region [47]. In the spin-splitting state of 2D Dirac fermion system such as graphene, the edge state with a pair
of inverse spins exists at near the edge of sample as shown in Figure 12. According to the theory by Osada [47], the saturation of data at low temperatures and high magnetic fields in Figure 11a
strongly indicates the existence of the edge state. Thus, ν= 0 quantum Hall state is expected in this region. Note that in this special state, the characteristic in-plane transport (σ[xx] and σ[xy])
phenomena in conventional quantum Hall state is not detected.
Edge state Spector in ν = 0 quantum Hall state.
In conclusion of this section, we succeeded in detecting the zero-mode Landau level. The characteristic feature of zero-mode Landau carriers including the Zeeman effect was clearly seen in the
inter-layer transport. The experimental data suggest that with increasing magnetic field or decreasing temperature, the system changes from a “Dirac fermion” state to a “quantum limit state”, then to
a “spin-splitting” state and then to a “ν = 0 QHE” state as shown in Figure 13, where the boundaries between the states in the B–T plane are depicted using the data in Figure 11. Stepwise changes in
the states between boundaries, on the other hand, are observed in in-plane magnetoresistance as shown in Figure 14.
Schematic diagram of boundaries in B–T plane depicted using the data in Figure 11. “Dirac fermion” state is the low magnetic field region, “quantum limit” state is the region observed negative
magnetoresistance, the region that obeyed an exponential law is “spin-splitting” state and “ν = 0 quantum Hall effect” state is the saturation region at low temperature and high magnetic field.
(a) Magnetic field dependence of the in-plane resistance down to 0.5 K. The low-field region below 1 T is shown in the inset; (b) Temperature dependence of the in-plane resistance under a magnetic
field of up to 15 T. The arrows shows the boundaries between “Dirac fermion”, “quantum limit”, “spin-splitting” and “ν = 0 quantum Hall effect” states. Reproduced with permission from [5].
α-(BEDT-TTF)[2]I[3] under high hydrostatic pressure is an intrinsic zero-gap conductor with a Dirac type energy band. The carrier density, expressed as , is a characteristic feature of the zero-gap
system. On the other hand, the resistance remains constant from 2 to 300 K. The sheet resistance can be written in terms of the quantum resistance h/e^2 as R[s] = gh/e^2, where g is a parameter that
depends weakly on temperature. Moreover, we succeeded in detecting the interband effects of the magnetic field on the Hall conductivity when the chemical potential passes the Dirac point. In graphite
systems, a monolayer sample (graphene) is necessary to realize a zero-gap state. On the other hand, in the present system, it was demonstrated that bulk crystals can be 2D zero-gap systems. We can
observe this effect in the inter-layer magnetoresistance. The existence of the zero-mode Landau level is one of characteristic features in the zero-gap system. The effect of Landau degeneracy, which
is proportional to the strength of the magnetic field, gives rise to the large negative magnetoresistance. We anticipate the ν = 0 quantum Hall state in the spin-splitting state of zero-mode at low
temperature and high magnetic field. Lastly, we mention the robustness of zero-gap structure in this compound. The temperature dependence of the carrier density that ruled out law and log T law of R
[s] at low temperature below 7 K gives us the impression that the zero-gap structure is unstable. The detection of zero-mode Landau carriers including its spin splitting down to 0.07 K, however,
strongly suggests that the zero-gap structure is robust. Further investigation for the anomalous phenomena at low temperature should lead us to interesting phenomena.
This system offers a testing ground for a new type of particles, namely, massless Dirac fermions with a layered structure and anisotropic Fermi velocity.
We are grateful to M. Sato, S. Sugawara, M. Tamura, R. Kato and Y. Iye for fruitful collaborations which helped us understand the subjects discussed in this paper. We thank T. Osada, A. Kobayashi, S.
Katayama, Y. Suzumura, R. Kondo, T. Morinari, T. Tohyama and H. Fukuyama for valuable discussions. This work was supported by Grant-in-Aid for Scientific Research (No. 22540379 and No. 22224006) from
the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Katayama S. Noguchi K. Suzumura Y. 2004 73 3135 3148 10.1143/JPSJ.73.3135 Katayama S. Kobayashi A. Suzumura Y. Pressure-induced zero-gap semiconducting state in organic conductor α-(BEDT-TTFET)[2]I
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pressure 2007 80 Kajita K. Ojiro T. Fujii H. Nishio Y. Kobayashi H. Kobayashi A. Kato R. Magnetotransport Phenomena of α-Type (BEDT-TTFET)[2]I[3] under High Pressures 1993 61 23 26 Tajima N. Tamura
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conductor, α-(BEDT-TTFET)[2]I[3] 2001 62 393 395 10.1016/S0022-3697(00)00173-6 Wojciechowski R. Yamamoto K. Yakushi K. Inokuchi M. Kawamoto A. High-pressure Raman study of the charge ordering in α-
(BEDT-TTFET)[2]I[3] 2003 67 Moldenhauer J. Horn C.H. Pokhodnia K.I. Schweitzer D. Heinen I. Keller H.J. FT-IR absorption spectroscopy of BEDT-TTF radical salts: Charge transfer and donor-anion
interaction 1993 60 31 38 10.1016/0379-6779(93)91180-A Kakiuchi1 T. Wakabayashi Y. Sawa H. Takahashi T. Nakamura T. Charge ordering in α-(BEDT-TTFET)[2]I[3] by synchrotron X-ray diffraction 2007 76
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K. Nishio Y. Iye Y. Temperature independent conductivity of α-(BEDT-TTFET)[2]I[3]-compensation of the change in the carrier density and the mobility 1995 70 771 774 10.1016/0379-6779(94)02645-F Kondo
R. Kagoshima S. Crystal and electronic structures of α-(BEDT-TTF) I under uniaxial strains 2004 114 523 525 Kino H. Miyazaki T. First-principles study of electronic structure in α-(BEDT-TTFET)[2]I[3]
at ambient pressure and with uniaxial strain 2006 75 Goerbig M.O. Funchs J.N. Montambaux G. Piechon F. Tilted anisotropic Dirac cones in quinoid-type graphene and α-(BEDT-TTFET)[2]I[3] 2008 78 Mott
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The role of a broken supersymmetry 1998 80 3113 3116 10.1103/PhysRevLett.80.3113 Nomura K. MacDonald A.H. Quantum transport of massless dirac fermions 2007 98 Tajima N. Kato R. Sugawara S. Nishio Y.
Kajita K. Inter-band effects of magnetic field on Hall conductivity in the multilayered massless Dirac fermion system α-(BEDT-TTFET)[2]I[3] 2012 85 Chen J. Li L. Cullen W.G. Williams E.D. Fuhrer M.S.
Tunable Kondo effect in graphene with defects 2011 7 535 538 10.1038/nphys1962 Kanoda K. Private Communication Department of Applied Physics, University of Tokyo Tokyo, Japan 2011 Murata K. Tokumoto
M. Anzai H. Bando H. Saito G. Jajimura K. Ishiguro T. Superconductivity with the Onset at 8 K in the organic conductor β-(BEDT-TTFET)[2]I[3] under Pressure 1985 54 1236 1239 10.1143/JPSJ.54.1236
Salameh B. Nothardt A. Balthes E. Schmidt W. Schweitzer D. Strempfer J. Hinrichsen B. Jansen M. Maude D.K. Electronic properties of the organic metals θ-(BEDT-TTFET)[2]I[3] and θT-(BEDT-TTFET)[2]I[3]
2007 75 Fukuyama H. Anomalous orbital magnetism and hall effect of massless fermions in two dimension 2007 76 Kobayashi A. Suzumura Y. Fukuyama H. Hall effect and orbital diamagnetism in zerogap
state of molecular conductor α-(BEDT-TTFET)[2]I[3] 2008 77 Iimori S. Kajita K. Nishio Y. Iye Y. Anomalous metal-nonmetal transition in α-(BEDT-TTFET)[2]I[3] under high pressure 1995 71 1905 1906
10.1016/0379-6779(94)03101-B Nakamura M. Orbital magnetism and transport phenomena in two-dimensional Dirac fermions in a weak magnetic field 2007 76 Ando T. Theory of electronic states and transport
in carbon nanotubes 2005 74 777 817 10.1143/JPSJ.74.777 Tajima N. Sugawara S. Kato R. Nishio Y. Kajita K. Effect of the zero-mode landau level on interlayer magnetoresistance in multilayer massless
dirac fermion systems 2009 102 Tajima N. Sato M. Sugawara S. Kato R. Nishio Y. Kajita K. Spin and valley splittings in multilayered massless Dirac fermion system 2010 82 Osada T. Negative interlayer
magnetoresistance and Zero-Mode landau level in multilayer dirac electron systems 2008 77 Morinari T. Himura T. Tohyama T. Possible verification of tilted anisotropic dirac cone in α-(BEDT-TTFET)[2]I
[3] using interlayer magnetoresistance 2009 78 Sato M. Miura K. Endo S. Sugawara S. Tajima N. Murata K. Nishio Y. Kajita K. Transport phenomenon of multilayer zero-gap conductor in the quantum limit
2011 80 Osada T. Anomalous interlayer hall effect in multilayer massless dirac fermion system at the quantum limit 2011 80 Sugawara S. Tamura M. Tajima N. Kato R. Sato M. Nishio Y. Kajita K.
Temperature dependence of inter-layer longitudinal magnetoresistance in α-(BEDT-TTFET)[2]I[3]: Positive versus negative contributions in a tilted dirac cone system 2010 79 Osada T. Magnetotransport
in organic Dirac fermion system at the quantum limit: Interlayer Hall effect and surface transport via helical edge states 2012 249 962 966 10.1002/pssb.201100587 | {"url":"http://www.mdpi.com/2073-4352/2/2/643/xml","timestamp":"2014-04-16T17:44:13Z","content_type":null,"content_length":"115238","record_id":"<urn:uuid:eea699d4-534b-4b4d-917e-f8d367995602>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00663-ip-10-147-4-33.ec2.internal.warc.gz"} |
EXAMPLE SHEET Revision Exercises AUTUMN A long bridge with
EXAMPLE SHEET 1 AUTUMN 2008
Revision Exercises
A long bridge with piers 1.5 m wide, spaced 8 m between centres, crosses a river. The depth
of water upstream is 1.6 m and between the piers is 1.45 m. Calculate the volume flow rate
under one arch, assuming that the river bed is horizontal, the banks are parallel and frictional
effects are negligible. Find the maximum height to which water rises at the front of the piers.
A pipe carrying oil with specific gravity 0.75 tapers from diameter 0.5 m at A to 0.3 m at B,
which is 5 m above A. The velocity may be assumed uniform at both points and friction may
be neglected. If the velocity and gauge pressure at A are 2 m s–1 and 120 kPa respectively,
calculate the gauge pressure at B.
Q3. (White) 1
For the reducing section shown, diameters
D1 = 80 mm and D2 = 50 mm, whilst p2 is water
atmospheric pressure. If u1 = 5 m s–1 and the
manometer reading is h = 580 mm, estimate the total
force resisted by the flange bolts. h
(Take the density of mercury as 13600 kg m–3.) mercury
Water flows through the horizontal elbow shown at a weight flow
rate of 150 N s–1. The inlet pipe diameter D1 = 100 mm and the
nozzle exit has internal diameter 30 mm. The absolute pressure at
inlet is 330 kPa. Neglecting the weight of water and elbow, estimate
the force on the pipe bend.
(Take atmospheric pressure as 101 kPa.)
The velocity profile V(R) between coaxial cylinders, where the inner
cylinder (of radius R1) is fixed and the outer cylinder (of radius R2)
is rotating with angular velocity ω, is given by
1 − R12 /R 2
V = ωR
1 − R 2 /R 2
R1 R2
If the cylinders have length 0.5 m, the inner and outer radii are
20 mm and 40 mm respectively, the outer cylinder is rotating at a
steady speed of 120 rpm and the intervening fluid has viscosity
5×10–2 kg m–1 s–1, find:
(a) the shear stress on the outer cylinder;
(b) the power required to maintain steady rotation.
Hydraulics 2 E1-1 David Apsley
Q6. (Massey)
The air supply to an oil engine is measured by being taken directly from the atmosphere into
a large reservoir through a sharp-edged orifice 50 mm diameter. The pressure difference
across the orifice is measured by an alcohol manometer set at an angle of arcsin (0.1) to the
horizontal. Calculate the volume flow rate of air if the manometer reading is 271 mm, the
relative density of alcohol is 0.80, the coefficient of discharge for the orifice is 0.602 and
atmospheric pressure and temperature are, respectively, 775 mm Hg and 15.8 °C.
Continuity and Momentum Principle For Non-Uniform Velocity Profiles
Q7. (Examination, January 2006)
A gate is used to control the flow of water in a square-section duct of side h = 0.3 m
(Figure below). The upstream flow is uniform, with velocity u0. A short distance downstream
of the gate the velocity profile can be adequately approximated by
2, if y ≥ h / 2
u ( y) =
3 + cos(2πy / h), if y ≤ h / 2
where u is the velocity in m s and y is the distance from the floor of the duct in m.
y u(y)
(a) Find the upstream velocity, u0.
(b) If the hydrodynamic force on the gate is 400 N, find the pressure drop between the
upstream and downstream sections shown (neglecting drag on the duct walls).
(c) Hence deduce a pressure loss coefficient for the gate in this position.
Q8. (Examination, January 2001)
Water flows through a contraction at the end of a horizontal pipe of initial diameter 1.8 m.
The diameter at the exit is 1.0 m. The flow exhausts into atmosphere. The velocity
distribution across a radius of the pipe immediately upstream of the contraction is as follows:
Radius (mm) 0 250 500 675 800 875 900
Velocity (m s–1) 3.0 2.92 2.67 2.34 1.89 1.14 0.0
The velocity at the exit is uniform.
(a) What is the velocity at the exit?
(b) If the pressure upstream is 30 kPa what is the force exerted on the contraction?
Hydraulics 2 E1-2 David Apsley
Q9. (Examination, January 2001)
Velocities were measured in the wake of a cylinder spanning a 400 mm square test section of
a wind tunnel. The cylinder was placed in the middle of the test section and the flow was
two-dimensional. The velocity distribution at a downstream cross-section was found to be
symmetrical about the centre plane and given by
u = 20.0 1 − cos
0 .2
where u is the velocity (in m s–1) at a distance y (m) from the axis of the cylinder. (Take the
density of air as 1.2 kg m–3.)
(a) Find the upstream velocity, which is uniform over the test section.
(b) If the force on the body is 3.0 N, determine the pressure drop between upstream and
downstream sections.
An axisymmetric jet carrying 20 L s–1 of water impinges normally on
a plane wall. The velocity profile in the jet may be approximated by
U = 1 + (r / r0 ) 2
, if r < 3r0
where r is the distance from the axis and r0 = 0.025 m. Find:
(a) the maximum velocity, U0;
(b) the force on the wall;
(c) the maximum pressure on the wall.
A hydraulic jump occurs in an open channel of width 1.0 m (see figure). Upstream of the
jump the depth is 0.1 m and the velocity is uA (uniform). The velocity profile just
downstream of the jump is of the form
u πy
u = B [1 + cos ]
2 D
where u is the velocity at a distance y from the bed of the channel, uB is the velocity near the
bed and D (= 0.8 m) is the depth downstream of the jump.
0.8 m
0.1 m
(a) Determine uB, leaving your answer as a function of uA.
(b) Calculate the difference between the hydrostatic pressure forces on the fluid cross-
sections upstream and downstream of the jump.
(c) Neglecting viscous stresses on the channel bed or the free surface, use the momentum
principle to find the upstream velocity uA.
Hydraulics 2 E1-3 David Apsley
Non-Uniform Force Distributions
Q12. (Examination, January 2007)
The drag and lift on a long-span aerofoil (see figure) are to be found by a wake traverse and
by measurement of the surface pressure, respectively. The free-stream velocity is
U∞ = 50 m s–1 and the aerofoil has chord c = 0.5 m.
y pupper
x U(y)
The approach velocity is uniform (U∞), whilst the velocity deficit in the wake may be
approximated by
∆U [1 + cos( πy / d )] ,
U∞ −U =
if y < d
0 otherwise
where ∆U = 5.0 m s , d = 0.1 m and y is the distance from the chord line. The flow may be
regarded as two-dimensional, and both the free-stream-velocity and static-pressure
differences between upstream and downstream sections may be neglected.
(a) Find the drag force per unit span on the aerofoil.
(b) Define a suitable drag coefficient for the aerofoil and calculate its value.
The pressure distributions on the upper and lower surfaces of the aerofoil are given by
plower = 1500 − 3000 x − 48000 x(0.5 − x) 2
pupper = 1500 − 3000 x − 192000 x(0.5 − x) 2
where p is the pressure in Pascals and x is the distance from the leading edge in metres.
(c) Find the lift force per unit span on the aerofoil.
(d) Define a suitable lift coefficient and calculate its value.
Hydraulics 2 E1-4 David Apsley
Tank Filling and Emptying
A conical hopper of semi-vertex angle 30º contains water to a
depth of 0.8 m. If a small hole of diameter 20 mm is suddenly
0.8 m
opened at its point, estimate (assuming a discharge coefficient
cd = 0.8):
(a) the initial discharge (quantity of flow);
(b) the time taken to reduce the depth of water to 0.4 m.
A steep-sided reservoir has a constant surface area 0.3 km2 and discharges over a weir of
length b = 4 m. The discharge relationship (discharge Q against height over the sill, h) for the
weir is
Q = 1.8bh 3 / 2
where Q is in m3 s–1 and b and h are in m. The initial depth of water over the sill of the weir is
0.4 m. If there is no inflow to the reservoir, find the time (in hours) to reduce this depth to
0.2 m.
Hydraulics 2 E1-5 David Apsley
1. 23.9 m3 s–1
1.78 m
2. 73.1 kPa gauge
3. 164 N
4. Fx = 2082 N, Fy = 213 N
5. (a) 1.05 N m–2
(b) 0.066 W
6. 21.8 L s–1
7. (a) 2.5 m s–1
(b) 4.9 kPa
(c) 1.6
8. (a) 7.3 m s–1
(b) 48 kN
9. (a) 20 m s–1
(b) 259 Pa
10. (a) 4.42 m s–1
(b) 34.6 N
(c) 9790 Pa
11. (a) u B = 1 u A
(b) 3090 N
(c) 6.2 m s–1
12. (a) 51 N
(b) 0.068
(c) 750 N
(d) 1.0
13. (a) 0.996 L s–1
(b) 177 s
14. 15.2 hours
Hydraulics 2 E1-6 David Apsley | {"url":"http://www.docstoc.com/docs/5531224/EXAMPLE-SHEET-Revision-Exercises-AUTUMN-A-long-bridge-with","timestamp":"2014-04-20T15:30:50Z","content_type":null,"content_length":"64580","record_id":"<urn:uuid:7f63484f-31f4-495f-b125-43b449064086>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00119-ip-10-147-4-33.ec2.internal.warc.gz"} |
Department of Mathematics, Applied Mathematics and Statistics
Friday, October 25, 2013
Title: The theory of valuations and what it can do for you! Speaker: Franz Schuster
(Vienna University of Technology)
A function $\phi$ defined on convex (or more general) compact sets in $\mathbf{R}^n$ and taking values in an Abelian semigroup is called a valuation if $$\phi(K) + \phi(L) = \phi(K \cup L) + \phi(K \
cap L)$$ whenever $K \cup L$ is convex. The theory of valuations on convex sets is a classical part of (convex) geometry with traditionally strong relations to integral geometry. However, in the last
15 years there have been dynamic developments that have led to enormous progress both conceptual and technical. Even the notion of valuation itself has evolved in different directions (e.g. \emph
{finitely additive} smooth functionals on smooth manifolds or operators on function spaces) and the ties of valuation theory to other areas of both pure and applied mathematics have become much more
The purpose of this talk is to give a (by no means complete) survey on the recent developments in the theory of valuations and the new connections to other branches of mathematics, like differential
geometry, harmonic analysis, and the theory of isoperimetric and analytic inequalities. To be more specific, we discuss the recent breakthrough in the structure theory of invariant valuations which
in turn was the starting point for what is now called \emph{algebraic integral geometry}, we explain recent \emph{classifications of affine invariant notions of surface area} by Ludwig, Reitzner and
Haberl, Parapatits, and present \emph{new geometric inequalities} for convex body valued valuations which strengthen several classical isoperimetric and analytic inequalities. The latter results also
build a bridge to harmonic analysis as these valuations are closely related to Radon, cosine, and other convolution transforms. | {"url":"http://www.case.edu/artsci/math/Abstracts2013_2014/AbstractOct24_Schuster.html","timestamp":"2014-04-17T13:11:21Z","content_type":null,"content_length":"10581","record_id":"<urn:uuid:67f8c606-c8f1-4bdc-aecf-81fd53e46031>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00404-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sammamish Geometry Tutor
...I've also programming in VBA most recently for an Excel update function. I was also responsible for our network at our satellite office working for GE as well as web-based instruction on parts
of the GE system. I taught spiral math at the high school level in the Peace Corps.
39 Subjects: including geometry, reading, English, algebra 1
...These are skills and lessons that I apply to every lesson when I work with students. Feel free to contact me if you have any questions or want more information - I look forward to hearing from
you!I have worked as an Algebra teacher in Chicago and I thoroughly enjoy teaching the subject. I have...
27 Subjects: including geometry, chemistry, reading, writing
...I have had several years of classical training in piano with two of those at University of Puget Sound and Oregon State. I am an excellent sight reader and have been paid as an accompanist and
have training in music theory. I was employed for many years in the field of both local and wide area ...
43 Subjects: including geometry, chemistry, calculus, physics
...And then, I give the student sample problems to solve independently and coach them further as needed. My main goal is to make sure the student is self-sufficient, and capable of using the
methods on quizzes or tests. With respect to my educational background and work experience, I'm a Physiology major, and I just graduated from the University of Washington.
26 Subjects: including geometry, chemistry, calculus, physics
...I hold a PhD in Aeronautical and Astronautical Engineering from the University of Washington, and I have more than 40 years of project experience in science and engineering. I am uniquely
qualified to tutor precalculus, with a PhD in Aeronautical and Astronautical Engineering from the University...
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